Astronomical Photometry Handen Kaitchuck

ASTRONOMICALPHOTOMETRY

A Text and Handbookfor the Advanced Amateur

and Professional Astronomer

Arne A. HendenRonald H. KaitchuckDepartment of AstronomyThe Ohio State University

Published by

Willmann-BellJnc. «-—gO. Box 35025Richmond, Virginia 23235 © (804)United States of America 320-7016 *"-

Published by Willmann-Bell, Inc.P.O. Box 35025, Richmond, Virginia 23235

Copyright ©1982 by Van Nostrand Reinhold Company Inc.Copyright ©1990 by Arne A. Henden and Ronald H. Kaitchuck

All rights reserved. Except for brief passages quoted in a review, no part ofthis book may be reproduced by any mechanical, photographic, or electronicprocess, nor may it be stored in any information retrieval system, trans-mitted, or otherwise copied for public or private use, without the writtenpermission of the publisher. Requests for permission or further informationshould be addressed to Permissions Department, Willmann-Bell, Inc. P.O.Box 35025, Richmond, Virginia 23235.

First published 1982Second printing, with corrections 1990

Printed in the United States of America

Library of Congress Cataloging-in-Publication DataHenden, Arne A.

Astronomical photometry : a text and handbook for the advancedamateur and professional astronomer / Arne A. Henden, Ronald H.Kaitchuck

p. cm.Includes bibliographical references.ISBN 0-943396-25-51. Photometry, Astronomical. I. Kaitchuck, Ronald H. II. Title.

QB135.H44 1990 90-11908522'.62-dc20 CIP

PREFACE

Most people who do astronomical photometry have had to learn thehard way. Books for the newcomer to this field are almost totally lack-ing. We had to learn by word-of-mouth and searching libraries for whatfew references we could find.

The situation improved markedly after we began our graduate stud-ies in astronomy at Indiana University. We then had access to profes-sional astronomers with many years of experience in photometry. Indi-ana has a very good astronomical library, and the copy machine wasused heavily by both of us. Nevertheless, information was still gleanedin a piecemeal fashion. It became obvious to us that, as tedious as oureducational process had been, it must be a frustrating experience forthose with more limited reference resources. We were also aware of themany "tricks" we had learned which somehow never found their wayinto print. With this in mind, we set out to write a reference text bothto spare the beginner some of the hardships and mistakes we encoun-tered, and to encourage others to share in the satisfaction of doingmeaningful research.

Our basic approach was to create a self-contained book that could beused by the interested amateur with little or no college background, andby the astronomy major who is new to photometry. By self-contained,we mean the inclusion of sections on observational techniques, construc-tion, and reference material such as standard stars. In addition, weadded substantial theoretical background material. The more esotericmaterial was placed in appendices at the back of the book, therebyretaining the beginning level throughout the bulk of the manual yet pro-viding heavier reading for the most advanced student.

Photoelectric photometry is a relatively small field of science, andtherefore does not have the large commercial suppliers of instrumen-

vi PREFACE

tation. We have tried wherever possible to indicate sources of equip-ment, not to recommend any particular brand but to indicate startingpoints for any equipment selection procedure. Any implied endorsementis unintentional.

Similarly, we advocate certain techniques in both the data acquisitionand reduction. There are as many methods in photoelectric astronomyas there are observers and we will certainly have made some arguablestatements. We have tried to only present techniques that we have usedand found successful.

This book would not have been possible without the dedicated helpof Professor Martin S. Burkhead, who instructed us in observationaltechniques and acquainted us with Indiana University facilities, andProfessor R. Kent Honeycutt, who provided much of our theoreticalbackground knowledge by course material and stimulating conversa-tions. Both these professors, Russ Genet and Bob Cornett have proof-read much of the text for which we are very grateful. We would alsolike to thank Thomas L. Mullikin for writing the section on occultationtechniques. Our wives contributed more time and effort into proofread-ing and correcting than we would like to admit!

We hope that reading this book will instill in you the excitement andsatisfaction that we have found in astronomical photometry. Good luck!

Arne A. HendenRonald H. Kaitchuck

CONTENTS

Preface / v

1. AN INTRODUCTION TO ASTRONOMICAL PHOTOMETRY / 1

1.1 An Invitation / 11.2 The History of Photometry / 51.3 A Typical Photometer / 91.4 The Telescope / 111.5 Light Detectors / 13

a. Photomultiplier Tubes / 13b. PIN Photodiodes / 18

1.6 What Happens at the Telescope / 231.7 Instrumental Magnitudes and Colors / 251.8 Atmospheric Extinction Corrections / 281.9 Transforming to a Standard System / 291.10 Other Sources on Photoelectric Photometry / 30

2. PHOTOMETRIC SYSTEMS / 33

2.1 Properties of the UBV System / 342.2 The UBV Transformation Equations / 372.3 The Morgan-Keenan Spectral Classification System / 382.4 The M-K System and UBV Photometry / 42

*2.5 Absolute Calibration / 502.6 Differential Photometry / 522.7 Other Photometric Systems / 54

a. The Infrared Extension of the UBV System / 55b. The StrOmgren Four-Color System / 55c. Narrow-Band H/? Photometry / 57

v.i

viii CONTENTS

3. STATISTICS / 60

3.1 Kinds of Errors / 603.2 Mean and Median / 613.3 Dispersion and Standard Deviation / 643.4 Rejection of Data / 663.5 Linear Least Squares / 68

*a. Derivation of Linear Least Squares / 69b. Equations for Linear Least Squares / 70

3.6 Interpolation and Extrapolation / 73a. Exact Interpolation / 74b. Smoothed Interpolation / 76c. Extrapolation / 77

3.7 Signal-to-Noise Ratio / 773.8 Sources on Statistics / 78

4. DATA REDUCTION / 80

4.1 A Data-Reduction Overview / 804.2 Dead-Time Correction / 814.3 Calculation of Instrumental Magnitudes and Colors / 854.4 Extinction Corrections / 86

a. Air Mass Calculations / 86b. First-order Extinction / 88

*c. Second-order Extinction / 904.5 Zero-Point Values/ 914.6 Standard Magnitudes and Colors / 924.7 Transformation Coefficients / 934.8 Differential Photometry / 95

*4.9 The (U-B) Problem / 98

5. OBSERVATIONAL CALCULATIONS / 101

5.1 Calculators and Computers / 1015.2 Atmospheric Refraction and Dispersion / 104

a. Calculating Refraction / 104b. Effect of Refraction on Air Mass / 106c. Differential Refraction / 107

5.3 Time / 108a. Solar Time/ 108b. Universal Time / 109c. Sidereal Time / 110

CONTENTS ix

d. Julian Date/ 112*e. Heliocentric Julian Date / 113

5.4 Precession of Coordinates / 1165.5 Altitude and Azimuth / 119

*a. Derivation of Equations / 119b. General Considerations / 122

6. CONSTRUCTING THE PHOTOMETER HEAD / 124

6.1 The Optical Layout / 1246.2 The Photomultiplier Tube and Its Housing / 1286.3 Filters/ 1346.4 Diaphragms / 1386.5 A Simple Photometer Head Design / 1416.6 Electronic Construction / 1476.7 High-Voltage Power Supply / 149

a. Batteries / 149b. Filtered Supply/ 150c. RF Oscillator / 153d. Setup and Operation / 155

6.8 Reference Light Sources / 1556.9 Specialized Photometer Designs / 157

a. A Professional Single-beam Photometer / 157b. Chopping Photometers/ 159c. Dual-beam Photometers / 161d. Multifilter Photometers / 163

7. PULSE-COUNTING ELECTRONICS / 167

7.1 Pulse Amplifiers and Discriminators / 1677.2 A Practical Pulse Amplifier and Discriminator / 1707.3 Pulse Counters / 1727.4 A General-Purpose Pulse Counter / 1737.5 A Microprocessor Pulse Counter / 1787.6 Pulse Generators / 1817.7 Setup and Operation / 182

8. DC ELECTRONICS / 184

8.1 Operational Amplifiers / 1858.2 An Op-Amp DC Amplifier / 1888.3 Chart Recorders and Meters / 193

x CONTENTS

8.4 Voltage-to-Frequency Converters/ 1958.5 Constant Current Sources / 1968.6 Calibration and Operation / 197

9. PRACTICAL OBSERVING TECHNIQUES / 202

9.1 Finding Charts / 202a. Available Positional Atlases / 203b. Available Photographic Atlases / 204c. Preparation of Finding Charts / 205d. Published Finding Charts / 206

9.2 Comparison Stars / 207a. Selection of Comparison Stars / 208b. Use of Comparison Stars / 209

9.3 Individual Measurements of a Single Star / 210a. Pulse-counting Measurements / 210b. DC Photometry/ 213c. Differential Photometry/ 216d. Faint Sources/ 218

9.4 Diaphragm Selection / 220*a. The Optical System / 220*b. Stellar Profiles / 223

c. Practical Considerations / 224d. Background Removal / 226e. Aperture Calibration / 227

9.5 Extinction Notes / 2289.6 Light of the Night Sky / 2299.7 Your First Night at the Telescope / 231

10. APPLICATIONS OF PHOTOELECTRIC PHOTOMETRY / 238

10.1 Photometric Sequences / 23810.2 Monitoring Flare Stars / 24010.3 Occultation Photometry / 24510.4 Intrinsic Variables / 248

a. Short-period Variables / 249b. Medium-period Variables / 250c. Long-period Variables / 254d. The Eggen Paper Series / 254

10.5 Eclipsing Binaries / 256

CONTENTS xi

10.6 Solar System Objects / 27010.7 Extragalactic Photometry / 27210.8 Publication of Data / 273

APPENDICES / 279

A. First-Order Extinction Stars / 279B. Second-Order Extinction Pairs / 286C. UBV Standard Field Stars / 290D. Johnson UBV Standard Clusters / 297

D.I Pleiades/298D.2 Praesepe / 298D.3 1C 4665 / 302

E. North Polar Sequence Stars / 305F. Dead-Time Example / 308G. Extinction Example / 311

G.I Extinction Correction for Differential Photometry / 311G.2 Extinction Correction for "All-Sky" Photometry / 313G.3 Second-Order Extinction Coefficients / 320

H. Transformation Coefficients Examples / 322H.I DC Example / 322H.2 Pulse-counting Example / 327

I. Useful FORTRAN Subroutines / 3351.1 Dead-Time Correction for Pulse-Counting Method / 3361.2 Calculating Julian Date from UT Date / 3361.3 General Method for Coordinate Precession / 3371.4 Linear Regression (Least Squares) Method / 3381.5 Linear Regression (Least Squares) Method Using the UBV

Transformation Equations / 3391.6 Calculating Sidereal Time / 3401.7 Calculating Cartesian Coordinates for 1950.0 / 341

J. The Light Radiation from Stars / 342J.I Intensity, Flux, and Luminosity / 342J.2 Blackbody Radiators / 349J.3 Atmospheric Extinction Corrections / 351J.4 Transforming to the Standard System / 355

K. Advanced Statistics / 358K.I Statistical Distributions / 358K.2 Propagation of Errors / 361K.3 Multivariate Least Squares / 363

xii CONTENTS

K.4 Signal-to-Noise Ratio / 366a. Detective Quantum Efficiency / 367b. Regimes of Noise Dominance / 370

K.5 Theoretical Differences Between DC and Pulse-CountingTechniques / 372a. Pulse Height Distribution / 372b. Effect of Weighting Events on the DQE / 373

K.6 Practical Pulse-DC Comparison / 377K.7 Theoretical S/N Comparison of a Photodiode and a

Photomultiplier Tube / 378

INDEX / 389

ASTRONOMICALPHOTOMETRY

CHAPTER 1AN INTRODUCTION TO ASTRONOMICAL

PHOTOMETRY

1.1 AN INVITATION

In the direction of the constellation Lyra, at a distance of 26 light years,there is a star called Vega. Unknown to the ancients who named thisstar, its surface temperature is almost twice that of the sun and eachsquare centimeter of its surface radiates over 175,000 watts in the vis-ible portion of the spectrum. This is roughly 100 times the power of allthe electric lights in a typical home, radiating from a spot a littlesmaller than a postage stamp. After traveling for 26 years, the lightfrom Vega reaches the neighborhood of the sun diluted by a factor of10~39. Of this remaining light, approximately 20 percent is lost byabsorption in passing through the earth's atmosphere. Approximately30 percent is lost by scattering and absorption in the optics of a tele-scope. A 25-centimeter (10-inch) diameter telescope pointed at Vegawill collect only one half-billionth of a watt at its focus. Of this, only afraction is actually detected by a modern photoelectric detector. Thisincredibly small amount of energy corresponds to one of the brighteststars in the night sky. The amazing thing is that the stars can be seenat all! Perhaps even more amazing is that starlight can be accuratelymeasured by a device which can be constructed at a cost of a fewhundred dollars. Such is the nature of astronomical photometry.

The photometry of stars is of fundamental importance to astronomy.It gives the astronomer a direct measurement of the energy output ofstars at several wavelengths and thus sets constraints on the models ofstellar structure. The color of stars, as determined by measurements attwo different spectral regions, leads to information on the star's tern-

2 ASTRONOMICAL PHOTOMETRY

perature. Sometimes these same measurements are used as a probe ofinterstellar dust. Photometry is often needed to establish a star's dis-tance and size. The Hertzsprung-Russell diagram, the key to under-standing stellar evolution, is based on photometry and spectroscopy.

Finally, many stars are variable in their light output either due tointernal changes or to an occasional eclipse by a binary partner. In bothcases, the light curves obtained by photometry lead to important infor-mation about the structure and character of the stars. The photometryof stars, especially at several wavelengths, is one of the most importantobservational techniques in astronomy.

Astronomy differs from other modern sciences in an importantaspect. Vast commercial laboratories or university facilities are not nec-essary to undertake important research. An amateur astronomer witha modest telescope or persons with access to a small college observatoryequipped with a simple photoelectric photometer can make a valuableand a needed contribution to science. The number of stars, galaxies, andnebulae vastly outnumber the professional astronomers. As an example,there are less than 3000 professional astronomers in the United Statesand yet there are over 25,000 catalogued variable stars. On any givennight, almost all of them go unobserved! Furthermore, an observer witha large telescope will concentrate on faint objects for which a largeinstrument was intended. Very little time is given to the brighter starseven though they are no better understood than the faint ones! A smalltelescope is well suited for these objects and even a simple photometercan produce first-class results in the hands of a careful observer. Thisbook is, in part, an invitation and a guide to the amateur astronomer orpersons with access to small college observatories to share in the satis-faction of astronomical research through photoelectric photometry. InChapter 10, research projects for a small telescope are discussed. How-ever, to give the reader a "feel" for what can be done, we cite twoexamples now. Figure 1.1 shows a light curve for the short-period eclips-ing binary V566 Ophiuchi. This light curve was collected over severalnights using a homemade photometer on a 30-centimeter (12-inch) tel-escope. One result of this study was the discovery of a change in theorbital period, the first such change seen for this binary in 13 years.These changes are believed to be related to mass transfer, from one ofthe two stars to the other, which in turn is related to changes in thestars. Thus, indirectly, photometry allows stellar evolution to be seen!

AN INTRODUCTION TO ASTRONOMICAL PHOTOMETRY 3

1.5r-

0.7 0.8 0.9 0.1 0.2 0.3

PHASE

0.4 0.5 0,7

Figure 1 . 1 Light curve of an eclipsing binary.

Figure 1.2 shows the light curve of the double-mode cepheid TU Cas-siopeiae taken with a 40-centimeter (16-inch) telescope. The scatter inthe light curve is not due to poor photometry, but rather to the beatingof two pulsations, with different periods, that are occurring in this star.Careful determination of both periods allows theorists to determine itsmass, and the temporal variations in the light curve amplitude give con-straints on its evolutionary behavior. Monitoring this type of star overlong periods of time is essential, but difficult for the professional astron-omer to do without preventing colleagues from using the telescope fortheir research.

This book is also directed toward a second type of reader. Oftenundergraduate or graduate students in astronomy are faced with theprospect of beginning their research only to discover that the "how-to"of photometry is lacking in textbooks. Much of the necessary infor-mation, such as lists and finding charts of standard stars, is scatteredthroughout the literature. Hopefully, this book will go a long waytoward solving this problem. This reader will be more interested in theobserving techniques and data reduction and less interested in construc-tion details than the amateur astronomer. In like manner, there are the-oretical sections that will be of less interest to the amateur. These sec-tions have been either marked by an asterisk or placed in theappendices. The amateur astronomer may read or skip these sections as


7.20 h

7-40

7.60

7.80

8.00

0.40

I 0.60

0.80

0.30

0.40

0.50

_l L I *l I 1 L

I I 1 I I I I I I I I I I-0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20

PHASE

Figure 1.2 Light curve of a double-mode cepheid.

they are not mandatory for the construction and successful use of aphotometer.

Overall, this book is intended to be a thorough guide from theorythrough practical circuits and construction hints to worked examples ofdata reduction. As a first step, we discuss the history of photometry andthen consider a layout of a photoelectric photometer.


1 .2 THE HISTORY OF PHOTOMETRY

A person need not own a telescope or a photometer to know that starsdiffer greatly in apparent brightness. It is therefore not surprising thatthe first attempt to categorize stars predates the telescope and wasbased solely on the human eye. Over 2000 years ago, the Greek astron-omer Hipparchus divided the naked eye stars into six brightness classes.He produced a catalog of over 1000 stars ranking them by "magni-tudes" one through six, from the brightest to the dimmest. In about A.D.1 80, Claudius Ptolemy extended the work of Hipparchus, and from thattime, the magnitude system became part of astronomical tradition. In1856, N. R. Pogson confirmed HerchePs earlier discovery that a firstmagnitude star produces roughly 100 times the light flux1 of a sixthmagnitude star. The magnitude system had been based on the humaneye, which has a nonlinear response to light. The eye is designed to sup-press differences in brightness. It is this feature of the eye which allowsit to go from a darkened room into broad daylight without damage. Aphotomultiplier tube or a television camera, which responds linearly,cannot handle such a change without precautionary steps. It is this samefeature which makes the eye a poor discriminator of small brightnessdifferences and the photomultiplier tube a good one. Pogson decided toredefine the magnitude scale so that a difference of five magnitudes wasexactly a factor of 100 in light flux. The light flux ratio for a one-mag-nitude difference is 1001/5 or 102/5 or 2.512. This definition is oftenreferred to as a Pogson scale. The flux ratio for a two magnitude dif-ference is (102/5)2 and a three magnitude difference is (102/5)3 and soon. In general,

F,/F2 = (102/T'-"" (1.1)

where F,, F2 and m}, m2 refer to the fluxes and magnitudes of two stars.This can be rewritten as

log(F,/F2) = %(m2- m}) (1.2)

or

ml- m2= -2.51ogF,/F2. (1.3)tSee Appendix J for a discussion of flux, intensity, luminosity, and blackbody radiation.


Note that the 2.5 is exact and not 2.512 rounded off. Equation 1.1 tellsus that the eye responds in such a way that equal magnitude differencescorrespond to equal flux ratios. Pogson made his new magnitude scaleroughly agreewith theold one by defining the stars Aldebaran and Altairas having a magnitude of 1.0.

The human eye can generally interpolate the brightness of one starrelative to nearby comparisons to about 0.2 magnitude. This is anacceptable error for certain programs such as the monitoring of long-period, large-amplitude variable stars. Because of the speed of mea-surement, visual photometry can be performed in sky conditions unsuit-able for other forms of measurement. However, many problems existwith visual photometry, not the least of which are systematic errors suchas color sensitivity differences between observers, difficulty in extrapo-lating to fainter stars, and lack of accuracy. The latter can be reducedsomewhat by mechanical means introduced in the nineteenth century,so that the light from a variable artificial star visible to the observer canbe adjusted to the same brightness as the object being measured. Thistype of photometer was invented by ZOllner and reduced the error toabout 0.1 magnitude. A brief description of this device can be found inMiczaika and Sinton.'

Photography was quickly applied to photometry by Bond2 and othersat Harvard in the 1850s. The density and size of the image seemed tobe directly related to the brightness of the star. However, the magni-tudes determined by the photographic plate are not, in general, thesame as those determined by the eye. The visual magnitudes are deter-mined in the yellow-green portion of the spectrum where the sensitivityof the eye reaches a peak. The peak sensitivity of the basic photographicemulsion is in the blue portion of the spectrum. Magnitudes determinedby this method are called blue or photographic magnitudes. The morerecent panchromatic photographic plates can yield results whichroughly agree with visual magnitudes by placing a yellow filter in frontof the film. Magnitudes obtained in such manner are referred to as pho-tovisual. Photographic photometry quickly showed that the old visualscale was not accurate enough for photographic work. What was neededwas a new system, based on photographic photometry, and defined bya large number of standard stars. The unknown magnitude of a starcould then be found by comparing it to the standards and applyingEquation 1.1, where the flux ratio is determined by the image densitieson the film. Because the brightest stars are not always well positioned


for observation, a group of standard stars was defined in the vicinity ofthe north celestial pole. For a Northern Hemisphere observer, thesestars would always be above the horizon. The group became known asthe North Polar Sequence. Their magnitudes were defined so that thebrightest stars in the sky would still be close to the photovisual magni-tude of one. This system became known as the International System.At Mount Wilson Observatory, stars in 139 selected regions of the skywere established as secondary standards by comparison with the NorthPolar Sequence. Some of these stars were as faint as nineteenth mag-nitude. However, a large departure from Pogson's scale had occurredfor the fainter stars because the nonlinearity of the photographic platewas not properly taken into account. Photographic photometry is still inuse today, but primarily as a method of interpolating between nearbycomparison stars, giving an error of about 0.02 magnitude. Photographyoffers a permanent record with a vast multiplexing advantage: thou-sands of images are recorded at one time.

Because of the difficulties inherent in visual and photographic meth-ods, the application of the photoelectric method of measuring starlightin the late 1800s ushered in a new era in astronomy. Most early worksuch as that of Minchin3 used selenium photoconductive cells whichchanged their resistance upon exposure to light. These cells are similarto the photocells found in some modern cameras. A constant voltagesource was applied to the cell and the resulting variable current wasmeasured with a galvanometer (a very sensitive current indicator.) Agalvanometer is not used very often at present, primarily because of itsbulk and its difficult calibration and operation. Joel Stebbins and F. C.Brown4 were the first to use the selenium ceil in the United States. Steb-bins and his students were involved in most of the later development ofphotoelectric photometry (see Kron5 for more details).

Some of the major disadvantages of the selenium cell were its lowsensitivity (only bright stars and the moon were measured), narrowspectral response, and lack of commercial availability. Each cell had tobe made individually, and it often took dozens of trials to produce asensitive cell. Even so, in 1910 Stebbins6 published a light curve of Algolof far greater precision than ever before, showing for the first time theshallow secondary eclipse that had eluded visual observers.

The discovery of the photoelectric cell in 1911 promised more sensi-tive measurements. These cells were similar to a tube-type diode usingsodium, potassium, or other alkaline electrodes. A voltage of approxi-


mately 300 volts was applied, and when the cell was exposed to light,electrons liberated by the photoelectric process created a small current.This response was linear, that is, a source twice as bright gave twice thecurrent. Schultz,7 working with Stebbins, used the photoelectric cell torecord light from Arcturus and Capella. Similar systems were beingdeveloped in Europe by Guthnick8 and Rosenberg.9 For many years, theproblems associated with selenium cells plagued the newer design.Commercial photoelectric cells were not available until the 1930s. Gal-vanometers hung directly on the telescope and had to be kept level. Thelimit of detectability for the photoelectric cell-galvanometer combina-tion was about a seventh magnitude star for a 40-centimeter (16-inch)telescope. The reader is referred to Stebbins10 for details about theseearly measurements.

The electronic amplifier was introduced into astronomy by Whit-ford," stepping up the feeble photocurrents to the point where lessexpensive meters and, more importantly, chart recorders could be used.At the same time, however, tube thermionic noise and amplifier insta-bilities were now problems and became the limiting component of aphotoelectric system. The late 1920s and the 1930s also saw the adventof wide-band filters and the increasing adoption of the photoelectricphotometer.

The invention of the electron multiplier tube or photomultiplier in thelate 1930s was an important advance for astronomy. This tube is essen-tially a photocell with the addition of several cascaded secondary elec-tron stages which allow noiseless amplification of the photocurrent.Whitford and Kron12 used a prototype photomultiplier for automaticguiding. RCA introduced the 931 photomultiplier just prior to WorldWar II and the 1P21 during the war. Kron13 was the first to use thesetubes for astronomical purposes. With the prototype tubes and a gal-vanometer, eleventh magnitude stars were measured on the Lick 36-inch refractor.

It became clear with the development of photoelectric techniques thatthe North Polar Sequence had not been established with enough accu-racy. The new photoelectric magnitude systems are now defined by thechoice of filters, photomultiplier tube and a network of standard stars.The definition of these systems are taken up in detail in Chapter 2.

Recent years have seen improvements on existing photometric sys-tems, but no major changes. Various filter combinations and newer pho-tocathode materials extending measurements from the near-ultraviolet


to the near-infrared are being used. Less noisy amplifiers and pulsecounting techniques have been developed to retrieve the feeble pulsedcurrent. Innovative new designs that are in the prototype stage at thiswriting promise a bright future for the photoelectric measurement ofstarlight.

1.3 A TYPICAL PHOTOMETER

The heart of any photometer is the light detector. This device isexplained in detail in Section 1.5. For now, it is sufficient to say it is adevice that produces an electric current which is proportional to thelight flux striking its surface. The output of the detector must be ampli-fied before it can be measured and recorded by a device such as a strip-chart recorder. The detector is mounted in an enclosure on the telescopecalled the head, which allows only the light from a selected star to reachthe light-sensitive element. Figure 1.3 shows the principal componentsof a photoelectric photometer, when the light detector is a photomulti-plier tube. The telescope shown is a Cassegrain type, but any type maybe used. The components enclosed by a dashed line are contained in thehead, with its relative size exaggerated for clarity.

The first component is a circular diaphragm whose function is toexclude all light except that coming from a smal! area of sky surround-

DIAPHRAGM PHOTOMULTIPL1ERTUBE

~1

)

H)(l>AMPLIFIER

PULSE COUNTEROR

CHART RECORDER

Figure 1.3 A typical photometer.


ing the star under study. The sky background between the stars is nottotally dark for a number of reasons, not the least of which is city lightscattered by dust particles in the atmosphere. Some of this backgroundlight also enters the diaphragm. The telescope must be offset from thestar in order to make a separate measurement of the sky background,which can then be subtracted from the stellar measurement. The sizeof a stellar image at the focal plane of the telescope will vary withatmospheric conditions. Some nights it may seem nearly pinpoint in sizewhile on other nights atmospheric turbulence may enlarge the imagegreatly. For this reason, a slide containing apertures of various sizesreplaces the single diaphragm. To keep the effects of the sky back-ground at a minimum, it is advantageous to use a small diaphragm. Onthe other hand, this puts great demands on the telescope's clock driveto track accurately for the duration of the measurement. On any givennight, a few minutes of trial and error are necessary to determine thebest diaphragm choice.

The next component is a diaphragm viewing assembly. This consistsof a movable mirror, two lenses, and an eyepiece. Its purpose is to allowthe astronomer to view the star in the diaphragm to achieve proper cen-tering. When the mirror is swung into the light path, the diverging lightcone is directed toward the first lens. The focal length of this lens isequal to its distance from the diaphragm (which is at the focal point ofthe telescope). This makes the light rays parallel after passing throughthe lens. The second lens is a small telescope objective that refocusesthe light. The eyepiece gives a magnified view of the diaphragm. Oncethe star is centered, the mirror is swung out of the way and the lightpasses through the filter. As with the diaphragm, this is part of a slideassembly that allows different filters to be selected. The choice of filtersis dictated by the spectral regions to be measured and is discussed inChapter 6.

The next component is the Fabry lens. This simple lens is very impor-tant. Its purpose is to keep the light from the star projected on the samespot on the detector despite any motions the star may have in the dia-phragm because of clock drive errors or atmospheric turbulence. Thisis necessary because no photocathode can be made with uniform lightsensitivity across its surface. Without the Fabry lens, small variationsin the star's position would cause false variations in the measurements.The focal length of this lens is chosen so that it projects an image of theprimary mirror, illuminated by the light of the star, on the detector.


The final component in the photometer head is the photomultipliertube. It is usually housed in its own subcompartment with a dark slideso that it can be made light-tight from the rest of the head. The tube issurrounded by a magnetic shield that prevents external fields fromdeviating the paths of the electrons and hence changing the output ofthe tube. Details on the construction of the photometer head are dis-cussed in Chapter 6.

1.4 THE TELESCOPE

Before the reader rushes out to buy parts and start construction of thatshiny new photometer, there is a very important practical considerationto be tackled. Take a good, hard look at your telescope. Most amateur-built telescopes, and even those commercially made for amateurs, arenot directly suitable for photometry. The problem is usually not optical,but rather mechanical. These telescopes are seldom designed to carrythe weight of a photometer head at the focal plane. Even the simplesthead containing an uncooled detector weighs in the neighborhood of 4.5kilograms (10 pounds). The telescope should be capable of being rebal-anced to carry this load and the clock drive must still be capable oftracking smoothly. Furthermore, the mount must be sturdy enough sothat small gusts of wind do not shake the telescope and move the starout of the diaphragm.

If your telescope has a portable mount, there should be some provi-sion for attaining an equatorial alignment to better than 1 °. There areseveral techniques of alignment that have been discussed in theliterature.14'15'16'17 The clock drive must have sufficient accuracy to keepa star centered in a diaphragm long enough to make a measurement.Typically, this means 5 minutes when using a diaphragm size of 20 arcseconds. Many clock drive systems have difficulty doing this. It is notuncommon for amateur drive systems to suffer from periodic trackingerror. This is because of cutting errors in making the worm gear, andresults in the telescope oscillating between tracking too slowly and toofast. The cure is to use a large worm gear of good quality. It is essentialto have slow-motion controls on both axes. It is nearly impossible tocenter a star in a small diaphragm by hand. For right ascension, theslow-motion control can be the standard variable frequency drive cor-rector in common use today. The mechanical declination slow-motioncontrols supplied by most telescope manufacturers are far too coarse.


The declination motion should be as slow as the right ascension slowmotion. It may be possible to gear down an existing system that is toocoarse. An especially convenient method is to motorize the declinationmotion and then operate both axes by pushbuttons in a single handcontrol.

There are some requirements of the optical system as well. First ofall, a large F-ratio is preferred. A small F-ratio produces a light conethat diverges very rapidly inside the head. This means that the compo-nents must be placed uncomfortably close together near the focal point.Photometers have been placed on telescopes with F-ratios as small asfive. However, an F-ratio of eight or larger is recommended. A large F-ratio has a second advantage. It is highly desirable that the angulardiameter of a diaphragm on the sky be kept as small as possible. Thisreduces the sky background light that enters the photometer. With ashort F-ratio telescope, this becomes difficult since the diaphragm holescannot be drilled small enough with a conventional drill press.

Another important consideration is the location of the focal point.Some telescopes are designed so that the prime focus never extends out-side the drawtube. However, the diaphragm must be placed at theprime focus (see Figure 1.3). It may be necessary to move an opticalelement in the telescope to accomplish this.

Finally, the choice of the optical system itself is important. Refract-ing telescopes have very serious disadvantages. The glass of the objec-tive lens does not transmit ultraviolet light. Hence, the U magnitude ofthe UBV system cannot be measured. Note that this problem alsoapplies to Schmidt-Cassegrain telescopes (like the Celestron) though toa lesser extent, since the lens is very thin. A second problem with refrac-tors is chromatic aberration. No matter how well the lens is made, notall wavelengths have a common focal point. The modern achromaticlens minimizes this effect, but perfect correction is not possible. Whenthe diaphragm is at the focal point of blue light, some of the red lightis excluded from the photometer, because the red light cone is too wideto pass through the diaphragm. The only solution is to use very largediaphragms that allow a large amount of sky background light to reachthe detector. This makes the measurement of faint stars very difficultbecause the detector sees more sky background light than star light.

Thus, the Newtonian and Cassegrain telescopes are preferred. How-ever, there is still a potential problem. Most small reflecting telescopescome with mirrors which have been overcoated with silicon monoxide.


As the coating ages, it converts to silicon dioxide, which does not trans-mit ultraviolet light as well. The solution is to keep the overcoatingalways fresh, or not to overcoat the mirrors, or simply plan not to doany ultraviolet measurements.

We recommend that you modify or improve your telescope beforeyou spend a very frustrating night of attempting photometry with aninadequate telescope. Lest we end this section on too negative a note, itshould be emphasized that these modifications are well worth the effortand will result in a much better telescope.

1.5 LIGHT DETECTORS

Since the late 1940s, the most commonly used light detector in astron-omy has been the photomultiplier tube. However, a solid-state detectorknown as the photodiode may well become important in the near future.We discuss each of these devices in turn in this section.

1.5a Photomultiplier Tubes

The key to the operation of the photomultiplier tube is the photoelectriceffect, discovered in 1887 by Heinrich Hertz. He found that when lightstruck a metal surface, electrons were released, with the number of elec-trons released each second being directly proportional to the light inten-sity. The photoelectric effect is perfectly linear in this regard. Thekinetic energy of the released electrons depends on the frequency of thelight source and not on its brightness. For a given metal, there is a cer-tain minimum frequency below which no electrons are released no mat-ter how intense the light source may be.

The explanation of the photoelectric effect was given by Albert Ein-stein in 1905 for which he was later awarded the Nobel Prize. He pic-tured light as a stream of energy "bullets" or photons, each containingan amount of energy directly proportional to the frequency andinversely proportional to the wavelength of the light. Because electronsare bound to the metal by electrical forces, a certain minimum energyis required to free an electron. When an electron absorbs a photon, itgains the photon's energy. However, unless the frequency is above acertain value, the energy is insufficient for the electron to escape themetal. For frequencies higher than this threshold value, the electron canescape and any excess energy above the threshold becomes the kinetic


energy of the electron. For all frequencies above the threshold value,the number of electrons released is directly proportional to the numberof photons striking the metal surface.

There are other ways of releasing electrons from a metal surfacewhich are also of interest. Thermionic emission is essentially the sameas the photoelectric effect except that the energy that releases the elec-trons comes from heating of the metal rather than from light. Second-ary emission is the release of electrons because of the transfer of kineticenergy from particles that hit the metal surface. Finally, field emissionis the removal of electrons from the metal by a strong external electricfield. All of the above effects come into play in a photomultiplier tube.

Most photomultiplier (PM) tubes are about the size of the old-fash-ioned vacuum tubes used in radios and televisions. The components ofthe tube are contained by a glass envelope in a partial vacuum, so thatthe electrons can travel freely without colliding with air molecules. Fig-ure 6.2 shows a photograph of an RCA 1P21 PM tube. The heart ofthe tube is the metal surface that releases the photoelectrons. Since thissurface is at a large negative voltage with respect to ground, it is calledthe photocathode.

Photocathodes are not constructed of simple, common metals butrather a combination of metals (antimony and cesium in the case of the1P21). The metals are chosen to give the desired spectral response andlight sensitivity. For a typical photocathode material, the quantumefficiency is about 10 percent. (Of every 100 incident photons, only 10will be successful in releasing a photoelectron. The energy from theremaining 90 photons is absorbed by the metal and dissipated in otherways.) The current produced by the photoelectrons is very weak anddifficult to measure even for bright stars. For this reason, the early useof photocells met with limited success.

The PM tube differs from the photocell in that the PM tube amplifiesthis current internally. In order to accomplish this, the photoelectronsreleased by the photocathode are attracted to another metal surface byan electric potential. This metal surface is called a dynode and in the1P21 it is at a potential 100 V less negative than the photocathode. Asa result, this dynode looks positive compared to the photocathode. Pho-toelectrons are accelerated toward its surface, and the impact of eachreleases about five more electrons by the process of secondary emission.These electrons are in turn accelerated toward another dynode that is100 V less negative than the previous dynode. Once again, the processof secondary emission releases about five electrons per incident electron.


This process is then repeated at other dynodes. The 1P21 has ninedynodes, so for each photoelectron emitted at the photocathode thereare 59 or two million electrons emitted at the last dynode. This tube issaid to have an internal gain of two million. These electrons are thencollected at a final metal surface, called the anode, from which theyflow through a wire to the external electronics.

Figure 1.4 shows the arrangement of the photocathode, dynodes, andanode inside a 1P21. The arrows show the paths of the electrons (forsimplicity, not all the electron paths are shown). There are other PMtube designs and Figure 1.4 shows, schematically, the "Venetian blind,"and the "box-and-grid" types. The 1P21 is called a "squirrel-cage"design. Note that the 1P21 is a "side-window" design while the otherspictured in Figure 1.4 are examples of "end-on" tubes.

The current amplification produced by the dynode chain is anextremely important characteristic of the PM tube. This amplificationis essentially noise-free. Unlike the early photocells, far less externalamplification is required. As a result, the external amplifier noise is rel-atively unimportant.

While the amplification process of a PM tube is noise-free, there are,unfortunately, noise sources within the tube. Noise is defined as anyoutput current that is not the result of light striking the photocathode.With the PM tube sitting in total darkness, with the high voltage on,there is a so-called "dark current" which is produced by the tube. Thiscurrent is a result of electrons released at the dynodes by thermionicand field emission. Even at room temperature, the dynodes are warm

1P21 TOP VIEW LIGHT

TRANSPARENTPHOTOCATHODE

LIGHT

ANODE

PHOTOCATHODELIGHT

GRILLANODE

BOX-AND-GRIDTYPE

VENETIAN-BLINDTYPE

Figure 1.4 Photomultiplier tube designs.


enough for an electron to be released occasionally. When this happens,the electron is accelerated and amplified by the remaining dynodechain.

The obvious solution to large dark currents is to reduce the temper-ature of the tube. Most professional astronomers cool the PM tube withdry ice, almost totally eliminating thermionic emission. The amateurastronomer need not exert this much effort as an uncooled tube is stillvery useful. The only problem is that very faint stars are difficult todetect because the current they produce at the anode may be as smallor smaller than the dark current. There is little that can be done toeliminate field emission because the tube must contain strong electricfields. However, in practice this noise source is very small compared tothermionic emission.

The current that leaves the anode is still very weak and requiresamplification before it can be easily measured. There are two generalways to accomplish this. Because each photoelectron produces a burstof electrons at the anode, a pulse amplifier can be used to amplify eachburst and convert it to a voltage pulse that can be counted electroni-cally. The number of pulses counted in a given time interval is a mea-sure of the number of photons that strike the photocathode in the sametime interval. (We use the terms pulse counting and photon countinginterchangeably whenever referring to the technique of counting indi-vidual photoelectron pulses caused by an incident photon on the pho-tocathode of a photomultiplier tube.) The second technique is to use aDC amplifier and to smooth the bursts to look like a continuous current.This current is amplified and measured by a meter or a strip-chartrecorder. Both techniques are discussed in detail in Chapters 7 and 8.

No photocathode material releases the same numb.er of electrons atall wavelengths even when the light source is equally bright at all wave-lengths. The spectral response of a photomultiplier is an important char-acteristic to know. Figure 1.5 shows the spectral responses of a fewtypes of photocathode materials. The most common in astronomical usetoday is the cesium antimonide (Sb-Cs) surface, used in the first mass-produced photomultiplier, the RCA 931 A. The RCA 1P21 is the suc-cessor to this tube and is used to define the VBV photometric system.The spectral response of this surface is labeled "S-4" in Figure 1.5 (the"S" numbers refer to different spectral responses). The light sensitivitypeaks near 4000 A, cutting off at the blue end near 3000 A while on thered end there is a long tail to 6000 A. Individual tubes vary and some-

AN INTRODUCTION TO ASTRONOMICAL PHOTOMETRY 1 7

2000 4000 6000 8000 10,000

WAVELENGTH (ANGSTROMS)

Figure 1.5 Photocathode spectral response.

times the response goes beyond 7000 A, producing a problem when blueor ultraviolet filters are used. These filters transmit some light in thered and the tube detects red light passed by these filters. For red stars,this "red leak" can cause an error in a blue magnitude of a few percent.This problem is discussed later.

In Figure 1.4, two types of photocathodes are illustrated. The 1P21is a side-window device. The light strikes the front surface of the pho-tocathode and electrons are released from the same front surface. Thisis called an opaque photocathode. With a semitransparent photocath-ode, used with "end-on" PM tubes, the light strikes the front surfaceand the electrons are released from the back surface. These two typesof cathodes, even if made of the same material, have a slightly differentspectral response. The semitransparent photocathodes tend to be morered-sensitive. For this reason, a semitransparent photocathode made ofSb-Cs is designated S-ll, not S-4.

Another important photocathode material is the so-called "tri-alkali"


designated as S-20. Figure 1.5 shows that this material covers much thesame spectral range as the S-4 but with much useful sensitivity in thenear-infrared. This material is extremely sensitive in blue light and hasa quantum efficiency of 20 percent. By contrast, the Ag-O-Cs, S-l sur-face has a very low quantum efficiency of a few tenths of a percent.However, it has an extremely broad response up to 11,000 A. There isa wide dip in its sensitivity centered at 4700 A. The advantage of theS-l photocathode is that a single tube can be used to measure from theblue to the infrared. The disadvantage, of course, is that you are limitedto fairly bright objects.

Noise is a problem with PM tubes designed for infrared work.Infrared photons carry very little energy, which means the photocath-ode must be made of a material in which the electrons are bound veryloosely. Unfortunately, this means they are very easy to release ther-mally. Hence all infrared tubes are cooled with dry ice, and detectorsfor the far-infrared are cooled to an even lower temperature with liquidnitrogen or liquid helium.

1.5b PIN Photodiodes

To date, very little experimentation with photodiodes for astronomicalphotometry has been published. However, these devices look verypromising.18-19-20-2'122-23 A well-designed photodiode photometer is nowcommercially available from Optec, Inc.24 To understand the photo-diode, we will review the operation of an ordinary diode briefly. Morecomplete explanations can be found in most elementary electronicstexts.

In an isolated atom, electrons are confined to orbits about thenucleus, which correspond to sharply defined energy levels. When atomsare linked in a crystal of a solid, the energy level structure is quite dif-ferent. In a simplified view, the energy levels become two distinct bands.The lower band "contains" all the electrons (at least at very low tem-peratures) while the upper band is empty. There is a gap between thetwo energy bands that represents energy states unavailable to the elec-trons. If an electron somehow receives sufficient energy to reach theupper band, it can move freely through the crystal, unattached to anyone atom. An external electric field easily can cause these electrons tomove. For this reason, the upper band is called the conduction band.The electrons in the lower band are involved in the chemical bonds to

AN INTRODUCTION TO ASTRONOMICAL PHOTOMETRY 1 9

neighboring atoms in the crystal, so this band is called the valence band.For solids that are insulators, the gap between the two bands is verylarge. It is very unlikely that an electron from the valence band willreceive enough energy to promote it to the conduction band. Therefore,insulators are poor conductors of electric current. Likewise, a conductoris a material in which the two bands merge and electrons can easilymove into the conduction band. Semiconductors have a small gapbetween energy bands. Germanium (Ge) and silicon (Si) are the twomost commonly used semiconductor materials for making diodes andtransistors.

Semiconductors without impurities are called intrinsic semiconduc-tors. They have a rather low conductivity, but not as low as an insulator.If a semiconductor is "doped" with impurities, its conductivity can beincreased markedly. Si and Ge each have four valence electrons peratom, which are used in bonding to four adjacent atoms when makinga crystal. The process of doping involves replacing a few of these atomswith atoms that have one fewer (three) or one more (five) valence elec-trons. Suppose a Ge crystal is doped with arsenic, which has five valenceelectrons. Four of these electrons are used to bind the atom in the crys-tal with four neighboring Ge atoms. However, the fifth electron isloosely bound with an energy just below the conduction band. This elec-tron cannot be in the valence band since this band is "full." A smallamount of energy promotes this electron into the conduction band.Thus, doping has greatly increased the electrical conductivity. Theimpurity atom, arsenic, in this case is referred to as a donor because itsupplied the extra electron. A semiconductor doped in this way isreferred to as an n-type because a negative charge was donated.

The conduction of the crystal is also increased if it is doped withatoms that have only three valence electrons. These atoms are one shortof completing their bonds with neighboring atoms. Thus, a "hole"exists. This atom has an unfilled energy level and a nearby valence elec-tron can move into this location. Of course, this electron leaves a holebehind. In this way, it is possible for holes to migrate through the crys-tal. This impurity is labeled an acceptor because it accepts a valenceelectron from elsewhere in the crystal. Acceptor-doped crystals arereferred to as p-type semiconductors because the current carriers areholes, or a lack of electrons, which look positive by comparison to theelectrons.

A diode is made by bringing p-type and n-type material together. The


0©

©_ © ©_© ©_ ©0_ ©_ ©_

p-TYPE

0 = ACCEPTORS

+ = HOLE CARRIER

n-TYPE

0 = DONORS

- = ELECTRON CARRIER

'JUNCTION

©+ © +©+ 0 +© 0

0Q0

'©©©

©-©-©^

©_Q©_

p-TYPEDEPLETION

REGION

Figure 1.6 P-N junction.

n-TYPE

surface of contact is referred to as the junction. Holes from the p-sideand electrons from the n-side diffuse across the junction until an equi-librium is reached. The result is a region on either side of the junctionwhere there are no charge carriers because the electrons and holes havecombined to annihilate each other. This is called the depletion region,as shown in Figure 1.6. An electrostatic field is produced across thejunction because the p-side now has an excess of electrons filling theholes and the n-side now has an excess of holes because it has lost elec-trons to the p-side. This results in a situation where an electron fromthe n-side is not likely to cross the junction because it is faced with anexcess of electrons on the p-side that repel it. Similarly, a hole from thep-side will no longer cross to the n-side. If an external electric circuit isconnected to the diode, such that the n-side is connected to a positivepotential and the p-side to a negative potential (called reversed biased),no current will flow. This is because the external potential only increasesthe potential difference across the depletion layer. If the contacts arereversed, the current from the external circuit will tend to neutralizethe charge difference across the junction. The potential difference dropsand current flows. It is in this manner that alternating current can beconverted into direct current, because during only one half of the cycle,when the diode is forward biased, will current be allowed to flow.


Normally, when electronics texts discuss the operation of a diode, aswe have done above, they fail to mention one additional "complication."A graphic illustration of this is seen in the following experiment. Go toyour local electronics store and find a glass-encapsulated diode. With aknife, scrape off the black paint that coats the glass. Connect a volt-meter capable of reading a few tenths of a volt across the diode. Shinea bright light on the diode and watch the meter. The added "compli-cation" is just what makes diodes interesting to astronomers; they arehighly light-sensitive. Light energy absorbed at the p-n junction raisesan electron from the valence band to the conduction band. Such an elec-tron is repelled by the p-side of the depletion region and attracted bythe n-side. The opposite is true for the hole left behind. This process,when repeated over and over as light continues to strike the junction,results in the voltage detected in the above experiment. A diode used tomeasure light in this manner is said to be used in the photovoltaic mode.

In practice, diodes designed for light detection are constructed dif-ferently from ordinary diodes. In the so-called PIN photodiode a p-typelayer (P) is separated from the n-type layer (N) by an intrinsic layer(I). The light is absorbed in the intrinsic layer, creating an electron-holepair. The hole is attracted to the p-material and the electron to the n-materiai after drifting through the I layer. The function of the I layeris to reduce noise current produced by such effects as electron-hole pairscreated by thermal processes. Nevertheless, this is still the major sourceof noise in a photodiode.

There are numerous advantages to a photodiode as a detector in aphotometer. One advantage is seen in Figure 1.7, which shows that ablue-enhanced photodiode is an efficient detector from the ultraviolet tothe infrared. Furthermore, the quantum efficiency of the photodiode ismuch better than the photomultiplier, reaching 90 percent in the near-infrared. Even though an S-l photomultiplier can also span this rangeof wavelengths, it has a quantum efficiency of only a few tenths of apercent. Compared to photomultiplier tubes, photodiodes are also lessexpensive, much smaller, and do not require a high-voltage supply. Itwould thus appear that the professional astronomers should rush toreplace the photomultipliers with photodiodes. The reason this has notoccurred is that the photomultiplier tube still has one very importantadvantage. The dynode chain of the photomultiplier yields an internalcurrent amplification (gain) of about 106. This is not the case for thephotodiode. Therefore, the external electronics must amplify an addi-


4000 6000 8000


10,000J

12,000

Figure 1.7 Approximate quantum efficiency of a blue-enhanced photodiode.

tional factor of 106 and this introduces noise. For a photodiode to becompetitive with a photomultiplier tube, a well-designed amplifier isrequired and both the photodiode and the amplifier should be cooled.The internal gain of a photodiode is unity, which means that pulse-counting techniques cannot be used, so that DC photometry is required.In Appendix K, it is shown that DC is inferior to pulse counting whenit comes to measuring faint stars, but for bright stars the photodiodeworks well. This fact, combined with its convenience, makes the pho-todiode a detector to be seriously considered. Also in Appendix K, atheoretical comparison of the photodiode and the photomultiplier is pre-sented in order to help one decide on the best light detector for one'sobserving program and budget.

The size of the active area (light-sensitive area) of a photodiodeshould be kept fairly small to minimize the noise introduced by ther-mally produced electron-hole pairs. Thus, unlike the photomultiplier,the photodiode is placed at the focus of the telescope. This necessitatessome design changes in the photometer head. Figure 1.8 illustrates theoptical layout schematically when a photodiode is used. The first dif-ference is that no diaphragms are used. The light-sensitive area of thephotodiode is so small (typically 0.5 millimeter across) that it acts as itsown diaphragm. It is not possible to place a viewing eyepiece behind thediaphragm. Instead, the eyepiece must be placed in front of the photo-diode and equipped with a cross hair for centering the star on the pho-todiode. The placement of the photodiode as shown eliminates the need


RECORDER

PHOTODIODE

FILTER MIRROR AMPLIFIER

Figure 1.8 Photodiode photometer.

for a Fabry lens. The spectral response of the photodiode necessitatessome special considerations when choosing filters. These are discussedin Chapter 6.

1.6 WHAT HAPPENS AT THE TELESCOPE

In later chapters, we discuss observing techniques and data reductionin detail. However, for the benefit of the novice, we now outline theobserving procedure and define some terms. The actual observing pat-tern depends on the goal of the project and the form in which the finaldata are needed. In general, one of two techniques is followed. The sim-plest observing scheme, and the one highly recommended to the begin-ner, is differential photometry. In addition to its simplicity, it is the mostaccurate technique for measuring small variations in brightness. Thistechnique is widely used on variable stars, especially short-period vari-ables and eclipsing-binary systems.

In differential photometry, a second star of nearly the same color andbrightness as the variable star is used as a comparison star. This starshould be as near to the variable as possible, preferably within onedegree. This allows the observer to switch rapidly between the two stars.Another extremely important reason for choosing a nearby comparisonstar is that the extinction correction (Section 1.8) can often be ignored,because both stars are seen through nearly identical atmospheric layers.All changes in the variable star are determined as magnitude differ-ences between it and the comparison star. It is important that the com-parison star be measured frequently because the altitude of theseobjects is continuously changing throughout the night. This type of pho-tometry can be extremely accurate (0.005 magnitude) and is highly rec-


ommended where atmospheric conditions can be quite variable, such asthe midwestern United States. Any star that meets the criteria can bea comparison star. However, it is a good idea to pick a second star,called the check star, as a test of the nonvariability of the comparisonstar. The check star need be measured only occasionally during thenight.

The observational procedure is to alternate between the variable andcomparison stars, measuring them a few times in each filter. A "mea-surement" consists of centering the star in the diaphragm and thenmoving the flip mirror out of the light path so that the light can fall onthe detector. You then record the meter reading on your amplifier alongwith the time. If you are using pulse-counting equipment, you recordthe number displayed on your counter. If you are very lucky, yourmicrocomputer can record it for you! Once this has been done for eachfilter, the star is moved out of the diaphragm and the sky backgroundis recorded through each filter. This is necessary since the measure-ments of the star really include the star and the sky background.

The magnitude differences between the variable and comparisonstars in each filter can then be calculated using the expression

mx- mc = -2.5\og(dx/dc) (1.4)

where dx and dc represent the measurement of the variable and the com-parison stars minus sky background, respectively. If different amplifiergains were used for the two stars, this must also be included. An advan-tage of differential photometry is that no calibration to the standardphotometric system is necessary for many projects. The disadvantage isthat your magnitude differences will not be exactly the same as thosemeasured on the standard system. However, if you are using the spec-ified detector and filters, and have matched the color of the comparisonand variable stars, your results will not differ very much (see Section2.6), A further disadvantage is that your final results will be in differ-ences. You will not be able to specify the actual magnitudes or colors ofthe variable star unless you standardize the comparison star. However,these results are good enough for many projects such as determininglight curve shapes or the times of minimum light of an eclipsing binary.

The second technique is the most general and commonly used byprofessional astronomers. It is also the most demanding on the qualityof the sky conditions. In this scheme, numerous program stars, located


in many different places in the sky, are to be measured to determinetheir magnitudes and colors. As before, each star and its sky back-ground are measured through all filters. However, because each star isobserved at a different altitude above the horizon, each is seen througha slightly different thickness of the earth's atmosphere. Therefore,observations must also be made of another set of stars of known mag-nitudes and colors to determine the atmospheric extinction corrections.Finally, a set of standard stars must be observed to determine the trans-formation coefficients so the measurements of the program stars can betransformed into magnitudes and colors of a standard system, such asthe UBVsystem. This procedure often involves less observing time thanit would appear at first glance. This is because it is often possible to usesome of the same observations to determine the extinction correctionsand the transformation coefficients. Furthermore, the transformationcoefficients need only be determined occasionally. Details of the proce-dures are treated in Chapters 4 and 9.

1.7 INSTRUMENTAL MAGNITUDES AND COLORS

It would appear to the beginner that the determination of a star's mag-nitude is fairly simple and, furthermore, that magnitude can be simplyrelated to the star's light flux. Unfortunately, the latter is far from true.To see this more clearly, we rewrite Equation 1.3 as

mt = m2- 2.5 log F, + 2.5 log F2. (1.5)

Suppose star "2" is a reference star of magnitude zero and star "1" isthe unknown. Then

mt = q- 2.5 log Fl (1.6)

where q is a constant. Since there is now only one star the subscript "1"can be dropped in favor of lambda (A) to remind us that the magnitudedepends on the wavelength of observation. Thus,

mx= 4,-2.5 log/v (1.7)

Again, this equation only seems to verify the simple relationshipbetween magnitude and flux. However, the above equation refers to the


observed flux. The observed flux is related to the actual flux in a verycomplicated way. The problems can be broken into two groups: (1)extinction because of absorption or scattering of the stellar radiation onits way to the detector and (2) the departure of the detecting instrumentfrom one with ideal characteristics. We now discuss these two problemsin turn.

There are two sources of absorption of the stellar flux: interstellarabsorption because of dust and absorption within the earth's atmo-sphere. The former is generally neglected for published observations,but the latter is usually taken into account. The earth's atmosphere doesnot transmit all wavelengths freely. For example, ultraviolet light isheavily absorbed. Human life can be thankful for that! Observatoriesat higher elevations have less of the absorbing material above them,while others located near large bodies of water have more water vaporabove them. In addition, the atmosphere scatters blue light much morethan red light.

Not all telescopes transmit light in the same manner and this can bea function of wavelength. For instance, glass absorbs ultraviolet lightheavily, and various aluminum and silver coatings have different wave-length dependences of the reflectivity. Also, in practice it is not possibleto measure the flux from a star at one wavelength. Any filter transmitslight over an interval of wavelengths. Despite the best efforts of themanufacturers, no two filters or light detectors can be made withexactly the same wavelength characteristics. As a result, no two observ-atories measure the same observed flux for a given star.

A calibration process is necessary to enable instruments to yield thesame results. The observed flux, Fx, is related to the actual stellar flux,F*x, outside the earth's atmosphere, by

where

= fractional transmission of the earth's atmosphere= fractional transmission of the telescope

<l>f(\) = fractional transmission of the filter</>o(M = efficiency of the detector (1.0 corresponds to 100 percent).


This expression can be very complicated and the many factors are usu-ally poorly known. It is for this reason that stellar fluxes are very diffi-cult to measure accurately. Fortunately, the determination of stellarmagnitudes does not require a knowledge of most of these factors,except in an indirect manner. The magnitude scheme requires only thatcertain stars be defined to have certain magnitudes, so that magnitudesof other stars can be determined from observed fluxes that are correctedonly for atmospheric absorption. This is why the seemingly awkwardmagnitude system has survived so long.

The only remaining problem is to account for the individual differ-ences among telescope, filter, and detector combinations. This is wherethe set of standard stars comes into use. By observing a set of knownstars, it is possible for each observatory to determine the necessarytransformation coefficients to transform their instrumental magnitudesto the common standard system.

In practice, a star is not measured in flux units. The detector producesan electrical output that is directly proportional to the observed stellarflux. In DC photometry, the amplified output current of the detector ismeasured, while in pulse-counting techniques the number of counts persecond is recorded. In either case, the recorded quantity is only propor-tional to the observed flux. Symbolically,

F x = Kd* (1.8)

where dx is the practical measurement (i.e., current or counts per sec-ond), and K is the constant of proportionality. Equation 1.7 can be writ-ten as

mx = ft- 2.5 log K - 2.5 log rfx (1.9)

or

m,= q\ -2.5 log d, (1.10)

This then relates the actual measurement, dx, to the instrumental zeropoint constant q(, and to the instrumental magnitude, mx. The colorindex of a star is defined as the magnitude difference between two dif-


ferent spectral regions. If the subscripts 1 and 2 refer to these tworegions, then a color index is defined as

WAI - Wx2 = q'X] - q\2 - 2.5 log dM + 2.5 log dx2 (1.11)

or

mM - mx2 = qM2 - 2.5 log (dM/d^} (1.12)

where the zero point constants have been collected into a single term,qM2. Again the quantity (mxl — mX2) is in the instrumental system. Thetransformation from the instrumental system to the standard system isdiscussed shortly. Before that transformation can be made, it is neces-sary to correct for the absorption effects of the earth's atmosphere.

1 .8 ATMOSPHERIC EXTINCTION CORRECTIONS

Even on the clearest of nights, the stars are dimmed significantly byabsorption and scattering of their light by the earth's atmosphere. Theamount of light loss depends on the height of the star above the horizon,the wavelength of observation and the current atmospheric conditions.Because of this complex behavior, the measured magnitudes and colorindices are corrected to a location "above the earth's atmosphere." Inother words, they are corrected to give the same values an observer inspace would measure. In this way, measurements by two differentobservatories can be effectively compared.

A measured magnitude, mx, is corrected to the magnitude that wouldbe measured above the earth's atmosphere, mxo, by the followingequation,

where k'x is called the principal extinction coefficient and k'\ is the sec-ond-order extinction coeffcient. This second-order term is often smallenough to be ignored in practice. Here c is the observed color index and


X is called the air mass. At the zenith, X is 1.00 and it grows larger asthe altitude above the horizon decreases. To a good approximation,

X = secz, (1.14)

where z is the zenith distance (90° - altitude) of the star.Just as the sun grows red in color as it sets, the atmospheric extinction

process affects the color indices of stars. A measured color index, c, istransformed to a color index as seen from above the earth's atmosphere,CQ, by the following expression:

c0 = c - k'cX- \»Xc. (1.15)

as above, k'c and k"c represent the principal and second-order extinctioncoefficients, respectively. The subscript c is a reminder that the value ofthe coefficient depends on the two wavelength regions measured. Thatis to say, the extinction coefficient for a color index based on a blue anda yellow filter is not the same as that based on a yellow and red filter.The extinction coefficients, k\, k"K, A^ and k"c are determined obser-vationally. The details of this technique will be discussed in Chapter 4.The derivation of the above extinction equations can be found in Appen-dix J.

1.9 TRANSFORMING TO A STANDARD SYSTEM

A system of magnitudes and colors, such as the UBVsystem, is definedby a set of standard stars measured by a particular detector and filterset. In order for observers at different observatories to be able to com-pare observations, the observations must be transformed from theinstrumental systems (which are all different) to a standard system. Itis important for the observers to match the equipment used to definethe system of standard stars as closely as possible. However, no twofilter sets or detectors are exactly the same. Hence, it is necessary forall observers to measure the standard stars in order to determine howto transform their observations to the standard system.

A derivation of the transformation equations can be found in Appen-dix J. Only the results are stated here. Once the observed magnitude


has been corrected for atmospheric extinction, it can be transformed toa standardized magnitude (Mx) by

Mx = mxo + /3xC + 7* (1.16)

where C is the standard color index of the star, 0X and -yx are the colorcoefficient and zero-point constant, respectively, of the instrument. Thestandardized color index is given by

C = <5c0 + Tc (1.17)

where c0 is the observed color index which has been corrected for atmo-spheric extinction. Again, 5 is a color coefficient and 7^ is a zero-pointconstant. These coefficients and zero-point constants are determined foreach photometer system by the observation of standard stars. Thedetails of this are taken up in Chapter 4.

1.10 OTHER SOURCES ON PHOTOELECTRIC PHOTOMETRY

There are several sources relating to photoelectric photometry that areavailable in good astronomical libraries. Some of these are obscure andare difficult to locate. Most of the references listed below are out ofprint or are sections of expensive texts. However, if you are interestedin more detail than can be found in this text, we recommend looking atthose references available in your area.

• Irwin, J. B., ed. 1953. Proceedings of the National Science Founda-tion Astronomical Photoelectric Conference. Flagstaff, Arizona: Low-ell Observatory. This book has considerable detail on sky conditionsand site selections for observatories.

• Wood, F. B., ed. 1953. Astronomical Photoelectric Photometry.Washington, D.C.: AAAS. This is the proceedings of a symposiumon December 31, 1951. Contains many references of early photom-etry and describes DC, AC, and pulse-counting techniques as prac-ticed at that time.

• Hiltner, W. A., ed. 1962. Astronomical Techniques. Chicago: Univ.of Chicago Press. Three chapters of this book are of particular inter-est: Lallemand ("Photomultipliers"), Johnson ("Photoelectric Pho-


tometers and Amplifiers"), and Hardie ("Photoelectric Reduc-tions").

• Whitford, A. E. 1962. "Photoelectric Techniques." In Handbuch derPhysik. Berlin: Springer-Verlag Co. Edited by S. Flugge, p. 240. Thischapter is a well-rounded description of photomultiplier tubephotometry.

• Wood, F. B. 1963. Photoelectric Astronomy for Amateurs. NewYork: Macmillan. This text is low level and understandable, but isincomplete and contains out of date circuitry.

• AAVSO, 1967. Manual for Astronomical Photoelectric Photometry.Cambridge: AAVSO. The AAVSO has a short manual to startobservers on photometry.

• Golay, M. 1974. Introduction to Astronomical Photometry. Holland:D. Reidel. For complete theoretical descriptions of wide-band pho-tometry, this text is hard to beat. Requires extensive mathematicsand astronomy background.

• Young, A. T. 1974. In Methods of Experimental Physics: Astro-physics vol. 12A. Edited by N. Carleton. New York: AcademicPress. This is extremely complete in the problems arising in photom-etry and should be required reading.

In addition, some professional observatories have their own smallmanuals that can be obtained directly from them.

Amateur and professional astronomers interested in photometry arestrongly encouraged to join the International Amateur-ProfessionalPhotoelectric Photometry (IAPPP) association. The goal of this groupis to foster communication on the practical aspects of photometry. Thisis accomplished through annual IAPPP symposia and the IAPPP Com-munications. Interested persons should contact either of the followingpeople:

Dr. Terry D. OswaltDept. of Physics and Space SciencesFlorida Institute of TechnologyMelbourne, FL 32901

Mr. Robert C. ReisenweberRolling Ridge Observatory3621 Ridge ParkwayErie, PA 16510U. S. A.


Amateur astronomers are encouraged to coordinate their photometricobserving programs with those of other amateurs by contacting one of

the following organizations.

American Association of Variable Star Observers (AAVSO)25 Birch Street

Cambridge, MA 02138

Royal Astronomical Society of New ZealandVariable Star Section

P. O. Box 3093 GreentonTauranga, New Zealand

REFERENCES

1. Miczaika, C. R., and Sinton, W. M. 1961. Tools of the Astronomer. Cambridge,Mass.: Harvard Univ. Press, p. 156.

2. Bond, W. C. 1850. Annals of the Harvard College Observatory, I, I , CXLIX.3. Minchin, G. M. 1895. Proc. Roy. Soc. 58, 142.4. Stebbins, J., and Brown, F. C. 1907. Ap. J. 26, 326.5. Kron, G. E. 1966. Pub. A.S.P. 78, 214.6. Stebbins, J. 1910. Ap. J. 32, 185.7. Schultz, W. F. 1913. AP. J. 38, 187.8. Guthnick, P. 1913. Ast. Nach. 196, 357.9. Rosenberg, H. 1913. Viert. der Ast. Gesell. 48, 210.

10. Stebbins, J. 1928. Pub. Washburn Obs. XV, 1.11. Whitford, A. E. 1932. Ap. J. 76, 213.12. Whitford, A. E., and Kron, G. E,, 1937. Rev. Sci. Inst. 8, 78.13. Kron, G. E. 1946, Ap. J. 103, 326.14. Davis, F. W., Jr. 1973. Griffith Observer (May), 8.15. Custer, C. P. 1973. Sky and Tel. 46, 329.16. Souther, B. L. 1978. Sky and Tel. 55, 78.17. Souther, B. L. 1978. Sky and Tel. 55, 173.18. De Lara, E., Chavarria, K. C., Johnson, H. L. and Moreno, R. 1977. Revislia

Mexicana de Astron. y Astrof. 2, 65.19. Schumann, J. D., 1977. In Astronomical Applications of Image Detectors with

Linear Response. I. A. U. Colloquium No. 40, 31-1.20. Fisher, R. 1968. Appl Optics 7, 1079.21. Masek, N. L. 1976. South, Stars 26, 175.22. Corney, A. C. 1976. South. Stars 26, 177.23. McFaul, T. G. 1979. J. AAVSO 8, 64.24. Optec Inc., 119 Smith, Lowell, Ml 49331.

Chapter 2Photometric Systems

The basic goal of astronomical photometry sounds simple enough: tomeasure the light flux from a celestial object. So it would seem thatsimply placing a light detector at the focus of a telescope is all that isneeded. The problem begins when different observers using differentlight detectors and telescopes try to compare or combine their data.Even though they may have been observing the same star at exactly thesame time, their measurements will not necessarily be the same. Thisdifference is due to the different spectral response of the telescope anddetector. To take an extreme example, suppose a detector is mostly sen-sitive to blue light while a second is mostly sensitive to red. Stars arenot equally bright at all wavelengths so the two detectors cannot pos-sibly give the same results for the same star.

The obvious first step toward a uniform data set would be to have allobservers use the same kind of detector. It would also be extremely val-uable to isolate and measure certain portions of the spectrum containingfeatures that indicate physical conditions of the star. This can beachieved by using a detector with a broad spectral response with indi-vidual spectral regions isolated by filters transmitting only a limitedwavelength interval to the detector. Every observer should match thedetector and filters as closely as possible to a common system. However,even this is not enough to yield strict uniformity as it is impossible tomanufacture identical light detectors and filters. Thus, a third and finalcomponent is necessary: standard stars. Observations of the same, non-variable (hopefully!) stars, of known magnitudes and colors, will alloweach observer to determine his own coefficients for Equations 1.16 and1.17. It is then possible to measure any unknown star and use theseequations to transform the results to a common photometric system.

33


This is how a photometric system is denned: by specifying the detector,niters, and a set of standard stars.

Photometric systems can be broken into three rough categories basedon the size of the wavelength intervals transmitted by their filters.Wide-band systems (such as the UBV system) have filter widths ofabout 900 A, while intermediate-band filter widths are about 200 A.Narrow-band systems are used to isolate and measure a single spectralline and may have widths of 30 A or less. While the narrow-band sys-tems give very specific spectral information, they transmit only a smallfraction of the light of the star. Unless a large telescope is used, theiruse is limited to very bright stars. A discussion of various photometricsystems can be found in Golay.1

At the end of this chapter, an intermediate-band and a narrow-bandsystem are considered. However, in what follows, and throughout theremainder of this book, the wide-band UBV system will be discussed.By adopting a single system, specific examples of observing techniquesand data reduction can be used, avoiding a very general discussion thatwould be of much less benefit to the beginner. However, many of thoseprocedures can be applied to any system. The choice of the UB V systemas "the system" for this book is based on a number of considerations.The UBV system has become popular among astronomers, and thereexists a considerable data base of UBV observations in the literature.Being a wide-band system makes it especially suitable for users of smalltelescopes. The photomultiplier tube and filters used to define the sys-tem are readily available and relatively inexpensive. There is also afairly extensive set of standard stars. The reader should not assume thatthe choice of the UB V system for this book means that the other systemsare less important or yield measurements that tell us less about thestars. As you will see later, the UBV system is not a perfect system, andfor some research projects, other systems are preferred.

2.1 PROPERTIES OF THE UBV SYSTEM

The t/flKsystem was defined and established by H. L. Johnson and W.W. Morgan.2-3 They desired to establish a photoelectric system thatwould yield results comparable to the yellow and blue magnitudes ofthe International System (see Section 1.2), to have a third color forbetter discrimination of stellar attributes, and to be closely tied to theMorgan-Keenan (M-K) spectral classification system. The UBVsystem

3000 4000 5000 6000


7000

Figure 2.1 Typical response function of a 1P21 photomultiplier tube.

was developed around the RCA 1P21 photomultiplier tube and threebroad-band filters that give a visual magnitude (K), a blue magnitude(fi), and an ultraviolet magnitude (V). The response function of the1P21 is shown in Figure 2.1 and the transmission curves of the filtersare shown in Figure 2.2.

The V filter is yellow with a peak transmission around 5500 A. Thisfilter was chosen so that the F magnitude is almost identical to the pho-

3000 4000 5000 6000WAVELENGTH (ANGSTROMS)

7000

Figure 2.2 Normalized transmission function of the UBV filters.

35


tovisual magnitude of the International System. The long wavelengthcutoff is determined by the response of the 1P21 and not the filter. Theblue (B) filter is centered around 4300 A but has some transmissionover most of the sensitivity range of the 1P21. The B magnitude cor-responds well with the earlier blue photographic magnitudes. This filteractually consists of two: a blue filter and an ultraviolet blocking filter.This latter filter prevents the B magnitude from being affected by theBalmer discontinuity, which is discussed later. The U filter is centeredon 3500 A and has two problems. This filter has a red "leak," that is,it transmits some light in the near infrared. This red light must beblocked by a second filter or the red leak must be measured and sub-tracted from the U measurement. The second problem is that the shortwavelength cutoff is not set by either the filter or the photomultiplier,but instead by the earth's atmospheric ultraviolet transmission. This isa function of the observatory's altitude and can be variable dependingon atmospheric conditions. Thus the VBV system is not totally filter-defined.

The VBV standard stars were measured by Johnson's original pho-tometer without any transformation. In other words, except for someadditive constants, the UBV system is the instrumental system of thatphotometer. The zero points of the color indices, (B V) and(U — B), are defined by six AO V stars. These stars are a Lyr, 7 UMa,109 Vir, a Crb, 7 Oph, and HR 3314. The average color index of thesestars is defined to be zero, that is

(B - V) = (U - B) = 0.

The system was originally defined with 10 primary standard stars.Just 10 stars, spaced over the entire sky, is an insufficient number toallow other observatories to calibrate their photometers. Johnson andMorgan established a more extensive list of secondary standards thatare closely tied to the ten primary stars. Appendix C lists these stars.Secondary standards were also established within three open-star clus-ters. These clusters are especially valuable for UBV calibration sincethe uncertainties in atmospheric extinction are less important becauseof the proximity of the stars. The names of these clusters along withfinder charts are found in Appendix D.

PHOTOMETRIC SYSTEMS 37

2.2 THE UBV TRANSFORMATION EQUATIONS

Through the observations of standard stars, an observer can take instru-mental measurements of program stars and transform them to the stan-dard UBV system. In Chapter 1, the transformation equations are pre-sented in a general form to be applied to any photometric system. It iscustomary to change the symbols used in the transformation equationsto indicate the use of the UBV system. Equation 1.10 is replaced by

v = -2.5 log dv (2.1)b = -2.5 log db (2.2)u = -2.51ogrfu (2.3)

where v, b, u and dv, db, du represent the instrumental magnitudes andmeasurements through the K, B, and U filters, respectively. The con-stants, <?', have been dropped because they can be "absorbed" by thezero-point constant in the transformation equations. Equation 1.12 isreplaced by

(b- v) = -2.5 log db/dv (2.4)(u- b} = -2.5\ogdu/db (2.5)

The lower case w, b, v refer to the instrumental system while U, B, Vrefer to the standard system. The magnitude and colors corrected foratmospheric extinction, Equations 1.13 and 1.15 become

Vo = v - k'v X (2.6)(b - v)0 = (b - v)(l - kSv X) - k'bv X (2.7)(u - b)0 = (u - b) - k'ubX (2.8)

In the UBV system, k"ub is defined to be zero (more about this later),and experience has shown that k" is very small so it is not included inEquation 2.6. Equations 1.16 and 1.17 become

K = v0 + «(*- V) + r, (2.9)(B- V) = n(b- v)0 + fto (2.10)([/- B) = $(u- b)Q + {ub (2.11)


where e, M, $ are the transformation coefficients and fv, ffrv, fufc are thezero-point constants. These six values are found by observations of thestandard stars in Appendices C and D. The details of this calibrationare given in Chapter 4.

2.3 THE MORGAN-KEENAN SPECTRAL CLASSIFICATION SYSTEM

Spectral classification is a very important topic in stellar astronomy andthe reader can find an elementary review of this topic in the books byAbell,4 Swihart,5 and Smith and Jacobs,6 to name but a few. A moreadvanced discussion is given by Keenan.7 A brief review is given herebecause of the close relationship of the UBV system to the Morgan-Keenan (M-K) spectral classification system.

The first large-scale classification of stellar spectra began in the1920s at Harvard College Observatory and became known as the HenryDraper Catalog. Over 400,000 stellar spectra were classified. At first,the stars were broken into a few groups based on the strength of thehydrogen absorption lines. The groups were designated A through P,from strongest to the weakest lines. In time, it became clear that someof these classes did not exist but were a product of poor quality spectro-grams. Furthermore, simply arranging the spectrograms so the hydro-gen lines varied from strong to weak did not produce a continuous andlogical pattern in the remaining lines. Consequently, some classes weredropped and the remainder were rearranged. The result was a scram-bled alphabetic sequence (O, B, A, F, G, K, M) but a logical and con-tinuous variation in strength of all spectral lines. Better quality spectro-grams have led to the development of 10 subclasses indicated by anumber (zero through nine) following the letter. The sun is designatedas a G2 while Vega is an AO star. Figure 2.3 shows several spectra ofmain sequence stars. At the bottom of the figure, a typical filter plusphotomultiplier tube response function for the UBV system is shown. Itis customary to refer to stars near the beginning of the sequence as earlytype and those near the end as late type. That is, an AO star is an earliertype than an F5, and a KO is earlier than a K5.

We now know that the spectral sequence is an ordering by stellarsurface temperature. For instance, O stars are approximately 50,000 Kwhile M stars are 3000 K. The changing pattern of spectral lines inFigure 2.3 is a direct result of the change in the stellar temperature. AnO type star shows few lines because most atoms are totally ionized.


3000 4000 5000 6000

Wavelength (Angstroms)

Figure 2.3 Spectra of some main sequence stars.

7000


However, lines of He II (singly ionized helium) are fairly strong andare sometimes seen in emission. As one progresses towards the B class,the He II lines grow weaker and He I (neutral, un-ionized helium) andhydrogen lines grow stronger. By class B2, the He I lines dominate thespectrum. Hydrogen and ionized metal1 lines grow stronger until earlytype A. Hydrogen and many ionized metals reach maximum strengthat AO. By late A and into early F, the ionized metal lines grow whilehydrogen lines decrease rapidly. Through classes F to G the spectrallines of Ca II strengthen reaching a peak at G2. Neutral metals con-tinue to gain in strength as their ionized counterparts disappear. By lateK, molecular bands appear and neutral metal lines dominate. Thehydrogen lines are essentially gone and the calcium lines are still strong.By late K and into M, the bands of titanium oxide become prominent.Lines of the neutral metals are still stronger.

As stated earlier, the K filter was chosen to match the old visual mag-nitudes and the B filter to match the photographic magnitudes. The Ufilter was chosen to measure a spectral feature. In Figure 2.3, it is easyto see that the hydrogen lines dominate the early spectral types. Thespacing between these lines becomes closer and closer until at the Bal-mer limit they merge and the absorption becomes continuous. There-fore, at the Balmer limit (3647 A) there is a sharp drop in the continumlevel, called the Balmer discontinuity. Figure 2.3 also shows that the Ufilter straddles this discontinuity. Thus, the (U — B) color index is sen-sitive to the strength of the discontinuity, which in turn is a function ofthe star's spectral type. Note that the effective wavelength of observa-tion through the U filter depends on the strength of the Balmer discon-tinuity. If the discontinuity is strong, very little light is received short-ward of 3647 A. The light measured through the U filter is that whichpasses through the "red wing" of the filter, longward of 3647 A. Thus,we are effectively looking at a wavelength that is longer than the middleof the filter bandpass. On the other hand, a star that has a very weakdiscontinuity supplies light roughly equally across the bandpass of thefilter. Then the effective wavelength of observation is near the center ofthe bandpass. An important consequence of this effect is that the sec-ond-order atmospheric extinction coefficient for U — B has a compli-cated behavior with spectral type. That is to say, unlike the behavior of

t Astronomers use the term "metal" in a very different sense than do chemists. The term is usedto designate any element other than hydrogen or helium.


& 6 v » k"ub does not vary smoothly with spectral type, but rather shows adouble sawtoothed variation from type O to M. To avoid the time-con-suming process of correcting k"ub, Johnson and Morgan defined it to bezero. Since second-order terms are small, the error introduced by thisdefinition is of the order of 0.03 in U — B. A more detailed discussionof this problem can be found in Section 4.9.

The Harvard System is a one-dimensional classification scheme.However, it was realized rather early that a second dimension might benecessary. Some stars showed narrower absorption lines than other starsof the same class even though the pattern of spectral lines matched well.Between 1914 and 1935, Mount Wilson Observatory ordered spectra ofthe same class by the strength of certain spectral features. In time, itwas realized that narrower lines resulted from a lower density in theatmosphere of these stars. Because the temperatures are the same, theiratmospheres are much larger than those of normal main sequence stars.Therefore, these narrow-line stars are brighter than their main sequencecounterparts. (See Equation J.23.)

These spectral "anomalies" are in fact luminosity indicators. W. W.Morgan, P. C. Keenan, and E. Kellman8 developed a second dimensionto the spectral classification now in general use. This M-K system intro-duces luminosity classes as follows:

I: SupergiantsII: Bright giantsIII: GiantsIV: SubgiantsV: Main sequence (dwarfs)VI: Subdwarfs

The location of these groups in the Hertzsprung-Russell (H-R) diagramis shown in Figure 2.4. Classes I to V may be subdivided by using thesuffix a (brightest), or ab, or b (dimmest). The luminosity criteria arebased on line strengths, ratios of line strengths, and widths of hydrogenlines. The low density in the atmospheres of the larger stars alters thepercentage of atoms that are ionized. This in turn alters the linestrengths and makes the spectrum appear to belong to a hotter star, andtherefore to an earlier spectral type. Note in Figure 2.4 that the spectralclasses of the giants and supergiants occur to the right of the sameclasses for stars on the main sequence. However, the color index is


-10 i-

10

Ih

III

-0.5 0.5 1.0

B-V

1.5 2.0

Figure 2.4 The H-R diagram and luminosity classes.

largely unaffected by the changes in the lines and therefore indicates adifferent temperature.

2.4 THE M-K SYSTEM AND UBV PHOTOMETRY

One of the most important aspects of the UBV system is its close ties tothe M-K spectral classification system. As stated earlier, the zero pointsfor the color indices were denned by stars classified as AO V on the M-K system. This allows the colors of the UBV system to be relateddirectly to an M-K spectral type and temperature. Figures 2.5a and band Table 2.1 show these relationships for main sequence stars. Theseapply to stars that are not viewed through significant quantities of inter-stellar dust. This dust selectively absorbs more blue light than red lightmaking a star appear redder than it actually is.

TABLE 2.1. Color Indices and Temperatures for MainSequence Stars

Spectral Type

O5BOB5AOA5FOF5GOG5KOK5MOM5

(B- V)

-0.32-0.30-0.16

0.00+ 0.14

0.310.430.590.660.821.151.411.61

([/ — B) Effective Temperature ( °K)

-1.15-1.08-0.56

0.00+ 0.11

0.060.000.110.200.471.031.261.19

54,00029,20015,200

9600831073506700605056605240440037503200

SOURCE: Novotny, E. 1973. Introduction to Stellar Atmospheres and Interiors. NewYork: Oxford University Press, p. 10.

-0.5 r-

0.5

1.0

1 5

0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0

B A F G K M

SPECTRAL TYPE

Figure 2.5a B-V versus spectral type.

43


-0.5 -

0.5 -

1.0 -

0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 0B A F G K M

SPECTRAL TYPE

Figure 2.5b U-B versus spectral type.

Johnson and Morgan established the relationship between colorindices and absolute magnitude by a two-step process. The first step wasto measure the color indices of nearby stars with accurately known par-allaxes. From the parallax and apparent magnitude, the absolute mag-nitude can be calculated directly. Unfortunately, there are very fewearly-type stars with accurately measured parallaxes because thesestars are relatively rare. Even the A, F, and G stars are not as commonas one would like for calibration purposes. The second step was to fillthese gaps in spectral types using stars in nearby galactic clusters. Acolor versus apparent magnitude plot can be made for these clustersafter correcting the magnitudes for interstellar absorption. Because allthe stars within the cluster are nearly the same distance away, theapparent magnitude for each star differs from the absolute magnitudeby some additive constant. If it is assumed that there is no differencebetween the main sequence of nearby field stars and that of a cluster,


then the plot for the cluster can be slid vertically (in magnitude) on topof the plot for the field stars until the main sequences match. Then, theabsolute magnitude of the cluster stars and the distance to the clusteris defined. The clusters used for this process were NGC 2362, thePleiades, and the Praesepe. The completed diagram appears in Figure2.6, which is an H-R diagram using a color index instead of the M-Ktype. Because of the effect discussed at the end of Section 2.3, the rela-tion between color index and spectral class depends slightly on theluminosity class.

Figure 2.7 shows a plot of (U — B) versus (B -• V), for (unred-dened) main-sequence stars. This is a so-called color-color diagram.Note that the (U — B) color gets smaller as you move upwards in theplot (the star is becoming brighter in U than in B). Blackbodies of var-ious temperatures follow a nearly linear relation (upper curve). How-ever, stars (lower curve) deviate significantly from a blackbody. Thesetwo curves differ because of the absorption lines in stellar spectra. Fromtype O to AO, the hydrogen absorption lines increase in strength and sodoes the Balmer discontinuity. The flux seen in the U filter decreasescausing (U — B)io become larger. (Remember, magnitudes are largerif a star is fainter.) After AO, the Balmer lines (and the discontinuity)grow weaker and (U — B) begins to decrease. After class F5, however,the metal lines and molecular bands become strong. Many of these

-5

15

.NGC 2362

PLEIADES

PRAESEPE-

0 0.5 1.0

B-V

Figure 2.6 Main sequence matching.


-1.2 i-

-0.5

Figure 2.7 Color-color diagram.

absorption features are in the ultraviolet and cause the (£/ — B) colorto become large again. There is a complication for stars near the bumpin the color-color plot at F5. Because the value of (U — B) depends onthe amount of absorption by metal lines, an abnormal metal abundancecan have a significant effect on this color. A low metal abundancecauses the star to plot higher than a normal star.

Figure 2.8 shows the color-color plot again, but this time to illustratethe effect of interstellar reddening. Reddening causes a star to move tothe right nearly parallel to the reddening line in the figure. As an exam-ple, if an observed star plots at point A in the diagram, it can beassumed that extrapolating to the left, parallel to the reddening line,yields its intrinsic colors. The amount of color change produced by thedust is called the color excess and is denoted by E(B - V) andE(U — B), as labeled in Figure 2.8. The slope of the reddening line isgiven by

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4-0.5


REDDENINGLINE

0.5

(B-V)

1.0 1.5

Figure 2.8 Reddening displacement on the color-color diagram.

E(U - B)E(B - V)

= 0.72 - 0.05(5 - K) (2.12)

For early type stars (B — K) is nearly zero, so the second term is verysmall and

E(U- B)E(B - V)

^ 0.72 (2.13)

For stars that are later than type AO, it is not possible to use a color-color plot to determine the intrinsic color unambiguously. This is truebecause the color-color curve turns upward at AO. Extrapolating to theleft along the reddening line will result in two intersections of the color-color curve. For stars later than AO, the color excesses must be obtainedby comparing the spectral class implied by the observed colors to that


obtained by spectroscopy. The latter is not affected by the reddeningbecause it is based on the pattern of spectral lines.

For stars from type BO to AO, there is yet another way to deal withinterstellar reddening. The quantity Q is defined as

Q = (U - B) - 0.12(B - V) (2.14)

where (U — B) and (B — K) are the observed colors. Q is independentof reddening. To see this, note that

E(B - V) = (B - K) - (B - V). (2.15)

and

E(U - B) = (U- B) - (U- B)i (2.16)

where (B — V), and (U — 5), are the intrinsic colors of the star. Nowwe solve these two equations for (B — K) and (U — B), respectively,and substitute into Equation 2.14. Then,

Q = E(U - B) + (U - B), - 0.72 [E(B - K) + (B - K)(](2.17)

Q = (U - B)i - 0.72(5 - V), + E(U - B) - 0.72£(* - V).(2.18)

Now substitute Equation 2.13, which results in

Q = (U - B), - 0.72(5 - K),, (2.19)

independent of reddening. Equation 2.19 is then used to produce Figure2.9. The observed colors of a reddened star can be used to calculate Qby Equation 2.14. Figure 2.9 then yields the intrinsic spectral type.

The total absorption in the visual magnitude can be estimated in thefollowing way. Define a quantity R as

R = -r~ (2.20)/IB /\y

where A^and ABare the absorption, in magnitudes, in Kand B, respec-


-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

BO B1 B2 B3 B4 B5 B6 B7 B8 B9 AOSPECTRAL TYPE

Figure 2.9 Q versus MK spectral type.

tively. The observed magnitudes are related to the intrinsic magnitudesby

B = B,+ AB (2.21)V = K(. + Av. (2.22)

Substituting these two expressions into Equation 2.15 gives

E(B - V) = [(B, + A8) - (K,. + Ay)] - (B - V), (2.23)E(B - V) = AB- Ay. (2.24)

Thus, Equation 2.20 becomes

R =E(B - V)

or

Av = RE(B - V).

(2.25)

(2.26)

The value of R has been found to be about 3.0 for most directions inour galaxy. However, there appear to be some regions where the nature


of the interstellar dust is different and R may reach values as high as12. If E(B — V) is determined by one of the above methods and R isassumed to be 3.0, it is then possible to calculate Av and correct theapparent visual magnitude by

K, = V- Av (2.27)

to obtain the intrinsic visual magnitude.

•2.5 ABSOLUTE CALIBRATION

It is sometimes helpful or necessary to convert a measured magnitudeinto an actual flux measurement. The process of absolute calibration isboth difficult and tedious. The process has recently been discussed byLockwood et al.9 No attempt to explain the process is made here, buta means of approximately transforming VBV measurements into flux isshown. Recall Equation 1.7:

mx= qx- 2.5 log /v

Assume that the atmospheric extinction correction has been made.Denote this by an added subscript on m, that is,

mxo = ft - 2.5 log FA, (2.28)

or more explicitly,

V0 = qv - 2.5 log Fv (2.29)

B0 = qb- 2.51ogFfi (2.30)U0 = <?u - 2.5 log Fu. (2.31)

Johnson10 has determined the </'s as they appear in Table 2.2. Thus, theflux can be determined by

Fv = W-W'-M (2.32)FB = \Q-°«*-rt (2.33)Fv = Kr04^-'-' (2.34)


TABLE 2.2. Absolute Zero-Point Constants

Filter

UBVRIJKLMN

Approximate Equivalent Wavelength (Angstroms)

36004400550070009000

12,50022,00034,00050,000

102,000

Qi

-38.40-37.86-38.52-39.39-40.2-41.2-43.5-45.2-46.6-49.8

NOTE: Filters / through N will be explained in Section 2.7a.

The constants that appear in Table 2.2 are simply 2.5 times the loga-rithm of the flux of a zero magnitude star, in watts per square centi-meter per Angstrom. Because of the difficulties of the calibration pro-cess, these constants may contain errors between 10 and 20 percent.

Example: What is the flux reaching the earth from a star which hasK0 = 3.0? From Table 2.2, gv = -38.52.

_ 1^-0.4(3.0 + 38.52)

watts

V

V

Fv = 2.47 X 10-'7cm2 Angstrom

This is the flux at the equivalent wavelength of observation. (See Equa-tion J.53.) The total flux measured in the V filter can be found,approximately, by multiplying this number by the width of the filter'sbandpass (1000 A). Thus,

Fv =5 2.5 X 10"14 watts/cm2.

The total power collected in the V filter by the telescope is obtainedby multiplying by the collecting area, that is

/V« 2.5 X 10-'4(TT*2) watts,


where R, is the radius in centimeters of the primary mirror or lens ofthe telescope.

2.6 DIFFERENTIAL PHOTOMETRY

The concept of differential photometry is outlined in Section 1.6. Wenow proceed to a more detailed discussion. The actual observations con-sist of a series of measurements, which are given in counts per second(pulse counting) or percent of full-scale deflection (DC) through eachfilter of both the variable and the comparison star. We represent themeasurements through the K, B, and U filters by dv, db, and du, respec-tively. We add a second subscript to indicate the variable (x) or thecomparison star (c). The magnitude difference between the variableand the comparison star in each filter is given by

Av = -2.5 log -j* (2.35)

A6 = -2.5 log -~ (2.36)

AM = -2.5 log-r (2.37)duc

if pulse-counting electronics are used. If DC electronics are used, it ispossible that the two stars may require a different amplifier gain. Theabove equations are then modified to read,

Av = -2.5 log -~ + Gvx - Gvc (2.38)

A6 = -2.5 log + Gbx - Gte (2.39)f*bc

Aw 2.5 log + Gux - Guc (2.40)"uc

The additional terms give the difference in amplifier gain (in magni-tudes) between the variable and the comparison star. In Chapter 8, theprocedure of gain calibration is discussed.

It is also possible to use these same measurements to form differences


in color indices between the variable and the comparison star. To seethis, note that

A(/> - v) - (bx - v,) - (bf - v,)= (bx - bc) - (vx - vc)= A6 - Av. (2.41)

Likewise,

A(w - 6) = Au - A/>. (2.42)

The beginner need not carry the data reduction beyond this point. Thereare many worthwhile observing projects that can be done with differ-ential photometry, some of which are discussed in Chapter 10.

In rare circumstances, it might be necessary to apply a small extinc-tion correction to differential photometry. This should seldom be nec-essary, because the variable and comparison star are close together inthe sky and have been viewed through essentially the same air mass.However, in practice, this sometimes does not occur. It may be that asuitable comparison star was not found within 1 " of the variable or thatthe comparison star was not observed frequently. In the latter case, theearth's diurnal motion causes the two stars to be viewed through signif-icantly different air masses. Equations 2.35 through 2.37 (or 2.38through 2.40) must be corrected to give the magnitude difference abovethe earth's atmosphere by making use of Equation 2.6. Thus,

(Av)0 = Av - k&X, - Xf) (2.43)(A6)0 = A/> - k'b(Xx - Xc) (2.44)

(Aw)o = AM - ktt*, - *,), (2.45)

where Xx and Xc are the air masses of the variable and comparison star,respectively, at the time of observation. The color index differences canbe corrected using Equations 2.7 and 2.8. That is,

- v)0 = A(/> - v) - k'bv(Xx- Xc) - k"bvk(b - v)* (2.46)

and


A(« - *)0 = A(u - b) - k'ub(Xx - X). (2-47)

where is the average air mass of the variable and comparison star.It must be stressed that all of the above magnitude and color differ-

ences are on the instrumental system of the photometer in use. It ispossible to do differential photometry on the standard £/5Ksystem. Theprocedure is to observe your comparison star along with some UBVstandards. You can then determine V, (B — V\ and (U — B) of thecomparison star. This need only be done on one night if it is done well.On all other nights, you only need observe your variable and the com-parison star. The magnitude and color differences between your varia-ble and comparison star on the standard system can be found by rewrit-ing Equations 2.9 through 2.1 1 to obtain

AV = (Av)0 + cA(fi - F) (2.48)A(5- F) = n&(b - v)0 (2.49)A(C7- B) = ^A(u - 6)0 (2.50)

Note that if the two stars have nearly the same color, the second termon the right of Equation 2.48 is nearly zero. Furthermore, /i and \f/ areapproximately equal to one for most photometers using the 1P21 pho-tomultiplier tube and standard UBV filters. This justifies the earlierstatement that an uncalibrated photometer gives nearly the same mag-nitude and color difference as a calibrated one. The real advantage ofcalibrating the comparison star is that you can use your observations tocompute the actual standardized magnitude and colors of the variablestar by

Vx = Vc + A K (2.51)(B-V)X = (B- V)c + A(5 - K) (2.52)(U- B)x = (U- B)c + A(t/- B). (2.53)

2.7 OTHER PHOTOMETRIC SYSTEMS

By no means is the UBV system the only photometric system. There aremany other valuable systems. A complete discussion of all these systemsis beyond the scope of this text. However, we describe briefly an inter-


mediate-band and a narrow-band system following the discussion of anextension to the VBV system.

2.7a The Infrared Extension of the UBV System

In order to expand the usefulness of the VBV system to the classifica-tion of cool stars, the system has been extended with bandpasses in theinfrared. Table 2.2 lists the letter designation of each filter and itsapproximate effective wavelength. Photometry with filters U, B, V, R,I can be accomplished using an S-l, an extended-red S-20 photomul-tiplier," or a photodiode as a detector. Wavelengths in the range of Jthrough N require specialized detectors, such as those using lead sul-fide, and cooling to liquid-helium temperatures. These techniques arebeyond the scope of this text. Interested readers are referred to a reviewby Low and Rieke.12

2.7b The Stromgren Four-Color System

The Strflmgren system13 is an intermediate-band width system thatovercomes many of the shortcomings of the UBV system and providesastrophysically important information. Table 2.3 contains the filter des-ignations, central wavelengths, and bandwidths of the four filters.

Unlike the t/flFsystem, the Strtfmgren system is almost totally filter-defined. The y (yellow) filter matches the visual magnitude and corre-sponds well with V magnitudes. This filter transmits no strong spectralfeatures in early-type stars. The red limit is set by the filter and not bythe detector as in the case of the UBV system. The b (blue) filter iscentered about 300 A to the red of the B filter of the UBV system toreduce the effects of "line blanketing." For stars of spectral types later

TABLE 2.3. Filters Used in the Stromgren System

Filter

ybV

u

Central Wavelength(Angstroms)

5500470041003500

Full Width at Half Transmission(Angstroms)

200100200400


than AO, absorption lines of metals become strong. A filter that is cen-tered in a wavelength region where such lines are common transmitsless flux than it would if the lines were absent. This blanketing effect isa temperature indicator in that it becomes strong in later spectral types.To get a clear measure of its strength, it is necessary to measure a star'sflux in a region relatively free of blanketing and compare it to a regionwhere blanketing is strong. For early type stars, the b and y filters arefree from blanketing. In later type stars, the two filters are affectedalmost equally. The violet (v) filter is centered in a region of strongblanketing but longwards of the region where the hydrogen lines begincrowding together near the Balmer limit. The u (ultraviolet) filter mea-sures both blanketing and the Balmer discontinuity. Unlike the U filterin the UBV system, this filter is completely to the short wavelength sideof the Balmer discontinuity. Yet, it is centered far enough from theatmospheric limit near 3000 A so that the observing site plays no rolein defining the wavelength region observed. Hence, the system is nearlyfilter-defined and insensitive to the detector used. There are essentiallyno effects due to the filter's bandwidth. That is, there are no second-order color terms in the extinction corrections or the transformationequations. This is a simplification compared with the UBV system.

Figure 2.10 schematically illustrates a stellar spectrum and the place-ment of the four filters. The color indices in the Stromgren system arevery useful quantities. Because both the b and y filters are relativelyfree from blanketing, the index (b — y) is a good indicator of color and

LINEBLANKETING

BALMERDISCONTINUITY

3100 3800 4500


5200

Figure 2.10 Placement of the StrOmgren filters.Note: For the sake of clarity, stellar absorption lines are not shown.


effective temperature. A color index is essentially the slope of the con-tinuum. In the absence of blanketing, the continuum slope would beroughly constant and (b — y) approximately equals (v — b). Because(v -- b) is affected by blanketing, the difference between these twoindices indicates the strength of blanketing. Hence a metal index, m,,can be defined as

m, = (v - b) - (b - y). (2.54)

To determine how the continuum slope has been affected by the Balmerdiscontinuity, the index c{ is defined as

c. = (« - v) - (v - b). (2.55)

This index measures the Balmer discontinuity, nearly free from theaffects of line blanketing. To see this, note that the u measurement con-tains the effects of both blanketing and the Balmer discontinuity. Thev filter contains only the effect of blanketing which is roughly one-halfas strong as in the u filter. Further note that c} has been defined so thatEquation 2.55 can be rewritten as

c, = (u - 2v + b). (2.56)

Subtraction of the 2v term essentially cancels the blanketing, leavingthe effects of the Balmer discontinuity.

In summary, the Stromgren system provides a visual magnitude, ameasure of the effective temperature, a measure of the strength of metallines, and a measure of the Balmer discontinuity. Furthermore, it is fil-ter-defined, independent of any one detector and requires no second-order terms in extinction or transformation equations. The only majordrawback of the system is that the smaller bandpasses make faint starsmore difficult to measure. The Astronomical Almanacl4 has a list ofstandard stars for the StrOmgren system.

2.7c Narrow-Band H/? Photometry

As our example of narrow-band photometry, we discuss briefly a fre-quently used extension of the four-color system, H0 photometry. In thissystem, a narrow interference filter that is centered on the H/3 line is


BO

M 60</>•5.y)z<

WIDEFILTER

0 -

4600 4700 4800 4900WAVELENGTH (ANGSTROMS)

5000 5100

Figure 2.11 Filter responses, H0 system.

used. In early type stars, this is a strong absorption line. The amount oflight flux passed by the filter is heavily dependent on the line strength.The strength of H/? is a luminosity indicator in stars of spectral type Oto A and a temperature indicator in types A to G.

This system actually requires two niters since a small amount ofdetected flux could mean a strong H/3 absorption line or, simply, a faintstar. Thus, a second, broader filter that measures much of the adjacentcontinuum is used. The ratio of the measurements through the two fil-ters indicates the strength of H/3 with respect to the continuum. Figure2.11 shows the response of the filters.

Obviously, effective narrow-band photometry requires a large tele-scope. Furthermore, interference filters are expensive and require ther-mostating at the telescope. It is for these reasons we do not go into fur-ther depth on this topic. Tfye interested reader is referred to Chapter 5in Golay.'

REFERENCES

1. Golay, M. 1974. Introduction to Astronomical Photometry. Boston: D. Reidel.2. Johnson, H. L., and Morgan, W. W. 1951. Ap. J. 114, 522.3. Johnson, H. L., and Morgan, W. W. 1953. Ap. J. 117, 313.4. Abell, G. 1978. Exploration of the Universe, New York: Holt, Rinehart and

Winston.


5. Swihart, T. L. 1968. Astrophysics and Stellar Astronomy. New York: John Wileyand Sons.

6. Smith, E., and Jacobs, K. 1973. Introductory Astronomy and Astrophysics. Phil-adelphia: W. B. Saunders Co.

7. Keenan, P. C. 1963. In Basic Astronomical Data. Edited by K. Aa Strand. Chi-cago: Univ. of Chicago Press, chapter 8.

8. Morgan, W. W., Keenan, P. C., and Kellman, E. 1943. An Atlas of Stellar Spec-tra. Chicago: Univ. of Chicago Press.

9. Lockwood, G. W., White, N. M., and Tug, H. 1978. Sky and Tel. 56, 286.10. Johnson, H. L. 1965. Comm. Lunar and Planetary Lab. No. 53.11. Fernie, J. D. 1974. Pub. A.S.P. 86, 837.12. Low, F. J., and Rieke, G. H. 1974. In Methods of Experimental Physics. New

York: Academic Press 12, chapter 9.13. StrcSmgren, B., 1966. Ann. Rev. Astr. Ap. 4, 433. Palo Alto: Annual Review Inc.14. The Astronomical Almanac. Washington, D.C.: Government Printing Office.

Issued annually.

CHAPTER 3STATISTICS

If we measure 23,944 counts from a source in 10 seconds, will we mea-sure the same number of counts in the next 10-second interval? Howmany counts are needed to achieve 1 percent accuracy for the measure-ment? How do I analyze my data for errors? These are but a few of thequestions that need to be answered before any data reduction iscomplete.

Experimental observations always have inaccuracies. The role of theexperimenter is to know the extent of these inaccuracies and to accountfor them in the best manner. You must know how to combine obser-vations and errors to compute a result. If your observations are to becompared to theoretical predictions, it is necessary to know somethingabout the accuracies of both calculations if you want to make an intel-ligent comparison of their agreement.

This chapter attempts to answer some questions about errors and thefield of statistics in general. The derivations and advanced concepts canbe found in Appendix K. The majority of the material presented in thischapter comes from texts by Young and Bevington (see Section 3.8),both of which are available in paperback and are highly recommended.

3.1 KINDS OF ERRORS

Errors come in different types. Most errors occur in three major cate-gories: illegitimate, systematic, and random. These are discussed inturn.

Illegitimate errors are not directly concerned with the data itself.Instead, these include mistakes in recording numbers, setting up the

so

STATISTICS 61

equipment incorrectly, and blunders in arithmetic. They cannot be rep-resented by any theoretical model and must be eliminated by theobserver through careful work.

Systematic errors are errors associated with the equipment itself orwith the technique of using the equipment. For instance, if an analogamplifier has an offset voltage, the resulting chart recorder deflectionwill be in error. Another example is not removing the U filter's red leakfrom your data. Red stars will then appear brighter in the U filter thanthey really are.

Very often in experimental work, systematic errors are more impor-tant than random errors. However, they are also much more difficult todeal with. Always compare your results with the standard system valuesand other observers whenever possible to calibrate your equipment andobserving procedures.

Random or chance errors are produced by a large number of unpre-dictable and unknown variations in the experimental situation. Theycan result from small errors in judgment on the part of the observer,such as in reading a chart recorder record. Other causes are unpredict-able fluctuations in conditions, such as nearly invisible cirrus clouds orvariations in a photomultiplier tube's high-voltage power supply. It isfound empirically that such random errors are frequently distributedaccording to a simple law. This makes it possible to use statistical meth-ods to treat random errors.

Because random errors can be modeled, they form the basis for muchof the remaining material in this chapter. Illegitimate and systematicerrors must be eliminated by the experimentalist wherever possible.

3.2 MEAN AND MEDIAN

The actual value of what you are trying to measure is unknown. No oneknows the exact magnitude of a star, just as the speed of light, althoughwell measured, is not known exactly. If you determine the magnitudeof a star on five separate occasions, you are likely to get five differentvalues. Intuitively, you would suspect that the most reliable result forthe star's magnitude would be obtained by using all five measurementsrather than only one of them. You can approximate the true value bytaking these measurements (a sample) and determining the average or


sample mean by summing all of the measurements and dividing by thenumber of them. In a more general mathematical notation,

where xt are the values of the individual measurements and N is thetotal number of measurements taken. The summation sign in Equation3.1 is just a shorthand way of saying "the sum of x values from i = 1to TV." All such summations in the rest of the chapter have similar lim-its, and we may drop the i = 1 and N from the summation sign attimes.

Example: On five occasions, you measured the visual magnitude ofMizar to be 4.50, 4.65, 4.55, 4.45, and 4.60. What is Mizar's meanvisual magnitude from this data?

x = (4.50 + 4.65 + 4.55 + 4.45 + 4.60)

x = l- (22.75)

x = 4.55

Sometimes we want to compute the average of a set of values inwhich some of the numbers are more important than others. Forinstance, measurements taken on a cloudy night or at low altitudes areprobably less important than those taken on a crystal-clear night nearthe zenith.

A procedure that suggests itself is to assume that the clear zenithobservation was made more than once. Suppose we have a cloudy obser-vation, a low observation, and the clear zenith value. We include theclear zenith value twice to account for its supposed better accuracy.Then, of course, we must divide by the total number of observations,which is now four. More generally, if we have several observations withdifferent degrees of reliability, we can multiply each by an appropriate"weighting factor," and then divide the sum of these products by the

STATISTICS 63

sum of all of the weighting factors. This is the concept of the weightedmean, and can be represented mathematically by

* = -„ -- - (3-2)

Note that if all of the weights are unity (or more generally, if they areall equal), the weighted mean reduces to the mean as previously definedby Equation 3.1. The problem with weighted means is determining theweights in a rigorous manner without any observer bias. That is, is anobservation on a clear night two, three, or only one and a half times asgood as a cloudy night observation? Unless you can decide on a consis-tent scheme, it is probably better to just take a straight mean and useas many observations as possible.

The median of a sample (or set of observations) is defined as thatvalue for which half of the observations will be less than the medianand half greater. For our five-observation example, we first order theobservations in increasing order: 4.45, 4.50, 4.55, 4.60, and 4.65. Themedian is then the mid-value or 4.55. If we had six observations, themedian would fall between the third and fourth values. To compute themedian, we would average these two values.

Example: We now make a sixth observation of Mizar and obtain avisual magnitude of 4.90. What is the mean and median of oursample?

mean = \ (4.50 + 4.65 + 4.55 + 4.45 + 4.60 + 4.90)6

mean = - (27.65)6

mean = 4.61ordered values: 4.45, 4.50, 4.55, 4.60, 4.65, 4.90median = (4.55 + 4.60)/2median = 4.57


Note that the mean and median do not have to agree, as they areindependent estimates of the best value for the sample. Usually themean is used but there are cases in which the median is a better indi-cator of the sample.

3.3 DISPERSION AND STANDARD DEVIATION

Now that we have a method of determining the best value from oursample of observations, we need some indication of how much faith wehave in that value.

The deviation (d<) or residual of any measurement xt from the meanx is defined as the difference between x{ and jc. Mathematically,

dj = xt — x (3.3)

the deviation is a measure of the quality of the observations, so intui-tively you would think that taking their sum and dividing by the numberof values would give an average deviation. The problem is that some ofthe deviations are positive and others negative, and because of the waywe defined the mean and the deviations, their sum is exactly equal tozero. One method to get around this problem is to use the absolute valueof each deviation in the sum (i.e., all negative deviations are now posi-tive). This defines the average or mean deviation:

- 1 N

d = —-—T I*. - x l (34)" KT 1 / . I •*! •* I \J-TJ

We divide by N — 1 rather than N because we use at least one mea-surement to determine the mean, jc, and therefore get an unrealisticdeviation of zero for one measurement. The average deviation is a mea-sure of the dispersion (or spread or scatter) of the observations aroundthe mean. The presence of the absolute value sign makes the averagedeviation cumbersome to use in practice. It is not correct to call dt theerror in measurement x, because x is only an approximation to the truevalue. However, this is a fine point that few observers obey.

A parameter that is easier to use analytically and is theoretically jus-tified is the standard deviation, G. It is obtained by first squaring each

STATISTICS 65

deviation, thereby removing any minus signs, and obtaining the meansquared, which is the variance, a2:

The standard deviation (a or s.d.) is just the square root of the variance:

u - w. (3.6)N -

By rearranging, an easier computational form for ax can be obtained:

Thus, the standard deviation is the root mean square of the deviations.Note that the standard deviation is always positive and has the sameunits as xt.

The standard deviation defined in this manner tells us the amount ofdispersion to be expected in any single measurement. Clearly, a singlemeasurement in the sample has a larger deviation than the deviation ofthe mean of the sample. In other words, if you measure the magnitudeof a star on five separate occasions, you have confidence that the meanof these five observations is accurate. If you make a single additionalmeasurement, it may deviate significantly from the mean. However, themean of five additional observations lies close to the first mean.

We can therefore define the standard deviation of the mean (a-). Itcan be shown that ax is very closely related to vx, and is given by

(3-8)

or


Authors often quote a-f and thereby indicate greater accuracy in theirresults than is correct. In general, use ax unless you thoroughly under-stand the difference between these two values and know when to usea-..

3.4 REJECTION OF DATA

To understand the significance of the standard deviation, we must firstknow the expected distribution of observations. The probability distri-bution is found by taking a large number of observations and seeinghow probable it is to obtain any given value. This has been performedboth experimentally and analytically and it has been found that mostexperiments have a common probability distribution, called the normalor Gaussian distribution. It is defined by the equation

(3.10)

and is shown in Figure 3.1. We are not going to describe this distribu-tion in detail, but you should notice that it is symmetrical about themean and that the width of the peak depends on a, the standard devia-tion. A smaller a will yield a sharper peak. It can be shown that if thedata can be represented by a Gaussian distribution, then 68 percent ofthe data will fall within la of the mean, 95 percent within 2 CT, and 99.7percent within 3 tr. This says that if a data point falls more than 3 afrom the mean, there is a 99.7 percent probability that it is faulty.

One other term, called the most probable error (p.e.), is frequentlyused. If the data can be represented by a Gaussian distribution, 50 per-cent of the data falls within one probable error of the mean. Expressedin terms of the standard deviation,

p.e. = 0.675(7. (3.11)

Most scientific calculators are furnished with programs that calculatethe mean and standard deviation of a set of numbers. Learn always toquote an error when presenting data. A number by itself is almostuseless!

STATISTICS 67

-* .08-

13

Figure 3.1 Gaussian distribution for x = 5.0, a = 2.3.

Example: Using our set of six measurements of Mizar, compute thestandard deviation and present the mean.

al = I [(4.50 - 4.61)2 + (4.65 - 4.61)2 + (4.55 - 4.61)2

+ (4.45 - 4.61)2 + (4.60 - 4.61)2 + (4.90 - 4.61)2]

= i [0.0121 + 0.0016 + 0.0036 + 0.0256 + 0.0001

+ 0.0841]ffl = 0.0254<rx = VffJ = 0.16

visual magnitude of Mizar = 4.61 ± 0.16 (s.d.)

All of the discussion in this section has been leading up to the ques-tion of the rejection of data. Our sixth data point of 4.90 deviates by


2ff from the mean. It is tempting to regard this large deviation as ablunder or mistake rather than a random error. Should we remove itfrom the sample?

This is a controversial question and has no simple answer. It could bethat Mizar is an unknown eclipsing binary and you caught it duringeclipse. Throwing out that data value throws away the eclipse infor-mation! In any event, removing measurements constitutes tamperingwith or "fudging" the data.

Unless you can confidently state that a given measure is in errorbecause it was a cloudly night or some similar obvious problem, you willhave to make some sort of decision on data obviously out of bounds.There are two differing points of view that should be considered. It isyour choice as to what method should be employed.

At one extreme, there is the point of view that there is never anyjustification for throwing out data, and to do so is dishonest. If youadopt this point of view, there is nothing more to say. You can removemuch of the effect of one bad point by taking additional data, or youcould mention any extraneous points when you report your results.

The other point of view is to reject a measure if its occurrence is soimprobable that it would not be reasonably expected to occur. The usualcriterion is that data should be more than Iff or 3<r from the mean. Thebest way may be not to use the errant value in your calculations butreport it so others may make their own choice. In any case, never iter-ate, that is, remove data, calculate a new mean and standard deviation,and remove data again.

Of course, there is a third possibility. It is possible that the data arenot represented by a simple Gaussian, and that the wings of the distri-bution are larger than those of the Gaussian that fits the peak. Thus,the measure may be correct after all. You must decide this either bytheoretical considerations or by taking enough measures to map out thewings of the distribution.

3.5 LINEAR LEAST SQUARES

The method of least squares or regression analysis is almost exclusivelyused in fitting lines to experimental data. In examining a plot of exper-imental data, the human mind will "eyeball" a line that roughly splitshalf of the data above the line and half below. In a crude fashion, themind is approximating a least-squares line.

STATISTICS 69

*3.5a Derivation of Linear Least Squares

If a straight line is to be fitted to data, then the line has the functionalform

y = a + bx (3.12)

where y is the calculated y value for a given value of x. The fit is calledsimple linear regression, that is a linear function of only one variable,jc. The deviations of the individual data points from this line can bedefined as

(3.13)

or

A* -*-<*+ *>x,) (3.14)

Equation 3.14 is known as the equation of condition. Just as in the caseof standard deviation, we square the deviations and try to minimizetheir sum. This yields the line with the least error, or the least squareddeviation. If M is the sum of the squared deviations,

(3.15)/=i

M = L y] + b2 £x> + Na2 + 2a&Ex,- (3.16)

How do we minimize the sum? Mathematically, this is accomplishedby taking the partial derivative of the function M with respect to eachvariable of concern, and then setting these derivatives equal to zero. Thereader must take care to realize that the variables in this problem area and b (not x and y). So set

dM ^ dMda " db


The equations derived in this manner are called the normal equations.For our simple linear-regression example,

9M- = 2Na + 2b^xf - 2Ev, = 0

da

dM- = 2bL.x~ + 2aL,Xj — 2L,xly, =0

ob

or, rearranging,

« + <E *?>*-

3.5b Equations for Linear Least Squares

Equations 3.17 and 3.18 can be solved simultaneously to yield afterusing some identities:

Ex£yi £(*, - *)(>>,. - y)

,lntercept:a= ' - "^ -^ (3-20)

Fitting a straight line to experimental data is the most common use ofleast squares. A FORTRAN routine to perform simple linear regression isgiven in Section 1.4. This method can be generalized to any power of xor function M. For example,

y =

M =

or

STATISTICS 71

y = a cos (x)

M = Z[y, — a cos(jc,)]2.

Some comments are in order at this stage in the procedure. There aretwo basic reasons why the least-squares method is used rather than afreehand drawing. First, different people draw the freehand curveslightly differently because of observer bias. Second, the freehandmethod does not allow a quantitative measure of the goodness of the fit,an estimate of our confidence in the fitted line. The standard error, ae,of the least-squares estimate is given by

1 ^tf-=2£>'-' ) I <3"21>J V ^ i = i

This tells us the error expected at any point along the line. The goodnessof fit, r, is given by

- E(JC, - jc)(y, - y)(3.22)

The four values determined in the least-squares analysis (slope, /?; inter-cept, a; standard error, ae; goodness of fit, r) are used so commonly thatseveral calculators are preprogrammed to perform the analysis. To besafe, always plot the data and draw the calculated line. Any significantdeviations or a trend to the errors that may cause a lack of confidencein the fit then become obvious. A plot also serves as a check that all thedata were entered correctly.

Example: We have just built a DC amplifier. Because it is transis-torized, we suspect that its gain is temperature-sensitive. An experi-ment is devised, where a constant current is fed to the amplifier andthe temperature of the amplifier is changed, with the chart-recorderreading recorded at various temperatures. The values obtained arelisted below in the first two columns.


Temperature, °C(x) Amplifier Gain (y)

05101520253035404550

0.120.150.160.210.250.290.300.340.340.400.43

0.1170.1480.1790.2100.2410.2720.3030.3340.3650.3950.426

The data are shown in Figure 3.2. By visual inspection, it is obviousthat the gain is increasing with increasing temperature. A linearleast-squares line can be fit to the data and slope becomes the ampli-fier gain change per Celsius degree.

N = 11

EXi = 275 £>>, = 2.99

Ex,y, = 91.75 Ex* = 9625

1 1 X 9 1 . 7 5 - 2 7 5 X 2 . 9 911 X 9625 - 275 X 275

a = 7 (2.99 - 0.006182 X 275) = 0.1173

',. - y)2 = 0.001490

= 0.013

The plotted straight line is the linear least-squares fit, whose numer-ical values are listed under y in the table above.Multiple linear regression occurs when solving for more than one

slope. That is the case where

y,; = a + b}Xi + 62w, + &3z, + • • •

The solution to this problem is slightly more complicated because itinvolves putting the equation into matrix form and solving it by vector

STATISTICS 73

0.40

0.30

0.20

0.1010 15 20 25 30 35

TEMPERATURE (°C)

40 45 50

Figure 3.2 Amplifier temperature sensitivity.

differentiation and inversion. The solution is particularly useful in theleast-squares solution of the standard UBV transformation equationsand in the extinction calculation, but a discussion of this method is moreinvolved than warranted for this chapter. Appendix K gives a completediscussion of the multivariate least-squares method. The averageobserver may find it more accurate to treat the transformation equationproblem as a series of simple regression cases, thereby allowing bettercontrol over each step of the process.

3.6 INTERPOLATION AND EXTRAPOLATION

Often in photometry it is necessary to interpolate, that is, to find thevalue of some function between two base points. An example is whenperforming differential photometry in which several variable star obser-vations are sandwiched between consecutive comparison star measuresand you want the approximate comparison star reading at the time ofeach variable measure.

Consider the function in Figure 3.3. The solid line represents the truevalues of the function, with points identified at the base values *„, *,,x2, and X). The function y(x) might represent the star's intensity as itcrosses the meridian, and x the time of the observation. We want toknow its intensity between values x{ and x2. There are two usualapproaches: exact and smoothed interpolation.


Figure 3.3 Interpolating between points.

3.6a Exact Interpolation

Exact interpolation makes use of the fact that there is one and only onepolynomial of degree n or less which assumes the exact values y(x0),y ( x ] ) , . . . , y(xn) at the n + 1 distinct base points jc0, *,, . . . , xn.Therefore, to find y(x) between x, and jc2, we could use a linear poly-nomial with points x, and je2, a quadratic with x0, x}1 and x2, or a cubicwith all four points. Exact only means that the polynomial fits theknown data points exactly, not that it will interpolate exactly betweenthose points. For example, consider linear interpolation for our desiredvalue between x, and x2. If we use a straight line between xt and x2,we get a value reasonably close to the correct answer. If we use x0 and.x3 instead as our end points and draw a straight line between them, theresultant answer lies considerably below the correct one.

INTERPOLATED y|x}

— TRUEy(x )

Figure 3.4 High-order interpolation.

STATISTICS 75

Two problems arise in using high-order interpolating polynomials.First, increasing the order increases the number of "wiggles" betweendata values. This is shown in Figure 3.4, where we are using a cubicpolynomial to fit data that essentially lie on a straight line. Second,interpolating polynomials using unequally spaced base points get verycomplicated with higher order. Therefore, unless you know that theanswers should lie on a cubic or quartic line, use linear or at most qua-dratic interpolation.

To interpolate between points linearly, the function

y = a + bx (3.23)

is used, where b is the slope and a the intercept of the linear interpo-lating polynomial. To evaluate the slope and intercept:

b = Vl ~ yt (3.24)*2 - X}

a = y\ — bxi (3.25)

So,

y = a + bxy = y\ — bxi + bxy = y\ + b(x - x,)

m i + yi—£.(x-Xl) (3.26)

Example: The count rate of Mizar was measured to be 200,000counts per second at 03 : 00 UT and 300,000 counts per second at04 : 00 UT. What is the best linear guess as to the count rate at 03 : 30UT?

03:30 UT = 3.5 hours UT

( 3 0 0 0 0 0 0 0 0 0 0 )

y . 2oo,000 + M°<0.5)

y = 250,000 counts per second at 03:30 UT.


Working in a similar manner, we can derive the interpolating formulafor the second-order or quadratic polynomial between points XQ, jt,, andx2 as

y --/ y\ - y* (*-***-*>. (3.27)

You can see that even the quadratic interpolating polynomial is gettingcomplicated. Usually, higher-order polynomials are evaluated by com-puter. Note that Equation 3.27 looks like the linear form with an addedterm. This extra term can be considered the error that exists if linearinterpolation were used instead, and can be used to give an approximateerror when presenting the interpolated value.

3.6b Smoothed Interpolation

So far, we have investigated polynomials that passed exactly throughthe base values. As seen from Figure 3.4, this can cause large errors ifthe base values have some inaccuracies built into them. The bestmethod of interpolating under these circumstances is to use some sortof least-squares polynomial through the base points, and interpolatewith this approximate function. For photometric data, better accuracycan be achieved with smoothed interpolation, but with increased com-plexity. An example is to use several comparison star observations andfit a linear least-squares line to the observed count rate. Then this linecan be used along with the variable star measurements to derive theintensity differences for differential photometry. You usually achievegreater accuracy than if you use the comparison star observation closestto the variable star measure because you are using the information con-tained in earlier and later measurements.

In general, we suggest that you use exact linear interpolation when-ever interpolation is necessary. If you have a large number of base val-ues, smoothed linear interpolation could be used. Rememberhowever, that interpolation by nature is not exact, and requires data oneither side of the value to be calculated.

STATISTICS 77

3.6c Extrapolation

We have neglected the case of extrapolation, that is, the determinationof y(x) where x lies beyond any of the observed values. This is becauseextrapolation is very, very risky and should be avoided at all costs!

An example of the errors that can arise from extrapolation is themeasurement of the atmospheric extinction. If you follow an extinctionstar from the zenith to, say 45" above the horizon, determine extinctionfrom it, and use your values for an observation on the horizon, yourresults may be good. But there also may be a cloud bank or smoke layeron the horizon, making extinction there much different than near thezenith.

The rule of thumb for extrapolation is that if the data point is closeto the last base value, you can extrapolate, but should consider thisextrapolated value as having very low weight.

3.7 SIGNAL-TO-NOISE-RATIO

It is intuitively obvious that the longer we continue to gather data on astar during a single observation, the more accurate our results become.We would like a quantitative measure of this accuracy. Experimentalscientists commonly use a quantity known as the signal-to-noise ratio,or S/N, which tells us the relative size of the desired signal to theunderlying noise or background light. The noise is defined as the stan-dard deviation of a single measurement from the mean of all of themeasurements made on a star.

Astronomers typically consider a good photoelectric measurement asone that has a signal-to-noise ratio of 100, or in other words, the noiseis 1 percent of the signal. For photon arrivals, the statistical noise fluc-tuation is represented by the Poisson distribution, and for bright sourceswhere the sky background is negligible,

S total received countsN Vtotal received counts§- = Vtotal received counts (3.28)

N

Therefore, for a S/N of 100, we must acquire 10,000 counts. A S/N of100 means that the noise causes the counts to fluctuate about the mean


by an amount equal to one hundredth of the mean value. To computethis error in magnitudes, we compare the mean number of counts, c, tothe maximum or minimum values induced by the noise, that is

c±

Am = -2.5 log '

Am = -2.5 log 1 ±

c

1100

= ±0.01 magnitude.

In other words, a S/N of 100 implies an observational error of 0.01magnitude. More detail on both the signal-to-noise ratio and the Poissondistribution can be found in Appendix K.

3.8 SOURCES ON STATISTICS

Listed below is a sample of statistics and numerical analysis texts thatmay be of interest to the reader. This list is by no means complete aswe have not examined the dozens of available texts, but the sourceslisted do appear to present the material in a manner useful to theastronomer.• Bevington, A. R. 1969. Data Reduction and Error Analysis for The

Physical Sciences. New York: McGraw-Hill. Nice beginning college-level text with FORTRAN programs. Highly recommended.

• Bruning, J. L., and Kintz, B. L. 1977. Computational Handbook ofStatistics. 2d ed. Glenview, Illinois: Scott, Foresman & Co. No leastsquares, but takes a computational approach. Includes FORTRANprograms.

• Carnahan, B., Luther, H. A., and Wilkes J. O. 1969. AppliedNumerical Methods. New York: John Wiley and Sons. One of thebest FORTRAN numerical analysis books. Explanation is at collegelevel.

• Ehrenberg, A. S. C. 1975. Data Reduction: Analyzing and Interpret-ing Statistical Data. New York: John Wiley and Sons. Good begin-ning college text.

• Harnett, D. L. 1975. Introduction to Statistical Methods. 2d ed.Reading, Mass. Addison Wesley. Beginning college level.

STATISTICS 79

Meyer, S. L, 1975. Data Analysis for Scientists and Engineers. NewYork: John Wiley and Sons. Beginning college ievel.Young, H. D. 1962. Statistical Treatment of Experimental Data.New York: McGraw Hill. Very nice advanced high school level textwith lots of explanations.

CHAPTER 4DATA REDUCTION

There are three stages in the treatment of photoelectric data, which wecall acquisition, reduction, and analysis. The techniques for data acqui-sition are presented in Chapter 9 and should be thoroughly studiedbefore raw data are acquired. We treat data reduction now, rather thanafter Chapter 9, because intelligent data acquisition requires a knowl-edge of the types of observations necessary for the reduction process.The reduction of data from counts or meter deflections into magnitudestied to the standard system can be a complicated process, but one thatis required by many research projects. Careful reading of this materialand the examples in the appendices will enable you to reduce any UBVobservations and place them on the standard system. Most of the thirdstage, data analysis, is left up to the individual. Analysis involves thecalculation of such quantities as periods, orbital elements, and in gen-eral all calculations beyond the determination of magnitudes and colors.The analysis depends greatly on the purpose of the investigation andshould be obtained from other sources.

4.1 A DATA-REDUCTION OVERVIEW

You have some raw instrumental measurements of stars and sky back-ground. What are the steps necessary to complete the reduction? Thereare many different ways that data reduction can proceed. A generaloutline that fits most situations follows:

1. If you are pulse counting, correct your values to one consistent set,that is, counts per second rather than counts per various arbitrarytime intervals. The count rates should be corrected for dead time.

80

DATA REDUCTION 81

For DC photometry, the amplifier gain settings must be correctedto true gain using the gain table, which is discussed in Section 8.6.

2. Subtract the sky background from each stellar measurement. Thismust be done before the numbers are converted into logarithmicvalues (magnitudes).

3. Calculate the instrumental magnitude and colors. For differentialphotometry, calculate the magnitude differences between the var-iable and the comparison star.

4. Determine the extinction coefficients and apply the extinction cor-rection. This step is often unnecessary for differential photometry.If you intend to leave your differential photometry on the instru-mental system, skip to step 7.

5. Use the standard stars to determine the zero-point constants and,if necessary, the transformation coefficients.

6. Transform your instrumental measurements to the standardsystem.

7. Estimate the quality of the night by comparing the transformedstandard-star magnitudes and colors with their accepted values.For differential photometry, check the reproducibility of the com-parison star measurements after correcting them for extinction.

8. Perform any ancillary calculations such as time conversions thatare necessary to make your observations useful and publishable.

Steps 1, 2, 3, 5, 6, and 7 are illustrated by a worked example inAppendix H. An example of step 4 is found in Appendix G. Step 8 iscovered in detail in Chapter 5. In what follows, we review the conceptsand difficulties of some of these steps and present a worked example ofthe data reduction associated with differential photometry.

4.2 DEAD-TIME CORRECTION

One of the major drawbacks of a pulse-counting system is its inabilityto count closely spaced pulses with accuracy. After the photomultipliertube, preamp, or counter detects a pulse, there is a short time intervalin which the device is unable to respond to an additional pulse. If twoor more pulses arrive at any of the major components in an intervalshorter than the so-called dead time of the component, these pulses willbe detected as a single count. Incident photons from bright sources willon the average be more closely spaced in time than those from fainter


sources. But these photons do not arrive in evenly spaced time intervals.From a bright source, four pulses may arrive in the first 10 nanoseconds,none in the next 10 nanoseconds, etc. The manufacturer's specificationson the photomultiplier tube, preamp, or counter dead times should notbe relied upon, as those figures are based on evenly spaced, uniformpulses that are never found except in the testing laboratory.

The component with the longest dead time is the major contributorto the inaccuracy, so in general use a counter with at least a 100-MHzresponse and the fastest preamp possible. At Goethe Link Observatory,the Taylor preamp (see Chapter 7) is the slowest component. In gen-eral, the photomultiplier tube dead time is insignificant compared tothat of the preamp or counter. The dead-time problem makes pulsecounting nonlinear for bright sources.

The equation for the dead-time correction is simple in form, but dif-ficult to solve. The equation can be written as

n = Ne~'N (4.1)

where

n = observed count rate in counts per secondTV = "true" count rate for a perfect system in counts per second/ = dead-time coefficient defined as t = \/N when observed count

rate falls to \/e of the true count rate.

Equation 4.1 can be rearranged to yield

jj = *-"• (4.2)

Taking the natural logarithm of each side yields

\n(n/N) = -tN

or

In (AVn) = tN. (4.3)

If we graph In ( N / n ) versus N, then t is the slope of the best-fitted line.

DATA REDUCTION 83

Our problem, though, is that we do not know N and therefore cannotsolve for t.

The technique for finding t takes advantage of the fact that for lowcount rates, the dead-time correction is negligible. Suppose we havesome device that can attenuate the light reaching the photomultipliertube by a known factor, which we will designate as b. Then, when theattenuator is in place, only \jb of the light reaches the photomultipliertube. (The nature of the attenuator is discussed later.) If we observe alight source or star with the device in place, the observed count rate, nL,will be low and will very nearly equal the true count rate, NL. If theattenuator is removed, the true count rate, NH, will increase b times.That is,

NH = bNL =* bnL. (4.3a)

However, the observed rate increases by some smaller factor because ofdead-time losses. A comparison of the observed rate, nH, to bnL gives ameasure of the dead-time coefficient. Equation 4.3 can now be rewrittenas

In = tbn, (4.4)

If several light sources of different brightnesses are observed both withand without the attenuator, a plot of In (bnLjnH) versus bnL yields aline with slope t.

There are three methods of attenuation commonly used. Eachmethod is considered in turn.

Using aperture stops. In this method, the telescope is pointed at abright star and the front of the tube is covered by a piece of card-board with a small circular opening. The count rate recordedthrough this opening is nL. The count rate recorded through alarger opening in a second piece of cardboard is nH. The factor bis just the ratio of the areas of the two apertures. There is a dis-advantage with this method if a reflecting telescope is used. In thiscase, the apertures must be made small enough to be placed offthe optical axis so that the secondary mirror support does not


block incoming light. If this is not done, then one must be carefulto account for the area of the secondary support in the calculationof b. This may not be difficult to do if the mirror cell has a circularcross section and the support vanes are thin enough to ignore.

2. Using the photometer diaphragms. In this method, the light sourcecannot be a star. It must be an extended object with uniform sur-face brightness. For this purpose, a uniformly illuminated whitecard can be placed in front of the telescope. The diaphragms inthe photometer head can then be used, a small one to measure nL

and a large one to measure nH. The ratio of the diaphragm areasis b. The problem with this method is that the card must be uni-formly illuminated, the diaphragm sizes must be known accu-rately, and the light source must be variable if more than oneobservation is to be made with the available diaphragms. The day-time sky can be used as the light source if extreme care is used toprevent current overload of the photomultiplier tube.

3. Using a "neutral" density filter. This method uses a neutral den-sity filter placed in the filter slide. The star need only be measuredonce with and once without the filter in the light path. The densityof the filter is then b. The problem with this method is that nofilter is truly "neutral." This means that the amount of light trans-mitted by the filter depends, at least weakly, on the color of thestar. This would seem to make this method very cumbersome.However, once this color effect is calibrated, the neutral densityfilter offers a convenient way to find the dead-time coefficient. Tocalibrate the color dependence of the filter, several stars of widelydifferent colors are observed both with and without the filter. Thestars selected (see Appendix C) should be relatively faint so thatthe dead-time correction is negligible. For each star, a magnitudedifference (v, — v0) is calculated by

v1 - v0 = -2.5 log (/1,/rto), (4.5)

where n\ is the count rate with the filter in place and nQ is the ratewithout the filter. Then (v, — v0) is plotted versus (B — V) foreach star. The resultant graph should be a nearly horizontal line.In the case of a filter used at Indiana University, a least-squaresfit resulted in

V| - v0 = -0.008 (B - V) + 3.934.

DATA REDUCTION 85

It can be seen that this relation depends very weakly on color. Thefactor, £, is the ratio of light intensity without the filter to thattransmitted. That is,

v, - v 0 = -2.5 log (/,//„) (4.6)

or

b = /„//, = lo0-*"-^. (4.7)

The dead-time coefficient can then be found using the observationsof bright stars.

An example of a dead-time coefficient determination can be found inAppendix F. Even when t is known for a given count rate, an iterativetechnique is required to solve Equation 4.1. First substitute the observedcount rate, n0, for TV and calculate a corrected count rate, nv. This newvalue is then substituted for N and the process is repeated until nk

approaches «*+, to the accuracy required. A FORTRAN subroutine toiterate this equation is given in Section I.I, in addition to an exampleshown in Appendix H.

4.3 CALCULATION OF INSTRUMENTAL MAGNITUDES ANDCOLORS

Section 1.7 derives the equations necessary to formulate instrumentalmagnitudes and colors, along with a physical understanding of the con-stants involved. Writing Equations 1.10 and 1.12 explicitly for the VBVsystem, we have

v = cv - 2.5 log dv (4.8)

b- v = cbv- 2.5 log (</>/</,) (4.9)

u- b = cub- 2.5 log (djd& (4.10)

which relate the observed deflections or counts, d, to the instrumentalmagnitudes and colors. Because these instrumental values are used inSection 4.5 to evaluate the zero-point shifts in the transformation equa-tions, the constants, the c's, are arbitrary. For DC work, the usual formsof Equations 4.8 through 4.10 are:


v = -2.5 log (dv) + Gv (4.11)b - v = -2.5 log (db/dv) + Gb - G¥ (4.12)u - b = -2.5 log (du/db) + Gu- Gb (4.13)

where G is the relative gain for each filter and d is the chart-recorderdeflection at that gain setting. For pulse counting, the equationsbecome:

v = - 2.5 log (C) (4-14)b- v = -2.51og((VO (4-15)u- b = -2.51og(C/O (4.16)

where C is the count rate in counts per second through each filter.Worked examples of both forms of magnitude calculations can be foundin Appendix H.

4.4 EXTINCTION CORRECTIONS

Remember that extinction represents the loss of starlight while travers-ing the earth's atmosphere. All published photometric results correctfor this and essentially give the apparent magnitude of the star outsideof the earth's atmosphere, called the extra-atmospheric magnitude. Theequations discussed in Section 1.8 and derived in Appendix J are thebasis for the treatment of extinction. Much of the material in this sec-tion makes use of the results obtained by Hardie.' Most extinction cor-rections account for first-order extinction, along with the associated airmass calculation. For greater accuracy, second-order extinction shouldbe taken into account.

4.4a Air Mass Calculations

At altitudes more than 30° above the horizon, the simple plane-parallelapproximation, derived in Appendix J, for the amount of atmospherebetween an observer and a star is accurate to within 0.2 percent. Whena star's altitude is greater than 30° or correspondingly, the zenith dis-tance, z, is less than 60°, this approximation gives

X - secz (4.17)

DATA REDUCTION 87

where

sec z = (sin 0 sin 6 + cos<£ cos 6 cos //)"' (4.18)

where 0 is the observer's latitude, 5 the declination of the star, and His its hour angle in degrees. The mass of the air traversed by the star-light is X. This quantity is at a minimum when a star is directly over-head, or

sec 2 = sec 0° = 1.

This amount of air is called one air mass for convenience, rather thantrying to remember some large-numbered column density.

For zenith distances greater than 60°, the plane-parallel approxi-mation breaks down. An equation that more closely approximates theeffects of the spherical earth must be used. The most common polyno-mial approximation was made to data collected by Bemporad in 1904and is given in Hardie1 as:

X = sec z - 0.0018167 (sec z - 1) - 0.002875 (sec z - I)2

- 0.0008083 (sec z - I)3 (4.19)

where z is the apparent, not true, zenith distance. Equation 4.19 fitsBemporad's data to better than 0.1 percent at an air mass of 6.8. Thatis only 10° from the horizon and closer than you would want to observe.Because these data represent only average sky conditions at one locationat the turn of the century, we cannot expect that the actual accuracy ofthe air mass calculation is 0.1 percent.

Other methods of determining air mass involve the use of tables ornomographs and are not presented here because they are less practicalthan solving the equations above with a scientific calculator.

Example: An observer located at 40° north latitude locates SigmaLeo(RA = Ilh20m6s, Dec. = 6 °8'21") at an apparent hour angle of3h, which is 45°. What is the air mass between the observer andSigma Leo?


From Equation 4.18, we have:

sec z = [sin (40°) sin (6.1392°)+ cos (40°) cos (6.1392°) cos (45°)] ~'

sec 2 = 1.6466

From Equation 4.17, we have:

X = secz = 1.6466

For comparison, we can now use Equation 4.19 to improve ouraccuracy:

X = 1.6466 - 0.0018167(0.6466) - 0.002875(0.6466)2

- 0.0008083(0.6466)3

X = 1.6440

Notice that the two methods agree to about 0.1 percent.

4.4b First-order Extinction

Extinction is very difficult to model exactly because of the many vari-ables that play important roles in the absorption of light in the earth'satmosphere. To do a first-order approximation is to account for the larg-est contributor, the air mass variation. In this approximation, the fol-lowing equations hold:

v0 = v - k(X (4.20)(b - v)0 = (b - v) - k'bvX (4.21)(u - b)0 = (u- b) - k'ubX, (4.22)

where k' is the principal extinction coefficient in units of magnitudesper air mass and the subscript 0 is used to denote an extra-atmosphericvalue. Rearranging these three equations, we obtain

v = k(X + v0 (4.23)(b- v) = k^x+(b- v)0 (4.24)(«- b) = k'uhX+(u- b)0. (4.25)

DATA REDUCTION 89

The values of the extinction coefficients can then be found by followingone star through changing air masses and plotting the color index ormagnitude versus X. The slope of the line is the extinction coefficientand the intercept the extra-atmosphere magnitude or color index.

What has been presented is an ideal case. In reality, by the time theair mass in the direction of a star has changed appreciably, the sky mayhave undergone considerable change. Even if the atmosphere is static,the extinction in various parts of the sky is not constant. The changemay be due to a local fluctuation, giving rise to scatter about the meancurve, or it may be on a large scale, such as an east-west variation.Moreover, the atmosphere varies daily depending on its moisture anddust content. As is explained in Section A Ac another complication arisesas a result of the extinction itself being color-dependent.

Do not be discouraged by all of the problems discussed above. Theaccurate measurement of extinction is a tough problem and if deter-mined to high precision would leave little time for the actual observa-tions! Therefore, knowing the value of the extinction coefficients to anaccuracy of 2 or 3 percent is considered acceptable by most professionalastronomers. Observational recommendations for extinction can befound in Chapter 9. Several methods of determining the principalextinction coefficients are available. The two most common are usingcomparison stars, and using a sample of AO stars near your variables.These are discussed below.

Because of the spatial variation possible in extinction, the best pos-sible choice for your extinction star is the comparison star for theobserved variable. This choice has two advantages: the extinction mea-surements are never far in time or space from the variable star values,and if you are using comparison stars of the same color as your vari-ables, second-order effects become negligible. This method is onlysuited to observing programs in which a single variable is observed mostof the night. This is the only way to collect enough measurements of thecomparison star, over a wide range of air mass, in order to determinethe extinction coefficients.

Another method of determining the principal extinction coefficientsis to use a sample of AO stars covering the region of the sky in whichyou are observing. The extinction can then be determined by usingleast-squares analysis directly. This technique has the advantage ofrequiring a smaller amount of observing time. However, the analysis ismore complicated and if a computer program is used, there is a temp-


tation not to plot the measurements. In which case, one bad measure-ment because of haze moving in during your observations may not beapparent, yet it can affect your results drastically.

«4.4c Second-order Extinction

If we include the color dependence of the extinction coefficients, asexplained in Section J.3, then we can modify the principal coefficientsto

*C-» %+ k"v(b - v) (4.26)*£,-*£„+ k»bv(b- v) (4.27)

Equations 4.20 and 4.21 become

v0 = v - k'vX - k"v(b - v)X (4.28)(b - v)0 = (b - v) - k'bvX - k"bv(b - v)X. (4.29)

Equation 4.22 remains the same because of the definition of k"ab as zero.We can solve Equations 4.28 and 4.29 for the second-order coefficientsby using a close optical pair having very different colors. Because oftheir proximity, the air mass remains constant between them and wecan obtain

vo l- v02 = v,- ( v 2 - W- k"v(b- v)2X)

or

Av0 - Av - *? A(/> - v}X (4.30)Av = k"v A(/> - v)X + Av0.

Similarly,

- v) = k»bvk(b- v}X + A(6- v)0 (4.31)

where A indicates the difference in colors or magnitudes of the two starsat each air mass. The solution of Equations 4.30 and 4.31 is performed

DATA REDUCTION 91

easily by plotting Av or A(£ — v) versus A(6 — v}X; the slope is thenthe second-order extinction coefficient. Each pair so measured can alsobe used to determine the principal coefficients once the second-ordervalues are known.

From experience and theory, the second-order coefficient for v isessentially negligible. In addition, the values found through the use ofEquation 4.31 have been found to be relatively constant and probablydo not need to be determined any more often than are the color trans-formation coefficients.

An extinction example for both principal and second-order extinctioncan be found in Appendix G. Suitable pairs of stars can be found inAppendix B. Most of these pairs are from a list prepared by Kitt PeakNational Observatory and are near the equator. Note that the red starsare quite often faint and are more difficult to measure. Bright pairs arejust hard to find!

4.5 ZERO-POINT VALUES

From Chapter 2, we have the equations

y = €(B - V] + v0 + fv (4.32)(B - y) = p(b - v)0 + fa (4.33)(U- B) = $(u - b)0 + U. (4.34)

These three equations are the working equations for this and the follow-ing sections. Rearranging,

f, = V- v f l - t(B- V} (4.35)fa = (B - V) - n(b - v)0 (4.36)fa = (U- B) -*(M - 6)0. (4.37)

In other words, the zero point is equal to the standard value minus thetransformed extra-atmospheric value. The zero points are calculated bysolving Equations 4.35 to 4.37 for each standard and then taking means.

Figure 4.1 plots the three zero points determined with pulse-countingequipment at Goethe Link Observatory over an 18-month period. Thebreak in £"„ occurred when the mirrors were realuminized. It indicatesan approximate 0.5 magnitude gain in sensitivity. The scatter is caused


11.4 r-

11.2

11.0

>ftril

10.8

10.6

10.4

1.30.-

1.10 -

0.90

'" • /

1 1 1 1

V ••

1 1 ' 1 1 * 1 1

-1.00

-1.20

-1.40

•

• • •* v A *• •* • w\ *f r. •

".

i i9 1 1 1 3 5 7 9 1 1 1 3 5 7

MONTHS

Figure 4.1 Example of zero points.

primarily by the small number of standard stars used in the zero-pointcalculation and by differing sky transparency. The zero-point valuesmust be determined nightly.

4.6 STANDARD MAGNITUDES AND COLORS

Once the transformation coefficients have been determined, along withthe nightly values of extinction and the zero-point shifts, Equations4.28, 4.29, and 4.22 can be used to determine the extra-atmosphericinstrumental magnitudes and colors. Substituting these values intoEquations 4.32 through 4.34 yields the transformed standard magni-tudes and colors. These values may not agree with accepted values for

DATA REDUCTION 93

a constant star because of statistical scatter and the quality of the night,but means determined over several nights should yield good numbers.

4.7 TRANSFORMATION COEFFICIENTS

The transformation coefficients defined in Chapter 2 could be deter-mined directly from Equations 4.32 to 4.34 by measuring several starswhose standard magnitudes and colors are known. In fact, that is theapproach used for determining the V coefficient, c:

V- v0 = t(B- V} + £,. (4.38)

The slope of the best-fitted line for a plot of (V — v0) versus (B — K)will be the coefficient e. Note that you are plotting the differencebetween the two magnitudes; this is more accurate than plotting V ver-sus v0 because it magnifies small variations in either number. Forinstance, the change in v from 8.79 to 8.80 is less than 1 percent. Butif V — 8.85, then V — v0 = 0.06 in one case and 0.05 in the other, adifference of 20 percent.

However, our two equations for the color indices are not in differen-tial form. They can be converted to differential measurements by solv-ing for the extra-atmospheric instrumental colors as shown for (b — v):

(b - v). = S*. (4.39)M

Subtracting the extra-atmospheric instrumental value, (6 — v)0, fromboth sides of Equation 4.33;

(B - V)-(b- v)0 = p(b - v)0 - (b - v)0 + ffa

= ( M - \)(b- v)0 + r*v

and then substituting Equation 4.39 into the right-hand side, yielding

(B- V) - ( b - v ) 0 = (\--\B- V) + , (4.40)V M/ M


and similarly for (U — B\

f \\ r *([/ - B) - (u - b)Q = 1 - - (£/ - B) + . (4.41)

Equations 4.40 and 4.41 along with Equation 4.38 are our workingequations for determining the transformation coefficients. Plots of theleft-hand sides of these equations versus either (B — V) for Equations4.38 and 4.40, or (V -- B) for Equation 4.41, yield slopes that arerelated to the transformation coefficients.

In practice, there are two methods commonly used to solve for thetransformation coefficients:

1. Choose several standards from the Johnson standard list inAppendix C, measure them, and determine the transformationcoefficients. With this method, you are able to select bright starsof widely differing colors. It has the disadvantage of requiringaccurate knowledge of the extinction coefficients. Select at least10 and preferably 20 or more standard stars with a wide range of(B — F) and (U — B) colors and try to observe them when theyare near the zenith. Note that the Johnson list has internal errorsof ± OT02, so do not expect to achieve results that are significantlymore accurate.

2. Use one of the standard clusters from Appendix D for your stan-dards. This eliminates the extinction problem, but adds the prob-lem of faintness. Not only are most clusters fourth to sixth mag-nitude or fainter, but also the red stars are fainter than the hotblue stars of the cluster and add more error to the determination.Another problem is systematic errors in the cluster standard val-ues that can make the coefficient determination from one clusterdifferent from another. This is a minor effect, but it should be keptin mind.

We suggest that you use method 1 with a telescope aperture of 25centimeters (10 inches) or less. For larger telescopes, use method 2. Thenormal procedure is to determine carefully the transformation coeffi-cients at the beginning and end of your observing season, as well as onceor twice in between. Resolve to spend half of a good night for each ofthese determinations. The coefficients change very slowly with time, andmean values are generally sufficient. For method 2, the different deter-

DATA REDUCTION 95

minations should be made using different clusters, if possible. Transfor-mation examples using method 1 for DC photometry and method 2 forpulse counting are given in Appendix H.

4.8 DIFFERENTIAL PHOTOMETRY

The reduction of differential photometry data is treated somewhat dif-ferently from the descriptions found in previous sections. Because dif-ferential photometry is the starting point for most newcomers to pho-tometry, we explain the reduction process in detail. We assume that theobservations were made in accordance with the recommendations ofSection 9.3c. Variable star observations are bracketed by those of thecomparison stars.

Table 4.1 contains a partial list of observations of the eclipsing binaryUZ Leonis. In this example, a DC photometer was used. Columns 1, 2,and 3 contain the object's name, universal time of observation, and thefilter designation. Column 4 contains the amplifier gain in magnitudes.Columns 5 and 6 contain the chart recorder pen deflection of the starand sky through each filter.

1. The first step is to subtract the sky background from each stellarmeasurement. The results appear on column 7. If a pulse-countingphotometer had been used, and if the stars were fairly bright, adead-time correction would be applied (Section 4.2) before sub-tracting the sky background.

2, The observations in column 7 follow a pattern such as

cw... wcw... wcw... wcw... we...-v

Block

Block 3 Block 4

where C and V represent a measurement through each filter ofthe comparison and variable star, respectively. The data have ablock structure where each string of several variable star mea-surements is sandwiched between two comparison star measure-ments. Each block is reduced separately by averaging the com-parison star measurement at the beginning and end of the block.Equations 2.38 through 2.40 or 2.35 through 2.37 are used to pro-

TABLE 4.1. Differential Photometry Data

Object UT Filter

i ; Comp. VBU

UZ Leo V2:40 B

UV

2:42 B£ U

V

\

\

2:45 BU

Comp. KB

t ^— UZ Leo V

2:59 BU

N VX

£ 3:01 BUV

3:04 BU

Com p. V

BU

Gain

10.50510.50511.9381 1 .0341 1 .03412.432

10.50510.50511.9381 1 .0341 1 .03412.432

10.50510.50511.938

Star

40.250.044.341.248.450.741.148.450.341.548.851.041.051.246.241.849.251.442.049.552.542.150.052.541.251.045.3

Sky

12.012.024.015.115.133.6

12.012.024.515.315.334.6

11.811.824.0

Net Av A6 Au AK A(5 - V)

28.238.020.326.1 0.628 0.62833.3 0.689 0.05717.1 0.71726.0 0.633 0.63333.3 0.689 0.05216.7 0.74326.4 0.616 0.61633.7 0.676 0.05617.4 0.69829.039.221.726.5 0.634 0.63433.9 0.687 0.04916.8 0.76226.7 0.626 0.62634.2 0.677 0.04717.9 0.69326.8 0.622 0.62234.7 0.661 0.03617.9 0.69329.439.221.3

A(I/- B)

0.033

0.064

0.026

0.075

0.016

0.038

Average: A V = 0.626 ± 0.007 (s.d.)A(B - V) = 0.049 ± 0.007A(f / - B) = 0.042 ± 0.024

DATA REDUCTION 97

duce a value of Av, A6, Aw for each variable star measurementwithin the block. The process is then repeated until all the blockshave been reduced. Note that each comparison star measurementdoes "double duty" because the last comparison star measurementin one block is also the first one in the next block. Columns 8, 9,and 10 contain the calculated magnitude differences. Check a fewof these entries with your calculator.

In many cases, reduction stops at this point because extinction cor-rections are often ignored in differential photometry. Conversion to thestandard photometric system is unnecessary for many types of researchprojects. However, if the variable and comparison are separated bymore than a degree it may be wise to apply an extinction correction. Aworked example of this correction to differential photometry can befound in Appendix G. The data in Table 4.1 do not require thiscorrection.

If the transformation coefficients are known, it is possible to convertthe magnitude differences from the instrumental to the standardsystem.

3. In this example, e = -0.004, ju = 0.927, and $ = 1.178. Equa-tions 2.41 and 2.42 were used to compute A(6 — v) and A(w -b) for each stellar measurement. Equation 2.49 and 2.50 can thenbe used to compute A(fl — V) and A(t/ — B). Finally, Equation2.48 can be used to compute AK Check a few of the entries incolumns 11 through 13. Note that the difference between the mag-nitude and colors in the instrumental and standard systems ispractically negligible. This means that the detector and filtersused match the standard system well.

If the comparison star has been standardized, Equations 2.51 through2.53 can be used to calculate the actual magnitude and color of thevariable. In this particular example, the comparison star was standard-ized on a previous night with the following results:

V = 8.950 ± 0.038 (s.d.)(B - V) = 0.287 ± 0.012(U - B) = 0.039 ± 0.058.


If we average the values of AK, A(5 — K), and A(C7 — B) in columns11, 12, and 13 we can compute the magnitude and colors of UZ Leonisto be

V = 8.95 + 0.63 = 9.58(B - V) = 0.29 + 0.05 = 0.34(U - B) = 0.04 + 0.04 = 0.08.

The probable errors for these values are found by adding the standarddeviations of the comparison and variable star in quadrature, that is

p.e. = 0.675 VffLv + <r™-

The final quoted results are then

V = 9.58 ± 0.03 (p.e.)(B - F) = 0.34 ± 0.01(U - B) = 0.08 ± 0.04.

The larger error in (U — B) reflects the fact that both the comparisonstar and UZ Leonis are very faint in the U filter.

•4.9 THE (U - B) PROBLEM

In Chapter 2, the second-order extinction coefficient for ([/ — B) wasarbitrarily defined as zero. However, k"ub can be a larger correction thank"bv, because the u extinction depends on:

1. the Balmer discontinuity2. the second-order color term3. systematic nonlinear deviations resulting from the assumption

that fc*b was constant in the the original UBVdaia.

Because of these problems, the (U — B) color term is inaccurate andpoorly defined. Unfortunately, use of the existing system is so tradi-tional that it would be extremely difficult to redefine the VBV system.The best solution is to calculate your (u - - b)Q values correctly,accounting for all first- and second-order effects, and then transformyour ubv data to the existing, but nonideal, standard VBV system.

DATA REDUCTION 99

The remainder of this section presents one such method of transfor-mation as derived by Moffat and Vogt.2 This kind of correction is com-plicated and should only be attempted by those who are experienced inphotometry.

Moffat and Vogt found that, for any given star, the residuals in(U — B) vary linearly with air mass, X. That is,

A[(£/ - B) - (u - b)] = 7, + 72* (4-42)

A plot of this equation indicates that 7, and y2 are linearly related. Thatis,

where

0 -0.27.

We can therefore rewrite Equation 4.42 as

- * ) - ( M - b)] = 7,0 +/WO (4-43)

where (3 is a constant and 7, is a function of spectral type or color. Theproblem of correcting for (U — B) differences is then reduced to oneof determining 7,. However, 7, is a nonlinear function o f ( U — B) and,in addition, is not a unique function of (U — B). But it is a linear andunique function of another parameter, <?, that is similar to the redden-ing-free parameter of Johnson and Morgan:3

7i = P? (4-44)

where

g = (u - B) - \.Q5(B - V). (4.45)

The procedure to follow in correcting (U — B} is:

1. For each standard star, determine q from Equation 4.45.


2. For each standard, determine 7, from

_ (U- B)-(u- b)1 + &X

3. Plot 7i versus q and determine p from Equation 4.44.4. For all future stars, correct (U — B) by

(4.46)

(4.47)

This procedure will reduce the mean external error in (U — B) toabout Om.02. Without it, the mean error would be approximately threetimes higher. Include k"ub in all equations in a similar manner as k"v isincluded in (B — V} equations.

REFERENCES

1. Hardie, R. H. 1962. In Astronomical Techniques. Edited by W. A. Hiltner. Chi-cago: Univ. of Chicago Press, chapter 8.

2. Moffat, A. F. J., and Vogt, N. 1977. Pub. A.S.P 89, 323.3. Johnson, H. L., and Morgan, W. W. 1953. Ap. J. 117, 313.

CHAPTER 5OBSERVATIONAL CALCULATIONS

There are a number of calculations that are useful for obtaining andreducing observational data. These include determining when an objectappears above the horizon on a given night, precession of coordinates,and the calculation of date and time quantities. These and other cal-culations are discussed in this chapter. The subjects are recommendedreading even if photometry is not attempted, as they are also involvedin most visual observations.

5.1 CALCULATORS AND COMPUTERS

Observational calculations and data reduction are very tedious withoutthe use of a calculator or computer. The scientific calculator has nowbeen in existence for a decade. Since the introduction of the Hewlett-Packard HP-35, calculators have made great strides in capability witha reduction in cost. Programmable varieties are excellent and card-pro-grammables are the ultimate, as programs and data can be stored forlater recall. The mode of operation (whether reverse Polish notation(RPN), algebraic, or even a high-level language such as BASIC) is unim-portant as long as you are comfortable with your choice. We suggestthe use of a programmable calculator with seven to 10-digit accuracyand the capability of converting degrees-minutes-seconds into decimaldegrees, a real blessing to astronomers! The well-known brands, such asHewlett-Packard and Texas Instruments, should be your first choice. Areview of calculators is given in Sky and Telescope.'

A calculator is all that is required to perform the data reduction.However, some people may prefer something more advanced. The nextstep up from a programmable scientific calculator is the microcompu-

101


ter. Large-scale integration has advanced sufficiently that the purchase of amicrocomputer should be considered if much data reduction or instrumen-tation control is anticipated. The 16-bit microcomputer is the industry stan-dard at this time, though eight-bit and 32 bit systems are also available. Asthe industry seems to change with about a one-year time scale, only use thisinformation as a general guide.

We strongly recommend the use of a bus-type computer. Examples ofthese are the Apple II, the IBM PC and the Macintosh II. These computersgive you the easy ability to expand as your needs warrant. For example, youcan add a pulse counting board or image processing capability of any of thesecomputers. Other, non-expandable computers are usually cheaper and canbe cost-effective if you know beforehand what your computer needs will be inthe next few years.

At the time of this writing, the IBM PC is the industry standard computer.You can buy a better computer at lower cost with more peripherals and soft-ware programs for this system than any other. We recommend you look at aPC clone as your first choice.

Arithmetic operations used in data reduction can be performed in soft-ware at the expense of speed and space. Since most microprocessors do notinclude floating point operations in hardware, all such operations must beemulated in software. However, for many microprocessors, numeric copro-cessor chips are available. For example, the Intel 8088 has the 8087 copro-cessor, and Motorola 68000 has the 68881. Such chips perform floating-pointoperations a hundred times faster than can be accomplished in software. Wehighly recommend the purchase of a numeric coprocessor for your computerfor astronomical data processing.

Hard disks are another useful peripheral. Only a few years ago the floppydisk was king of the mountain, and 360Kb of storage on one disk was thoughtto be enormous. Now 20Mb hard disks are so cheap that few computers aresold without at least this much on-line storage. You will still want to archiveyour data on removable media such as floppy disks or cartridge tape, but thehard disk gives you speed during processing and the ability to easily look atmany data sets.

Some sort of graphics capability is essential. Most PC clones comewith Hercules-compatible monochrome graphics, which is perfectly adequate.Color is useful to overlay several data sets on the same plot, for memos andwindows in data reduction programs, and in general any place where featureseparation or visibility is important. Keep in mind, however, that color sys-tems cost more, and getting multicolored printouts can be difficult.

OBSERVATIONAL CALCULATIONS 103

A printer is essential. Inexpensive dot-matrix printers can be purchasedfor a few hundred dollars. Often these printers have built-in bit-mappedgraphics which enable you, with the proper software package, to incorportatedrawings, and graphs directly into your final print-out. Other printer choicesinclude color ink-jet printers (very nice for multicolored graphics) and laserprinters which are both fast and have high-quality output. When first intro-duced, the H-P LaserJet was priced well over $3,500 but within several years,and several model changes, it is possible to purchase a 4 page per minutemodel for under $1,000.

A nice option is the modem. This device links your computer to the tele-phone f ine so that you can communicate with other computer systems. We usea modem to access database services such as CompuServe, where other userspost messages, catalogs, data and computer programs that are of interest toastronomers. You should not only consider the purchase price when choosinga modem but the differential in connect time between it and a faster device.Almost every time you use a modem you will be either paying long distanceor user fees which are based on time.

Equally as important as the computer hardware is the software necessaryto run it. There are two approaches to computer software: buying commercialprograms and developing your own. There are many astronomical softwarepackages available, including star catalogs, astronomical utilities {time/date,rise/set, etc.), image processing and much more. For example, there is anoptional companion suite of software for this book that requires no program-ming knowledge to do photoelectric data reduction. Look through the ads inmajor astronomical magazines or find reviews to give you more informationon these commercial programs.

The other approach is to write your own programs. If you take this ap-proach, we recommend using a high-level language like FORTRAN, BASIC,FORTH or C, preferably compiled rather than interpreted and with relocat-able object code. There are many such compilers available, so look for reviewsto help you decide which one to purchase. In addition, your computer shouldhave the capability of being programmed in machine language for dedicatedtasks or input/output operations. Often, the high-level language comes withassembly language programming capability built-in.

A very good microcomputer system built around an IBM-PC clone with1Mb memory, 20Mb hard disk, numeric coprocessor, dot-matrix printer,2400-baud modem, monochrome graphics and software can be purchasedfor around $2,500 (1990). This may seem expensive, but the cost is stilldecreasing and some of it is defrayed by the fact that the computer can


replace considerable equipment, such as the counter and magnetic tapeunit of the photon-counting system. Find a computer store and ask thedealer to help you in your computer selection.

Another approach to data reduction involves the use of a computingcenter found at many universities. Here all of the hardware is main-tained for you, and high-speed, reliable, and efficient programming lan-guages and equipment are available. In addition, computing centersmay have programming support and plotting or graphics capabilities onhaiid. Some centers allow outsiders to use the system at minimal costif it is being used for research. Approach the center directly, or talk tosomeone in the astronomy, physics, or computer science departmentsabout your needs and desires.

5.2 ATMOSPHERIC REFRACTION AND DISPERSION

When we observe the sun and stars near the horizon, the atmospherebends the rays and makes the object appear higher in the sky than itreally is, as shown in Figure 5.1. This effect reaches a maximum on thehorizon, where an object appears to be 35 arc minutes above its actuallocation. This means that when the sun appears to touch the horizon, inreality it has already set! Atmospheric refraction affects images in threeways: it changes the measured zenith angle and therefore the air mass,it disperses the image so that each star looks like a miniature spectrum,and it changes the apparent right ascension and declination of a star.These effects must be accounted for accurately when viewing objectsnear the horizon.

5.2a Calculating Refraction

The simplest method of calculating refraction is to assume a plane-par-allel atmosphere made of layers, each with a differing index of refrac-

APPARENTLOCATION

TRUELOCATION

Figure 5.1 Refraction.


lion decreasing uniformly outward. Using Snell's law at each boundary,we find the angle of refraction, r, is approximately equal to

r = 60?4tanz ( r , (5.1)

where zlf is the true zenith distance.This equation has an error of about 1 arc second at z = 60°. An

empirical improvement was derived by Cassini and Bessel in the sev-enteenth century and is of the form

r = 60T4 tan ztr - 0"06688 tan3 z,,. (5.2)

Equation 5.2 is accurate to better than 1 arc second at z = 75°, or 15°above the horizon. A more accurate equation which accounts for atmo-spheric pressure, temperature, and the observer's elevation is given byDoggett et al.2 A short listing of the refraction correction is presentedin Table 5.1. Remember that this correction is subtracted from the truezenith distance to get the apparent zenith distance, zap.

TABLE 5.1. AtmosphericRefraction (760 mm Hg, 10°C,

55OO A)

V510152025303540454648505254555657

r"

510162127344149596065697480838689

V

5859606162636465666768697071727374

r"

9397101105109114119124130136143151159168177188200

V7576777879SO81828384858687888989.590

r"

214229247267291319353393444508592704S65

1105149417902189


Example: The true zenith distance of BX And was determined tobe 69°. What is its apparent location?

1. From Equation 5.1 we have:

T = 6074tan(69°)= 15773

r = 2'37"2ap = zlr - r

= 69" - 2'37"zap = 68D57'23"

(5.2a)

2. From Equation 5.2:

r = 6074 tan (69°) - 0"06688 tan3 (69°)r = 15672

= 68°57'24"

3. From Table 5.1:

r = 143"zap = 68°57'37"

Because zap approximately equals zfr, you can use either value inEquation 5.1 or 5.2 with minimal error.

5.2b Effect of Refraction on Air Mass

If you use the apparent zenith distance, the air mass calculated fromeither Equation 4.18 or 4.19 will be correct. However, you must assumethat the hour angle setting circle is correct, or you must include therefraction numerically in the hour angle calculation. Table 5.2 showsthe error involved in neglecting refraction at several zenith distances.For z greater than 60°, the error is significant enough that it cannot beignored. Generally, you can neglect refraction above an altitude of 30°,but be sure to include the correction at lower altitudes.


TABLE 5.2. Refraction Air MassErrors

030606570758085

1. 0001.1541.9942.3562.9043.8165.59810.211

1.0001.1541.9962.3592.9103.8305.64510.468

0.000.000.100.130.210.370.832.46

5.2c Differential Refraction

The index of refraction for glass, air, or any material is not constantwith wavelength. This spreads the light from a star into a miniaturespectrum, as if the earth's atmosphere were a prism. This dispersion isobvious when looking at stars near the horizon, as they appear blue onthe top and red below. It also gives rise to the "green flash" of the set-ting sun. Table 5.3 gives the separation angle of the red and blue imagesat various zenith distances. For z greater than 75 °, the images are farenough apart that they are no longer centered in a small diaphragm.They cause erroneous measurements unless a correction is made. Inother words, do not observe within 15° of the horizon unless it is abso-lutely necessary! A secondary effect of differential refraction is that thered and blue rays that make up the observed stellar image are separatedby the earth's atmosphere and thus give rise to a color-dependent scin-tillation, manifested in red and blue flashes. More detail on various

TABLE 5.3. Red and Blue ImageSeparation

0 0.0030 0.3545 0.6060 1.0475 2.2490 29.00


refractive effects can be found in Tricker3 or Humphreys.4 Both makequite interesting reading.

5.3 TIME

Time is the most accurate piece of data the astronomer has and shouldbe treated accordingly. A 1 percent error in the determination of themagnitude of a star is considered excellent, yet a digital watch has anaccuracy of 1 second in a day (0.001 percent), or 1000 times moreaccurate.

Because astronomers are located worldwide and have been observingfor centuries, certain conventions are observed in order that measure-ments made by two different observers can be correlated with minimaleffort.

This section assumes some knowledge of the various time systemsinvolved: solar time, sidereal time, and so on. More detail can be foundin any good introductory astronomy text such as Abell.5

5.3a Solar Time

An apparent solar day is defined as the length of time between twosuccessive transits of the sun, from astronomical noon until astronomi-cal noon the following day. The length of time is dependent on threefactors: the rotation of the earth on its axis, the obliquity of the ecliptic,and the speed of the earth in its revolution around the sun. Because thelatter two items have effects which are variable throughout the year,the length of the apparent day is not constant. Mean solar time, thelength of an average day, one year divided by 365/4 days, eliminates thisproblem. Apparent solar time is measured by sundials; mean solar timeis the time measured by clocks, with which everyone is familiar.

Another problem exists because noon does not occur simultaneouslyat all places on earth. Time zones were created to eliminate this prob-lem, with each zone being about 15" wide or approximately & of theearth's daily rotation. Greenwich, England is defined as the arbitraryzero point; an observer located at 15° west longitude is one hour behindGreenwich, an observer at 30" west longitude is two hours behindGreenwich, and so forth. This is convenient for daily living but not forthe astronomer. If an eclipse was observed at 3 p.m. local time in


Hawaii, what time was it where you live? All astronomical observationsmust be recorded in Universal Time (UT), the local mean time in Green-wich, England. For an observer in San Francisco, eight hours are addedto Pacific Standard Time (PST) before recording data. UT is kept ona 24-hour clock to eliminate a.m.-p.m. ambiguities.

Example: A meteor is seen at 10:30 pm EST in New York on Jan-uary 1, 1981. What UT should be recorded?

1. Convert EST to 24-hour time.10:30 pm = 22:30 EST

2. Add 5 hours for time-zone difference.22:30 EST + 05:00 = 27:30 UT

3. Subtract 24 hours (it is the next day in Greenwich).27:30 - 24:00 = 03:30 UT

4. Add 1 day to the date because of step 3.January 1 + 1 = January 2

The observation was made on January 2, 1981 at 03:30 UT.

5.3b Universal Time

Observations should be recorded to within the nearest minute, or moreprecisely for rapidly varying objects. This accuracy cannot be reliablyobtained from your AM radio or local bank sign. The best method is touse a shortwave receiver and listen to one of the national time signalsbroadcast by WWV in the United States, CHU in Canada, and similarservices in other countries. A complete list is published by the BritishAstronomical Association.6 WWV broadcasts at 2.5, 5, 10, and 15MHz, and CHU broadcasts at 3.330, 7.335, and 14.670 MHz.

To receive these signals, buy a new portable multiband radio or asingle-frequency radio such as the Radio Shack TIMEKUBE® or buya used general-coverage receiver. Check local ads, amateur radio deal-ers and clubs, or the classified ads of a magazine such as QST or HamRadio. If the radio you obtain works on batteries only, consider buyingan AC adapter as batteries are adversely affected by cold weather andmay become inoperative. A word of warning: if you intend to use amicrocomputer with your system, the harmonics generated by its inter-


nal clock will interfere with the WWV frequencies, but generally notthose of CHU.

A digital 24-hour clock is extremely convenient for the observatory,as no conversion is necessary to record the time. Several commercialclocks are available on the market at reasonable cost. Instructions forbuilding your own can be found in back issues of many electronics mag-azines. Look for those using direct drive to each digit to eliminate RFmultiplexing noise. Clocks with internal calendars, such as the CT7001by Cal-Tex, or with BCD output, such as the MM5313 by National,may be used in microcomputer applications.

5.3c Sidereal Time

Sidereal time (ST) is the time kept by the stars, or more precisely, theright ascension of a star currently on the meridian. The length of asidereal day is defined as the time between successive transits of thevernal equinox. A sidereal day is about 4 minutes shorter than a solarday. Knowledge of this time is essential to be able to point your tele-scope to the right region of the sky.

For many telescopes, the normal setting circles measure declinationand hour angle, the distance from the celestial meridian. Hour anglecan be expressed as

Hour angle (HA) = Local sidereal time (LST)- Right ascension (RA). (5.3)

We use the terms local sidereal time and sidereal time interchangeablyin this chapter. The RA of a star in this equation is for the presentepoch, that is not 1900, 1950, and so forth. Given the HA of an objectand its coordinates, you can then point your telescope to the correctdirection in space. Correspondingly, given the ST and RA for an object,the hour angle can be calculated for use in computing air mass.

Calculation of sidereal time can be accomplished in four basic ways:(1) from the HA and RA of some easily found object such as a brightstar; (2) from tables given in The Astronomical Almanac;1 (3) throughuse of a programmable calculator if the UT and date are known; or (4)from a sidereal rate clock. These four methods are explained below.


1. From HA and RA. Pick some bright star and set your telescopeon it. By reading the HA from the setting circles, set a solar rateclock to ST from the equation

ST = RA + HA (5.4)

This method is as accurate as your setting circles. The solar rateclock keeps nearly sidereal time for about a night's observations.

2. From tables. The Astronomical Almanac1 gives the sidereal timeat Oh UT for every day of the year. Abell5 includes a coarser table.Converting this time to the sidereal time at another location is afairly complicated procedure. Examples are given in the Almanac.You need to know the day, UT, and your longitude, which can befound from Goode's World Atlas, or from a road atlas, a topo-graphic map, or a plot survey of your observatory.

Example: What is the ST at 03:00 UT on July 7, 1973, for anobserver at longitude 86°23'7 west?

LST at Greenwich from the Almanac 18" 59m 16s

- your longitude (86°23f7 at 15° per hour -5 h 45m 35s

+ UT difference from Oh +3 h 00m 00B

+ ST/UT difference over 3h period at 10s + 308

per hourSidereal Time at 03:00 UT: 16 h 14m 11s

3. With a calculator. Given the longitude, UT, and Julian date (seeSection 5.3d), there is a simple equation to obtain sidereal time:

ST = 6.6460556 + 2400.0512617 (JD - 2415020)/36525+ 1.0027379 (UT) - longitude (hours). (5.5)

This equation takes into account the fact that there is approxi-mately one extra sidereal day in a year, or 2400 extra hours in acentury. We want 0 < ST < 24h, so after solving the above equa-


tion, we must subtract off that multiple of 24 hours (extra siderealdays) that leaves a remainder in this range. A FORTRAN subroutinethat calculates sidereal time can be found in Section 1.6.

Example: Calculate ST for 03:00 UT on July 7, 1973, from lon-gitude 86° 23'.7 west (5h45m35s).The Julian date is 2441870.5 at Oh UT from the Almanac. Then

ST = 6.6460556 + 2400.0512617(2441870.5- 2415020)/36525 + 1.0027379(3) - 5.75972

= 1768.23611 hours (73 X 24 = 1752 hours)= 1768.23611 -- 1752.0

ST = 16h14m10s

4. From a sidereal rate clock. There are two varieties of this spe-cialized clock: an electric clock, with a sidereal rate motor that isvery expensive, and an electronic digital clock. Both can be pur-chased commercially. The electronic version can also be built fromplans published in Sky and Telescope.8 It is basically the same asa pulse counter, counting one pulse per sidereal second. This canbe achieved either by using a crystal oscillator or by adding extrapulses to a 60-Hz clock. Setting the sidereal rate clock should beperformed using methods 2 or 3 above and should be checkedoften in case of power failures or oscillator drift.

5.3d Julian Date

Just as time zones cause problems between widely spaced observers,differing dates of observations can be a real headache when using thestandard calendar. How many days have passed since you were born?The simplest approach is to use a running count of the number ofelapsed days. This count was proposed by J. J. Scaliger in 1582 and iscalled the Julian date (JD) of an observation or event. The zero pointwas set far enough in the past that all recorded astronomical eventshave a positive JD. Scaliger suggested the use of Julian date 0 = 12h

UT on January 1, 4713 B.C. because several calendars were in phase onthat day. The JD begins at noon because most active observers in the


sixteenth century were in Europe and no date change would occurduring a night's observations for them.

The Julian date for any given day can be found in the Almanac orthrough use of the following equation:

JD (Oh UT) = 2415020 + 365 (year - 1900)+ (days from start of year) + (no. of leap years since 1900) — 0.5.

(5.6)

A FORTRAN subroutine for this calculation can be found in Section 1.2.Note that most observations are recorded in JD units including frac-tions of a day, instead of separate JD and UT.

Example: An observation was made on June 15, 1973, at 11:40 UT.What Julian date should be recorded?

JD (Oh UT) = 2415020 + 365 (1973 - 1900) + 166 + 18 - 0.5

JD(Oh UT) = 2441848.5

11:40 UT = 11.6667 hours UT/24 = 0.4861 day UT

JD = 2441848.5 + 0.4861

JD = 2441848.9861.

*5.3e Heliocentric Julian Date (HJD)

Any time recorded in the process of observing is geocentric, that is,made from a site on the earth. Because the earth revolves around thesun, an observer is closer to or further from a particular star at differenttimes of the year. Six months after the earth's closest approach to thestar, it is two astronomical units farther away (or less if the star is noton the ecliptic). Because of the finite speed of light, up to an additional16 minutes are required for its light to traverse this extra distance. ThisHght-travel-time effect causes scatter around the mean light curve of avariable, as compared with observations made from a relatively station-ary object like the sun. Astronomers therefore prefer to record all obser-


vations as though made from the sun by adding or subtracting thelight's travel time, depending on whether the earth is farther or closer,respectively, to the object than the sun is. The date derived in this man-ner is called the heliocentric Julian date (HJD).

Before the advent of the small computer, the heliocentric correctionwas made through the use of laboriously precomputed tables. Examplesof these are Prager,9 which is difficult to find, Landolt and Blondeau,10

and Bateson," which is coarser and less accurate than the others. It isnow simpler to compute the correction than to use tables.

The most thorough description of the geometry involved is presentedby Binnendijk,12 and the reader is encouraged to glance at his figuresand derivations. Basically, the problem is one of projection. There aretwo planes involved: the earth's equatorial plane, where right ascension,declination, and solar X, Y, Z Cartesian coordinates are defined; andthe earth-sun-object plane, where we wish to know the projection of theearth's distance from the sun on the sun-object line. If the projectionsare carried through properly, one arrives at

HJD = JD + A/ (5.7)

where

Ar(days) = - 0.0057755 [(cos 5 cos a)X-f (tan « sin 5 + cos 6 sin a) Y] (5.8)

and X, Y are the rectangular coordinates of the sun for the date inquestion; a, 5 are the right ascension and declination of the star for thatdate, respectively; and c is the obliquity of the ecliptic. 23" 11'. Equation5.8 holds as long as epochs are consistent. The values of X and Y canbe obtained from the Almanac1 or by the trigonometric series given byDoggett et al.2

The method of Doggett et al.2 is shown below because it is relativelyeasy to program on a small computer or programmable calculator. SeeDoggett et al. for definitions of the various terms used in this method.

1. Determine the relative Julian century by

T = (JD - 2415020)/36525 (5.9)


2. Obtain the mean solar longitude from

L = 279!696678 + 36000.768927 + 0.00030372 - p (5.10)

where

p = [1.396041 + 0.000308(7-1- 0.5)] [7- 0.499998] (5.11)

The value p is the precession from 1950 to date and therefore issubtracted from the present epoch longitude in Equation 5.10 toobtain 1950.0 longitude.

3. Obtain the mean solar anomaly by

G = 358!475833 + 35999.049757 - 0.0001572 (5.12)

4. Finally, obtain A'and Kfor 1950.0 through the expansions

X = 0.99986 cos I - 0.025127 cos (G - L)+ 0.008374 cos (G + L)+ 0.000105 cos (2G + L) + 0.0000637 cos (G - L)+ 0.000035 cos (2G - L) (5.13)

Y = 0.917308 sin L + 0.023053 sin (G - L)+ 0.007683 sin (G + L)+ 0.000097 sin (2G + L) ~ 0.0000577 sin (G - L)- 0.000032 sin (2G - L). (5.14)

Only the first few terms of the X and Y expansions need to be kept forthe accuracy required in this application. A slightly more accurate FOR-TRAN routine that calculates X, Y, and Z coordinates is presented inSection 1.7.

Example: On June 15, 1973, at 11:40 UT, an observation was madeof V402 Cygni. What is the appropriate heliocentric correction?

1. T= (2441848.9861 - 2415020)/365257 = 0.7345376 century

11 6 ASTRONOMICAL PHOTOMETRY

2. L = 279T696678 + 36000.76892(0.7345376)+ 0.000303(0.7345376)2 - [1.396041+ 0.000308(0.7345376 + 0.5)](0.7345376 - 0.499998)

L - 26723!2879 (subtract multiples of 360" to get 83:2879)3. G= 358:475833 + 35999.04975(0.7345376)

- 0.00015(0.7345376)2

G = 26801:1315 (subtract multiples of 360 ° to get 161! 1315)4. Compute A"and Kfrom Equations 5.13 and 5.14

X = 0.108021Y = 0.926683

5. Calculate coordinates

6 = 37'<K33 = 37:00556(1950.0)a = 20h07m15s = 20*12083 = 301:8125 (1950.0)

6. Finally, calculate A/ and the heliocentric date

A; = -0.0057755{[cos (37:00556) cos (301:8125)]0.108021+ [tan (23'27') sin (37:00556) + cos (37:00556)sin(301!8125)] X 0.926683}

A* = -0.0057755(0.045473 - 0.386915)A/ = 0'.00197

HJD = 2441848.9861 + 0.00197HJD = 2441848.9881

This calculation is much easier when programmed on a calculator orcomputer!

5.4 PRECESSION OF COORDINATES

The coordinates of a star or other celestial object do not remain constantwith time. Listed below are some of the major contributing factors,along with their maximum values. This should give you an idea howdifficult it is to set a telescope accurately.

1. Precession: 50 arc seconds per year.2. Nutation: 9 arc seconds over 19 years.

OBSERVATIONAL CALCULATIONS 11 7

3. Aberration: 20 arc seconds over 1 year.4. Heliocentric parallax: 0.75 arc second over 1 year.5. Refraction: 60.4 tan z arc seconds daily.6. Variation in latitude: 0.3 arc second.7. Proper motion: 10 arc seconds per year.8. Barycentric parallax of solar system: a few arc seconds.9. Geocentric parallax: at most a few arc seconds even for solar sys-

tem objects.

Most of these are short-term, cyclic aberrations, except for precessionand proper motion. Proper motion is not discussed because it has a sig-nificant effect only on the closest stars.

Precession is caused by the gravitational pull on the earth's equatorialbulge by the sun and planets. The coordinates go through a completecycle in 26,000 years, so the change per year is small, but the effect iscumulative. Catalogs and atlases give the mean coordinates of stars,usually at the beginning of some year, such as Equinox 1950.0. How-ever, to identify a star on the Bonner Durchmusterung (BD) atlas (seeSection 9.la), which is Equinox 1855.0, with 1950 coordinates can bedifficult unless one processes the coordinates to the equinox of the atlas.More importantly, a telescope measures the present equinox coordi-nates and not those of 1900.0, 1950.0, or any other equinox given instandard catalogs.

Because the precession of the earth's axis is well known, equations toprecess the coordinates are available. Their basic form is

Aa = (m + n 1 t a n 6 m s i n a m ) ( r / — t0) (5.15)

A5 = (n* cos am)(tf- t0) (5.16)

where Aa is in seconds, A5 is in arc seconds and

t0 = equinox of the known coordinates (years)tf = equinox to precess to (years)

m = 3S.07234 + Of00001863rm (5.17)ns = 1V336457 - 0!00000569?m (5.18)

n" = 20?04685 - Ov0000853?ffl (5.19)


tm = mean time with respect to 1900:

<„ = ('/+ *o)/2- 1900 (5.20)

This value is negative for equinoxes before 1900. The subscript m fora and 3 indicate that the coordinates should be for the midpoint of theprecession. That is, if you are precessing from 1950 to 1980, the coor-dinates on the right-hand side of the equations should be for 1965. Also,because you are taking the sine and cosine of right ascension, a shouldbe converted to degrees.

For most calculations, the above equations will give reasonably goodresults, even if am and 5m are replaced by a0 and 50 (the originalequinox).

Example: Precess y Aql from 1953.0 to 1975.0.

7 Aql 1953.0: a = 19h44m01s.458 = 191733738 = 296:006075 = 10'29'50?83 = 10:497453

7 Aql 1975.0:

( , - / „ = 1975 - 1953 = 22 years

tm = (1975 + 1953)/2 - 1900 = 64 years

m = 3.07234 + 0.00001863(64) = 3!073532

ns = 1.336457 - 0.00000569(64) = 13360928

n" = 20.04685 - 0.0000853(64) = 207041391

A« = [3.073532 + 1.3360928 tan (10.497453) sin(296.00607)](22)Aa = 62*723

A3 = [20.041391 cos (296.00607)] (22)A5 = 193732


«(1975) = o(1953) + Aa= 19h44m01!458 4- 62!723

o(1975) = 19h45m04!1815(1975) = 5(1953) + A5

= 10'29'50r83 + 19373255(1975) = 10°33'4715

For comparison, here are the values from the 1975 Almanac:

y Aql (1975.0): a = 19h45m4s.25 = 10°33'5"

Note that within the accuracy of the Almanac, our answers are cor-rect even though the mean coordinates were not used.

When high accuracy is needed, an analytical method of obtaining thecoordinates by integrating the precession equations is used rather thanusing the mean coordinates with iteration. This more rigorous methodis presented in the explanation in the Almanac,1 and a FORTRAN sub-routine using it is given in Section 1.3.

5.5 ALTITUDE AND AZIMUTH

One of the most common calculations made by astronomers is deter-mining where an object is above the horizon and where along the hori-zon it lies. These two coordinates are called altitude and azimuth, andconstitute the most natural coordinate system.

The equations for determining altitude and azimuth can be found inSmart,13 who gives diagrams, and Doggett et al.,2 who give the equa-tions used in the Almanac. Neither source gives a complete explanationof how the equations are derived. Though these equations can be usedby themselves, it is useful to know their derivation.

*5.5a Derivation of Equations

Figure 5.2 shows the situation where the altitude and azimuth of a staris unknown. This is a problem in spherical trigonometry because we are


NCR

ZENITH

HORIZON

Figure 5.2 Altitude-azimuth sphere.

measuring angles on the celestial sphere. We are looking for h, thealtitude above the horizon, and A, the azimuth measured east from duenorth. Three great circle lines are shown, one representing the celestialequator and two passing through the star from the zenith to the horizonand from the north celestial pole (NCP) to the celestial equator. Thelatter two lines and the celestial meridian define the spherical triangleneeded to obtain h and A.

A spherical arc is defined by

D = RB (5.21)

90

Figure 5.3 Close-up of the altitude-azimuth triangle.


Figure 5.4 A general spherical triangle.

where 8 is the angle subtended by the arc and R is the spherical radius.In the case of a unit sphere (R = 1), the sides of the spherical triangleare equal to the angle subtended by the arc. Therefore, the three sidesare: (1) from the NCP to z, an angle of 90" -- 0, 0 = latitude; (2)from the zenith to the star, the zenith distance or 90° - h\ and (3)from the NCP to the star, an angle of 90° - 5, 5 = declination. Theenclosed angles of concern are the hour angle, //, measured from thecelestial meridian to the star, and the coazimuth, 360° - A. See Figure5.3.

The solution of the triangle makes use of the law of cosines, and canbe found in books such as the CRC tables.14 Referring to Figure 5.4,

cos a = cos b cos c + sin b sin c cos A. (5.22)

This works with any appropriate permutation. For altitude, we note:

cos (90° -- h) = cos (90° - 0) cos (90° - 5)+ sin (90° - 0) sin (90° - 5) cos H

or

sin A = sin 0 sin 5 -f cos 0 cos 5 cos H (5.23)

where H is given in degrees by

H = 15(LST - a). (5.24)

For azimuth, we use the other known enclosed angle:


cos (90° - 5) = cos (90° - 0) cos (90° - A )+ sin (90° - 0) sin (90° •- h) cos (360° - A)

or

sin 5 = sin 0 sin h + cos 0 cos h cos A. (5.25)

Solving for the azimuth,

cos A = (sin 5 — sin 0 sin /z)/(cos 0 cos h). (5.26)

5.5b General Considerations

Equations 5.23 and 5.26 are our working relations for altitude and azi-muth. One problem exists in solving for A. All calculators and com-puters return values of cosines only between 0 and 180°, yet azimuthextends over the entire range of 0 to 360°. We can remove the ambi-guity by noting that when H is greater than 0 °, A is greater than 180 °.Therefore, for -180" < H < 180°,

A — A when H < 0, andA = 360" - A when H > 0.

Another solution, by Doggett et al.z solves for tan A. While this givessome computational simplicity in that the altitude is not needed, it addsthe complexity that tan A is undefined at +90° and —90°. It is sug-gested that Equation 5.26 be used, unless there is good reason to useanother method.

Example: What is altitude and azimuth for AS Cas on February 5,1979, at 01:00 UT from Goethe Link Observatory?For that date and time, LST = 4h12m = 4h.2000.

a(l979) = Oh24m23s = OM0645(1979) = 64'6:5 = 64:10830 = 39° 18' = 39?30H = 15 (4.2000 - 0.4064) = 56! 9042)


sin h = sin (39:3) sin (64! 1083)+ cos (39:3) cos (64? 1083) cos (56:9042)

sin h = 0.75432

h = 48:97

cos A = [sin (64M083) - sin (39:3) sin (48!97)]/[cos (39:3) cos (48:97)] =0.83037

A = 33:86.

But, since H = 56° (H > 0),

A = 360° - 33:86

A = 326M4.

REFERENCES

1. Staff, 1979. Sky and Tel. 58, 25.2. Doggett, L. E., Kaplan, G. H., and Seidelmann, P. K. 1978. Almanac for Com-

puters for the Year 1978. Washington, D.C.: Nautical Almanac Office.3. Tricker, R. A. R. 1970. Introduction to Meteorological Optics. New York: Amer-

ican Elsevier.4. Humphreys, W. J. 1940. Physics of the Air. New York: McGraw-Hill.5. Abell, G. O. 1975. Exploration of the Universe. New York: Holt, Rinehart and

Winston.6. The Handbook of the British Astronomical Association. England: Sumfield and

Day, Ltd. Published yearly.7. The Astronomical Almanac. Washington, D.C.: Government Printing Office.

Issued annually.8. Reid, F., and Honeycutt, R. K. 1976. Sky and Tel. 52, 59.9. Prager, R. 1932. Klein. Veroff. Univ. Sternw. Berlin-Babelsberg. no. 12.

10. Landolt, A. U., and Blondeau, K. L. 1972. Pub. A. S. P. 84, 784.11. Bateson, F. M. 1963. In Photoelectric Astronomy for Amateurs. Edited by F. B.

Wood. New York: Macmillan, p. 97.12. Binnendijk, K. L. 1960. Properties of Double Stars. Philadelphia: Univ of Penn-

sylvania Press, pp. 228-232.13. Smart, W. M. 1962. Text-Book on Spherical Astronomy. Cambridge: Cambridge

Univ. Press.14. S. M. Selby, ed. CRC Standard Mathematical Tables. Cleveland: The Chemical

Rubber Co. Published yearly.

CHAPTER 6CONSTRUCTING THE PHOTOMETER

HEAD

Careful design and construction of the photometer head is very impor-tant and it requires substantial comment. The goal of this chapter is tosupply you with enough background information to allow you to designand construct a photometer head intelligently. We have not includeddetailed construction plans because the requirements of individualobservatories and researchers varies greatly. Amateur and professionalastronomers approach the construction of photometers somewhat dif-ferently. The professional intends to mount the completed photometerhead on a rather large telescope. Hence the components can be madefrom heavy metal stock and the tube can be totally enclosed in a coldbox. The total weight of the finished photometer can be as much as 45kilograms (100 pounds), which exceeds the weight of many an ama-teur's telescope! Obviously, the amateur must keep weight and size asprimary restrictions on the design. Our emphasis in this chapter is onthe needs of this small telescope user. We first make some comments ondesign and construction. We then describe a simple, lightweight designin Section 6.5.

We discuss briefly designs utilizing a photodiode as a detectorbecause such designs are difficult to find in the literature. We referinterested readers to papers by Persha1 and De Lara et al.2 Finally, aphotodiode photometer is available commercially from Optec, Inc.3

6.1 THE OPTICAL LAYOUT

The first step in designing a photometer head is to position the opticalelements on a drawing showing the correct relative sizes and spacing.

124

CONSTRUCTING THE PHOTOMETER HEAD 125

This layout depends on the F-ratio of the telescope to be used, becausethis determines the rate at which the light cone formed by the tele-scope's objective diverges from the focal point. For instance, if you areusing an F/8 telescope, the light cone will have a 1-centimeter diameterat a distance of 8 centimeters from the focus, a 2-centimeter diameterat a distance of 16 centimeters, and so forth. A small F-ratio telescopecauses special problems because the cone diverges very rapidly, forcingthe photometer components to be placed within an uncomfortably smalldistance from the focal point. If you plan to use a small F-ratio tele-scope, say F/5 or less, consider the design described by Burke andPippin.4

The optical layout procedure is best accomplished by using a largesheet of graph paper so that the drawing can be made full size. Pick aspot near the left side of the paper to be the focal point. Draw a hori-zontal line through the focal point across the page to represent theoptical axis of the photometer. Now draw the expanding light cone fromthe focal point according to your F-ratio as described above. The dia-phragm is of course located at the focal point. The filters are positionedat a distance from the focal point where the light cone has a diameterof 5 to 10 millimeters. This is a compromise between the desire to usesmall filters and the need to illuminate an area of the filter large enoughso that dust or small defects in the filter do not affect the light conesignificantly. The amount of available space between the filter and dia-phragm is now fixed. For an F/8 light cone, this distance is 80 milli-meters for a 10-millimeter spot size on the filter. Within this space, thephotometer builder must find room for the filter assembly and the flipmirror with its associated lenses. In Figure 6.1, we have positioned thediaphragm and filter. Consult this figure as you read the remainder ofthis section.

The next element to be positioned is the Fabry or field lens. This lenshas a very important function. It focuses an image of the telescope'sobjective, illuminated by the light of the star on the photocathode. Thisspot of light remains at the same place on the photocathode no matterwhere the star drifts within the diaphragm. This is important becauseno photocathode can be made with uniform sensitivity. Without theFabry lens, the photomultiplier output could change considerablydepending on the position of the star in the diaphragm. The desired spotsize depends on the size of the photocathode of the photomultiplier. Ifthe spot is too large, light will be thrown away because some of it will


PHOTOCATHODE

DIAGONALMIRROR (OPTIONAL)

40 mm

Figure 6.1 Optical layout.

miss the photocathode. If the spot is too small, local defects in the pho-tocathode may have a degrading effect on the tube's output. For the1P21 photomultiplier, a spot size of about 5 millimeters is about right.To calculate the Fabry lens's focal length, recall from elementary opticsthat for a thin lens the ratio of the object size to the object distanceequals the ratio of the image size to the image distance. The object sizeis the diameter of the telescope's objective, D, and the object distanceis essentially the telescope's focal length, /, so the relation becomes

D

f , image distance

where a is the image size. Because the focal length of the telescope isvery large compared to that of the Fabry lens, the image distance isvery nearly equal to the focal length, /, of the Fabry lens. That is,

D _ a

f , = = /

or

(6.1)


where F is the focal ratio of the telescope,^ divided by D. For the F/8telescope used in our example, the Fabry lens must have a focal lengthof 40 millimeters for a 5-millimeter spot size. The spot size can beadjusted slightly to give a focal length of a common lens that can befound easily in optics catalogs. The spacing between the Fabry lens andthe photocathode is now fixed as the focal length of the Fabry lens. Thediameter of the lens depends upon its distance from the diaphragm.This distance is not critical. We have arbitrarily placed it 12 millimetersbehind the filter in Figure 6.1. Once the lens is positioned on your draw-ing, the size of the light cone at this point gives the size of the lens. Thelens should actually be made a little larger to allow for small misalign-ment after construction and the fact that the lens mount covers theedges. The lens itself is just a simple double convex lens. For properultraviolet transmission, this lens should be made of crown glass or,preferably, quartz.

The flip mirror can now be added to the drawing. It is placed betweenthe diaphragm and the filter at a 45" angle to the optical axis. Thismirror directs light into the first lens of the diaphragm-viewing optics.This lens is an achromat of sufficient diameter to accept the total lightcone. It is positioned at a distance from the telescope's focal point equalto its own focal length. This results in a collimated beam of light afterpassing through the lens. If you cannot afford the luxury of custom-made optics, it is best to consult an optics catalog to find which focallengths are available before positioning this lens in your drawing. A 60-millimeter focal length lens is available from Edmund Scientific5 or A.Jaegers.6 We have positioned this lens at a distance of 60 millimetersfrom the diaphragm. The next lens is an identical achromat that can bepositioned at any comfortable distance from the first lens. The secondlens reconverges this light for the viewing eyepiece. This opticalarrangement gives a focused image of both the diaphragm and the starwithin it. Figure 6.1 shows the optical elements with the dimensionsindicated.

A look at Figure 1.8 should convince you that the head for a photo-diode photometer is somewhat simpler. Because the detector is at thetelescope's focus, the flip mirror can direct the light cone directly to aviewing eyepiece without the need of the two lenses discussed above.There is also no Fabry lens or diaphragm. The size of the active area ofthe photodiode defines the "diaphragm size." Unfortunately, this cannot be adjusted. Without a diaphragm, it is necessary to have illumi-


nated cross hairs in the viewing eyepiece. They are aligned such that ifa star is centered, when the flip mirror is removed the star's light willfall directly on the photodiode. The mechanical design of the head mustallow for the components to be adjusted to achieve this alignment. It isalso necessary for the eyepiece focus to be adjusted and locked so thatwhen a star appears focused in the eyepiece, the photodiode will be atthe telescope's focus. These design problems are no more difficult tosolve than those encountered in a photomultiplier-type photometer.

There is quite a large jump between laying out the optical compo-nents and performing the actual mechanical construction. We now offersome specific comments to make that jump seem a little smaller.

6.2 THE PHOTOMULTIPLIER TUBE AND ITS HOUSING

There are many manufacturers of photomultiplier tubes, each offeringa wide array of devices. Suppliers of tubes frequently used by astrono-mers are EMI Gencom,7 ITT,8 and RCA.9 The tube you select dependsupon the spectral response required by your research. For example, Fig-ure 1.5 shows that you should not try to use a tube with an S-4 responseto measure stars at 8000 A. An S-20 or S-l tube response should beused instead. If you wish to obtain measurements on the UBV system,the choice of tube response is very crucial. As discussed in Chapter 2,the UBV system is defined by both the tube response and the filter trans-mission. For example, the U filter, and to a lesser extent the B and Vfilters, transmit light beyond 7000 A. This means that these filters donot isolate the single spectral region they were meant to measure. How-ever, the 1P21 has very little sensitivity beyond 7000 A and the filter'sred leaks, except for a small amount from the U filter, can be ignored.If the 1P21 is replaced with a tube with an S-l or S-20 response, thefilter set must be changed to "plug" the red leaks. Fernie10 has used asingle red-extended S-20, EMI 9658R tube to make UBVRl measure-ments. However, the UBV filters had to be altered from the type usuallyused with the 1P21. The point to be made here is that care must betaken to match detectors and filters to insure proper transformation tothe standard system.

There are many tubes newer than the 1P21 that have been success-fully used to make UBV measurements. The EMI 6256 has an S-l 1response that is only slightly more red-sensitive than an S-4 response.This tube has a peak quantum efficiency of 21 percent compared to 13


percent for the 1P21. Tubes such as the ITT FW-118 (S-l), and FW-130 (S-20) and the RCA 7102 (S-l) have also been used successfullyfor UBVR1 observations, with the appropriate filter modifications.Another tube of interest is the EMI 9789, which has a bialkali (cesium-potassium antimonide) surface with a spectral response very similar toS-4. However, it has a peak quantum efficiency of 20 percent and asmaller photocathode than the 1P21. A smaller photocathode has lessarea for thermionic emission, in this case resulting in about one-fifth ofthe dark current of the 1P21 at room temperature. The tube appears totransform well to the UBV system with the standard filters.

There are many different kinds of photomultipliers in successful usetoday. Despite all the advances in photomultiplier tube technology sincethe introduction of the 1P21, there are some good reasons to use thistube for UBV observations. The first is the fact that this tube was usedto define the UBV system and requires little experimentation with fil-ters. While many astronomers have used other tube and filter combi-nations successfully to make UBV observations, it may not always beapparent to the novice when there is a problem with the transformation.Another important reason for using the 1P21 is cost. The EMI 9789and 6256 cost over $300 and $600, respectively. The 1P21 costs lessthan $ 100. The 1P21 has a companion tube called the RCA 931 A, iden-tical in all respects except that the 931A has a lower sensitivity andlower cost. This tube sells for less than $20 and is ideal for testing anew photometer and learning observational techniques. If the tube issomehow damaged, it will not cost hundreds of dollars to replace. Onceconfidence has been gained in both the photometer and the observer'sabilities, the 931A can be replaced with a 1P21 without any modifica-tions to the photometer. The 931A represents a good investment for thenewcomer to astronomical photometry. If you are constructing a pho-tometer of small size, the Hamamatsu Corporation" has introduced aline of miniature photomultipliers. Their R869 tube is equivalent to theRCA 1P21 but at one-half the size and about the same cost. In thefollowing discussion of photometer construction, we assume that thedetector isa 1P21 ora931A. Modifications for other tubes can be madeby the reader, based on the manufacturer's specification sheets.

Figure 6.2 shows a photograph of the 1P21. The photocathode is justbehind the grid of wires seen near the front of the tube. Below the glassenvelope is a base with 11 electrical pin contacts and a center post forpositioning the tube in the socket. This center post has a key that must


Figure 6.2 A 1P21 photomultiplier tube.

face the incident light for proper tube orientation. Figure 6.3 shows thephysical dimensions of the tube taken from the RCA specificationsheets. The photocathode is about 5 millimeters in front of the centralaxis of the tube. You should account for this displacement when youposition the Fabry lens.

The 1P21 has nine dynodes, a photocathode, and an anode. The pho-tocathode is operated at a potential of about —1000 V with respect toground. The first dynode is at a potential of about —900 V, or about100 V more positive than the photocathode. This potential differenceprovides for the acceleration of the electrons released at the photocath-ode to the first dynode. Each succeeding dynode is 100 V less negativethan its predecessor. Finally, the last dynode is —100 V with respect to


3.tW

3.18 M/\X.

2Ot.

1.94 ^.09

MAX.

I 1

1I 1' II 1L_J

1.31

PHOTOCATHODE-*"" 0.94 X 0.31 (MINIMUM)

^^GLASS ENVELOPE

PIN^-*- CONTACTS

. ____ POSITIONING POSTWITH KEY

Figure 6.3 Physical dimensions of the 1P21 (all dimensions are in inches).

the anode that collects the cascade of secondary electrons. The pins onthe tube base are provided so that the proper voltage can be applied tothese tube elements. The voltage differences between dynodes areachieved by a simple voltage divider circuit that is wired directly to thetube socket. The tube socket is an Amphenol 7851 IT or equivalent. Fig-ure 6.4 shows the tube socket with its voltage divider resistor string asviewed from the bottom. If you are pulse counting, it may be necessaryto put a capacitor (0.01 mf, 1000 V, ceramic) between each pin (Ithrough 9) and ground. This reduces the instrumental sensitivity toexternal noise pulses.

1P21 (AND931A)

PINS 1-9:DYNODES 1-9

PIN 10:ANODE

PIN 11:PHOTOCATHODE

ALL RESISTORS1/4 WATT

_ TO HIGH-VOLTAGESUPPLY

INCIDENT LIGHT

Figure 6.4 Wiring of tube socket (bottom view).


Note that the high voltage is applied to the photocathode on pin 11,which is next to pin 10, the anode pin from which the signal is collected.In order to prevent high voltage from leaking across to pin 10 and enter-ing your amplifier, the socket must be made from a very high-resistancematerial such as mica or Teflon. Under no circumstances should a plas-tic socket be used.

After assembly, the socket should be cleaned with isopropyl alcoholto remove any fingerprints that may lead to leakage current. Becausemoisture at the tube socket can lead to leakage current, many astron-omers seal the entire voltage divider string with silicon rubber. Pre-wired, sealed-tube sockets can be purchased from EMI and Hama-matsu. A hand-wired tube socket that is kept clean and dry is veryadequate for a simple photometer.

The photomultiplier is a delicate device that should be handled withcare. To avoid leakage-current problems, do not touch the tube at thebase of the connector pins or the glass envelope. The photocathode isextremely sensitive and can be permanently damaged by exposure tobright light. The rule of thumb is never to allow light brighter thanstarlight to strike the tube when the high voltage is on. However, brightstars, especially on larger telescopes, can temporarily damage the tube,causing a condition known as fatigue. When a tube reaches the fatiguelevel, the tube seems to lose sensitivity and the output current drops.For the 1P21, fatigue occurs at an output current of about 1 nA. Thesensitivity returns after the tube has been allowed to "rest" in totaldarkness. Fatigue can be avoided by lowering the high voltage appliedto the tube, allowing brighter stars to be measured. Experience hasshown that for most 1P21 tubes the transformation coefficients are notaffected by a voltage change. There is no guarantee that this is true forall 1P21 tubes and it is certainly not true for many other tube designs.Even exposure to room light with the high voltage off can temporarilyincrease the dark current. Therefore, it is a good idea to handle the tubein subdued light during its installation in the tube housing. From thenon, the dark slide should be kept closed except while observing. Anotherimportant precaution is to make sure that your high-voltage supply isconnected properly so that negative high voltage is applied to the tube.The tube cannot be damaged by cold, but exposure to high tempera-tures for prolonged periods of time can lead to degraded performance.The best storage conditions for a photomultiplier tube are cool, dry, anddark.


The photomultiplier tube housing can be as simple as a pair of brasscylinders or as complex as a hermetically sealed thermoelectricallycooled chamber. The choice of housing depends on your commitment toserious observing and your budget. For the user of a small telescopewho plans to observe brighter objects, we strongly recommend the sim-ple approach. The function of the housing is to keep the tube in a dry,light-tight environment with the photocathode aligned to the opticalaxis of the photometer. The housing contains a dark slide which can beopened to allow the starlight to reach the photocathode or closed to keepthe tube in total darkness when it is not being used. In a simple housing,one brass cylinder holds the tube socket, voltage divider resistors, andthe cable connectors. This cylinder then snugly slides over the other cyl-inder, making a light-tight housing except for the small hole that allowslight to reach the photocathode. The dark slide mechanism is mountedin front of this hole. A simple housing of this type will be shown inSection 6.5. The next step in complexity is to add a magnetic shield.This is a cylinder of mu-metal that fits into the housing and surroundsthe photomultiplier. It shields the tube from external magnetic fieldsthat might interfere with electrostatic focusing of the secondary elec-trons emitted from each dynode. External fields that would affect thetube significantly are rarely encountered. However, such a shield is arather inexpensive precaution. Magnetic shields for the 1P21 can besupplied by the Hamamatsu Corporation" or Perfection MicaCompany.12

The ultimate in photomultiplier tube housings incorporates a coolingsystem. Cooling the photomultiplier to dry-ice temperature can elim-nate most of the tube's dark current. This has significant advantages formeasuring faint stars that may produce a tube current comparable onlyto the dark current. Among professional astronomers, cooled tubes arethe rule, not the exception. Cooling the 1P21 reduces the dark currentfrom about 200 counts per second, for pulse counting, to less than onecount per second. Essentially any output from the tube then results fromstarlight. A cooled tube has less sensitivity to red light, reducing theamount of the red leak through the U filter. This also means that thetransformation coefficients of a tube change when it is cooled. If youdetermine these coefficients when the tube is uncooled, they will not bevalid for observations made when the tube is cooled.

The usual means of cooling the tube is to replace the simple housingdiscussed previously with a cold box. The layout of a cold box is nicely


illustrated in an article by Johnson.'3 A cold box consists of three boxes.The photomultiplier tube is sealed in an airtight container that has asmall window allowing light to reach the photocathode. This inner con-tainer is surrounded by a larger box which holds one-half to one kilo-gram of dry ice. This box is, in turn, surrounded by the outermost boxthat holds styrofoam or polyurethane foam to insulate the dry-ice con-tainer. The light entrance to the cold box is a window and/or the Fabrylens. It is sometimes necessary to mount a small heating element nearthese glass components to prevent them from developing a coating offrost. Cold boxes can add considerable weight to the photometer. Forthis reason, special lightweight designs are required for telescopes lessthan 40 centimeters (16 inches) in aperture. Recently, thermoelectriccooling systems have begun to be used in astronomy. They can reducethe tube temperature typically by 20 to 40 Celsius degrees below theoutdoor temperature. These cooling systems are large, heavy, andexpensive. They probably work as well as a dry-ice cold box but mayrequire a water supply or a fluid-circulation system. It is possible topurchase simple uncooled housings and dry-ice cold boxes. Suppliersare EMI Gencom,7 Hamamatsu," Pacific Precision Instruments,14 EG& G Princeton Applied Research,15 and Products for Research.16 Thereis a word of caution lo note before you place an order for a cooled pho-tomultiplier tube housing. Most of the housings built by these compa-nies are intended for laboratory use. Consequently, not every model cansupport its own weight properly if it is held to the photometer by a sim-ple mounting flange around the light input port. Before placing anorder, it is advisable to call the company and make certain that themodel of your choice can be mounted to a telescope, that it can work inany position, and that it can function in the sometimes hostile environ-ment of your observatory.

6.3 FILTERS

Most of the wide-band filters used in optical astronomy are made byCorning Glass Works17 or Schott Optical Glass.18 Table 6.1 lists a rec-ommended filter set for VBV photometry using the 1P21. Note that itis important to order the specified thickness. Filters of different thick-nesses have slightly different transmission curves. The B filter isactually a sandwich of two filters. The GG13 filter looks like clear glass


TABLE 6.1. Recommended UBV Filters

Bandpass Filter and Thickness (Schott Filters)

U UG2* (2 mm)B GGI3 (2 mm) + BG12 (I mm)V GG14(2mm)Red leak UG2' {2 mm) + GG14 (2 mm)

•These Iwo fillers arc from the same melt.

but it is designed to prevent transmission of light shortward of the Bal-mar discontinuity. Some observers have reported difficulty in makingthe V filter transformations unless the filter matches the one used todefine the t/BKsystem. This filter is a Corning 7-54 made from Corning9863 glass. Unfortunately, this filter has a larger red leak than theSchott UG2. The red-leak filter listed in Table 6.1 is just a sandwichmade from a second V filter and a V filter. The V filter does not trans-mit ultraviolet light normally passed by the V filter. However, it doestransmit any red light which the U filter transmits. The combination ofthe two filters transmits only the red light "leaked" by the U filter. Aftera star has been measured with the ordinary U filter, it is measured againwith the red-leak filter. The red-leak measurement can then be sub-tracted from the U measurement to obtain a corrected U measurement.When ordering these U filters, you should request that they both comefrom the same melt or order a single piece from which you can cut twofilters. This helps to insure that the red-leak properties of the two filtersare as nearly identical as possible.

If you are using a photometer that places the filters inside the tele-scope's focal point, be sure to add a Schott cover glass to make all filtersthe same total thickness. The passage of the telescope's light conethrough glass alters the focal point slightly. If the filters are of differentthicknesses, they will each cause the light to focus at a different point.This is not of concern for any photometer design discussed in this book,with the exception of the photodiode photometer.

Because of its different spectral response, a photodiode must use adifferent set of filters to match the UBV system. De Lara et al.2 used anEG & G Electro-Optics Division19 SGD-040L PIN photodiode with thefilters listed in Table 6.2. The thicknesses of the Corning filters are notspecified because they are adjusted by the manufacturer to achieve the


TABLE 6.2. Filters Used with a Photodiode by De Lara et al.

Bandpass Filters

B Corning 5030 + Corning 9782 + Schott GG13 (2 mm)V Corning 9780 + Corning 3384R Corning 3480 + Corning 4600/ Corning 2600 + Corning 3850

required bandpasses. It should be noted that De Lara et al. were notsatisfied with this filter set. It is presented here as a starting point forthose who wish to experiment with photodiode and filter combinations.

The VBV filters can be ordered in a 2.5 X 2.5 centimeter (1 X 1inch) size at a cost of a few dollars per filter. They should be handledwith care because they are very thin and made from soft glass that canbe scratched easily. The surface of the U filter may appear to developa water-spotlike pattern on its surface with age. This can be removedby polishing the surface with rouge or barnesite.

Normally, filters are mounted in a photometer in one of two ways.One technique uses a filter slide. The filters are held side by side in along rectangular holder. The holder slides lengthwise so that the filterscan be positioned one at a time in the light beam. This design presentsa minor problem for a small photometer. The slide must be long enoughto accommodate each filter and the width of the slide walls. For a four-filter photometer, the slide is at least 11.4 centimeters (4.5 inches) longfor 2.5 centimeter (1 inch) wide filters. With either end filter in the lightpath, the slide sticks out about 10 centimeters (4 inches) to either sideof the optical axis. That requires a total linear span of more than 19centimeters (7.5 inches). This makes it very difficult to contain the filterslide within the main chassis of a small photometer head, which wouldsimplify the design and make it easier to keep light-tight. One is forcedto adopt a design that is frequently used by professional astronomers.Figure 6.5 illustrates the idea schematically. The filter slide fits withina rectangular housing which is light-tight except for holes on the topand bottom plates that allow the telescope's light cone to enter, passthrough the filter, and exit. The outer housing is inserted into the pho-tometer chassis and allows the slide to move its entire length, whileeliminating the need to enlarge the photometer chassis. It is importantto incorporate some sort of detent device so that each filter "clicks" into


OUTERFILTER SLIDEHOUSING

FILTERS

MAINPHOTOMETERCHASSIS

FILTER SLIDE

Figure 6.5 Filter slide.

place and is held in the light path during a measurement. This is usuallyaccomplished by notching the positioning rod and mounting a spring-loaded ball bearing in the rod's bearing at the end of the outer housing.

The second approach to filter mounting is the filter wheel. A flat cir-cular disk has filters mounted around its periphery. There is a smallhole under each filter to allow light to pass through the disk. The filterwheel is aligned perpendicular to the optical axis of the photometer withits rotation axis offset from the optical axis so that each filter passes intothe light beam as the wheel rotates. Looking ahead to Figure 6.16, youfind a sketch of a filter wheel. Once again, a detent mechanism is nec-essary to insure proper positioning of the filters. Of course, there areother methods of mounting filters. In a design by Dick et al.20 the filtersare mounted on a four-sided carousel that rotates around the photo-multiplier. This very simple design has the advantages of compactnessand ease of construction.


6.4 DIAPHRAGMS

Photometer diaphragms are usually made by drilling small holes in ametal plate. The first step is to decide on the desired sizes. We speakcommonly of the size of a diaphragm in terms of the angular size of thefield of view it permits us to see in the telescope's focal plane. That is,a diaphragm which is said to be 20 arc seconds exposes a portion of thesky 20 arc seconds in diameter to our detector. To translate an angularsize to the physical diameter of the hole we need to drill requires aknowledge of the plate scale of the telescope. If we took a photographicexposure of the full moon (0. 5 " across) at the focus of our telescope andit appeared 1 centimeter wide on the developed film, the plate scalewould be 0.5" per centimeter. In other words, this is simply a statementof how angular sizes in the sky project to the telescope's focal plane.The plate scale depends upon focal length. Short focal length telescopeshave large plate scales; that is, they are wide field instruments. Longfocal length telescopes have very small plate scales, hence narrow fieldsand high magnification. The plate scale can be computed in units of arcseconds per millimeter by

ifplate scale = —

where K is 20626 (81 20) if the focal length of the telescope is expressedin centimeters (inches).

Let us suppose that the F/8 telescope discussed in Section 6.1 has a20.3-centimeter (8-inch) diameter. Its focal length is then 162.4 centi-meters (64 inches) and by the above equation it has a plate scale of 127arc seconds per millimeter. Table 6.3 contains a list of twist drill num-bers in the U.S. system and their corresponding diameters. We can usethis table together with the plate scale to predict the angular size of adiaphragm made with any of these drills. For instance, a number 75drill has a diameter of 0.533 millimeters, which gives a diaphragm of0.533 X 127 or 67.7 arc seconds for this telescope. It is desirable tomake the diaphragms rather small to minimize the amount of sky back-ground light seen. Typically, professional astronomers use a diaphragmwhich is 20 arc seconds or less. This is practically impossible with smalltelescopes because their focal lengths are shorter, making the necessarydrill sizes impossibly small. Even with a number 80 drill, which is only


TABLE 6.3. Twist Drill Diameters

Drill No.

404!424344454647484950515253545556575859

Inches

0.09800.09600.09350.08900.08600.08200.08100.07850.07600.07300.07000.06700.06350.0595O.U5500.05200.04650.04300.04200.0410

Millimeters

2.4892.4382.3752.2612.1842.0832.0571.9941.9301.8541.7781.7021.6131.5111.3971.3211.1811.0921.0671.041

Drill No.

606162636465666768697071727374757677787980

Inches

0.04000.03900.03800.03700.03600.03500.03300.03200.03100.02930.02800.02600.02500.02400.02250.02100.02000.01800.01600.01450.0135

Millimeters

1.0160.9910.9650.9400.9140.8890.8380.8130.7870.7430.7110.6600.6350.6100.5720.5330.5080.4570.4060.3680.343

a few times the width of a human hair, the diaphragm is 44 arc secondsfor our 20.3-centimeter (8-inch) telescope. The same diaphragm usedwith a 76.2-centimeter (30-inch) diameter F/8 telescope is 12 arc sec-onds. This points out a basic disadvantage that the small telescope usermust face. The larger plate scale of the small telescope means that pho-tometry must be done with diaphragms that admit a fairly large amountof sky background light. This leads to a poorer signal-to-noise ratio,with the result that faint stars cannot be measured as well.

Even a simple photometer should contain at least three different dia-phragm sizes, one of which is fairly large. This allows the observer touse a larger diaphragm on nights of poor seeing conditions, or a smallone when the moon is bright. An intermediate size probably is used themost and should be of such a size that your clock drive can keep a starwithin this diaphragm for at least 5 to 10 minutes. Additional obser-vational considerations for diaphragm selection are discussed in Section9.4.

Section 6.5 contains a description of a photometer designed for a20.3-centimeter (8-inch) F/8 telescope discussed in this section. The


diaphragm selection in this design is based on the above considerationsand on the availability of small drills. The largest diaphragm was madewith a number 47 drill, which for a plate scale of 127 arc seconds permillimeter yields 253 arc seconds, or 4.2 arc minutes. The remainingtwo diaphragms were made with number 61 and 76 drills, yielding sizesof 126 and 64.5 arc seconds, respectively.

The diaphragm holes can be drilled in a brass or steel plate. A drillpress must be used to insure that the holes are drilled straight and tominimize drill breakage. This latter point becomes important becausethese drills are very thin and break easily. Drills of this size usuallymust be purchased at large hardware stores or dealers in machinists'supplies. Purchase at least two of the smaller drills as you are bound tobreak at least one. The chucks on many drill presses do not close downfar enough to hold drills this small. It may be necessary to find a localmachinist who can drill these holes for you. It is important to counter-bore the holes to avoid a tunneling effect when looking through the dia-phragm viewing eyepiece. The drill for the counterbore should be sev-eral times the size of the diaphragm drill. The counterbore should bemade as deep as possible without enlarging the diaphragm hole. It ispossible to make holes about half the size of a number 80 drill by prick-ing aluminum foil with a sharp needle. The foil is placed on a flat metalsurface, and the hole is made by turning the needle between your fin-gers. Make several holes and inspect them with a magnifying glass. Theone which appears most round can be used as a diaphragm by gluingthe foil to a piece of metal that has a larger hole in it.

Like filters, the diaphragm position usually is selected by a dia-phragm slide or wheel. A diaphragm slide is usually used because,unlike the case of filters, it can be very short and completely containedwithin the main photometer chassis. The diaphragm holes are drilled ina flat rectangular piece of metal. It is also necessary to provide somesort of detent system so each diaphragm "clicks" into position on theoptical axis. A diaphragm slide assembly is described in Section 6.5.

There are two final points to be made. First, when you view a star inthe diaphragm through the viewing eyepiece it may be very difficult totell if it is centered. This is because the sky background light does notoutline the diaphragm sufficiently. As a result, everything except thestar looks black. An exception to this occurs if you are observing nearan urban area. In this case, the sky background is such that the dia-phragm appears as a gray circle in the eyepiece. This is about the only


benefit that light pollution provides for astronomy! Hopefully, you willhave the opportunity to observe from a site that requires an internalsource of diaphragm illumination. The simplest method is to mount avery tiny light bulb or a light-emitting diode (LED) just above the dia-phragm slide. The electric current to this lamp is controlled by a contactswitch attached to the flip mirror. When the flip mirror is moved out ofthe light path, the lamp is automatically shut off. It is a good idea tohave a potentiometer in the circuit in order to adjust the lampbrightness.

Second, make sure that the focus of the diaphragm viewing eyepieceis at least initially adjustable. A quick look at Figure 6.1 should con-vince you that the spacing and size of the optical components dependson the assumption that the diaphragm is placed at the telescope's focalplane. This is a condition that must be established at the beginning ofeach observing session. This is accomplished first by focusing the dia-phragm eyepiece until the diaphragm appears with sharp definition.Then center a star in the diaphragm. If this star appears out of focus,the diaphragm plane and focal plane do not coincide. This is remediedby adjusting the telescope's focus until the star's image appears sharp.Many photometers are designed so that the eyepiece focus is adjustedonce during construction and locked in place. When observing, it is thenonly necessary to adjust the telescope's focus to make the stellar imageclear. The disadvantage of this procedure is that eyeglass wearers mustuse their glasses when looking through the eyepiece. If they focus thestar without their glasses on, the diaphragm will not appear focusedbecause this was adjusted in the workshop by someone who, presum-ably, had normal vision. For the eyeglass wearer who does not use hisor her glasses at the eyepiece, it is best to leave the eyepiece focusadjustable.

6.5 A SIMPLE PHOTOMETER HEAD DESIGN

In this section, we describe a simple photometer head design suitablefor use by amateur astronomers or any user of a small telescope.Detailed construction plans are not presented. Instead, the sketches anddiscussion presented are intended as a guide and a source of ideas forthe photometer builder. We recommend that this person look at thedesigns of the AAVSO photometry manual,21 Allen," Burke and Pip-pin,4 Code,23 Dick et al.20, Grauer et al.24 and Nye.45


BRASS PHOTOMULTIPLIER'TUBE HOUSING

FILTER-POSITIONKNOB

FILTERDIAPHRAGM COMPARTMENTVIEWING COVEREYEPIECE

3

DARKSLIDE

FLIP-MIRRORCONTROL

.1.25-INCHO.D.TUBING

TLIGHT

Figure 6.6 Exterior view of photometer head.

The design presented below was built originally for a 20.3-centimeter(8-inch) F/8 Newtonian telescope. Because it is difficult to counterbal-ance objects placed at a Newtonian focus, this photometer head wasmade as lightweight as possible, 1.4 kilograms (3.1 pounds). It has beenused successfully for years and has produced a lot of observational datawith the 20.3-centimeter (8-inch) and larger telescopes. The basicdesign is patterned after one by D. Engelkmeir.21 Figure 6.6 shows asketch of the exterior of the photometer head. The central box is madefrom a standard 12.7 X 10.2 X 7.6-centimeter (5 X 4 X 3-inch) alu-minum electrical chassis. This choice was made to avoid machining andto provide a lightweight box at a cost of a few dollars. The interior wasspray painted with flat black paint to reduce scattered light. The bottomof the box is mounted on a 19.0 X 19.0-centimeter (7.5 X 7.5-inch)base that is oversized to help mount the photometer head to the tele-scope securely. This is illustrated later. The starlight enters a 8.18-cen-timeter (1.25-inch) O.D. brass tube that fits the standard focusing


drawtube of U.S.-made amateur telescopes. The diaphragm viewingeyepiece is attached to a diagonal mirror for ease of viewing. This isespecially helpful when using Newtonian telescopes. The filters are con-tained in a separate compartment above the main box for easy access.The photomultiplier tube, a 1P21, is mounted in a simple, uncooledhousing made from brass tubing. There are four external controls: thedark slide, the filter position, the diaphragm slide, and the flip mirror.Figure 6.7 shows a photograph of this head, seen in a perspectiveslightly different from Figure 6.6. Obviously, a photometer head shouldbe made light-tight except for incident starlight.

Figure 6.8 shows a cut-away view of the interior of the photometerhead. Wherever possible, plastic and aluminum have been used toreduce weight. The flip mirror is a quarter-wave optical flat. This mirroris mounted to a flat metal plate with double-sided adhesive tape. Themetal plate is soldered to a rod that attaches to a knob on the outsideof the box. The mirror directs light to the collimating and imaging len-ses used for viewing the diaphragm. These lenses are identical achro-mats mounted in a single tube that helps to insure their mutual align-

Figure 6.7 The photomultiplier head.


BRASSTUBING

H.V. DIVIDERRESISTORS

SHV H.V.CONNECTOR

MAGNETICSHIELD

FILTERWHEEL

DARK SLIDE

FABRY LENS

FLIP MIRROR

DIAPHRAGM SLIDE

IMAGINGLENS

4 INCHES

Figure 6.8 Cutaway view of photometer head.

ment. This tube is held in place by two plastic blocks. The tube passesthrough a hole in each block and is held in place by three adjustmentscrews similar to the way finder telescopes are mounted. In the work-shop, the screws can be adjusted until the image of the diaphragmappears centered in the eyepiece. The ability to make this adjustmentis valuable especially if you are unable to build components to machine-shop precision.

The top plate of the box is removable. On its interior side, the Fabrylens is held by a small plastic block. The exterior side holds a filter com-partment and the tube housing. The left half of the housing, as shownin Figure 6.8, contains a magnetic shield and the photomultiplier tube.The right half of the housing is made from slightly larger tubing so thatit slides over the left half. It holds the tube socket, voltage divider resis-tors, and cable connectors. The tube socket is held in place by a shortlength of tubing, which in turn is held snugly in place by a spacerbetween it and the interior housing wall. It is important to establish an


electrical connection between the walls of the tube housing and electri-cal ground. This is necessary to provide good electrical shielding for thetube. The recommended BNC and SHV connectors should be installedso that the outside of the connector makes electrical contact with boththe housing walls and the cable shield.

Just below the housing is the dark slide. This is simply a narrow com-partment in which a small metal plate can be moved by a rod over thelight opening. The floor of this compartment is lined with felt to makethe slide move smoothly and to enhance the light seal.

Below the dark slide is the filter compartment. In this design, thefilters were placed behind the Fabry lens rather than in front. This waspurely a matter of convenience. It is preferable to place filters in a lightbeam that is nearly parallel so they do not deviate the beam. Both thetelescope and Fabry lens have about the same F ratio, so it makes littledifference where the filters are placed. Figure 6.9 shows some details ofthe filter assembly. The filters are mounted on a circular disk 6 centi-meters (2.4 inches) in diameter. Because of this small size, the filtershad to be cut smaller than their normal one-inch size. Access to thefilter wheel is gained by removing a cover plate that is held in place bya single wing nut. The filter wheel is turned by two identical gears. Oneis mounted on the central shaft of the filter wheel, high enough to clearthe filters. The second gear is mounted on a shaft that comes down fromthe top of the filter cover. This gear system was necessary since the filterwheel shaft is covered partially by the tube housing, leaving no roomfor a positioning knob. The two gears move this knob about 1.5 centi-meters (0.6 inch) to the right in Figure 6.9. The detent positioning isaccomplished by a scheme suggested by Burke and Pippin4 andStokes25. The shaft that holds the gear on the filter cover is actually theshaft from an electronic rotary switch. Such switches are inexpensiveand contain an accurate detent system. All that is necessary is toremove the wafers containing the switch contacts. Usually theseswitches have a stop that prevents 360° rotation. This stop is often justa metal tab that can be bent out of the way. The switch positions shouldbe spaced evenly around the knob. The number of switch positionsshould be a whole number times the number of filters. In this case, a12-position switch was used with four filters.

Figure 6.9 also shows the diaphragm assembly. This slide consists offive pieces made from flat steel stock. The top and bottom plates areboth 6.4 X 5.1 X 0.3 centimeters (2.5 X 2.0 X K inches) with a 0.64-


PHOTOCATHODE

FILTER SELECTORKNOB

ROTARY SWITCH

GEARS

FILTER WHEEL

FILTER COVER

FABRY LENS18 mm dia.f.l. 40mm

DIAPHRAGM ASSEMBLY

TOP VIEW

%-INCH HOLE IN BOTH TOPAND BOTTOM PLATE

END VIEW

TAP LOWER PLATE

SLIDE

POSITION DETENTS

COUNTERBORE

17 V tf I I "~1

DRILLSIZE #76 61 47 14 INCH

SCALE: 10CENTIMETERS

FILTER WHEEL 4 INCHES

Figure 6.9 Detail of filter assembly and diaphragm slide.

centimeter (0.25-inch) central hole. There are two 6.4 X 1.3 X 0.3-centimeter (2.5 X 0.5 X %-inch) metal strips separating the top andbottom plates and forming the 2.54-centimeter (1-inch) cavity for thediaphragm slide. The fifth piece is 6.4 X 2.5 X 0.3 centimeters (2.5X 1.0 X J6 inches) and contains the diaphragm holes. There are threediaphragms, as discussed in Section 6.4 and a 0.6-centimeter (0.25-inch) hole for a wide field of view. Detent positioning is accomplishedby a spring-loaded ball bearing mounted in the bottom plate. This ballpushes against the bottom of the slide. As seen in Figure 6.9, there is aseries of holes drilled just above each diaphragm position. When a dia-phragm is on the optical axis, the spring pushes the ball into the detenthole and the slide locks into position. Because the detent holes aresmaller than the ball bearing, pressure on the diaphragm rod pushes theball bearing down and the slide moves until the next detent hole isaligned.


While the photometer described here has worked well for manyyears, we recommend two design changes. First, the filter wheel andfilter compartment should be made larger in order to accept standard1-inch filters. The second change concerns the flip mirror. A look atFigure 6.8 reveals the problem. When the mirror is swung back to makea measurement, it reflects any room light entering the eyepiece into theFabry lens. It has been found necessary to place a cap over the eyepiecewhen a measurement is being made. The solution is to hinge the mirrorabout a point in the upper left of Figure 6.8 so that when the mirror isswung out of the light cone it covers the front of the collimating lens.A foam-rubber ring placed around the front of the lens housing wouldcushion the mirror.

It is very important to mount the photometer head solidly to the tel-escope. Any flexure can cause misalignment and move the star out ofthe diaphragm. Supporting the photometer head solely with the draw-tube of the typical small telescope is insufficient unless the telescope'sfocuser has been redesigned. Figure 6.10 shows the photometer headjust described mounted at the Cassegrain focus of a 30.5-centimeter(12-inch) telescope. The large base plate of the head is attached to amounting rail on the telescope. A similar mounting rail could be usedwith Newtonian telescopes. The Cassegrain telescopes used by profes-sional astronomers usually focus by moving the secondary mirror ratherthan the rack-and-pinion devices used by amateurs. This allows the pho-tometer to be bolted firmly to the tail piece of the telescope, eliminatingmost flexure. This focusing arrangement should be considered moreseriously by amateur astronomers.

6.6 ELECTRONIC CONSTRUCTION

In this and succeeding chapters, we present designs for electronic cir-cuitry. These designs represent in most cases prototype units that havebeen constructed and tested. However, we would like to make threepoints:

1. You should not attempt to construct these circuits unless you arefamilar with high-frequency, high-voltage, and high-gain con-struction techniques. Otherwise, we recommend purchasing com-mercial units.

2. While the prototype units work as reported, different layouts andcomponents may require minor modifications to perform properly.


Figure 6.10 The photometer head mounted on a telescope.

3. Because you are building high-performance circuits, you shoulduse the best quality components that you can afford.

Finding sources of components can be complicated. Other than RadioShack, you can generally find components at large electronic distribu-tors. These companies usually have a $25 minimum order. Other excel-lent sources are hamfests, especially the larger ones in Dayton andAtlanta. These are advertised in QST and other amateur radio maga-zines. For most of us, the primary and cheapest source of electroniccomponents is the mail-order house. The Radio Amateur's Handbook26

lists many sources of parts in the chapter on construction techniques.We suggest reading this chapter as it contains much information thatthe amateur astronomer can use.


6.7 HIGH-VOLTAGE POWER SUPPLY

The high-voltage power supply for the photomultiplier tube is one of themost important components of the photometer. It should be adjustable,in order that the correct operating point for each tube can be found,and it should be well regulated. A 1 percent change in the output volt-age can make a much larger change in the photomultipiier tube outputbecause the gain of each stage is multiplicative. This kind of systematicerror should be kept to a value smaller than anticipated from observa-tional error (photon statistics); that is, the power supply should be reg-ulated to better than 0.1 percent. Most variations are caused by voltagefluctuations from the power line, and may occur on a very short timescale.

The requirements listed above would be relatively simple with mod-ern technology if not for the magnitude of the voltage required. A 1P21tube uses approximately 1000 V (1 kV) and some other tubes, such asthe FW-118, may require 2 kV. No integrated circuit (1C) voltage reg-ulator and few transistors can handle these voltages.

Of course, one can always purchase a commercial, adjustable high-voltage supply. Appropriate used units can be obtained from manyfirms that deal in reconditioned test equipment, such as the Ted DamesCo.27 New units are available from companies such as EMI,7 Kepco,28

Lambda,29 Ortec,30 or Princeton Applied Research.15 Be advised, how-ever, that some of these new units cost $200 to $1000.

You can construct your own power supply. There is a dearth of up-to-date circuits and we try to present those that we could find or derive.However, these circuits handle lethal voltages, so be extremely carefulin working with them.

There are three general approaches used in building adjustable high-voltage supplies. The simplest is to use a bank of high-voltage batteries.The easiest electronically regulated supply is to obtain approximately2 kV from a filtered supply and then use a voltage divider to pick offthe appropriate voltage. The third method is also the most elegant:amplitude-modulate an RF oscillator and rectify the output voltage.Each of these approaches is discussed in more detail.

6.7a Batteries

The use of a battery bank has the advantage of excellent regulation andnoise immunity. In addition, the bank is portable and the current drain


WV-10k/10W

— 10-90 V BATTERIES-=- (NEDA 204)

T

Figure 6.11 Battery supply.

is low enough that the batteries last their normal shelf life. The twodisadvantages are that batteries are temperature-sensitive (keep themwarm!) and expensive. Expect to pay about $1 per 10 V. Typical avail-able batteries are the PX18 (45 V) and the NEDA 204 (90 V). Somehigh-voltage batteries used by the military can be obtained from surplussupply houses at a much lower cost; check electronics magazines fornames and addresses. The U200 batteries (300 V) were available for-merly and may still be found in local stores.

A typical battery-operated system is shown in Figure 6.11. The 10 Kseries resistor acts as a current limiter when an external short occurs.The case for this system should have insulation to prevent leakage pathsto ground, moisture proofing to prevent leakage paths and damage tothe batteries, and shielding on the outside for safety and noiseimmunity.

6.7b Filtered Supply

A schematic for a typical voltage divider supply is shown in Figure 6.12.This involves an AC transformer with a voltage doubler consisting ofdiodes Dl through D4. Each diode is bypassed with an RC combinationto equalize voltage drops across the diodes and to guard against tran-sients. Current-limiting resistors Rl and R4 protect the diodes againstthe initial turn-on surge and to allow some current to flow through thezener chain at all times. The output is filtered by capacitors C7 throughCIO, with equalizing resistors R7 through RIO doubling as bleederresistors when the supply is shut off. The filtered output is fed to thezener bank. These diodes regulate the voltage and by switch or jumper

117 VAC

T1: 500-600 VAC (CalectroD 1-761,StancorP-8173/6358,Allied 6K53VG)

D5-D8: 200V/1W Zener (1N3051)D9-D12: 50V/1W Zener (1N3037)D1-D4: 1000V/1A (1N4007)

C7-C10: 20 mf/450V {Calectro A1-179)C1-C6: 0.01/1kVdisc

Figure 6.12 Zener voltage supply.

*RCA

Figure 6.13 Current-regulated supply. (Copyright 1981 National Semiconductor Corporation)

CONSTRUCTING THE PHOTOMETER HEAD 1 53

selection can allow some voltage adjustment. A simplified chain of five200 V zeners could be used if adjustment is not desired.

The regulation from this supply is not as good as might be expected.Because the zener knee is not infinitely sharp, there is some voltagechange with current. In addition, zeners are very temperature-sensitive.The amount of regulation can be determined by comparing the current-limiting resistor to the dynamic resistance of the zeners, as this circuitis basically a voltage divider. If there is 20 percent ripple on the unre-gulated supply and the resistance ratio is 100 (typical values), then theregulation is approximately 0.2 percent. The advantage of this supplyis that it is the least expensive to build of the electronically regulatedsupplies. If you do build it, try to thermostat the supply.

A novel variation of the filtered high-voltage supply was designed byElkstrand31 and is shown in Figure 6.13. Instead of trying to regulatethe voltage, this circuit uses an LM100 1C regulator and regulates thecurrent passing through the photomultiplier tube. A full-wave rectifieroperating off one winding of the power transformer Tl provides a 15-Vbias voltage for the LM100. The other winding is used in a voltagedoubler as in Figure 6.12 with the output passing through the photo-multiplier tube divider chain that develops the operating voltages forthe cathode and dynodes. Five cascaded transistors, Ql through Q5, areused as the pass transistors, each therefore passing one-fifth of the totalvoltage. This is the most economical solution to the problem of handlingthe required voltage levels. Base drive is provided for the cascade stringby R3 through R7 in a manner that does not affect regulation. Capac-itors Cl through C5 suppress and equalize transients across the passtransistors, and clamp diodes across the sensitive emitter-base junctionsof the transistors prevent damage from voltage transients.

6.7c RF Oscillator

This approach was used by Code.23 A schematic of this circuit is repro-duced in Figure 6.14. Because of the high voltages involved, the onlyconvenient oscillator involves vacuum tubes, and therefore the design iscumbersome. Basically, part of the rectified output voltage from an RFoscillator running at 100 kHz is used in a feedback loop to control theamplitude of the oscillator. By changing the amount of feedback volt-age, the output voltage is adjustable. Because the current drain from aphotomultiplier tube is negligible (less than 1 jiA), the RF oscillatordoes not need much power capability.

AN-3102-20-27P

I—lF

6.3V

5651

AN-3102-18-1P

R Plus the resistance of the choke is to be 25 ohmsC is sufficient to produce resonance in the tuned circuit.

Figure 6.14 RF oscillator supply. (Reproduced from Photoelectric Astronomy for Amateurs, ed. F. B. Wood, Macmillan)


While obtaining the proper photomultiplier tube voltage in one stepis impossible with transistors, there is another approach. Electronicflash units use an oscillator type of DC-DC supply that takes 4 to 6 Vand transforms it to voltages in the 300 V range. These DC-DC suppliesare available on the surplus market, and can be tied together in a seriesfashion to obtain the 1 kV voltage necessary for photomultiplier tubes.Regulation is then provided in the input voltages to the supplies, whichcan be obtained from standard 1C regulators.

Commercial DC-DC supplies are available from companies such asRCA,9 Venus,32 Ortec,30 EMI,7 and Hamamatsu," These supplies rangefrom $100 to $500, and offer a small, convenient method of generatingthe high voltage for the photomultiplier tube. The input voltage is gen-erally between 6 and 20 V to provide 900 to 1500 V output, with outputregulation again controlled by the input voltage regulation. A DC-DCsupply is an alternative to batteries for a small, portable supply.

6.7d Setup and Operation

To connect a high-voltage supply to the photomultiplier tube, coaxialcable such as RG58 can be used. For voltages above 1 kV, special high-voltage connectors of the SHV or MHV type should be used to gainimmunity from possible dielectric breakdown. Always remember theseprecautions:

1. Never work on the high-voltage power supply with the powerturned on.

2. Never disconnect the photomultiplier tube or the high-voltagecable to the photomultiplier tube with the high voltage turned on.

3. Bring the high voltage up and down slowly to avoid rapid changes.4. Observe the correct polarity of the high-voltage power supply.

6.8 REFERENCE LIGHT SOURCES

As we mention in this and subsequent chapters, electronic equipment isgenerally temperature-sensitive. This means that if you use a set of AOstars at the beginning of a night to determine the zero-point coefficientsin the transformation equations, and then use these coefficients toreduce data taken many hours later, the results can be in error. Thereare other factors, such as telescope position dependence as a result of


equipment flexure or magnetic fields, that also contribute errors to yourresults. To calibrate these irregularities out of the observations, astron-omers use standard light souces. These come in three varieties: ground-mounted standard lamps, small radioactive sources, and stars.

Standard lamps are usually filament-operated devices with operatingparameters kept constant. They cannot be mounted on the telescopebecause irregularities in the light output are common because of fila-ment sag. In addition, they are difficult to use for astronomical purposesbecause of their relatively large output energy. To image the lightsource through the telescope requires that the energy collected by thetelescope from the source must be equivalent in intensity to an averagestar. In general, such a lamp must be placed more than 100 metersaway from the telescope. In addition, the filaments operate at a cooltemperature in comparison to most stars and therefore temperature-sensitive errors tend to propagate in opposite directions.

Radioactive sources are usually weak beta emitters that are placedin contact with a phosphor. The fast electrons enter the phosphor crys-tals and produce light in one or more bands whose width is usually onthe order of a hundred Angstroms. These sources can be made verysmall and are usually placed in the filter side or near the photomulti-plier. This allows the observer to test the photometer for positional vari-ations in addition to temperature and aging effects. The light output ofthese sources is usually blue-green, making calibrations a little easierthan for the standard lamps. The phosphor in radioactive sources suffersfrom temperature effects equal to or greater than those of the photom-eter, and cannot be used for calibration unless thermostatted. In addi-tion, the phosphor is highly light-sensitive, and there are long-termbrightness variations that preclude calibrations over periods longer thana few months.

For the reasons listed above, few astronomers regularly use standardlamps or radioactive sources. They are used in initial calibration of thetelescope and photometer and in cases where long-term accuracy(months for radioactive sources, longer for other types) is needed. Analternative to the ground-based calibration techniques is to use standardstars to monitor any equipment changes. This method has the advan-tage that it also measures atmospheric transparency changes. However,it suffers from increased complexity (extinction must be taken intoaccount and often several stars are used), the problem of finding stan-dard stars over the entire sky to measure flexure errors, and that cali-brations take up observing time.


In general, we recommend using differential techniques in photome-try. This eliminates almost all instrumental variations as they occurequally to both the comparison and the program star. If you intend tomeasure stars over the entire sky, to standardize your comparison starsfor example, then use the North Polar Sequence stars to check trans-parency and temperature effects as their air mass (and therefore extinc-tion) changes very little in the course of a night. In all cases, includingextinction determinations, observe standards throughout the night, notjust at the beginning or end.

6.9 SPECIALIZED PHOTOMETER DESIGNS

Earlier in this chapter we detailed the construction of a simple photom-eter that is certainly adequate for amateur and small telescopes. It islight, compact, and requires few specialized tools to construct. This sec-tion discusses other, more complicated designs that observatories haveconstructed. While the treatment is brief, it should be sufficient toinform you of other ideas and direct you to references where more detailcan be found.

6.9a A Professional Single-beam Photometer

The photometer used at the Morgan Monroe Station of Goethe LinkObservatory is a prime example of a professional instrument. Based ona design pioneered by William Hiltner, the photometer is shown in Fig-ure 6.15. All external framework components are made of 0.635-cen-timeter (^i-inch) aluminum stock, milled with rabbet joints to formlight-tight boxes. The assembled unit including the photomultiplier'scold box, weighs about 32 kilograms (70 pounds).

As seen from the top, where light enters the photometer from thetelescope, we can identify the three major subdivisions of the instru-ment: guide box, filter- and diaphragm assembly, and the cold box. Theguide box has a military surplus eyepiece mounted on a movable X-Yplatform for offset guiding and accurate star centering. This eyepieceand its associated mirror are placed in front of the diaphragm slide. Theeyepiece has an illuminated reticle, with 30 arc second arms on the crosspattern. Once having centered the star in the diaphragm and adjustedthe eyepiece position, the observer can repeatedly center stars in a 10arc second diaphragm using the cross hairs without checking the dia-phragm viewing eyepiece. Once a star is centered on the reticle the


Figure 6.15 The Goethe Link Observatory photometer.

viewing mirror is slid, parallel to its surface, until a hole in the mirroraligns with the optical axis and the starlight can enter the photometer.The eyepiece can be moved using its X-Y motion to find a guide starbecause the rest of the field is still available for viewing. The Erfle view-ing eyepiece was chosen to provide sufficient eye relief so that anobserver with eyeglasses can use the photometer easily.

The filter and diaphragm assembly begins with a detented filter slidethat under normal circumstances contains an open hole and two neutraldensity filters, allowing the observer to perform photometry on stars asbright as zero magnitude. This slide can also be used to hold polarizing


filters. The light beam then encounters a detented diaphragm slide withholes ranging from 10 to 120 arc seconds. Next the beam passes to arotating filter wheel. Electronics control the motion of this wheel tomove from one filter to another under programmed control. Manualmotion is available with an externally mounted pushbutton, and the fil-ter position number is shown on a seven-segment LED readout. Thefilter assembly is carefully machined and is removable with one screw,allowing convenient filter wheel and/or mechanism replacement.

The cold box has a self-contained dark slide and Fabry lens, allowingthe observer to change photomultipliers during the night by switchingcold-box assemblies. Through careful design, the Fabry lens does notrequire heating to remain frost-free. A single dry ice charge, about 1kilogram (2.2 pounds), usually lasts an entire night. The pulse preampand discriminator is mounted on the cold box, making good ground con-tact and matching the cold-box tube assembly with a preamp tailoredto the photomultiplier tube's best operating conditions.

6.9b Chopping Photometers

The next level of complexity is to provide two light paths and to switchthe detector back and forth between them. For instance, you could usetwo diaphragms in the focal plane and pick out the program star anda nearby comparison star. By directing the program or comparisonstar's light to the photomultiplier by the use of a mirror, you can"move" from one star to the other in a fraction of a second, nullifyingany atmospheric changes. This setup allows differential photometry inone color during very poor sky conditions, such as uniform cirrus orbroken cloud cover. An example of the dual-diaphragm arrangement isthe photometer designed by Taylor33 and shown in Figure 6.16. Herethe photometer only chops between star and sky, which makes themechanism simpler because the only moving element is the opticaldetent system; the filter and photometer assemblies remain the same.Because you are chopping between star and sky, you can only obtainaccurate color indices. You must "intercompare" two stars to obtaindifferential magnitudes.

A similar system is used at Skibotn Valley in the Arctic Circle.34 Inthis photometer the secondary mirror of the telescope is moved to passthe light of two separate stars, or the star and sky through a single dia-phragm. This arrangement allowed the photometry of an eclipsing


Stepper motor200 steps/re volution

Portion ofdiaphragmslide showingone hole pair2 steppersteps apart

Chopperwheel

2 steps rotationto skv, then 23more steps to'star' of nextfilter

2-infrared (9400 A)light emittingdiodes

Note: Samearea onfilter usedfor star asfor sky.

OpticalChopper detent holeholes, indicatingone for first filter |each positionfilter

Opticaldetentsystem

Optical detentholes one starand one skyhole for eachfilter

-photo transistors

25 stepsbetweenfilters

Figure 6.16 The Taylor dual-diaphragm photometer. (Courtesy of the Publications of the Astro-nomical Society of the Pacific)

binary star during an aurora that was as bright as a sixth-magnitudestar and that varied by five magnitudes in 1 minute!

Chopping photometers work because clouds are relatively neutral inextinction; Serkowski35 and Honeycutt36 have shown that the effect of


one magnitude of cloud extinction is only about 0.01 magnitude on theUBV colors. The biggest error is the variable transparency, easilyaccounted for by chopped photometers. Because only one detector isused, half of the time is usually spent observing the sky, decreasing thecollecting efficiency.

An interesting variation of the chopping photometer was designed atIndiana University by De Veny." This photometer uses two diaphragmsof different sizes on the same star. By switching between the two dia-phragms, you obtain a star and sky reading, and then a star and skyreading with a known additional amount of sky. You can then deter-mine the sky background around the star mathematically and subtractit. The advantage of this system is that you are always measuring skysurrounding the star, while continuously observing the star, effectivelymultiplexing in the sky observations.

6.9c Dual-beam Photometers

A dual-beam photometer in this context means any system where twoseparate detectors are used. It can be built in one of two ways: eitherdividing the light from a single star into two components, or using thelight from two stars separately.

The single-star photometer usually uses a dichroic beam-splitter todivide the beam into a blue and a red component. The response curvefor such a beam splitter was calculated by Morbey and Fletcher38 andis shown in Figure 6.17. Note that the division is not pure and sharp.This means that it is difficult to use a filter with the beam-splitter outputand match the UBV colors exactly. Also, two separate detectors areused and their transformation coefficients must be known very accu-rately to obtain color indices. Half-aluminized mirrors could be used forthe beam splitter, but then only half of the light in a given wavelengthwould be available, offering little gain over a chopping photometer. Byusing a dichroic splitter, you double throughput and prevent atmo-spheric variations from affecting the derived color index. An exampleof a single-star photometer as designed by Wood and Lockwood39 isshown in Figure 6.18.

Basically, the dual-star photometer is two separate photometersmounted at the focal plane of the telescope. Usually one photometer isfixed on the optical axis and the other is movable in angle and radius tomeasure sky or a nearby comparison star. Because there is no beamsplitter, no light is lost and no filter response change is involved. Usually


100

•30

40

Redtransmission

Dichroic filter/reflectorat 45 to incident beam

5 6 7

Wavelength

ax io3 A

Figure 6.17 Response curves for dichroic filter/reflector. (Permission granted from the NationalResearch Council of Canada.)

for accurate measurements, the role of the two photometers is inter-changed to eliminate the instrumental response differences. An exampleof such a photometer as designed by Geyer and Hoffmann40 is shown inFigure 6.19.

A novel approach to the dual-star photometer is the twin photometricreflector at Edinburgh.41 Here two telescopes are contained on the samemount, with one continuously adjustable with respect to the other byseveral degrees. By pointing at one standard star with one telescope, theother telescope can determine a photoelectric sequence in a cluster inshort order, or both telescopes can be used on a star for simultaneoustwo-color photometry.

PIN photodiodes would make an excellent dual-beam photometer.Small and lightweight, such a photometer would be feasible for mod-erate-sized amateur telescopes. This photometer would work well inareas with few photometric nights, such as the eastern U.S. The authorswould like to hear about the design of this type of photometer for pos-sible inclusion in future editions of this text.


Figure 6.18 Single star photometer by Wood and Lockwood. (By permission of the LeanderMcCormick Observatory)

6.9d Multifilter Photometers

A multifilter photometer can also be classified as a coarse spectrometer.Light from a single star is broken into several beams that are measuredindividually. Dichroic beam splitters are not used because of theirresponse curves; adequate separation of three or more colors isextremely difficult. Roberts42 used aluminum beam splitters to obtain athree-channel photometer. Since aluminum is essentially neutral, threeequal beams can be obtained with each passing through an appropriate


Telescope flange

Figure 6.19 Schematic drawing of the Greyer photometer. O is the wide angle offset eyepiece,D the diaphragm wheel, M the periscope flip-flop mirror, L the Fabry lens, F the color filterwheel, and P the photomultiplier tube. (Courtesy of Astronomy And Astrophysics)

fitter. However, each beam then contains only one-third of the light atany given wavelength so that the net result is the same as if youswitched from one filter to another, in terms of throughput or netcounts. However, the measurements are simultaneous.

A much more practical multifilter arrangement is the Walraven pho-tometer.43 Here a quartz prism is used to disperse the light, as in a spec-trograph, and the resultant spectrum is sampled by five filter and detec-tor combinations. It is impossible with such a combination to match theUBV system, as its wide-band response curves overlap each other.Medium- and narrow-band systems such as the Stromgren four-colorand the Walraven five-color systems are ideally suited to multifilterphotometers. The Mira group in California44 have a 512-channel "pho-tometer" covering the visible spectrum that they intend to use to acquirerapid spectrophotometry of the 125,000 stars of the Henry Draper Cat-alog visible in the Northern Hemisphere. As you can see, as the pho-tometric instrumentation becomes more complicated, the dividing linesbetween types of instruments become very nebulous.

REFERENCES

1. Persha, G. 1980. IAPPP Com. 2, 11.2. De Lara, E., Chavarria K., Johnson, H. L., and Moreno, R., 1977. Revista Mex-

icana de Astron. y Astrof. 2, 65.


3. Optec, Inc., 199 Smith, Lowell, MI 49331.4. Burke, E. W. Jr., and Pippin, D. M. 1976. Pub. A. S. P. 88, 561.5. Edmund Scientific, 101 E. Gloucester Pike, Barrington, NJ 08007.6. Jaegers, A., 691S Merrick Rd., Lynbrook, NY 11563.7. EMI Gencom Inc., 80 Express St., Plainview, NY 11803.8. ITT, Electro-Optical Products Division, 3700 E. Pontiac St., Fort Wayne, IN

46803.9. RCA, Electro Optics and Devices, Lancaster PA 17604. RCA photomultipliers

are available from electronics suppliers.10. Fernie, J. D. 1974. Pub. A. S. P. 86, 837.11. Hamamatsu Corp., 420 South Ave., Middlesex, NJ 08846.12 Perfection Mica Co., Magnetic Shield Division, 740 N. Thomas Drive, Bensen-

ville.IL 60106.13. Johnson, H. L. 1962. In Astronomical Techniques, Edited by W. Hiltner. Chi-

cago: Univ. of Chicago Press, p. 157.14. Pacific Precision Instruments, 1040 Shary Court, Concord, CA 94518.15. EG & G Princeton Applied Research, P.O. Box 2565, Princeton, NJ 08540.16. Products for Research, Inc., 88 Molten St., Danvers, MA 01923.17. Corning Glass Works, Houghton Park, Corning, NY 14830. Corning filters may

be ordered from Swift Glass Co., 104 Glass St., Elmira, NY 14902.18. Schott Optical Glass Inc., 400 York Ave., Duryea, PA 18642.19. EG & G Electro-Optics Division, 35 Congress St., Salem, MA 01970.20. Dick, R., Fraser, A., tossing, F., and Welch, D. 1978. J. R. A. S. Canada 72, 40.21. Photometry Committee. 1962. Manual for Astronomical Photoelectric Photome-

try, AAVSO, 187 Concord Ave., Cambridge, MA 02138.22. Allen, W. H. 1980. IAPPP Com. 2, 1.23. Code, A. D. 1963. In Photoelectric Astronomy for Amateurs. Edited by F. B.

Wood. New York: Macmillan.24. Grauer, A. D., Pittman, C. E., and Russwurm, G. 1976. Sky and Tel. 52, 86.25. Stokes, A., 1980. In paper presented at IAPPP. Symposium, Dayton, OH.26. The Radio Amateur's Handbook. Newington: The American Radio Relay

League. Published yearly.27. The Ted Dames Co., 308 Hickory St., Arlington, NJ 07032.28. Kepcolnc., 131-38 Stanford Ave., Flushing, NY 11352.29. Lambda Electronics Corp., 515 Broad Hollow Rd., Melville, NY 11747.30. EG & G Ortec Inc., 100 Midland Rd., Oak Ridge, TN 37830.31. Elkstrand, J. P. 1973. In Linear Applications Handbook, volume 1. National

Semiconductor Corp. (A.N. 8).32. Venus Scientific Inc., 399 Smith St., Farmingdale, NY 11735.33. Taylor, D. J. 1980. Pub. A. S. P. 92, 108.34. Myrabo, H. K. 1978. Observatory 98, 234.35. Serkowski, K. 1970. Pub. A. S. P. 82, 908.36. Honeycutt, R. K. 1971. Pub. A. S. P. 83, 502.37. De Veny, J. B. 1967. An Improved Technique for Photoelectric Measurement of

Faint Stars. Masters thesis, Indiana University.38. Morbey, C. L., and Fletcher, J. M. 1974. Pub. Dom. Ap. Obs. 14, 11.39. Wood, H. J., and Lockwood, G. W. 1967. Pub. Leander McCormick Obs. XV, 25.


40. Geyer, H., and Hoffmann, M. 1975. Ast. and Ap. 38, 359.41. Reddish, V. C. 1966. Sky and Tel. 32, 124.42. Roberts, G. L. 1967. Appl. Opt. 6, 907.43. Walraven, T. and Walraven, J. H. 1960. Bui. Ast. Inst. Neth. 15, 67.44. Overbye, D. 1979. Sky and Tel. 57, 223.45. Nye, R. A. 1981. Sky and Tel. 62, 496.

CHAPTER 7PULSE-COUNTING ELECTRONICS

Pulse-counting systems are rapidly becoming comparable in expense toany other method of photoelectric photometry. A typical but very gen-eral layout of such a system is shown in Figure 7.1. The output fromthe photomultiplier is fed to the preamp, which amplifies the pulse,shapes it, and rejects noise pulses. This conditioned pulse is the input tothe pulse counter, also known as a. frequency counter. The pulse counterconsists of three major parts: a counting circuit that counts every inputpulse, a gate that allows pulses to reach the counter only for a specifiedtime interval, and the timing circuit controlling gate.

The counts can be read directly from the counter or sent to a smallcomputer through an interface. There the counts may be transformedto a crude magnitude scale. The data can be printed out on a teletypeor displayed visually. It may be transferred to magnetic tape or disk toawait further data reduction or for permanent storage.

Preamps and pulse counters are described in this chapter, along withsome representative circuits. Some interfacing and testing proceduresfollow.

It should be emphasized that pulse counting cannot be performedwith PIN diode photometers. Pulse counting requires hundreds of thou-sands of electrons for each incident photon. An incident photon pro-duces only one electron-hole pair in a photodiode. Thus, with presentphotodiode technology, you must use DC methods.

7.1 PULSE AMPLIFIERS AND DISCRIMINATORS

The pulse amplifier increases the size and shapes the feeble pulse fromthe photomultiplier tube. The discriminator rejects pulses that are

167


PHOTOMULTIPLIERTUBE

PREAMP PULSECOUNTER

INTERFACE COMPUTER

VISUALRECORDING

Figure 7.1. Block diagram of pulse counting system.

inherent to the photomultiplier tube itself and not from the source. Theelectronics that accomplish these two purposes is often in a single pack-age, commonly called a preamp.

The amplification is necessary because each pulse contains on theorder of a million (106) electrons, a current of only 10~12 A if averagedover 1 second. Most frequency counters require inputs of 100 mV (0.1V) to count correctly. Therefore, using Ohm's law, we would have touse a series resistor of 10" ohms to yield adequate counting voltage.This is a very difficult value to obtain.

If you look at the output of a typical photomultiplier tube at hightime resolution, you would see something similar to Figure 7.2. Eachpulse represents the output from a photon event, and the signal betweenpulses is the background or dark current of the tube. If you counted allof the pulses of a certain size occurring in a fixed time interval and

TYPICAL ONEPHOTOELECTRON(P.EJ PULSE HEIGHT

TIME

Figure 7.2. Typical photomultiplier output.

PULSE-COUNTING ELECTRONICS 169

3P.E. PEAK

Figure 7.3. Pulse height distribution.

plotted your results, you would obtain a pulse height distribution. Atheoretical pulse height distribution is shown in Figure 7.3. This showsthat the number of background noise pulses decrease rapidly with theenergy of the pulse. There are several peaks in the distribution, corre-sponding to the ejection of one or more electrons from the photocathodeby the photon. Most events ejecting more than one electron are causedby cosmic rays and are few in number.

We do not want to amplify the noise pulses and count them alongwith photon events. Instead, we want to discriminate against them. Thisis usually achieved by setting a minimum threshold level below whichno output pulse results. You can never eliminate all of the noise pulsesbecause some arise on the photocathode itself and look like photonevents, but by setting the threshold near the minimum between thenoise and the one-photoelectron distribution, you will reject the maxi-mum noise and accept the maximum signal. To be highly accurate, youwould also reject the two and higher photoelectron pulses, because theyare caused primarily by cosmic rays, and create a window discrimina-tor. However, this trade-off is unnecessary because only a few pulseswould be rejected with a large increase in circuit complexity.

A good pulse amplifier and discriminator should:

1. Have an output pulse no more than 50 nanoseconds wide, therebyproviding a counting rate of about 20 MHz.


2. Have minimal temperature sensitivity.3. Have stability and high noise immunity.4. Be small, simple, and require only one operating voltage.5. Be able to amplify a 0.5-mV pulse and provide a TTL-compatible

output.

There are readily available commercial preamps. These includemodels from Princeton Applied Research,1 Hamamatsu,2 and Productsfor Research.3 However, be prepared to spend several hundred dollarsfor one of these commercial devices. Amptek4 has recently introducedhybrid charge-sensitive preamps that are the size of a dime. These couldbe mounted on the tube base and provide a very compact package.However, the current models are not sensitive enough for most astro-nomical applications. DuPuy5 has recently published a circuit based onthe MVL 100 single-chip amplifier. One limitation of this chip is thatit may not have enough gain for some photomultiplier tubes.

7.2 A PRACTICAL PULSE AMPLIFIER AND DISCRIMINATOR

A preamp circuit that has been used by many observatories wasdescribed in 1973 by Taylor,6 who recently revised the original circuit.7

This enhanced preamp is presented in this section. The Taylor preampwas designed with simplicity and low cost in mind. The circuit is shownin Figure 7.4. It consists of nine 2N4124 transistors ($0.30 each) anda 1N3717 tunnel diode (about $10).

The first six transistors comprise the amplifier section and are con-nected as shunt-series feedback pairs, cascaded for an overall gain ofabout 1000. The tunnel diode monostable oscillator acts as the discrim-inator. When triggered, it generates a standard —0.5V pulse, which isbuffered by an emitter follower, amplified and inverted, and fed to asecond emitter follower to drive a 50-ohm cable to + 5V. The shape ofthe output pulse is similar to the positive half of a sine wave with a basewidth of 20 nanoseconds. The discriminator level is adjusted by meansof a current bias potentiometer. An 1C regulator is included in the cir-cuit to improve stability and eliminate the need for a separate zenerregulator for the tunnel diode.

The circuit should be constructed on a single board. Point-to-pointwiring is recommended, using a double-sided printed-circuit board asthe chassis. Layout of the parts is not critical and no shielding between

PREAMPLIFIER DISCRIMINATOR OUTPUT AMPLIFIER FAIRCHILD7815 +18V

TO

PULSE OUTPUT TO 50OHM COAX & LOAD

ALLTRANSISTORS2N4124

GROUND TO CASE HERE FOR STABILITY

Figure 7.4. Schematic diagram of the improved pulse and amplifier circuit. All transistors are 2N4124's, L, is 1 pH (30 turns on a 1 Meg % wattresistor as a coil form). (Courtesy of the Publications of the Astronomical Society of the Pacific.)


stages is necessary, but a layout resembling the circuit diagram is sug-gested. After construction, the board should be mounted in a metal boxfor shielding. This can be a commercial box such as Pomona Electron-ics8 model 3302 or constructed out of double-sided printed-circuit boardmaterial. The board should be grounded to the case in several places toprevent pulse doubling. BNC-type connectors should be used for theinput and output.

The Taylor preamp is somewhat temperature-sensitive, losing sensi-tivity as the temperature decreases. The 1973 version had up to a 20percent variation in the count rate with a 40 Celsius degree change. Forthis reason, obtaining accurate measurements requires thermostattingthe circuit. Problems with temperature sensitivity can be minimized byusing differential photometry and/or never observing near sunset whentemperature variations are at their maximum.

With the pulse resolution of this preamp, dead-time corrections startto become important at around 100,000 counts per second. Use themethods discussed in Chapter 4 to correct for this error.

7.3 PULSE COUNTERS

A frequency or pulse counter for astronomical purposes has certainrequirements:

1. Counting ability to 100 MHz or higher.2. Selectable time base, with at least 1- and 10-second gating times

for manual use, and 0.001- and 0.01-second gating for occultationobservations.

3. Capability of external gate triggering and counter reset.4. BCD or binary output for computer interfacing.

The last two items may not be necessary immediately when the counterpurchase is contemplated, but should be considered for future applica-tions. A rule of thumb is to be able to count 30 times faster than themost rapid anticipated nonuniform rate; 100 MHz gives a large marginof error in most cases. For astronomical purposes, a counting accuracyof 0.1 percent is entirely adequate. This is a condition met by all com-mercial frequency counters.

Examples of adequate commercial frequency counters are the Opto-electronics9 7010, Heathkit10 SM-2420, and Hal-Tronix" HAL-600A;


all are 600-MHz counters using the newly released ICM 7216 fre-quency counter 1C. Introduced by Intersil,12 the 7216 has enormouspotential for astronomical use because of its simplicity and low cost(under $25). The only additional major parts required for a 100-MHzcounter with a selectable timebase are a 10-MHz crystal, a divide-by-ten prescaler (11C90 or 95H90), and a LED display of up to eight dig-its. Its only disadvantage is the very complicated interface for computerapplications. Still, the ICM 7216 has allowed manufacturers to supplyadequate frequency counters for manual use in the $100 to $200 pricerange.

If you want to build your own counter from scratch, consider theICM 7216 and consult the data sheets supplied by Intersil. Other ICsare available to perform the major functions, such as a six-decadecounter. The use of discrete ICs in building a counter makes for a cum-bersome design, but has the advantage of easy computer interfacing asall signals are present continuously.

Normal quoted accuracies of the time bases for frequency countersare around 10ppm/C°.This means that a change of 10 Celsius degreescauses a 0.01 percent change in the gating time, which is insignificantfor overnight use. This error could become important if a less accuratetime base were used. In addition, most commercial grade ICs quit work-ing at 0"C (32"F), and the frequency response of both the pulse shap-ing input and the counter degrade as the temperature approaches zero.In other words, for best results the counter should be thermostatted toa constant temperature in summer and winter, just like the preamp. Ifyou must operate without thermostatting, use the military (5400 series)instead of the commercial grade (7400 series) ICs.

7.4 A GENERAL-PURPOSE PULSE COUNTER

In this section, we describe a general-purpose frequency counter con-structed from discrete integrated circuits. It contains 19 ICs and wouldcost around $ 100 to construct. Though more complicated than a counterusing the ICM 7216 frequency counter 1C, its main advantage is easeof computer control. All outputs are latched and can be brought out toa connector for computer input, and time base select and reset functionsare simple to interface for computer control.

The counter is shown in Figures 7,5 through 7.7. The maximumcount rate is 100 MHz, controlled by the 74SOO gate and the 74S196

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decade counter. If lower count rates are acceptable, the 74S196 can bereplaced by a 74196 or another decade chip with some minor rewiring.

The input section amplifies the signal using a field-effect transistor(FET) and then conditions it into a square wave using device U18. Thissection can be eliminated if the counter's only use is for photometry,where there is a separate preamp in which the output pulse drives thegating circuitry directly. The pulse is routed through a timing gate andon to a series of decade counters. The 74143 is a combination counter,latch, and decoder-driver and would be used for all stages except thatit has a low counting rate, 18 MHz. The 74S196 is negative-edge trig-gered, and its output must be inverted to drive the positive-edge trig-gered 74143. All LED displays are common-anode MAN-1 equivalents.

A CMOS 4060 1C is used as the oscillator. It divides the 1.2288-MHz crystal frequency by 210 (1024) to provide a 1200-Hz square-waveoutput. This is further divided by the dual-decade 4518 counters, andfour frequencies (1200, 120, 12, and 1.2 Hz) are fed to a 4051 demul-tiplexer. The desired frequency is selected by a DP4T rotary switch androuted to a divide-by-12 circuit. This opens the gate for 10 pulses,latches for one, and resets for one. Therefore, the final output gatingtimes are 0.01, 0.1, 1, and 10 seconds, with 0.002, 0.02, 0.2, and 2 sec-onds, respectively, of dead-time between subsequent gatings. Note thatthe 4051 can accept up to 8 inputs, so that the second output stage ofU3 (100 seconds) and the 27 output of the 4060 (1.2 milliseconds) couldalso be included in the gating selection with a larger switch.

The power supply is conventional with a full-wave bridge and an 1Cregulator. The bridge could be replaced by a single unit instead of fourindividual diodes. Be sure that the + 5 V line on each board is bypassedby a 10-jiF tantalum capacitor and that each counter 1C is bypassedindividually by O-Ol-^F disc capacitors for noise immunity.

Our version of the counter was constructed on four boards: preamp,time base (and 74S196/74LS75), display, and power supply. Wire-wrap techniques were used and the final product was placed in a RadioShack 270-270 cabinet.

To interface this counter to a computer, bring the 24 latched BCDlines to a back 25-pin connector along with a ground lead. The latchsignal should also be available to the computer as you should not readthe data while latching occurs. The computer should control the timebase selection (add a DPDT toggle switch to change from manual toautomatic time base select) and a reset signal to start counting (use oneof the unused NAND gates in a similar manner to U7C).


7.5 A MICROPROCESSOR PULSE COUNTER

A high-speed pulse-counting board has been designed and built byKephart.13 This versatile board for the S-100 microcomputer bus satis-fies the need for high time resolution (1 millisecond) for lunar occulta-tion work and for moderate time resolution (0.1 second to several min-utes) for multichannel applications.

Because of the complexity of the board and the fact that it is designedaround a specific computer system (an 8080 with the S-100 bus), we donot give a complete schematic. Rather, Figure 7.8 shows a block dia-gram of the basic circuit in sufficient detail so that its logic can beimplemented in other designs.

Two 21-bit counters are used as data counters. They each use one-half of a high-speed 74S112 J-K flip-flop for the least significant bit(LSB), with the pulse input routed to the J input and a control selectsignal to the K input. The remaining 20 bits are obtained from a 74197and two 74393 binary counters. A third 16-bit counter using two 74393ICs is used for interval timing.

For the millisecond time resolution application, the two data countersare used in a double-buffer mode to reduce the dead-time to a few nano-seconds of gate propagation time. One counter is disabled, read, andcleared while the other counter is acquiring data; at the end of thecounting period the roles of the two counters are reversed. Dual-channelcounting is performed by disabling, reading, and clearing both counterssimultaneously, resulting in a dead-time of a few tens of microsecondsper readout.

Communication with the microcomputer is through two parallel I/Oports (implemented on the pulse-counting board with Intel 8212 chips)and an interrupt instruction latch. One of the output ports from themicroprocessor is used as a counting control latch. The other outputport is used to set a comparison latch (with 7485 comparators) for thethird onboard interval timer counter. An input port to the microcom-puter is used to transfer any selected eight-bit byte from either 21-bitdata counter to the CPU. The multiplexing is accomplished with 8T97tri-state buffers. The CPU can be interrupted by the board, indicatingto the CPU that service to the board is required.

The counting control latch byte from the CPU to the board is usedto select functions of the pulse-counting board. From information writ-ten into the counting control latch, either interrupts or pulse counters

8 BITSFROM CPU

8 BITSFROM CPU

INTREQ

INTALOW

8 BITSTO CPU

Figure 7.8. Block diagram of photon counting board. Published in the Proceedings of The Society of Photo-Optical InstrumentationEngineers, Volume 172, Instrumentation in Astronomy III, Bellingham, Washington.


or both can be disabled, counter selection (determining which counteris in the read mode) can be made, complete board reset or individualcounter resets can be performed, byte selection between the MSB andLSB of the interval comparison latch, and byte selection for reading the21-bit counters can be made.

The interval counter can be used to count either an external clockpulse or a signal from the S-100 bus. By setting the desired intervalthrough software into the interval counter comparison latch, time res-olution can be controlled by the user. The board creates an interruptwhen the interval counter reaches the number held in the intervalcounter latch. This counter has 16-bit resolution allowing up to 65,535clock pulses to be counted per interval. If a millisecond clock is used,then intervals from 0.001 to 65.535 seconds can be selected. This designallows the user to select the desired time resolution using interrupt con-trol. The CPU can be used to display real-time data, reduce a previousobservation, or perform any other desired task until an interrupt isissued from the pulse counting board, at which time the board is ser-viced and the CPU then returns to its previous task.

Figure 7.9. Microprocessor pulse counter.


Construction is straightforward, using wire-wrap techniques on aVector prototype design board. As constructed, the board would costabout $125. Figure 7.9 shows the finished pulse counter. A second-gen-eration counter would use the Intel 8255 triple-port I/O chips todecrease chip density and power consumption. Such a microprocessor-controlled pulse-counting board demonstrates the versatility that can beachieved with a computer-pulse counting marriage.

7.6 PULSE GENERATORS

A useful piece of test equipment for the photometrist is the pulse gen-erator. It produces pulses of known frequency, height, and duration thatcan be used to test frequency counters and preamps.

An example of a commercial pulse generator is the Continental Spe-cialties Corporation14 model 4001 (under $200). It has a frequencyrange of 0.5 Hz to 5 MHz, with pulses 100 mV to 10 V high and 100nanoseconds to 1 second wide. Other generators are available fromlarge manufacturers like Hewlett-Packard with tighter specificationsand more ranges.

However, for testing photomultiplier preamps, pulses of —5.0 mVare desirable. A simple pulser satisfying this need is shown in Figure7.10. Constructed at the Indiana University electronic shop, this pulserputs out a 5.0-mV, 500-nanosecond negative pulse to a 50-ohm load.This is a simple, inexpensive way to test a photomultiplier preamp. The

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transistors Ql and Q2 generate a ramp, with R2 and C2 controlling theramp frequency. The R4/C2 pair provide the decay time of the rampand Dl shapes the pulse. A voltage divider is formed by R5/R6, pro-viding the 0.5-mV output pulse.

7.7 SETUP AND OPERATION

Pulse-counting systems are very sensitive to stray capacitances andnoise. Stray impulses caused by heaters, motors, and relays turning onand off, along with other sources, are counted by the pulse counter andpreamp just as if they came from the photomultiplier tube itself. Toprevent this interference, connect the preamp solidly to the photomul-tiplier tube assembly. This keeps the interconnecting cable as short aspossible and creates a common ground plane. Bypass all power leadswith LC circuits to route all high-frequency interference to ground.

RG58 coxial cable should be connected between the tube and thepreamp, and again between the preamp and the pulse counter. RG58coax matches the input and output impedances of the preamp properly,and if connected to the 50-ohm input of a pulse counter will presentproper termination to the preamp. If the coax is not terminated with 50ohms, there will be a mismatch, and the fast pulses will be reflectedback and forth giving rise to "ringing," where the pulse counter willcount several pulses for each actual pulse from the preamp.

The discriminator level adjustment is one of trial and error. For anuncooled 1P21 tube, the final count rate for dark current should beabout 200 counts per second. For a dry ice cooled tube, adjust the dis-criminator to allow about one or two counts per second. Taylor suggeststhat his preamp can be adjusted by attaching the preamp to a counter(but with no photomultiplier attached) and increasing the discriminatorbias by the potentiometer until the discriminator oscillates. Then backoff on the bias until first the oscillation and then the stray counts fromamplifier-noise peaks cease. You still have to make final adjustments atthe telescope to get optimum noise discrimination.

Setting the high voltage for an optimum signal-to-noise ratio is easierthan in the case of DC amplifiers. First expose the photomultiplier tubeto a constant light source (starlight for example) and then increase thevoltage in steps of about 100 V. The observed count rate increases rap-idly until a plateau is reached at which the count rate from the sourceincreases only slightly with increased voltage. Further increases in volt-


age serve no useful purpose; generally the dark current rises with nosignificant corresponding increases in signal count. For the 1P21 pho-tometers in use at Indiana University, we have found that a high voltagearound —900 to —950 V is optimum. To avoid dead-time effects duringthe voltage increase, pick a source that should eventually yield about100,000 counts per second.

REFERENCES

1. EG & G Princeton Applied Research, P. O. Box 2565, Princeton, NJ 08540.2. Hamamatsu Corp. 420 South Ave., Middlesex, NJ 08846.3. Products for Research, Inc., 88 Holten St., Danvers, MA 01923.4. Amptek, Inc., 6 De Angelo Dr., Bedford, MA 01730.5. DuPuy, D. L. 1981. Pub AS.P., 93, 144.6. Taylor, D. J. 1972. Pub. A. S. P. 84, 379.7. Taylor, D. J. 1980. Pub. A. S. P. 92, 108.8. ITT Pomona Electronics, 1500 E. Ninth St., Pomona, CA 91766.9. Optoelectronics Inc., 5821 N.E. 14th Ave. Ft. Lauderdale, FL 33334.

10. Heath Co., Benton Harbor, MI 49022.11. Hal-Tronix, P.O. Box 1101, Southgate, MI 48195.12. Intersil Inc., 10710 N. Tantau Ave., Cupertino, CA 95014.13. Honeycutt, R. K., Kephart, J. E., and Henden, A. A., 1979. In Instrumentation

in Astronomy III. Edited by D. L. Crawford. Society of Photo-Optical Instru-mentation Engineers Proceedings, 172, 408.

14. Continental Specialties Corporation, P. O. Box 1942, New Haven, CT 06509,

CHAPTER 8DC ELECTRONICS

The photomultiplier tube is a high-gain current amplifier. For each elec-tron released at the photocathode by a detected photon, about one mil-lion electrons are collected at the anode. A stream of incident photonsgenerates a series of closely spaced bursts of current at the anode. Inthe pulse-counting mode, the goal is to count these bursts over a selectedtime interval. In the DC technique, the current is not resolved intobursts of current, but instead is averaged to give a continuous current.Despite the large amplification of the tube, the output current isextremely small and requires further amplification so that it can bemeasured easily. This current amplifier must be extremely linearbecause the photomultiplier's output current is directly proportional tothe incident light flux. The 1P21 photomultiplier has a typical dark cur-rent of 10~9 A at room temperature. The amplifier should be capableof raising this to an easily measurable value, of about 1 mA (10~3 A).Thus, the amplifier should have a current gain of 106. This is easilyachieved with some very simple electronics. On the other hand, the pho-todiode has an internal gain of unity and therefore requires an amplifierof much higher gain. This presents some special amplifier design prob-lems, as discussed by Persha.1 To date, very little has been publishedabout amplifier designs for photodiodes used as astronomical detectors.For this reason, we restrict this chapter to an amplifier designed for usewith a photomultiplier tube.

At this point, a very brief review of operational amplifiers (op amps)is needed. A more complete and lucid discussion of this topic can befound in the book by Melen and Garland.2 The following discussionassumes a background in elementary electronics.

184

DC ELECTRONICS 186

8.1 OPERATIONAL AMPLIFIERS

A few years ago, an operational amplifier was large, costly, fragile, andhad a rather large power consumption. A modern op amp can be fab-ricated on a tiny silicon chip at a cost of a few dollars. Each chip maycontain the equivalent of dozens of transistors, resistors, and capacitors.The details of the internal operation are not necessary for the presentdiscussion. Figure 8.1 shows the symbol for an op amp. The op amp isa high-gain voltage amplifier. The " " terminal is called the invertinginput and the "+" terminal is called the noninverting input. An increas-ing voltage applied to the inverting input results in a decreasing voltageat the output (£"„„,). The same voltage applied to the noninverting inputresults in an increasing voltage at the output. If the same signal voltagewas applied simultaneously to both inputs, the two amplified signalswould be 180° out of phase and would cancel each other completely.The output is the amplified voltage difference between the two inputs.The output is unaffected by voltage changes that occur at both inputs;only the difference is amplified. It is sufficient for most of this discussionto consider the op amp to have "ideal" characteristics, namely infiniteinput impedance, infinite voltage gain, and zero output impedance. Fig-ure 8.2 shows the op amp used as an inverting voltage amplifier. Theopen-loop gain, A, is defined as

A =E.

where £,„ and Eou, are the input and output voltages, respectively. Foran ideal op amp, A is infinite. For a practical op amp, A is 104 to 106.Real input impedances are typically 100 kli but the input impedance ofop amps utilizing FETs can reach 1012 ohms or more. A typical outputimpedance is 50 ohms.

Figure 8 .1 . Op amp symbol.


Figure 8.2. Voitage amplifier.

With the background developed above, it is now possible to discusshow the op amp is used as a current amplifier for DC photometry. Fig-ure 8.3 shows a simplified current amplifier circuit. This particular typeof circuit is not recommended. It is illustrated here as an example of acircuit design to be avoided. This type of circuit has been used in astro-nomical photometry in the past without a general appreciation of itsinherent inaccuracy. An example of this type of circuit can be found inWood.3 We do not discuss the operation of this circuit except to pointout the source of the problem. An input current from the photomulti-plier tube, /,, flows through RL to ground instead of entering the higher-impedance amplifier. This elevates the potential at point B to IfRL. Thismeans that the potential difference between the anode of the photo-multiplier and ground has been changed slightly. The amount of changedepends on /,, which in turn depends on the brightness of the star.Young4 has shown that this can result in a nonlinear tube response ofa few tenths of a percent. This is a small error, but it need not be tol-erated because there is a very simple solution. Circuits that use ananode load resistor should be avoided. The anode should always seeground potential directly.

Figure 8.4 illustrates the necessary circuit. In this type of circuit pointa, the input seen by the anode, is always very nearly at ground potential.To see this, suppose a small positive external voltage, E,-, is applied at

Figure 8.3. Current amplifier (not recommended).

DC ELECTRONICS 187

Figure 8.4. Current amplifier utilizing virtual ground.

the inverting input. This results in a negative output voltage, E0. Theoutput is connected, via the feedback loop, to the inverting input. Thetotal input voltage at point a, Ea, is then

E = (8.1)

The output voltage, Em is related to the input, Em and the voltage gain,A, by

E0 = ~AEa.

The minus sign results from the use of the inverting input. Combiningthe above two equations yields

By Equation 8.1,

£, = EJil + A).

0 = Ea- E,-».- EJil +A).

Thus the input and output voltage are related by

£, Ea- Ea(\ + A) _ ~AE, ~ Ea(\ + A) 1 + A'

As long as A is a large number, then

(8.2)


Combining Equations 8. 1 and 8.2, we obtain

Point a is said to be at virtual ground because it is essentially at groundpotential, but current does not flow to ground at this point. The anodeof the photomultiplier always sees ground potential when connected tothis circuit and the tube linearity is not affected.

Now consider specifically how this circuit is used as a current ampli-fier. An input current from the photomultiplier flows through resistorsRf and R0 to ground. Only a negligible amount of current enters theamplifier because of its very high input impedance. Because R0 isalways made very small compared to Rf, point a would seem to be ata potential of 7,^ with respect to ground. This cannot happen, by ourdiscussion above. By Equation 8.2, the voltage at point b must be— IjRf. This potential results in current flowing between the amplifieroutput, resistor R0, and ground. This output current, /0, must be relatedto the input current by

or

/ „ = - / i - - (8.3)

There is a linear amplification value that depends on the ratio of thesetwo resistors and not on the characteristics of the op amp itself. Whilethis is strictly true only for an ideal amplifier, it does show that a circuitthat is very insensitive to changes within the op amp can be built. Itwould be very difficult to find an op amp with amplification stableenough for photometry, especially with the large temperature changesfound in an observatory, if the feedback loop were not utilized. Even so,it is advisable to purchase a high-quality op amp to insure stability.

8.2 AN OP-AMP DC AMPLIFIER

Figure 8.5 shows a practical DC amplifier circuit. The op amp used isa Sylvania ECG 940 which has an FET input with an impedance of

DC ELECTRONICS 189

S1:

RESISTORS

1 MEG 1%10 MEG 1%100 MEG 1%

S2:

RESISTORS (ALL1%OR BETTER}

100012630.9 n398.1 SI251.2 £J158.5 J2100.0 £2

LABEL

2.5 MAG5.0 MAG7.5 MAG

J.ABEL

0.0 MAG0.5 MAG1.0 MAG1.5 MAG2.0 MAG2.5 MAG

O +12V

-1 2V

Figure 8.5. DC amplifier circuit.

about 1012 ohms and an open-loop gain of 106. This amplifier has a spec-ified operating range from 0° to 70°C (32° to 158°F), which meansthat the amplifier must be kept warm in the winter. This is not a bigdisadvantage because it is always a good idea to operate the electronicsat a constant temperature to avoid drift. Other op amps with a widertemperature range, such as the Analog Devices AD523K can be used.This latter device has a range from —55° to + 125°C, but costs twiceas much.

The resistors Rf and R0 have been replaced by switches that allowvarious combinations of resistors, and hence, various combinations of


current gain to be selected. To make the gain large, the resistors onswitch SI (Rf) must be made large and those on switch S2 (R0) rathersmall. The 1-megohm resistor of switch SI is used for the brightest starsand the 100-megohm resistor is used for the faintest stars. Asdiscussed in Section 6.2, the 1P21 shows fatigue effects if the tube cur-rent exceeds 10~6 A. Because our amplifier does not give a direct read-out of the tube current, it would be advantageous to have some safetymechanism to let us know when this level is reached. The simplestapproach is to design the amplifier so that a tube current of 10"6 Ayields a full-scale deflection on the amplifier meter when the lowest gainsetting is used. Because a full-scale reading on the meter is 10~3 A, thelowest current gain should be 103. This sets the value of the largest R0

resistor at 1000 ohms by Equation 8.3, since the lowest Rfis 106. Theremaining resistors in switch S2 decrease in 0.5 magnitude steps; thatis each is smaller than its predecessor by a factor of 0.6310. The resis-tors in switch Si change by a factor of 10, yielding 2.5 magnitude steps.With the highest current gain (Rf = 10s, R0 = 102), an input currentof 10~9 A, which is the dark current of the 1P21 at room temperature,gives a full-scale deflection.

If you intend to use this amplifier with a cooled photomultiplier tube,one or two more resistors should be added to switch SI with values of1000 and 10,000 megohms, respectively. These resistors are required totake advantage of the reduction (by a factor of 100) in dark current,which allows much fainter stars, and hence, much lower currents to bemeasured. Unlike the idealized case, an actual op amp draws a smallamount of input current during operation. This is referred to as theinput bias current. This current itself can be a noise source just like thedark current from the photomultiplier. The op amp used in this circuithas an input bias current of 10~'° A, which is 10 times less than thedark current from an uncooled 1P21 and is therefore negligible. If youuse a cooled tube, however, the bias current will become the dominantnoise source when measuring faint stars. If you plan to use a cooledtube, the op amp in Figure 8.5 should be replaced by one with an inputbias current of 10~12 A or less. Such op amps are available but are moreexpensive.

The feedback resistors have large values to achieve high amplifiergain and to minimize noise. Thermal (or Johnson) noise in these resis-tors varies with the square root of the resistance. Large values of Rf

make this noise small compared to the current to be measured. The

DC ELECTRONICS 191

feedback resistors, Rfy should be accurate to 1 percent or better. Vic-toreen Instrument Company5 can supply high-megohm resistors in glassencapsulation for the required accuracy. The R0 resistors have muchlower values of resistance, which makes them inherently more stable,and they need not be glass-encapsulated. However, they should beaccurate to 1 percent or better, because precision resistors are more sta-ble than ordinary carbon resistors. It is impossible to find 1 percentresistors that equal the listed values exactly, so the values should bematched within 10 percent. A perfect match is not necessary becausethe amplifier is calibrated after construction.

The 10 k!2 potentiometer in Figure 8.5 is used to balance the circuitso that a zero input current produces a zero output current. An ordinaryone-turn pot can be used but a five-turn pot makes circuit balancingmuch easier, especially at the high-gain settings. The purpose of selectorswitch Cl is to add a time constant that helps to smooth variationsresulting from atmospheric scintillation and tube noise. A disadvantageof this circuit is that the time constant varies with the SI switch posi-tion. The time constant, which is the product of /?7and Cl, is negligibleat the lowest gain setting and becomes significant only on the highestsetting. Table 8.1 lists the time constant for each combination of /?randCl. The capacitor values in this circuit can be changed to obtain othertime constants. The variation in time constant with Rf is not a seriousproblem because larger time constants are preferred when higher gainsettings are in use. It would be more convenient to be able to use thesame time constant for all gain settings. The circuit designed by Oliver6

avoids this problem by using a second op amp, of unity gain, connectedto the output of the first op amp. This second amplifier drives the meterand any external recorder. It also has its own adjustable RC time con-stant in its feedback loop, which is independent of Rf in the firstamplifier.

Selector switch Cl is wired so that each capacitor is shorted when

TABLE 8.1. Amplifier TimeConstants

0.01 jiF 0.02 nF 0.03

1 Meg 0,01 sec 0.02 sec 0.03 sec10 Meg 0.1 0.2 0.3

100 Meg 1.0 2.0 3.0


Figure 8.6. The DC amplifier, front view (top), and rear view (bottom).

not in use. This prevents some unwanted voltage spike from appearingat the amplifier input when a new capacitor is switched into the circuit.The 100-ohm resistor in the output circuit produces a voltage drop foran external chart recorder. An output of 1 mA produces a 100-mVdrop, which produces a full-scale reading on a 100-mV chart recorder.The value of this resistor may be changed for recorders with differentfull-scale sensitivity.

The power supply circuit is taken from Stokes.7 He found this simple

DC ELECTRONICS 193

zener-regulated design adequate for an earlier DC amplifier design.Both the amplifier and power supply can be built into one small chassis.Figure 8.6 shows a photograph of the completed unit, which is smallenough to be mounted directly on the telescope if so desired.

There are two important comments to be made about the operationof this amplifier. First, FET devices, such as the op amp in this circuit,are damaged easily by an electrostatic charge at the input. As a pre-caution, always turn the amplifier on before connecting the signal cable.This prevents damage from any charge accumulated by the cable. Also,be careful when handling the op amp in a dry-air environment. Always"discharge" yourself by touching a ground, such as a water pipe, beforehandling the op amp. The second comment concerns zero-point drift.During the first 20 minutes of operation, there is substantial drift on thehigh-gain settings. It is therefore a good idea to turn on the electronicsat least one-half hour before observing.

The circuit presented here is inexpensive and very simple to build. Amore advanced circuit has been designed by Oliver.6 As already men-tioned, this circuit handles the time constant problem nicely. It hassome other valuable features such as an internal constant current sourcefor calibration and sky background cancellation. Although these extrafeatures increase the cost, this circuit should be seriously considered bythe advanced observer.

8.3 CHART RECORDERS AND METERS

Once your amplifier, high-voltage supply, and photometer head arebuilt, you are ready to begin making some measurements. The questionthen arises as to the method of recording the data. The simplest tech-nique is to read the amplifier meter and record the measurement withpencil and paper. To achieve good photometric accuracy, you must beable to read the meter to an accuracy of at least 1 percent. This isobviously not possible with the tiny edge meter seen in Figure 8.6. Thismeter was intended to be used only to monitor the functioning of theamplifier. If you plan to "meter read," you must invest in a larger meterof good quality. The minimum size should be 7.5 centimeters (3 inches)or preferably larger. The meter should have a quoted accuracy of 1 per-cent or better at full scale. Meters with a mirrored scale are preferredbecause they minimize parallax. The meter can be mounted in theamplifier chassis or in its own separate chassis connected to the ampli-


fier with a cable. The later arrangement is convenient if you plan tomount the amplifier on the telescope. If you plan to take measurementswith a meter, invest in a good one.

The disadvantages to meter reading are obvious. Atmospheric scin-tillation causes the needle position to jitter, making estimates difficult.The problem is complicated by observer fatigue, especially after 3 a.m.!Above all, there is no permanent record of the actual meter output. Ifthe meter reading is written incorrectly, the observation is lost forever.Any DC observer who can afford it quickly invests in a strip-chartrecorder, A strip-chart recorder consists of a device that drives a longroll of chart paper under a pen whose transverse displacement acrossthe width of the paper is proportional to the input voltage. This devicereduces the amount of effort required at the telescope. The observerneeds only to write comments on the paper occasionally such as theamplifier gain or the time. Because the chart paper advances at a con-stant rate, the time of any observation can be found by interpolation.The chart record can be studied by an alert observer the following day.A permanent record exists that can be checked if an observation failsto reduce properly. Figure 9.9 illustrates the appearance of a chartrecording for a series of observations.

There are a number of characteristics you should look for in a chartrecorder to be used for photometry. There are a number of small andinexpensive models on the market. Unfortunately, these units must usenarrow paper. This limits the accuracy to which the pen tracing can beread. It is best to buy a recorder that uses chart paper at least 25 cen-timeters (10 inches) wide. The recorder should be a DC voltmeter typewith a range from zero to a few hundred millivolts. The latter value isnot critical because the resistor in the output circuit of Figure 8.5 canbe changed. A 100-mV input is used commonly by chart recorder man-ufacturers. The accuracy of the recorder at full scale should be at leastone percent. Recorders with 0.5 percent accuracy are readily available.Finally, the chart speed must be considered. Most recorders have anadjustable speed. Experience has shown that a speed of 1 to 5 centi-meters per minute is a good choice for most kinds of photometry. Besure that the recorder you consider has a speed in this range.

There are many companies that manufacture chart recorders to meetthe above criteria. Examples are the Markson Science Incorporated8

model 5740, the Cole-Parmer Instrument9 models C-8386-32 and C-8373-00, the Hewlett-Packard10 model 7131 A, and the Heath" model

DC ELECTRONICS 195

IR-18M. This is just a very short list and the inclusion or omission ofa company's name does not constitute an endorsement or criticism oftheir products. This list is merely a starting place for the would-be chartrecorder owner. Unfortunately, all of these recorders, with the exceptionof the Heath IR-18M, cost over $700. This price is certainly beyond alimited budget. Consequently, most amateur astronomers have turnedto the less expensive $230 Heath recorder. This recorder is not so strongmechanically as the other recorders but it does meet all of the selectioncriteria listed above. Experience has shown that this recorder, as mostothers, does not function well in the cold, winterlike environment of anobservatory. A small heated enclosure solves the problem nicely. If youplan to do a lot of DC photometry, a strip-chart recorder is a veryworthwhile investment.

8.4 VOLTAGE-TO-FREQUENCY CONVERTERS

One of the advantages of pulse counting over DC is the digital output,which frees the observer from making amplifier gain adjustments andcalibrations. There is another approach to DC photometry that hasthese same attributes. The traditional DC amplifier is replaced by avoltage-to-frequency converter (VFC) circuit. The basic idea is to con-vert the current output of the photomultiplier to voltage which can serveas the input of a voltage-controlled oscillator. This oscillator has a fre-quency that varies linearly with the input voltage. The output of thisoscillator is fed to a frequency counter in exactly the same way the out-put of a pulse amplifier is when pulse counting (see Sections 7.3 and7.4).

The VFC design of Dunham and Elliot12 is shown in Figure 8.7. Thecurrent from the photomultiplier tube is converted to a voltage by thefirst op amp. If the signal is weak, a second op amp is switched into thecircuit for additional amplification. This voltage is applied to the inputof a voltage-controlled oscillator. This single-chip device produces anoutput frequency of 10s Hz per volt at the input. The frequency counteris not shown in Figure 8.7. Before constructing this circuit, you shouldconsult Dunham and Elliot for valuable commentary. The purpose forshowing this circuit is to emphasize its simplicity. It is also possible touse a voltage-controlled oscillator and frequency counter with a conven-tional DC amplifier. This is certainly better than meter reading andmay be less expensive than a chart recorder.


INPUT FROMPHOTOMULTIPLIEFtANODE

rFigure 8.7. Voltage to frequency converter circuit diagram. The current to voltage converter isat left followed by the optional gain of 10.3 amplifier. The 470501 is the voltage controlled oscil-lator and the NOR gates on the right are line drivers. The filters at the bottom are lowpass powersupply filters. Courtesy of the Publications of the Astronomical Society of the Pacific.

The VFC approach has some definite advantages over the conven-tional DC amplifier. There are fewer gain switches to adjust, which isespecially valuable when light levels change very rapidly as occurs dur-ing occultation photometry. This also does away with the need for fre-quent amplifier gain calibration. Unlike pulse counting, there are nodead-time corrections to be made, and the digital output does away withthe tedium of reading chart recorder tracings. However, unless you havesome recording device such as a minicomputer with a disk drive and/ora printer, there is no permanent record of an observation. One approachthat has been used by McGraw et al.13 is to record the output frequencyon magnetic tape as an audio signal. Then, as with a chart recorder,observations can be reviewed the following day. Finally, it must be keptin mind that just because you get a digital output from a VFC, this doesnot mean that you are photon counting. This is still DC photometry andthe conclusion of Section K.5b still applies; the signal-to-noise ratio ofDC photometry is inferior to pulse counting.

8.5 CONSTANT CURRENT SOURCES

The most common method of calibration of a DC amplifier requires aconstant input current. In principle, the photomultiplier tube could be

TO AMPLIFIER6

Figure 8.8. Constant current source.

used by exposing it to a constant light source, but in practice such asource is difficult to find. For instance, the brightness of an ordinarylight bulb is very sensitive to changes in line voltage. Furthermore, anuncooled tube introduces noise that limits the accuracy of the calibra-tion. A much better approach is to build a constant current source. Fig-ure 8.8 shows such a circuit, which is extremely simple to build. Therotary switch and the potentiometer are used to adjust the current. Theresistors used on this switch are the same value as those used on switchSI of Figure 8.5. The rotary switch steps the current by factors of 10just as switch SI steps the amplifier gain by factors of 10. The 10 kftpotentiometer is for fine adjustment of the current. Because the ampli-fier input is a virtual ground, the calibration current is given by E/R,where E is the voltage at the potentiometer wiper, 0 to 3 V, and R isthe value of the rotary switch resistor. This circuit draws very little cur-rent, so two ordinary 1.5-V batteries comprise a sufficient power supply.We now describe how this circuit is used to calibrate the amplifier.

8.6 CALIBRATION AND OPERATION

The current gain of a DC amplifier is established by the resistors asso-ciated with its gain switches. The accuracy of your photometry depends,in part, on an accurate knowledge of the gain differences betweenswitch positions. The actual sizes of these gain steps must be measuredfor two other reasons. First, it is very difficult to find resistors thatexactly match the required values for 0.5 magnitude steps. Second,resistors tend to change value with time. This is especially true for thefeedback resistors with values exceeding 108 ohms. This means that thecalibration of the amplifier must be checked regularly. The frequency


of these calibrations depends upon the quality of the resistors and theirstorage environment. Initially, you should plan to do a calibration everyfew months. You may find this is too frequent if the calibration appearsto change little. On the other hand, you may find this is not frequentenough if significant calibration changes are seen. If calibration driftsseem to be associated with just one switch position, you may wish toreplace that resistor with a more stable one.

There are usually two approaches to the amplifier calibration. Thefirst measures the resistances of the amplifier resistors with a laboratoryWheatstone bridge. It is best to make these measurements with theresistors actually in place in the circuit. The process of cutting the resis-tor leads and soldering them to the switch may alter their valuesslightly. As a rule, most commonly found Wheatstone bridges cannotmeasure the large megohm resistors found in the feedback loop. Unfor-tunately, it is these large resistors that tend to be the most unstable.The second approach uses a constant current source. This has theadvantage that you actually measure the amplifier gain directly andboth gain switches can be calibrated by this process. We now describethe calibration using a constant current source in detail.

The actual calibration procedure is quite simple. Turn the amplifieron at least 30 minutes early to minimize drift while measurements arebeing taken. The fine gain steps are calibrated first. Place the coarsegain switch to its lowest position (2.5) and connect the constant currentsource to the amplifier input. Turn the fine gain switch to its lowestposition (0.0) and adjust the rotary switch of the current source to itshighest position, that is, the 1-megohm position. Turn the current sourceon and adjust its potentiometer to obtain an amplifier meter deflectionof about one-half of full scale. Record the reading, turn the currentsource off, and record the zero-point level. Turn the current source onand repeat the process. Once a half dozen measurement pairs have beentaken, increase the fine gain setting by one step (0.5 magnitude) andrepeat the entire process. The ratio of the net deflections at these twoswitch positions yields the actual magnitude difference.

With the fine gain set at 0.5 magnitude, adjust the current source toreduce the meter reading to about half scale and take another series ofreadings at 0.5 and 1.0 magnitude. The entire process is repeated untilmeasurements have been made at every fine gain switch position. InTable 8.2, we have listed a set of calibration readings. To save space,only one measurement per switch position is shown.

The next step is to subtract the zero-point readings to obtain the net

DC ELECTRONICS 199

TABLE 8.2. Data for Fine Calibration

Gain Position

0.00.5

0.51.0

1.01.5

1.52.0

2.02.5

Deflection

48.674.8

57.386.9

53.882.4

58.990.4

58.190.5

Zero Point

7.16.3

6.35.3

5.24.3

4.33.8

3.83.8

Net Magnitude Difference

41.568.5

51.281.6

48.678.1

54.686.6

54.386.7

0.544

0.506

0.515

0.501

0.508

deflection for each measurement. Finally, the magnitude differencebetween each switch position is calculated by

Am = 2.5 log (dHjdL\

where dH and dL are the net deflection at the high and low gains, respec-tively. These values are listed in the last column in Table 8.2. You haveabout a half dozen such values for each switch position pair. If youdetermine an average and compute the standard deviation of the mean,you will have the best estimate of the gain difference and its error. Youshould strive to obtain a standard deviation of less than 0.005 magni-tude. With the amplifier in Figure 8.5 and the current source of Figure8.6, a standard deviation of 0.002 magnitude was easily obtained in lab-oratory tests.

The next step is to use these magnitude differences to construct again table. This table is used during data reduction. It allows theresearcher to find the gain difference between any two switch positionsat a glance. Table 8.3 shows a gain table constructed from the magni-tude differences listed in Table 8.2. The horizontal and vertical axes arethe gain positions. For example, to find the gain difference between the1.5 and the 0.5 positions we simply look to where the "1.5 row" inter-sects the "0.5 column" and read 1.021 magnitudes. The entries in thistable were determined by simply summing the magnitude differences ofTable 8.2 between each combination of switch positions.


TABLE 8.3. Gain Table for FineAdjustment Switch

2.52.01.51.00.5

0.0

2.5742.0661.5651.0500.544

0.5

2.0301.5221.0210.506

1.0

1.5241.0160.515

1.5

1.0090.501

2.0

0.508

Once the fine gain switch positions have been calibrated, the coarsepositions can be calibrated with respect to them. The rotary switch ofthe constant current source is placed in the next position (10-megohmresistor) with both the coarse and fine gain set to 2.5. Again the seriesof measurements are made. The fine gain is then reduced to 0.0 and thecoarse gain increased to 5.0. Another set of measurements is then made.In Table 8.4, we list a sample measurement. If the amplifier had a per-fect set of resistors, these two sets of measurements would be identicalbecause the total gain of the two switches is the same (5.0). The rotaryswitch of the current source is moved to the last position (100-megohmresistor) and the procedure is repeated. The first set of measurementsis taken with the coarse gain at 5.0 and the fine gain at 2.5. For thesecond set, the coarse and fine gains are set to 7.5 and 0.0, respectively.

Once the net deflections have been calculated, the first step is to cor-rect for the fact that the fine gain difference is not exactly 2.5 magni-tudes. According to Table 8.3, the gain difference between the 0.0 and2.5 position is actually 2.574, Because this gain is larger than it shouldbe, the deflections taken when the fine gain was set to 2.5 need to becorrected downward. If DH is the net deflection, then the correcteddeflection, />£, is given by

* = l f -0 .4 (A/ - 2.5)—

where A/ is the actual magnitude difference of the fine gain control(2.574). The results appear in column 6 of Table 8.4. Finally, the truecoarse gain differences, Am, can be calculated by

Am = 2.500 - 2.5 log (D*/DL)

DC ELECTRONICS 201

TABLE 8.4. Data for Coarse Gain Calibration

GainCoarse Fine Deflection Zero Point Net DJ Am

2.55.0

5.07.5

2.50.0

2.50.0

72.275.4

83.483.2

14.821.2

15.221.4

57.454.2

68.261.8

53.62.512

63.72.467

where DL is the net deflection obtained when the fine gain is set to 0.0.The coarse gain differences appear in the last column of Table 8.4.

The operation of the amplifier is straightforward, requiring only alittle care and common sense. This is a sensitive device that should beused only to measure the output of the photomultiplier tube or the con-stant current source. As mentioned earlier, the amplifier should beturned on before connecting the signal cable as a precaution againstdamaging the FET input of the op amp. Finally, when measuring anunknown star for the first time, begin at the lowest gain setting. Grad-ually increase the gain until the desired deflection is reached. This pro-cedure avoids possible damage to your meter or chart recorder if abright star is measured with a gain setting that is too high.

REFERENCES

1. Persha, G., 1980. IAPPP Com. 2, 11.2. Melen, R. and Garland, H. 1971. Understanding 1C Operational Amplifiers.

Indianapolis: Howard W. Sams and Co.3. Wood, F. B. 1963. Photoelectric Astronomy for Amateurs. New York: Macmil-

lan, p. 70.4. Young, A. T. 1974. In Methods of Experimental Physics: Astrophysics, vol. 12A.

Edited by N. Carleton. New York: Academic Press, p. 52.5. The Victoreen Instrument Company, 10101 Woodland Ave., Cleveland, OH

44104.6. Oliver, J. P., 1975. Pub. A. S. P. 87, 217.7. Stokes, A. J., 1972. J. AAVSO \, 60.8. Markson Science Inc., 565 Oak St., Box 767, Del Mar, CA 92014.9. Cole-Parmer Instrument Co., 7425 N. Oak Park Ave., Chicago IL 60648.

10. Hewlett-Packard Co., 5201 Tollview Dr., Rolling Meadows, IL 60008.11. Heath Co., Benton Harbor, MI 49022.12. Dunham, E., and Elliot, J. L, 1978. Pub. A. S. P. 90, 119.13. McGraw, J. T., Wells, D. C, and Wiant, J. R. 1973. Rev. Sci. Inst. 44, 748.

CHAPTER 9PRACTICAL OBSERVING TECHNIQUES

Chapters 1 through 5 present the foundation for understanding photom-etry and starlight in general, along with the rudiments of data reduc-tion. Chapters 6 through 8 show how to construct or buy the necessaryequipment, set it up, and perform the necessary calibrations. We arenow ready to discuss using your photometer. This chapter explains inmore detail how to perform photometric measurements, from selectingcomparison stars and making a finding chart, through the actual acqui-sition of data. It ends with some comments about sources of error exter-nal to your equipment with which you must contend.

No book can replace actual experience with the equipment at hand.We can give you some practical advice and try to guide you past someof the pitfalls that we found, but you must learn much of photometryby trial and error. One suggestion we would like to make is to pick onevariable that is bright, short period, and very well observed as your firsttrial. In this manner, you can be sure that your data compares favorablywith previous results.

9.1 FINDING CHARTS

Sirius, Polaris, and other bright stars are easy to find in the sky. Fainterstars become increasingly difficult to find, not only because they areharder to see but also because there are more of them. With care, starsfainter than those visible by eye through a telescope can be measuredby photoelectric photometry. Fainter stars, being more difficult tolocate, require the use of a good finding chart.

The usual method of identifying program stars is through the prep-aration of a finding chart: a sketch or photograph of the region of the202

PRACTICAL OBSERVING TECHNIQUES 203

sky containing the object. You can prepare a finding chart from variousatlases, from your own photographs of the area, or by obtaining previ-ously prepared charts from published sources. Each of these methods isdescribed below.

9.1 a Available Positional Atlases

Positional atlases are drawn from catalogs of star positions. In manycases, stars are omitted for lack of data or were positioned incorrectly.Still, they can contain more information about the stars than a photo-graph. Several atlases include stars brighter than eighth magnitude.Most of these atlases have been reviewed by Larson' and are readilyavailable. For objects brighter than ninth or tenth magnitude, threeatlases are commonly available. They are:

1. Banner Durchmusterung (BD) and Cdrdoba Durchmusterung(CD) Atlases.2-3 The BD was produced by Argelander and Schon-feld in the period 1859-1886, covering the northern sky, and theCD was published between 1892 and 1932, covering the southernsky. Together, they contain approximately 580,000 stars to a lim-iting visual magnitude of 10 and have been the mainstay foralmost a century. These catalogs are available at existing librariesand observatories. New copies are available in magnetic tape formonly. Epoch 1855 coordinates are used and must be precessed.

2. The Smithsonian Astrophysical Observatory (SAO) Atlas.4 Thesecharts contain approximately the same stars as the BD and CDatlases, but the charts are smaller and stars are plotted closertogether on a smaller scale. Most variables brighter than ninthmagnitude are marked. All stars are identified in the accompa-nying catalog, available from the U. S. Government PrintingOffice. Transparent overlays allow the location of stars with arcminute accuracy. The coordinates are for epoch 1950.

3. Atlas Borealis, Eclipticalis, and Australis.5 These charts byBecvar cover the sky to approximately tenth magnitude and iden-tify variables by their variable star designations. One of the nicefeatures of these charts is the color coding of spectral type. Epoch1950 coordinates are used with transparent overlays. This atlas setis widely used by amateur astronomers and is relativelyinexpensive.


9.1b Available Photographic Atlases

For stars fainter than ninth magnitude, photographic atlases must beused because of the large number of stars involved. These atlases consistof either photo-offset charts from original plates or actual photographicprints. Ingrao and Kasperian6 review early photographic atlases. Themajor photographic atlases are listed below.

1. Photographic Star Atlas (Falkau Atlas),1 This atlas used platesthat were blue-sensitive and covers the entire sky in two volumes.The limiting magnitude is 13, the scale is 1 millimeter = 4 arcminutes, and each chart is about 10° on a side.

2. Atlas Stellarum 1950.0.* This atlas covers the entire sky in threevolumes using blue-sensitive plates. The limiting magnitude is14.5, the scale is 1 millimeter = 2 arc minutes, and a completeset of extremely useful transparent overlay grids is included. Thisatlas costs $225 in 1980.

3. True Visual Magnitude Photographic Star Atlas.9 This atlas isvery similar to Atlas Stellarum in that it covers the entire sky inthree volumes with the same scale. The limiting magnitude is 13.5,and a green-sensitive emulsion has been used. For finding charts,green sensitivity is a great advantage as it closely matches theresponse of the eye.

4. Lick Observatory Sky Atlas (North)10 and Canterbury Sky Atlas(South)." Rather than using the photo-offset methods of the pre-viously listed photographic atlases, these two atlases are actualprints of blue plates. The scale is 1 millimeter = 3.88 arc minutes,the limiting magnitude is 15, and each print covers about 18° ona side. No overlays exist and copies are no longer available exceptat existing libraries and observatories.

5. National Geographic-Palomar Observatory Sky Survey(POSS).]2 This survey with the Palomar 48-inch Schmidt is theRolls Royce of the astronomical atlases. Both red and blue plateswere used, with the atlas consisting of positive prints with a scaleof 1 millimeter =1.1 arc minutes and a limiting magnitude of 20(red) or 21 (blue). Each print is 6.6° on a side and a sequence ofoverlays exists, though less useful than most as no fiducial marksare found on the prints. The POSS is complete to —24° declina-


tion, with a red-plate extension to —45°. The plates were takenin the early 1950s, and Palomar intends to redo the survey startingsometime in 1985. A complete set of prints costs several thousanddollars.

6. The European Southern Observatory (ESO)fScience ResearchCouncil (SRC) Atlas of the Southern Sky.l3 In a similar mannerto that of the POSS, the southern sky is presently being photo-graphed by the two large Schmidt telescopes in the SouthernHemisphere.14 The ESO 1 meter at La Silla is taking red platesand the SRC 1.2 meter at Siding Spring is taking the blue surveyplates, both with a scale of 1 millimeter =1 .1 arc minutes and alimiting magnitude of 22. The atlas covers the sky from —90° to— 17" declination with plate centers at 5° spacing. This atlas isbeing released in limited quantities (150 copies) only on 36-cen-timeter (14-inch) Aerographic Duplicating Film.

9.1c Preparation of Finding Charts

The goal of a finding chart is to allow easy identification of the programobject at the telescope. Generally, two charts are prepared. A small-scale chart matching the field of view of the main telescope, typically15 arc minutes square, should be prepared carefully, including stars twoto three magnitudes fainter than the variable. Mark the program object,any nearby comparison stars, and an area with no stars to be used forsky background measurements. Many observers prefer these charts tomatch exactly the view of the telescope, that is, reversed and/orinverted. Cardinal directions should be indicated as well as the chartscale, perhaps by an angular measurement grid. A large field chartroughly matching the finder can provide pointing information for themain telescope, and at the same time identify photoelectric comparisonstars and readily identifiable patterns to help locate the field.

The best charts are Polaroid copies of photographic atlases, negativecopies of atlas prints, which can then be enlarged, or prints made fromyour original negatives of the sky. Direct tracings of atlases or xeroxesmay be acceptable provided that the limiting magnitude near the pro-gram object, comparison stars, and sky measurement position is suffi-ciently faint.


9.1d Published Finding Charts

In most cases, earlier observers have published finding charts for yourobject of interest. These charts may lie in obscure journals or sufferfrom poor quality. However, it is highly recommended to search forpublished finding charts of variables fainter than ninth magnitudebefore preparing your own.

The major source for finding charts is the General Catalog of Vari-able Stars (GCVS).'5 Its extensive reference list contains chart refer-ences for the vast majority of identified variable stars. However, manyof these finding charts are published in Russian journals, which areidentified only in the Cyrillic alphabet rather than the English transli-terated names under which they are cataloged in most major libraries.

There are several collections of finding charts available if the GCVSitself or its referenced charts are not available. These are listed below.

1. AAVSO Variable Star Atlas. '6 There are 178 charts in this atlas,measuring 11 X 14 inches with a scale of 15 millimeters perdegree. These charts contain all of the American Association ofVariable Star Observers' program stars and all variables thatreach 10.5 visual magnitudes or brighter at maximum. The chartsare very similar to the SAO Atlas.

2. The Sonneberg charts.11 Several thousand variables were discov-ered at Sonneberg during the early part of the twentieth century.The charts contained in reference 17 cover 3600 of the Sonnebergvariables.

3. The Odessa charts.l8 This reference contains light curves andfinding charts for 266 stars.

4. Atlas Stellarum Variabilium.19 These charts cover all variablesbrighter than tenth magnitude at minimum that were known by1930. The entire sky was covered, with each chart measuring 20°on a side and plotting stars to a limiting magnitude of 14.

5. Charts for Southern Variables.20 The 400 charts in this collectionwere sponsored by the International Astronomical Union andcover all long-period variables brighter than thirteenth magnitudeat maximum and south of — 30 ° declination. Later volumes in theseries have photoelectric sequences for nearby stars.

6. Atlas of Finding Charts of Variable Stars.2I This work is hard tofind in Western libraries.


Goddard Space Flight Center (GSFC) in Greenbelt, Maryland is thedistributor for the magnetic tape version of the GCVS. The tape alsoincludes a cross-reference list for the Sonneberg variables, which isextremely helpful. In addition, GSFC can generate POSS Atlas over-lays to identify faint variable stars. Write to Code 680 of GSFC formore details.22

9.2 COMPARISON STARS

Now that you have located your program star, you need to decidewhether or not to use a comparison star in your measurements. Theseare stars that are near to your variable and are observed immediatelybefore and/or after your program star. Usually, measurements arereported as differential observations, star A minus star B. This differ-ence in the magnitudes is less likely to vary, as conditions that affectone star usually affect the other. Use of comparison stars has severaladvantages as noted below.

1. Slow atmospheric variations are eliminated, as the variationaffects both stars equally.

2. First-order extinction corrections are nearly equal for both starsand becomes unimportant for differential measurements. Extinc-tion measures are not necessary except to place the comparisonstar on the standard system.

3. Zero-point differences between you and other observers areremoved if common comparison stars are used.

4. If a comparison star of similar color is used, errors in the trans-formation equations will have less effect.

5. If a comparison star of a similar magnitude is used, differentialdead-time corrections become unimportant in pulse counting, andgain correction errors are removed if the same gain setting is usedfor both stars in DC photometry.

6. The comparison star can serve a double purpose as an extinctionstar if several measurements over wide air mass differences aremade. In addition, if advantage 4 holds, second-order extinctionis automatically taken into account.

7. For high accuracy and long-term reproducibility, errors in differ-ential measurements with respect to a comparison star canapproach the equipment short-term accuracy, less than Om.01,


instead of the night-to-night transformation variation of Om.02 ormore. This is very important for small amplitude variables.

You can observe variable stars without using comparison stars. Manyextensive surveys of variables have been carried out in the past withoutthese aids, especially in the southwestern United States where photo-metric skies are common. However, for accurate low-amplitude pho-tometry they are almost a necessity.

9.2a Selection of Comparison Stars

Make every effort to use the same comparison star as previous observersof the variable star. This should be done to minimize systematic differ-ences between data sets of various observers. However, if this cannot bedone (because no one has observed this star before or previous observersdid not specify their comparison star), you will need to select your own.

Comparison stars should meet five criteria: (1) less than 1" from theprogram star, (2) of similar color, (3) equal in magnitude, (4) non-varying, and (5) not red in color. Red stars are almost always variable,and are quite likely flare stars. Rules 2 and 3 are not rigid, as few starswill be near the variable and be the same brightness and color. Butmake every effort to enforce rules 1 and 4! Pick a brighter comparisonstar rather than one which is fainter than the variable, as better statis-tics can be obtained in a shorter time.

Selection of similarly colored stars can be difficult. For stars brighterthan tenth magnitude, the Becvar atlases5 indicate spectral class by thecolor of the dot representing the star. A more exact method is to use theHenry Draper (HD) Catalog,23 which lists accurate spectral types forover 200,000 stars. This catalog is available at most colleges. A micro-fiche version is available from the University of North Carolina atGreensboro,24 and a magnetic tape version from GSFC. Modern spec-tral classification yielding more accurate spectral types for the HD starsis being carried out by the University of Michigan25 and by the Miragroup in California.26

Nearby comparison stars of similar brightness can be found by exam-ining the prepared finding charts for the variables. If you have a choice,pick a star listed in one of the major catalogs: BD, CD, HD, or SAO.This allows easy publication of your results without the necessity ofincluding a finding chart for your comparisons.


If all else fails, a few minutes of observation in the field surroundingyour variable will usually find a nearby comparison star candidate ofthe same magnitude. These observations can be performed quickly, asyou are looking for a star with similar deflections or count rate to thoseof your variable and can easily eliminate stars that differ widely.

9.2b Use of Comparison Stars

Two comparison stars for each variable are usually chosen. One is con-sidered to be the actual comparison star. The other is called a checkstar, and is used to test the stability of the comparison star. Differentialmeasurements between the comparison and check stars should remainconstant to within the nightly errors, usually Om.02 or less with goodskies. If variations larger than this consistently occur, either the com-parison or the check star is variable, a fairly frequent occurrence!Which one is the culprit can be decided by comparing the nightly stan-dardization of the comparisons or by deciding which star gives a lightcurve for the variable that has the smallest scatter. The check star isobserved once or twice a night; the remaining measurements use onlythe comparison star.

The recommended observing sequence for each variable star obser-vation is:

1. Sky2. Comparison3. Variable4. Comparison5. Variable6. Comparison7. Sky8. Check (optional)

Each step in turn is comprised by the (/, B, F, red leak, and perhapsdark-current measurements as discussed in Section 9.3. If the variableis faint, so that steps 3 and 5 would take several minutes each to reach1 percent accuracy, it is much better to perform several sequences withless accuracy and average them than to risk sky transparency changesduring the variable and comparison star observations.

If the comparison star is to be placed on the standard VBV system,


it should be done on several occasions: at the beginning of the program,in the middle, and at the end. This can be done through differentialmeasurements of nearby standard stars, or by using the transformationequations and nightly determination of the extinction.

9.3 INDIVIDUAL MEASUREMENTS OF A SINGLE STAR

An individual star observation really consists of four separate determi-nations in UBV photometry: U, B, V, and red leak. If you are doingonly BV photometry, the U and red-leak measurements are removed.Red leak is unimportant in blue stars because little of their energy fallsin the red. Sometimes observers also measure the dark current. This isusually negligible for bright stars or a dry-ice photomultiplier tube, andis subtracted automatically during sky background subtraction.

For pulse counting, the usual measurement sequence is K, B, U, redleak, and dark current. This is because V is the most commonly listedmagnitude and gives a check for proper star selection and equipmentoperation. If you are using a filter wheel, you will end on the dark filterposition, allowing movement to the next star, and then will be ready toperform the V observation with the next rotation of the wheel. The timeof the observation is the average of the starting and stopping times ofthe sequence.

For DC photometry, the usual sequence is V, B, U, red leak, {/, B,and V. This sequence allows checking for instrumental drift effects dur-ing the sequence.

Achieving one percent accuracy in a star measurement alwaysrequires a signal-to-noise ratio (S/N) of 100. How this is determined isdifferent for pulse counting and DC photometry.

9.3a Pulse-Counting Measurements

Remember from Chapter 3 that 10,000 total source counts are neces-sary from a star to approach one percent or Om.01 accuracy. Figure 9.1shows the typical count rate for telescopes of 20-, 40-, and 75-centi-meter diameters. For example, an 1 lm.7 star produces roughly the countrates shown in Table 9.1. For faint stars, measuring the U magnitudeis always a problem. For example, the count rate for cepheids in U isten times less than in V. You must decide in these cases if having the(U — B) color index is essential to your program and make allowancesfor the increased time if it is.


TABLE 9.1. Pulse-Counting Rates

Integration Time toTelescope Size Rate Achieve I % Accuracy(centimeters) (counts/second) (seconds)

204075

100380

1400

10026

7

The times quoted in Table 9.1 are only for measuring the star in asingle color. However, the sky must also be measured accurately to sub-tract its contribution from the star measurement. The ideal ratio of timespent measuring the sky background to time spent on star and sky is

3 4 5 6 7 8 9 10 11 12 13 14 15

V MAGNITUDE (B - V = 0)

Figure 9.1. Count rate versus apparent magnitude for three different telescope diameters-


100 r

10

0.1

0.01

0.0010.2 0.4 0.6 0.8 1-0 1.2

ti,

Figure 9.2. The optimum fraction of observing time to be spent on sky background.

found by taking the derivative of Equation K.41 with respect to thesource time. The result is

Ltr, (9.1)

where the subscript s refers to source measurements and the subscriptb refers to background measurements, C is the count rate, and t is thetime per measurement. This ratio is plotted in Figure 9.2. From aknown count rate ratio the corresponding time ratio can be read directlyfrom this figure.

Example: 1 lm.7 star produces roughly 100 counts per second in the20-centimeter telescope. Assume the sky background gives roughly50 counts per second in the chosen diaphragm. How long do we haveto observe the star and the background?


tjLm A 50/100/, V (50/100) + 1

Or, CbjCs = 50/100 = 0.5, and reading from Figure 9.2, the cor-responding time ratio of 0.56. Because we need to observe the starfor 100 seconds for 1 percent accuracy, we need to observe sky alonefor at least 56 to 58 seconds.

9.3b DC Photometry

When pulse counting, there is a fairly simple rule to follow to achievean accuracy of 1 percent. This requires an S/N of 100, which meansthat a total of 10,000 counts must be accumulated. If a star producesonly 1000 counts in one second, you simply observe it for at least 10seconds. The guidelines are not so simple in DC photometry. If you wereto watch the output of a DC amplifier on a chart recorder as starlightstrikes the detector, you would see the pen rise and then jitter aboutsome mean level.

Figure 9.3 shows the chart recorder tracing for the first observationof UZ Leonis in Table 4.1. The mean level of the pen represents thesignal from the star and the jitter is the noise. It is noise that preventsus from determining the stellar signal with perfect accuracy. Unlike

T I M E

Figure 9.3. Chart recorder tracing.


pulse counting, the S/N does not improve when we observe the starlonger, the chart tracing just gets longer and continues to look muchthe same. The same is true if we watch the amplifier meter. Increasingthe observing time improves the photometric accuracy to some extentbecause the longer chart tracing makes it easier to estimate the meanlevel using a straightedge. However, once the tracing is several centi-meters long, continuing the deflection brings diminishing returns(unlike pulse counting). The problem is that in DC photometry theintegration time is set by the RC time constant at the amplifier input.The obvious way to improve the S/N is to increase this time constant.Indeed, the tracing does become smoother when the time constant isincreased. However, there is a trade-off when using a capacitor tosmooth the signal. When the detector is exposed to light, the currententering the amplifier must charge the capacitor. The rate at which thecapacitor charges depends on the RC time constant. As the capacitorcharges, the pen makes an exponential rise to its final value. For the pento reach 99 percent of its final value requires a period of 4.6 time con-stants. If a large time constant is used, a significant amount of observingtime is spent waiting for the pen to reach its final level. For this reason,DC amplifiers seldom use time constants that exceed a few seconds. TheDC amplifier in Chapter 8 has an adjustable time constant. For brightstars that have a large S/N, a small time constant is used to saveobserving time, while for fainter stars a longer time constant is used toimprove the S/N.

For many stars, a time constant of less than a second is not enoughto achieve a S/N of 100. Figure 9.3 shows a case with a 0.5 second timeconstant. We can estimate the S/N by the amount of jitter about themean. The mean signal level is 30.3 units on the chart paper. The noisecauses variations of 0.9 unit to either side of the mean. If the sky back-ground is large, it would be necessary to subtract this from the star toobtain the net signal. In this particular example, the sky background islow so that the S/N is very nearly 30.3/0.9 = 34. By Equation K.25,we see that the S/N increases with the square root of the total integra-tion time, t. Therefore, we would need to increase the amplifier timeconstant by a factor of nine to achieve an S/N of 100. A time constantof this length is not available for this amplifier. Then the procedure isto take several deflections (each many time constants in duration) andform an average to make a single observation. A rough estimate of thenecessary number, «, of such observations is given by

PRACTICAL OBSERVING TECHNIQUES 21 5

The time between each of these deflections can be spent recentering thestar in the diaphragm (if necessary) or making a deflection in anotherfilter. For the chart tracing in Figure 9.3, the S/N implies that a singlemeasurement would have an error of 0.03 magnitude. The formulaabove implies that nine observations should be averaged for a 0.01 mag-nitude error. In fact, the actual standard deviation from the mean ofnine observations that night was 0.012 magnitude.

This example points out a disadvantage of DC compared to pulsecounting. The nine observations required would take several minutes ofobserving time. With a pulse-counting system, if we obtained a S/N of34 in 0.5 second (the time constant used that night), we would needonly to integrate nine times longer (because S/N oc t l /2) or 4.5 secondsfor a S/N of 100. For many observing projects, this difference in observ-ing time is unimportant. But there are rapid variable stars and short-period binaries that have measurable changes of brightness in just a fewminutes. We must observe the light curve on more nights to obtain thesame quality of data as that obtained with a pulse-counting system.Alternatively, you can retain the time resolution by not averaging asmany deflections but you obtain a noisier light curve.

Note that this disadvantage of DC photometry disappears if the chartrecorder is replaced by a voltage-to-frequency converter and a counter.A very small amplifier time constant can then be used to integrate onthe star until the desired S/N is reached. However, the S/N analysishas a further complication. Unlike pulse counting, the number thatappears on your counter is not equal to the number of detected photons.Instead, it corresponds to some level of current flowing in the feedbackloop of the amplifier. This in turn depends on both the brightness of thestar and the amplifier gain. In this case, an empirical method is thesimplest way to determine S/N. Take a series of short test integrationsand calculate the standard deviation from the mean. If ~c is the meancounts and s.d. is the standard deviation, then

S/N = - (9.3)


To obtain an S/N of 100, the required integration time, 71, is

where t is the total time of all the test integrations.The amount of observing time spent on sky measurements can be

estimated by Equation 9.1 just as it is for pulse counting. The countrates in that equation are simply replaced by net pen deflections. In theexample above, the sky background was &> of the stellar signal. Equa-tion 9.1 then tells us that 20 percent of our observing time should bespent on sky background.

There are also some differences in the observing procedure betweenpulse counting and DC photometry. These are discussed in Section 9.7.

In Sections 9.3a and 9.3b, much emphasis has been placed on theS/N as an indicator of the quality of an observation. However, a highS/N is a necessary but not a sufficient condition for an accurate obser-vation. There are many other factors that can come into play. Forinstance, electronic drift or the slow passage of cirrus clouds are notobvious in the noise level in a single measurement. However, theybecome apparent when measurements of the same object, such as thecomparison star, fail to repeat. Discrepancies that exceed the noise lev-els are indicators of a problem. Even if a single measurement has a veryhigh S/N, never assume that it will be reproducible; always take at leasttwo. There is no substitute for an alert, experienced observer who cantell when "things are not quite right."

9.3c Differential Photometry

Differential photometry is the simplest and potentially the most accu-rate of photometric techniques. The basic idea is to compare the bright-ness of the variable star to that of a nearby and constant comparisonstar. However, simple as it sounds, certain observing procedures mustbe followed strictly if differential photometry is to be done properly.

The golden rule of differential photometry is: interpolate, neverextrapolate. To illustrate the meaning of this rule, consider what hap-pens as we observe our two stars during the night. Suppose that earlyin the evening the variable and comparison star are near the easternhorizon. We measure the comparison star and then, for some reason,

PRACTICAL OBSERVING TECHNIQUES 21 7

delay measuring the variable for 20 minutes. During those 20 minutes,the stars have risen higher and the extinction is considerably less thanit was when the comparison star was measured. The result is that thevariable looks too bright with respect to the comparison star measure-ment. This is an example of extrapolation; we took a comparison starmeasurement and assumed it was valid 20 minutes later. Obviously, abetter procedure is to measure the comparison, variable, and then thecomparison star again. We can then interpolate to estimate the appar-ent brightness of the comparison star at the time of the variable starmeasurement. Our golden rule can be restated: always sandwich thevariable star measurements between comparison star measurements.

If the variable star is faint or varies slowly in brightness, the observ-ing sequence in Section 9.2b is recommended. However, if you areobserving a star that varies rapidly, such as an eclipsing binary with ahalf-day orbital period, a slightly different observing pattern is pre-ferred. If we let C and K represent an observation through each filterof the comparison and variable star respectively, then the observingsequence might look like the following.

CW... WCW... WCW... WCW...

The brackets mark a data group that we refer to as a block. Each blockbegins and ends with a comparison star measurement. The number ofvariable star measurements in a block depends on three factors. First,the required number of measurements needed so that when combined,a single observation with a S/N of at least 100 is produced. (This wasdiscussed in Sections 9.3a and 9.3b.) Second is the speed with whichthe variable changes. Obviously, if the star only varies by O.lm duringthe entire night, you need not look at it as often as one that changes bythe same amount in 30 minutes. In the latter case, it would be desirableto obtain several observations per block (i.e., spend a higher percentageof the observing time on the variable). The third factor is zenith dis-tance. When the air mass is large, variations in extinction can have alarge impact. Therefore, the comparison star must be observed morefrequently. There is no simple rule on how long to make a block, but ofcourse, there is no substitute for experience. However, it is certainlyadvisable to observe the comparison star as frequently as possible.Experience with short-period eclipsing binaries observed through thesomewhat variable skies of the midwestern United States suggests that


the comparison star should be observed at intervals of 20 minutes orless. If the observing sequence must be interrupted at any point, it isimportant to end with a comparison star measurement. If observingresumes later, you should begin with a comparison star measurement.

The block structure outlined above does not indicate sky backgroundmeasurements. The reason is that the amount of time spent measuringthe sky depends on the relative brightness of the star and the sky back-ground. The method of Section 9.3a can be used to estimate the per-centage of observing time spent monitoring the sky background. If, forinstance, it turns out that 25 percent of your time should be spent onthe sky, then every fourth measurement in the block should be of thesky. If you suspect the sky background is changing rapidly (for exam-ple, if the moon is rising), then you should measure the sky morefrequently.

Note that our block structure does not contain separate measure-ments of the dark current. Some authors recommend measuring thedark current frequently. However, our experience has been that withwell-designed amplifiers and fairly stable photomultiplier tube temper-atures, the dark current is very constant. Every time the sky backgroundis observed, we actually measure sky plus dark current (plus any zero-point shift if a DC system is used). When this is subtracted from thestellar measurement, the dark current (plus any zero-point shift) is sub-tracted automatically. There is no need to measure and subtract thedark current specifically from all the measurements. Therefore, thedark current need be measured only occasionally as a check on the sta-bility of the photometer.

9.3d Faint Sources

Photometry of faint sources can be a time-consuming and exasperatingproject that should only be undertaken by the experienced observer. Byfaint sources we mean objects that are comparable to the sky back-ground in brightness, or objects near the visual limit of the telescope.There are several points to consider when observing faint sources.

First, pulse counting with a cooled photomultiplier tube is the mostpractical method of observing faint sources. DC methods yield chartrecorder deflections that are not significantly greater than the randomfluctuations in the sky background. A smoother trace can be obtained


by increasing the time constant, thereby integrating over a longerperiod. However, the time required to reach a constant level is alsoincreased. Eye measurement of a star plus sky trace that is only a fewpercent greater than the sky trace alone is very difficult. Long integra-tions with pulse counting are very easy, requiring only the selection ofa longer gating time on the pulse counter.

Second, time is limited by the accuracy of your telescope drive andby sky conditions. Generally, never integrate on one star for more than5 minutes, including time to measure all colors. If you have insufficientcounts with the 5-minute limitation, move to the comparison star orbackground and then return to the program object for another 5-minuteobservation.

Third, never observe except under optimum conditions. This includesusing only moonless nights, observing near the zenith, and with the bestpossible seeing conditions. Under these conditions, you can use thesmallest diaphragm to reduce the sky background and increase the con-trast between the star and the sky.

For stars that approach the sky background in intensity, the timespent on observing the star and observing the sky should be about equal(see Figure 9.2). This means that you should alternate 5-minute inte-grations between star and sky. Always cycle through all filters on oneobject before moving the telescope to look at sky or a comparisonsource.

If you are observing one source for a significant amount of time, say30 minutes or more, plot the sky values versus time. You may find asignificant trend because of a brighter sky near the horizon or slowlyvarying sky brightness that allows you to interpolate between adjacentsky readings to give a better sky value at the time of observation of yourprogram star.

Stars near the visual limit or fainter can be measured photoelectri-cally, but are very difficult to place in the diaphragm. The usual pro-cedure is to have the guiding or finding eyepiece on a stage with X andY movements and offset to a brighter star in the same field. Thisrequires the ability to measure the amount of offset in both axes andthe knowledge of the plate scale of the telescope. For simple systems,the first requirement can be met by counting screw turns between twostars in the field with known positions. To perform offset photometry,first position the cross hairs on the center of the diaphragm, and then


move the eyepiece in X and Y the distance between the object to bemeasured and the nearby bright star. The positioning of the bright staron the cross hairs places the source to be measured in the diaphragm.

You must find a region near the star where no stars within five mag-nitudes of the program star's brightness exist. For example, a ninthmagnitude variable must have a sky reading with no stars brighter thanfourteenth magnitude in the diaphragm. Otherwise, the sky backgroundreading gives a sky value significantly higher than the sky reading atthe star itself, and the measurement for the star becomes fainter thanit actually is when you subtract the incorrect brighter sky value from it.This becomes troublesome particularly around tenth magnitude for theprogram object, as there are many fifteenth magnitude stars within anaverage 30 arc second area. It is difficult to find a clear region for thesky reading. Also, there is a large chance of including a faint companionnear the star itself. A secondary problem is finding charts that go faintenough; even using Atlas Stellarum with its fourteenth magnitude limit,you cannot observe any stars fainter than ninth magnitude without thechance of significant sky background subtraction errors.

9.4 DIAPHRAGM SELECTION

Chapter 6 presented the practical details of constructing the photometerhead and the diaphragms. However, there are some basic considerationsto be made when using these diaphragms in your observing program.

The most obvious reason for having more than one diaphragm is toprevent unwanted stars from contributing to the light entering the dia-phragm. In addition, a smaller diaphragm allows less sky backgroundradiation to pass through, while permitting most of the program star'slight to pass through unhindered.

It is tempting when using a photometer with several diaphragms touse a small one on one star, then to use a larger one on another star,because it is brighter or has no companions. There are several reasonsfor not doing this, and these are presented in this section.

*9.4a The Optical System

The image of a point source created by a telescope is not a point sourcebut rather a very complicated distribution. Because the telescope is not


infinite in size, only a portion of the total wave front of light from thestar is intercepted by the telescopic mirror or lens. The mirror is thencalled the entrance aperture of the optical system.

As the light is focused by this aperture, the path length of light raysfrom a given wave front is different for different parts of the image.Some of these rays add together in phase, resulting in constructive inter-ference; others add together 180° out of phase, resulting in destructiveinterference. The resultant image of a point source by a circular aper-ture looks like a bright central disk, the Airy disk, surrounded by fainterconcentric rings of light. This image can be seen with good optics on agood night and high magnification, and actually can be used to colli-mate the optical system. The important point to remember is that theimage is not concentrated at a point even if the source appears pointlike.The size of the central disk and the brightness of the secondary ringsare inversely proportional to the diameter of the telescope.

An additional problem arises when a secondary mirror is used,because part of the circular aperture is obscured and an annularentrance aperture results. A theoretical description of the resultantimage of a point source is discussed by Young.27 The major change froma strictly uniform, circular aperture is an increase in the amount of lightin the secondary rings.

If a diaphragm is placed in the focal plane of the telescope, a certainamount of the energy from the star will be removed, the amountdepending on how many of the secondary rings are larger than the dia-phragm. A good approximation to the excluded energy is

82,500X

dD(\ - | ) _

where X ( d ) is the fractional excluded energy as a function of dia-phragm diameter, X is the wavelength of light, D is the diameter of thelens or mirror, d is the diameter of the diaphragm in arc seconds, andt is the fractional obscuration of the primary mirror by the secondarymirror. The variables D and X must be in the same units. The totalenergy included by a diaphragm is then given by

/(</) = 1 - X(d) (9.6)


For a 40-centimeter (16-inch) telescope with a 40 percent centralobscuration and a wavelength of 5000 A (5 X 10~s centimeters),

82,500(5 X 10"s)40</(1 - 0.4)

0.1718(9.7)

Figure 9.4 shows the result of Equation 9.7. The approximationbreaks down for small diaphragms, which is why the included energyfrom the figure does not approach zero for small diaphragms. The pointto note, however, is that for a 40-centimeter telescope, changing froma 10 arc second diaphragm (2 percent excluded energy) on one star toa 20 arc second diaphragm (1 percent excluded) on another star causesat least a 1 percent error. However, the difference between a 20 arcsecond and a 30 arc second diaphragm is small enough to neglect unless

1.00 i-

0.95 ~

0.90 -

0.85 -

0.80

0.7510 15 20

d larcsec)

25 30

Figure 9.4. Total energy included as a function of diaphragm size, for a 40-centimeter aperturetelescope.


precision greater than ± 0.01 magnitude is desired. But reducing thesize of the telescope to 20 centimeters would then make the 20 to 30 arcsecond change as large as the 10 to 20 arc second change with the largertelescope. Therefore, use as large a telescope as possible to eliminatethis error; many observers even do not use diaphragms smaller than 20arc seconds with telescopes smaller than about 20 centimeters indiameter.

We have not included other effects that result from diffraction spikesfrom the secondary supports or the optical aberrations caused by themirror itself, all of which increase the amount of energy outside of thecentral disk. In other words, consider the above estimates to be lowerlimits on the errors involved from the optical system itself.

9.4b Stellar Profiles

Just as the light from the sun scatters, making the sky blue, the lightfrom any star scatters over the entire sky. The profile of a stellar imageon the sky is therefore not strictly pointlike, but rather spread out byrefraction, diffraction, and scattering in the atmosphere and diffractionand scattering within the telescope. The profile concentrates heavilytowards the center, producing a "seeing disk" typically 2 arc secondsacross and then decreases rapidly outside that diameter. The seeing diskor stellar profile is not constant for a given instrument. A hazy nightcan broaden the image greatly.

Figure 9.5 shows a typical stellar profile for a mv = 0 star based onthe results by King28 and Picarillo.29 Note that, although the intensityfalls off rapidly to a 10 arc second radius, the decreasing intensity soonapproaches an inverse square law drop. You might think because at a10 arc second radius, equivalent to a 20 arc second diaphragm, the star-light is ten magnitudes fainter than near the profile center that theremaining radiation could be neglected. However, the total light enter-ing the diaphragm is the product of the intensity per unit area and thetotal area of the diaphragm, which increases as the square of the radius.The resultant total light pattern is shown in Figure 9.6. Here you cansee that a 30 arc second radius circle, 60 arc second diaphragm,includes most of the light of a star, but that using a 20 arc second dia-phragm on one star and a 10 arc second diaphragm on another cancause significant error.


Or-

10,000

Figure 9.5. Typical stellar profile.

9.4c Practical Considerations

The fact that a stellar image is not concentrated at one point has severaldirect consequences to photometry:

1. Changing diaphragms between a comparison and the variable starcauses different fractions of each star's total light to reach thephotometer.

2. On hazy nights, even observations with the same diaphragm sizecan be in error as the haze scatters a changing percentage of thestarlight out of the diaphragm, depending on observation time oraltitude.

3. A miscentered image increases the chance for observational errorgreatly, because a significant fraction of one star's light may bemisplaced out of the diaphragm.

4. A misfocused image enlarges the seeing disk and allows less lightto reach the photomultiplier tube and increases the chance of mis-


1 0

09

0.8

O. /

0.0

O

1:0.5Oz2 0.4

O

SE 0.3

0.2

0.1

0.00.1 10 100

r (arcsecl

1000 10,000

Figure 9.6. Integral of starlight versus radius.

centering errors. This error is common when comparing early andlate evening observations if the telescope has not been refocusedfor temperature changes.

5. Never change diaphragms between star and sky measurements.Whereas the stellar contributions through a 20 or a 60 arc secondaperture is the same to within 1 percent, the 60 arc second aper-ture allows nine times more skylight to pass through. At times itis difficult to find a clear patch of sky for the background mea-surement. It is then the observer's discretion whether to make thesky measurement further from the program star, or to use asmaller diaphragm for all measurements.

6. There is a limit on the smallness of the diaphragm because of thestellar profile size. If apertures much less than 10 arc seconds areused, a significant fraction of the star's light is rejected. One rea-son the 2.1-meter Space Telescope (to be launched from the spaceshuttle by NASA in 1990) is expected to outperform the largestearth-based telescopes is that with no atmospheric seeing, a dif-


fraction-limited star profile is obtained, where 70 percent of thelight is concentrated within a 0.1 arc second circle. Then dia-phragms of 0,4 and 1 arc second are easily used, removing muchmore of the sky background from the measurement than is possi-ble from the ground.

7. The diameter of the seeing disk varies with altitude above thehorizon because of the differing air mass. Near the zenith the diskmay be 1 or 2 arc seconds in diameter, but near the horizon it hasexpanded to perhaps 10 arc seconds. Therefore, even using thesame diaphragm on stars at differing altitude, differing amountsof the total starlight are admitted. Young27 gives a good review ofthis and other seeing and scintillation effects.

At all costs, try to use one diaphragm size for all observations thatare to be compared on a given night. You do not have to measure 100percent of the light from a star, or even 95 percent, to get accurateresults. What you must try to do is measure the same fraction of lightfrom every star you want to compare.

9.4d Background Removal

Because the sky background acts like an extended source, a larger dia-phragm will admit more background radiation since a larger area of thesky is seen by the photometer. However, few more star photons areacquired as they come from a nearly pointlike image amply covered bythe diaphragm. The light from the sky passing through the diaphragmcannot be distinguished from the light of some faint star as the photom-eter knows only that a certain number of photons have reached it, notthe origin of those photons in the area covered by the diaphragm.Therefore, we can compute the magnitude of an equivalent star thatproduces the same number of photons as produced by the sky. This isshown in Figure 9.7 for a typical site where the sky brightness is 22m.6per square arc second at the zenith. You can see that if a 30 arc seconddiaphragm is used, as much light reaches the photometer from the skyas if a 15m.5 star were in the aperture. As Equation 9.1 indicates, thelarger the sky-to-star ratio becomes, the longer is the time required tocomplete an observation. Therefore, you should use the smallest dia-phragm feasible when measuring faint sources. A size of 15 to 20 arcseconds is usually considered the minimum size for amateur telescopes,


Figure 9.7. Sky brightness as seen through different diaphragms assuming a surface brightnessof 22.6 per square arcsec.

though diaphragms of 4 to 7 arc seconds have been used on the Hale 5-meter (200-inch) telescope with good seeing and the advantage of itssuperior drive.

The full moon can easily increase the brightness indicated in Figure9.7 by three or more magnitudes. Also, because of the increased scat-tering near the horizon, the sky is two to three times brighter there thannear the zenith. Faint star observations are therefore usually only per-formed under moonless conditions near the zenith and with good seeing.

9.4e Aperture Calibration

There are two main reasons why an accurate calibration of the aperturesize might be necessary. When photometry of extended objects such asgalaxies or comets is intended, the measurements must be reduced tomagnitudes per square arc second. When you want to compare mea-surements taken with different apertures, both measurements must bereduced to a common aperture size.

Because of the small size involved, not many methods are availableto measure the diameter or area. A direct measurement of a 0.01 inch


hole to 0.1 percent accuracy requires a precision of ± 10 microns, notavailable from your typical ruler! Usually this is performed with a largeinstrument called a measuring engine, and must be done at an astron-omy department possessing such an instrument. An indirect measure-ment of the area can be obtained (and the diameter from A = trD2/4)by measuring a uniform object like a white card on a dome througheach diaphragm and noting the intensity ratios. After the relative aper-ture sizes have been obtained, the size of one aperture can be obtainedby measuring an object with known surface brightness, such as a con-stant light source, or by direct measurement of the largest diaphragm.

9.5 EXTINCTION NOTES

Chapter 4 gave the types of stars to be used for extinction measure-ments. Listed below are some of the methods for obtaining these obser-vations during a program of all-sky photometry (i.e., not differentialphotometry).

1. Use more than one star to determine extinction. This reduceserrors due to the non-uniform sky.

2. Use stars near the celestial equator so that they move through airmass values (X) quickly. Also, measure them several times whilethey are near the horizon, as X varies quickly there and you wantto fill in the plot with more than one point at large X. Do not gettoo close to the horizon. Refraction and particulate problems aremuch more severe at low altitudes. A practical limit is X < 5.

3. If you are in a dry climate or are under an extended high-pressuresystem, you can average values from several consecutive nights toobtain mean extinction coefficients.

4. Observe in the following sequence: extinction stars, program stars,extinction stars, etc., spending about 80 percent of time on yourprogram stars but obtaining extinction measurements throughoutthe night.

5. To check on the temporal, that is, variation with time, consistencyof the sky, look at some of the North Polar Sequence stars fromAppendix E throughout the night, as the zenith distance of thesestars (and therefore X} changes very little. Temporal variationsoccur most frequently near sunrise, sunset, and when there is anapproaching atmospheric frontal zone.


6. Always interpolate, never extrapolate. In other words, make surethat you have extinction measurements at air masses larger andsmaller than those for your program stars.

9.6 LIGHT OF THE NIGHT SKY

Even when the sun disappears beneath the horizon, our world is filledwith light. It may be fainter than daylight or present in the infraredand therefore undetectable by the eye, but is a major contributor to theerrors of photometric observations. Roach and Gordon30 give a goodreview of the light of the night sky, which should be studied if moredetails are needed. In this section we will consider only the naturalsources of night sky light, neglecting man-made light pollution.

There are six general contributors to the night sky brightness: (1)integrated light from distant galaxies; (2) integrated starlight fromwithin our galaxy; (3) zodiacal light; (4) night airglow; (5) aurora; and(6) twilight emission lines. Night airglow, aurora, and twilight emissionlines are results of a planet with an atmosphere and magnetic field.Zodiacal light is a result of being within a solar system. The remainingtwo contributors would be present anywhere within our galaxy. We dis-cuss only the spectral region covered by the UBV system with somedigression into the near-infrared.

Background light from faint stars and galaxies is probably the lim-iting factor in photometry of faint sources. Miller31 notes that the skybackground can vary over distances of a few arc minutes. This meansthat if you observe a star and then offset to another location to measureonly the sky, the two sky values may not agree. However, in most casesthe light from these background stars and galaxies will not be importantunless the program object is very faint and difficult to observe even ifyou are using a very large telescope. In addition, these contributors arestatic; if you always offset to the same location to measure sky, the con-tribution from these faint sources will always be the same.

Zodiacal light is caused by sunlight reflecting off dust in the plane ofthe solar system. It increases in brightness as the observer looks closerto the sun, and is always confined to the ecliptic plane. Zodiacal lightmay or may not be important as a background source in your observa-tions depending on the location of your source with respect to the sunand the ecliptic. For instance, within 50° of the sun the zodiacal lightin the ecliptic is brighter than the brightest part of the Milky Way, and


it is brighter than integrated starlight over most of the sky. The zodiacallight is relatively uniform in the range of arc minutes, follows the solarspectrum, and is highly polarized like the blue sky.

Twilight emission lines are only important for a very short time aftersunset and seldom interfere with astronomical observations. A layer inthe upper atmosphere containing sodium atoms is illuminated by thesun after sunset as seen from the earth's surface. This illuminationexcites the atoms, causing them to emit the sodium D lines (5892 A).However, only at a solar depression angle (the distance at which sun isfound below the horizon) of 7° to 10" is this emission observable. If thesun is closer to the horizon, scattering overpowers the emission; below10°, the layer is no longer illuminated. A similar case is noted for thered lines of oxygen atoms (6300, 6364 A). Both of these effects contrib-ute to errors in the V magnitude, but are important for less than anhour. Note, though, that observing in twilight carries its disadvantages.

All of the effects discussed in detail so far are minor contributors thatcause errors only for short intervals or when very faint stars, thosewhose brightness is comparable to the sky brightness, are to beobserved.

The final two terrestrial contributors can cause larger errors, bothspatial and temporal. Night airglow is the fluorescence of the atoms andmolecules in the air from photochemical excitation. It occurs primarilyin a layer about 100 kilometers above the earth and is variable, depend-ing on sky conditions, local time, latitude, season, and solar activity.There is a component that is present at most wavelengths, called thecontinuum, primarily caused by nitrous oxide and other molecules, butthe major component is caused by distinct emission lines. Both compo-nents are always present, tend to increase in brightness near the hori-zon, and are not strongly affected by geomagnetic activity. The primarylines in the airglow are atomic oxygen (5577 A), sodium (5892 A),molecular oxygen (7619, 8645 A), and hydroxyl, OH~ (mostly in thenear-infrared). All of these emissions can be fairly strong, with someobservers seeing the 5577 A structure with the unaided eye at darklocations. Peterson and KiefFaber32 present photographs of the hydroxylnear-infrared emission showing the mottled structure of the emission.As observed with infrared sensors, the night airglow can look like bandsof cirrus clouds and move across the sky. Therefore, at least in the Vfilter and certainly in any filter redward of this, the airglow is a variablethat always reduces the consistency of the measurements during anygiven night.


The aurora again occurs with the same mechanisms and altitudes asairglow, but varies with the solar cycle. The primary excitation mech-anism is incoming charged particles from the sun. These particlesbecome trapped in the geomagnetic field and spiral towards the poles,where they excite the atoms and molecules in the air. These polar auro-ras differ from airglow primarily in their strength, being up to severalhundred times brighter, and their higher degree of excitation. The mainauroral lines are atomic oxygen (5577, 6300, 6364 A), hydrogen (6563A), and the red molecular nitrogen bands. These lines vary accordingto three factors: (1) solar activity, during which aurora occur more oftennear sunspot maxima when flares are more common; (2) latitude,because the particles concentrate near the poles, most of the radiationoccurs there; and (3) time of year, peaking in March and October. Oursuggestion is not to observe if an aurora is visible at your site. Thoughphotometry can be accomplished, auroras vary rapidly in intensity anddirection. Special techniques are necessary to achieve accurate results.This is one of the reasons few observatories are located above the ArcticCircle or in auroral zones. Myrabe33 gives an example of the possibleresults when a chopping technique is used to remove the effects of abright aurora that can vary five magnitudes within 1 minute.

The conclusions of this section are threefold. First, the light of thenight sky is not constant in time or space, and therefore will alwayslimit the accuracy of your measurements. Do not expect ± Om.001results! Second, do not observe during an aurora, near the horizon, orat twilight. And last, measurements redward of the V filter are stronglyaffected by the varying night sky and should be avoided until experienceis gained with the VBV system. Except for auroras or while observingvery faint stars, the night sky variations will probably never be notice-able in your photometry, but knowledge of the possibility of these errorsshould be filed in your mind for later reference.

9.7 YOUR FIRST NIGHT AT THE TELESCOPE

For the newcomer to photometry, the previous sections may seem help-ful but somehow cloud the answer to the question "What do I do at thetelescope?" In this section, we attempt to pull together the concepts ofearlier chapters to outline some procedures to follow at the telescope.

It is important to be prepared before going to the telescope. Unlessyou plan to observe some very bright objects, you should prepare a setof finding charts in order to identify your "targets" for the night. It is


important to mark the charts to indicate the orientation and the size ofthe field of view of your guide telescope or wide-field eyepiece of thephotometer. This can save hours of frustration when you try to identifythe star fields at the telescope. If you are doing differential photometry,be sure the comparison and check stars are marked on the chart inaddition to the variable star. If you are using a pulse-counting system,you should determine the dead-time coefficient as outlined in Section4.2. For a DC photometer, you should calibrate the gain settings asdescribed in Section 8.6.

It is a good idea to arrive at the telescope early to uncap the tube andlet the optics adjust to the outdoor temperature. All the electronicsshould be turned on at least 60 minutes prior to observing. This is nec-essary to allow the electronics time to stabilize. DC amplifiers, forexample, tend to drift rapidly for the first few minutes after they areturned on. The high voltage should also be applied to the photomulti-plier tube (with the dark slide closed) because the tube tends to benoisier than normal during the first few minutes of operation. If yourphotometer uses a cooled detector, this is the time to turn on the coolingsystem or to add dry ice to the cold box.

If you are using a photomultiplier tube, the next step is very impor-tant. In order for the photometer's optical elements to be in the properlocation within the telescope's light cone, it is important that the dia-phragm be positioned in the focal plane of the telescope. To do this, firstaim the telescope at some bright background (such as the observatorywall, or the twilight sky) so that the outline of the diaphragm can beseen clearly in the diaphragm eyepiece. This can also be accomplishedwith a diaphragm light. This eyepiece is then focused to make the dia-phragm as clear as possible. Do not adjust this eyepiece focus again therest of the night. Next, a bright star is centered in the diaphragm andfocused sharply using the telescope focus. Now both the star and thediaphragm appear sharp when viewed in the diaphragm eyepiece. Thediaphragm is now in the focal plane of the telescope.

This procedure is unnecessary for a photodiode photometer. In thiscase, the photometer head does not have a diaphragm. The head is con-structed so that when the stellar image appears focused in the eyepiece,it is also focused on the photodiode when the flip mirror is removed fromthe light path. The set-up procedure is concluded by choosing the dia-phragm to be used (see Section 9.4), setting the telescope coordinates,and setting the observatory clock.


There are some important comments to be made about the art of datataking at the telescope. Making a stellar measurement with a pulse-counting system is the simplest. The star is centered in the diaphragm,the dark slide is opened, and the counts are recorded. For this system,use the advice of Section 9.3a to estimate the total observing timethrough each filter. Measurements through the red-leak filter shouldhave the same duration as the U filter. If you are using a DC system,some additional advice is in order. The DC amplifier zero point maydrift slightly during the night. This drift may be either positive or neg-ative. Most chart recorders and meters do not follow a negative drift offthe bottom of the scale. For this reason, it is a good idea to put the zeropoint at about 10 percent of full scale. To do this, simply close the darkslide (with the high voltage on) and adjust the "zero adjustment" knobon the DC amplifier. Chart recorders and meters are most accurate attheir maximum (full-scale) reading. Therefore, whenever possible,adjust the amplifier gain so that a stellar measurement gives a nearlyfull-scale reading.

There are three additional rules to follow when doing DC photome-try. When you measure a star for the first time, be sure the amplifiergain is at its lowest setting. You then increase the gain to achieve thedesired meter deflection. This procedure is designed to protect the meterand/or chart recorder from damage if too high a gain setting is used.If this happens, the pen or meter needle will slam beyond full scale.With a little practice, you can estimate a safe gain setting simply byobserving the apparent brightness of the star in the eyepiece. The sec-ond rule is to use the same gain settings for the sky measurements aswas used for the stellar measurements. A much higher gain may beused for the U measurements than is used for B or K The U sky back-ground measurement also must be made at this higher gain setting. Thisis obviously necessary if we are to subtract the sky measurement fromthe stellar measurement directly. However, this is also necessarybecause it is possible for the amplifier zero point to change slightly withdifferent gain settings. The third rule is to record the coarse and finegain settings separately and not their total. The reason is that each gainposition requires a calibration correction as discussed in Section 8.6.More than one combination of switch positions can yield the same total.If your records only contain the total gain, you will not be able to deter-mine the proper gain correction without ambiguity.

The observing pattern used at the telescope depends strongly on the

Before Going to the Telescope:Find dead time coefficient, if pulse counting (Section 4.2 and Appendix F).Calibrate amplifier gains, if DC (Section 8.7).Make finding charts (Section 9.1).

Set Up:Turn equipment on one hour early.Uncap telescope so it thermally stabilizes.(Ice PM tube if you have a cold box).Find bright star to set telescope coordinates and to focus photometer (Section 9.71.Set clock.

Track 1: Differential Photometry Track 2. Determination of Transformation Coefficients Track 3: All Sky Photometry

^Instrumental System

tMake test measurementof comp. and var. siarto estimate S/N andlength of an observation(Sections 9.3a, 9.3b).

*Decide on block length(Section 9.3c) andpercent of time to bespent on sky(Section 9.3a).

|

Measure check starand begin observingpattern (Section 9.3c).

tData Reduction(Section 4.81 .

Standard System

*Follow Track 2 iftransformation coefficientsunknown. Then makeseveral high S/Nmeasurements ofcomparison star. Followreduction procedure ofTrack 3 to obtainstandardized magnitudesand colors of comparisonstar.

+Choose a set of AO, and standard stars(or cluster), (Section 4.4, AppendicesA and C or D).

tObserve each star with S/N > 100(Section 9.3b or 9.3c). Record timeor hour angle of each observation.

tIf second order extinction coefficientunknown, measure an extinction pairat several air masses (Section 4 4cand Appendix B).

!Data Reduction(Appendices G and H|.

If transformation coefficients unknown,follow Track 2 first.

*Choose a set of standard stars(Appendix C).

IIntermix measurements of standardsand variables keeping S/N > 100(Section 9.3b or 9.3c). Record timeor hour angle of each observation.

tData Reduction:1. Calculate instrumental magnitudes and

colors (eqs. 4 1 1 to 4 13 or 4 14 to4-16).

2. Correct for extinction (Appendix Gand eqs. 4-22. 4-28, 4-29).

3. Calculate transformation coefficientsif not already known (Appendix H).

4. Calculate standardized magnitudesand colors (eqs. 4-32 to 4-34).

Figure 9.8. Observing flowchart.


research program and the habits of the observer. For example, onemight begin the night with simple differential photometry of a rapidvariable but after an hour or two start a program of observing cepheidvariables on the UBV system scattered all over the sky. Clearly, thiscalls for a combination of observing techniques. However, on any givennight, the observing pattern can be broken into three categories. Thefirst category is differential photometry either on the standard or theinstrumental photometric system. The second is a sequence of obser-vations designed to yield the transformation coefficients to the standardsystem. The third category is what we call "all-sky" photometry inwhich many objects are observed throughout the night, located at var-ious positions in the sky. The cepheid observations mentioned above fallinto this category. We must stress that there are several other forms ofphotometry, such as occultation or galaxy surface photometry, that donot fit these categories directly. These categories are merely the mostcommon types and they are defined to help the novice to see how thingsare done. Figure 9.8 illustrates possible observing patterns with thethree categories labeled as tracks I, 2, and 3. This figure is a flowchartthat can be followed from observing preparations to data reductions.For your reference, we have labeled the appropriate sections where adiscussion or a worked example can be found.

A final comment about data recording is in order for those using DCphotometers. If you are using the amplifier meter to make your mea-surements, it is a good idea to follow a simple pattern. As you watchthe needle, it appears to jitter above and below some mean value.Rather than watching the meter for a long stretch of time, it is betterto watch the meter for, say, 10 seconds and then estimate and recordthe mean value. This procedure should be repeated several times andaveraged to make a single measurement. When using a chart recorder,the jitter of the needle is transformed to the jitter of a pen recording onstrip-chart paper. The main advantage of the chart recorder is that youhave a permanent record of the actual observations and you can trans-form the pen tracings to numbers the next day. Figure 9.9 shows a sam-ple chart tracing to illustrate the technique of extracting the data. Eachmeasurement is estimated by drawing a straight line through the middleof the jitter. The two sets of sky measurements illustrate that our goldenrule of differential photometry really applies to all photometry. Bydrawing a line connecting the sky measurements, we can interpolate thesky background to the time of each stellar measurement. In this exam-


100

90

80

70

60

50

40

30

20

10

013.5

Figure 9.9. Typical chart tracing.

pie, the sky background was increasing because of a rising moon. Notethat if we had made only the first sky measurement, it would haveappeared that the star was brightening steadily. With the two sets ofsky measurements and the graphical interpolation, you can see that thestar was essentially constant once the sky background is subtracted.

We wish you good luck on your first night of photometry.

REFERENCES

1. Larson, W. J. 1978. Sky and Tel. 56, 507.2. Argelander, F. W. A. and Schonfeld, E. 1859-1886. Astronomiche Beobachten

Sternwarte Konigl. 3, 4, 5, 8.3. Thome, J. M. and Perrine, C. D., 1892-1932. Resultados Observatories National

Argentina 16, 17, 18, 21.4. Staff, 1969. The Smithsonian Astrophysical Observatory Star Atlas. Cambridge:

MIT Press.5. Becvar, A., 1964. Atlas Borealis, Eclipticalis, and Australis. Cambridge: Sky

Publishing Co.6. Ingrao, H. C. and Kasperian, E. 1967. Sky and Tel. 34, 284.7. Vehrenberg, H. 1963. Photographic Star Atlas. DUsseldorf: Treugesell-Verlag

KG.8. Vehrenberg, H., 1970. Atlas Stellarum 1950.0. DUsseldorf: Treugesell-Verlag

KG.9. Papadopoulos, C. 1979. True Visual Magnitude Photographic Star Atlas. Elms-

ford, NY: Pergamon.10. Lick Observatory 1965. Lick Observatory Sky Atlas. Mt. Hamilton, California.11. Doughty, N. A., Shane, C. O., and Wood, F. B. 1972. Canterbury Sky Atlas

(Australis). New Zealand: Mount John University Observatory.


12. Palomar Observatory, 1954. National Geographic-Palomar Observatory SkySurvey. Pasadena: California Institute of Technology.

13. ESO/SRC. Atlas of the Southern Sky. In preparation.14. Overbye, D. 1979. Sky and Tel. 58, 30.15. Kukarkin, B. V., Kholopov, P. N., Efremov, Yu. N., Kukarakina, N. P., Kuro-

chkin, N. E., Medvedera, G. I., Peruva, N. B., Fedorovich, V. P., and Frolov, M.S. 1969-1974. General Catalog of Variable Stars. Moscow: Academy of Sciencesof theU.S.S.R.

16. Scovil.C. E. 1980. AA VSO Variable Star Atlas. Cambridge: Sky Publishing Co.17. Hoffmeister, C. Mitteilungen der Sternwarte zu Sonneberg. No. 12-22 (1928-

1933), No. 245-330(1957).18. Anon., 1954. Communications of the Observatory of the University at Odessa 4,

1-3.19. Hagen, J. G. 1934. Atlas Stellarum Variabiliutn. Vatican.20. Bateson, F. M., Jones, A. F., et al. 1958-1977. Charts for Southern Variables.

Wellington.21. Odessa Universitet Observatoriia Izvestiia, 1953, 1954, 1955, vol. IV, parts I, II,

III.22. Code 680, Goddard Space Flight Center, Greenbelt, MD 2077123. Cannon, A. J., and Pickering, E. C. 1918-1924. Henry Draper Caialog and

Extensions, Harvard Annals 91-100.24. Danford, S., and Muir, R., 1978. Bui. Amer. Astr. Soc. 10, 461.25. Houk, N. and Cowley, A. P. 1975. University of Michigan Catalogue of Two

Dimensional Spectral Types for the HD Stars. Ann Arbor: Univ. of MichiganPress, vol. 1.

26. Overbye, D. 1979. Sky and Tel. 57, 223.27. Young, A. T. 1974. In Methods of Experimental Physics: Astrophysics, vol. 12A.

Edited by N. Carleton. New York: Academic Press.28. King, I. R. 1971. Pub. A. S. P. 83, 199.29. Picarillo, J. 1973. Pub. A. S. P. 85, 278.30. Roach, F. E., and Gordon, J. L. 1973. The Light of the Night Sky. Boston: D.

Reidel.31. Miller, R. H. 1963. Ap. J. 137, 1049.32. Peterson, A. W. and Kieffaber, L., 1973. Sky and Tel. 46, 338.33. Myraba, H. K. 1978. Observatory 98, 234.

CHAPTER 10APPLICATIONS OF PHOTOELECTRIC

PHOTOMETRY

By now, you probably have built or purchased a photoelectric photom-eter and have measured the magnitudes of a few bright stars. You mayrightfully wonder what to do next! Happily, this is the least of yourworries, as there are thousands of interesting stars and projects withinthe range of most amateur and small college telescopes.

In this chapter, we present a few of these projects. By no means isthis an exhaustive compendium and we invite you to branch out andpursue other topics. However, the ideas mentioned here should give youa feeling of the breadth and diversity of the field of photometry. Notonly can it be as fun and interesting as photography or visual observing,but you can help in the advancement of scientific knowledge.

10.1 PHOTOMETRIC SEQUENCES

Are you unselfish and like to share your work? Then determining pho-tometric sequences should be right up your alley! The object is to deter-mine the magnitudes of several standard stars in a field surrounding ornear some interesting source. These sequences have to be determinedcarefully, as many observers following in your footsteps will be usingyour data as the foundation for their own research. For the purposes ofthis section, we identify three types of comparison star sequences: forvisual observers, for photoelectric observers, and for calibrating photo-graphic plates.

Visual variable star observers have two primary advantages over pho-toelectric observers: they are less influenced by light cloud cover

238

APPLICATIONS OF PHOTOELECTRIC PHOTOMETRY 239

because the comparison is made almost instantaneously, and they canwork faster because the object's light does not have to be centered in adiaphragm. For these reasons, among others, visual photometry stillflourishes in the ranks of amateur astronomers. However, the compar-ison star sequences around visual variables are generally not very accu-rate. The stars should cluster uniformly in space and brightness aroundthe variable, appear similar in color, and have their magnitudes mea-sured in a filter bandpass approximating the response of the unaidedeye. This latter requirement has been difficult to achieve. Landis1 men-tions that the V filter does not match the eye's response, as blue starslook fainter and red stars brighter in V than with the eye. Stanton2

experimented with filter responses and found that a Schott GG4 filteralong with an EMI 9789B (S-ll cathode) matched the eye's responsebest. Each visual observer's eye has a slightly different response becauseof inherent and dark adaptation variations. We suggest that you coop-erate with the American Association of Variable Star Observers(AAVSO)3 and help them in determining photoelectric sequences inpoorly defined fields and in finding good filter-photometer combinationsfor visual magnitude sequences.

Photoelectric observers need fewer comparison stars per variable thando visual observers, but the selection of suitable comparisons can bevery tedious. The color (especially) and magnitude should match asclosely as possible, and the star should be within 1" of the variable.Unlike visual observers, where 0.1 magnitude accuracy is acceptable,photoelectric observers need to determine their variables to 0.01 mag-nitude or better. This means the comparison stars must be at least thisstable. Two stars should be located and measured on several occasionsto ascertain whether they are truly constant in magnitude. It is difficultto obtain comparison stars for red variables because a large fraction ofcool stars are variable, often with long periods, so that they may beconstant during a few night's observations. Try to determine compari-son stars for a group of variables, such as bright flare stars, cepheidswith 5-day periods, etc. Another project is to reobserve comparison starsused in the literature to see if they remain constant or are long-periodvariables. Check with a local professional astronomer or the AAVSOfor additional or more specific ideas.

Photographic photometry is still valuable today because of its greatmultiplexing advantages. By calibrating a cluster plate accurately, forexample, you can measure the brightness of many hundreds of stars in


a few hours. In crowded fields, photographic photometry may be theonly practical method of determining magnitudes. It is also much fasterin measuring faint stars than photoelectric methods. Calibrating pho-tographic plates is impossible unless a good photoelectric sequenceappears in the plate field. This sequence should cover as wide a mag-nitude range as possible, and be obtained in the filter bandpass of theplate, which is usually blue. The plate calibration of the large surveyssuch as the National Geographic-Palomar Sky Survey is particularlyimportant, as the Space Telescope will require accurate magnitudesover the entire sky, obtainable only from such surveys. Argue et al.4

have gathered the known photoelectric sequences, which are usuallyaround galactic and globular clusters. Complete coverage of the sky islacking, though, and amateur observations are particularly needed hereto take some of the burden from the professional observatories. Thereare over 1000 fields in the National Geographic-Palomar Sky Surveyalone, each requiring 10 to 20 sequence stars to fifteenth visual mag-nitude for the fine guidance system of the Space Telescope. You can seethat there is plenty of opportunity here for your photometer to feelneeded!

10.2 MONITORING FLARE STARS

In 1924, the astronomer E. Hertzsprung5 was photographing an area ofsky in the constellation of Carina. He discovered that on one photographa star was two magnitudes brighter than it had been on previous pho-tographs. The rate at which the star brightened was much too fast tobe either a nova or any kind of intrinsic variable then known. Hertz-sprung tried to explain the event as the result of an asteroid falling intoa star. This is the first known observation of what we now call a flarestar. Later, this same star was seen to flare many more times and wasgiven the variable star designation DH Car. In the years that followed,other M-type dwarfs were seen to flare but they did not receive recog-nition as a new class of variable stars until 20 years after Hertzsprung'sinitial observation.

In 1947 the American astronomer Carpenter (Luyten6) was photo-graphing a red dwarf. This star was known to have a large propermotion, which implied that it was nearby. Carpenter was attempting tomeasure its parallax. On a single plate, he took a series of five 4-minute


exposures, moving the telescope slightly after each one. When the platewas developed, he expected to see five equally bright images. But thesecond image was much brighter than the first. The star faded in thelater images, returning to its normal brightness. The star had become12 times brighter in about 3 minutes. This rate of increase is even fasterthan a supernova, though the total luminosity is much less. This star isnow called UV Ceti and flare stars are now commonly referred to asW Ceti variables.

The first photoelectric recording of a flare was obtained by Gordonand Kron7 in 1949. They were observing a late-type eclipsing binarywith the 0.9-meter (36-inch) refractor at Lick Observatory. They weremeasuring the comparison star when the needle on the DC amplifierwent off the top of the scale. After several frantic minutes of experi-mentation, they realized their equipment was not malfunctioning andthe comparison star had flared. They were able to observe the starslowly return to its normal brightness. This star is known today as ADLeo. This story simultaneously points out the sudden nature of flare stareruptions and one of the dangers of using a red comparison star!

Flare stars are usually M-type dwarfs that have emission lines intheir spectra. K-type stars occasionally have been seen to flare but theydo so much less frequently. Flare stars are typically one-twentieth toone-half the mass of the sun and only 10~5 times as luminous. They areso intrinsically faint that they must be nearby to be seen at all. Theirsudden increases in luminosity have been likened to solar flares, onlythousands of times stronger. These are very remarkable events for suchsmall stars. Flare amplitudes can range from a few tenths of a magni-tude or less to several magnitudes. There are two general types of flares.A spike flare increases suddenly with a rise time of only a few secondsand then fades in a few minutes. A slow flare brightens gradually overan interval of minutes to tens of minutes and then fades at about thesame rate.

However, these classifications should not be taken too literally. Manyflares are a combination of both types and there is great variability fromflare to flare even in the same star. A more complete description of theseinteresting objects is beyond the scope of this section. The interestedreader is referred to the work of Moffett,8 Lovell,9 Gershberg,10 andGurzadyan."

The monitoring of flare stars can only be recommended to the begin-ner with some reluctance. Catching a flare photoelectrically can be


every bit as exciting today as it was for Gordon and Kron in 1949. How-ever, the observer must be prepared for possibly spending many tens ofhours at the telescope and recording nothing but a constant star. Flarescannot be predicted, and catching one is a matter of looking in the rightplace at the right time. For those who cannot resist the chance ofactually seeing a flare, we supply a list of flare stars, brighter thantwelfth visual magnitude, in Table 10.1. Finder charts for some of thesestars are available from the AAVSO at a modest cost.

Besides the requirement of patience, there are certain requirementsof your equipment. Unlike normal photometry, in flare star monitoringyou measure the same star for hours at a time, interrupted occasionallyby a measurement of the sky background. It is therefore important thatyour telescope can track well and keep the star centered in the dia-phragm as long as possible. Because the flare may occur very rapidlyand at some unknown time, the observations are usually made in a sin-gle filter. Flares are brightest in the ultraviolet, so an ultraviolet or bluefilter is preferred. Pulse-counting photometers cannot be used unless alarge digital memory that can store hundreds or thousands of measure-ments is available. Such a system has been described by Warner andNather12 and Nather.13 This instrumentation is fairly elaborate and amuch simpler system for the beginner is a DC amplifier and a strip-chart recorder. If the telescope tracks well, the equipment can be leftunattended for small intervals of time and the chart recorder will pre-serve a record of any flare activity. It is important to use the shortestpossible amplifier time constant. This is necessary so the amplifier canfollow any rapid flickering during the flare event. Unlike the normalprocedure, it is a good idea to adjust the amplifier gain so that the stardoes not read nearly full scale. After all, if the star flares it would behelpful if the amplifier and chart recorder do not go off the top of thescale.

As mentioned above, catching a flare involves a lot of luck. One ofthe authors (RHK) recalls accumulating over 23 hours of monitoringof a flare star only to collect what seemed like miles of chart paper witha constant ink line. In utter frustration, the telescope was moved to theflare star EV Lac. Within the first hour of monitoring the spike flareshown in Figure 10.1 was recorded. Note that this flare lasted only 82seconds. It would have been very unfortunate if during those 82 secondsthe sky background was measured instead!

TABLE 10.1.

(1950.0)Star

BD +43° 44CQAndBD + 66°34Butler's StarLPM63CCEri40 Eri CRoss 42V371 OriBD -21 '1377Ross 614

PZMonAC + 38*23616YYGemYZCMiBD + 33M646BAD LeoSZUMaRoss 128DT VirEQHerV645 CenDM +16° 2708DM +55° 1 823VI 054 OphRoss 868BYDraV1216Sgr

RA

00*15.5"00 15.500 29.30058.101 09.902 32.503 13.105 29.50531.206 08.506 26.80645.80706.70731.60742.108 05.71016-911 17.51 1 45.21258.31332.114 26.31452.116 16-01652.817 17.91832.718 46.8

Dec.

43 '441443 44.466 57.8

-73 13.4-17 16.0-44 00.6-07 44.1

09 47.301 54.8

-21 50.6-02 46.2

01 16.638 37.531 58.803 40.832 56.020 07.366 07.001 01.012 38.7

-08 05.1-62 28.1

16 18.355 23.8

-08 14.726 32.851 41.0

-23 53.5

Flare Stars

(1985.0)RA

OOM7.3"00 17.30031.300 59.201 11.602 33.803 14.805 31.405 33.006 10.006 28.506 47.60709.107 33.607 43.908 07.910 18.811 19.61 1 47.01300.013 33.914 29.01453.716 16.816 54.717 19.318 33.51848.9

Dec.

43° 560)43 56.067 09.3

-73 02.0-17 04.9-43 51.4-07 36.3

09 48.801 56.2

-21 51.0—02 47.6

01 14.238 34.131 54.203 35.732 49.819 56.765 55.500 49.312 27.4

-08 15.8-62 37.4

16 09.855 18-7

-08 18.026 30.751 42.7

-23 51.1

Vis. Mag. Comments

Visual doublel l . O j10.510.611.68.7

11.211.511.78.1

11. 110.811.59.1

11.211.09.49.3

11.19.89.3

11.110.2JO.O ,

I Visual Companion, Ross 867,also flare star (12.9 mag.)

8.610.6 '

TABLE 10.1.. Flare Stars (continued)

Star

V1285 AqlWOLF 1130ATMicAU MicAC +39 '57322BD +56' 2783L7 17-22EVLacDM +19-5116BD -4-T4774

(1950.0)RA Dec.

1853.019 20.120 38.72042.12058.122 26.222 36.022 44.723 29.523 46.6

08 20.354 18.2

-32 36.6-31 31.1

39 52.757 26.8

-20 52.844 04.619 39.702 08.2

(1985.0)RA Dec.

1854.719 20.920 40.920 44.220 59.422 27.522 37.922 46.22331.223 48.4

08 26.054 22.2

-32 29.1-31 23.5

40 00.957 37,5

-20 41,944 15.519 51.302 19.9

Vis. Mag.

10.111.910.88.6

10.39.9

11.510.210.49.0

Comments

Visual Companion, DO Ccp,[ also flare star (13.3 mag.)


EV LACERTAE9/27/74

START OF FLARE3:05:49 UT

DURATION 82 SEC.AMAG. = 1.11

1.0 MAG

TNiUAV^n^^V^A^v

15 SEC.

Figure 10.1. Flare of EV Lacertae.

10.3 OCCULTATION PHOTOMETRY

Occultations are among the oldest astronomical observations. With theadvent of modern photoelectric photometers, chart recorders, and accu-rate timekeeping via WWV, amateur astronomers now have the meansto make high-quality observations of occultation events. As more andmore observers couple microcomputers to their equipment, the limit tocomplex occultation projects will depend only on the ingenuity of theastronomer. Space does not permit a thorough coverage of the methodsof observing occupations. Instead, the reader is urged strongly to studythe appropriate references in this section for a more in-depth treatment.

This section describes several types of occultation observations.Occultation observers routinely seek information on the figure of thelunar limb, the separation of otherwise unresolvable binary stars, thediameter of an asteroid, or the angular size of a star. No matter whatsort of occultation project is undertaken, one theme underlies each ofthese observations: occupations yield very high angular resolution.Observers can resolve angles as small as a few arc milliseconds. Mea-suring, say, 5 arc milliseconds is like determining the angular size of anorange at a distance of 3200 kilometers (1920 miles). The reason forthis greatly enhanced resolution over the Dawes limit is that you no


longer use the telescope's optics to do the resolving. Instead, you takeadvantage of the geometry of the occultation by using the separationbetween your earth-bound location and the occulting body as a sort of"interplanetary optical bench" in probing the object of interest.

Requirements for observing occultations vary with the complexity ofthe project. For many observations, only a strip-chart recorder and anaccurate timepiece are needed. In order to obtain the high resolutionafforded by occultations, observations must be made at high time res-olution. Instrumentation such as that described by McGraw et al.14 pro-vides a great deal of versatility and high time resolution. Integrationsmust be less than 1 second and should be somewhat less for some of theprojects described. This may be accomplished by running the chartrecorder at high speed or for those with microcomputers, reading pho-tometer counts into a "circulating" memory at a rate controlled by thecomputer. By use of microcomputers, a time resolution of 0.001 secondis not difficult to obtain. This high time resolution is necessary forangular diameter studies of stars.

Grazing lunar occultations have long been of interest to amateurastronomers. Amateurs have banded themselves into groups that packup their portable equipment, travel to "graze lines," and through coop-erative efforts obtain lunar limb profiles. Many graze observers obtainvery good results with little more than a stopwatch and telescope. Useof photometry for the quick disappearances and reappearances char-acteristic of grazes yields an unbiased, more accurate recording ofgrazes. Harold Povenmire15 has devoted considerable effort to the obser-vation of grazing occultations.

From time to time, the planets occult stars. During such events,observers are afforded the chance to obtain accurate diameters of theseobjects or information on their upper atmospheres. Rather startling dis-coveries have recently come from occultations of stars by planets. Thediscovery of the Uranian rings in 1977 by Elliot et al.16 was made byoccultation. Almost all information on those rings continues to be fromoccultation observations. In April 1980, Pluto nearly occulted a star.Such an observation would be extremely valuable as a measure ofPluto's diameter; no such occultation has yet been reported. During that1980 appulse, Pluto's moon Chiron did occult the star as seen by oneobserver, A. R. Walker17 in South Africa. An upper limit of 1200 kilo-meters for the diameter was deduced from the observation. Occultationsof stars by planets is still fertile ground for new discoveries.


In the past few years, astrometric techniques have been pushedtoward the goal of providing better predictions of occultations by aster-oids. Accurate predictions are important because the path of observa-bility is equal to the diameter of the occulting body, and asteroids aregenerally less than a few hundred kilometers across. Results from thesemeasurements are impressive. Often, diameter determinations are accu-rate to within 1 percent.

The observations just described do not require microcomputer controlof the occupation event. However, as more observers couple computersto their telescopes, it is probably only a matter of time before an intrepidamateur attempts to determine a stellar angular diameter. During alunar occultation, the light from a star is diffracted by the lunar limb.What the photometer "sees" during the occultation is the passage of thefresnel diffraction fringes: an oscillation of the stellar flux just prior tooccultation. By analysis of the diffraction fringe spacing and height, theangular size of the occulted star may be determined. References con-cerning aspects of lunar occultations may be found in a two-articlesequence by Evans.18-19 A more rigorous development of the topic maybe found in Nather and Evans,20 Nather,21 Evans,22 and Nather andMcCants.23 The fresnel diffraction pattern is only seen at high time res-olution, that is, integrations of around 1 millisecond.

Related to occultations are mutual phenomena of planetary satellites,that is the mutual occultations, eclipses, and transits of satellites duringthe time the nodes of their orbital planes align with the earth. Accuratetiming of these events can give accurate shapes and sizes of thesesatellites.

Predictions of the described occultations may be found from severalsources such as Sky and Telescope, the Astronomical Almanac.2" or theObserver's Handbook,25 to name a few. Ambitious astronomers maywish to try their hand at predicting occultations for their particularlocation. These readers are referred to Smart's Spherical Astronomy,26

where an excellent treatment may be found. Amateurs interested in anypart of occultation observing are urged to join the International Occul-tation Timing Association.27 IOTA provides excellent predictions, infor-mation, and hints on observing all types of occultations. Furthermore,this organization serves as a clearing house for observations of almostall types of occultations.

[Note: Section 10.3 was contributed by T. L. Mullikin, Space Oper-ations and Satellite Systems Development, Rockwell International.]


10.4 INTRINSIC VARIABLES

Intrinsic variables are those stars that vary in brightness because ofinternal changes. Normally, this is evider.ced by pulsational behavior:the star periodically shrinks and expands. Sometimes the light varia-tions are highly regular, as in the case of cepheids and RR Lyrae vari-ables; sometimes it is quite erratic, as for Mira and RV Tauri variables.

A thorough description of the characteristics and pulsational mech-anisms for these stars would occupy more space than this text. There-fore, we refer the interested reader to Glasby,28 Strohmeier,29 andKukarkin30 for more detail. The major pcint of interest in this sectionis the fact that there are thousands of intrinsic variables, each oneunique though they are placed in general categories. Astronomers haveneither the manpower nor the telescope time to investigate even amajority of these stars with the thoroughness they deserve. The energiesof the amateur variable star observer should be channeled in the follow-ing directions:

1. Determining standard comparison stars and preparing findingcharts for those variables where no charts exist.

2. Obtaining light curves for stars with missing or incomplete curves.3. Observing a variable star at an epoch several years removed from

previous measurements, in order to detect temporal variations.4. Investigating stars mentioned in the Catalog of Suspected Vari-

ables^ or other publications to determine their nature and period.

Two rules should be followed for accurate data: use comparison starsfor all measures, and obtain as many observations as possible, coveringthe entire light curve.

In this section, several tables of stars are presented. In each case, the1950 coordinates are as accurate as could be found. Those variables forwhich coordinates are less accurate can be identified by their 1950 dec-lination, where the arc seconds are multiples of six. The magnitudes area general range, with the particular color indicated with a letter: P forphotographic, B for the B filter of the t/BX system, and Kfor the visual.Each list contains approximately 30 bright variables in each periodrange, as culled from the magnetic tape version of the General Catalogof Variable Stars. For those of you with access to large computer facil-ities, we highly recommend acquiring a copy of this magnetic tape


through Goddard Space Flight Center in Greenbelt, Maryland.32

Another good source of bright variables is the Atlas of the HeavensCatalog,™ which has 13 pages of variables with 1950 coordinates.

Note that as the periods of the stars mentioned in this sectionincrease, the accuracy with which they are determined decreases. Thisis because photoelectric data are missing on most of the long-periodvariables. They have not been observed for nearly as many periods asthose variables with short cycles. The use of the terms epoch and periodis explained in Section 10.5.

10.4a Short-Period Variables

As with the other classifications, as soon as researchers find more thanone star with similar characteristics, they invent a new group of vari-ables. The short-period intrinsic variables have therefore been dividedinto four major groups:

1. 8 Scuti stars. These have low amplitude, sinusoidal light curves.Periods are less than 0.3 day.

2. Dwarf cepheids. These have larger amplitudes (though less thanone magnitude), varied light curve shapes (either sinusoidal orasymmetric with a faster rise to maximum brightness than theensuing decline), and periods also less than 0.3 day.

3. RR Lyrae variables. Named after their prototype, these are sim-ilar to dwarf cepheids but with periods ranging between 0.3 and1.0 day. They are commonly called cluster variables because oftheir affinity to globular clusters.

4. Cepheids. Again similar to dwarf cepheids, these variables haveperiods ranging between 1 and approximately 50 days.

Eggen34 and Tsesevich35 give more information on these stars. For theamateur, 6 Scuti stars are very difficult to observe as the amplitude isseldom greater than 0.10 magnitude in the visual. Therefore, we con-centrate on the latter three groups.

Dwarf cepheids brighter than twelfth visual magnitude and with aAKof 0.5 or less have been listed by Percy et al." The coordinates thatthey give are for 1900.0 and will have to be precessed to be usable.Fourteen stars are included on their list. A complete list (but withoutcoordinates) is given by Eggen,34 including all 5 Scuti and dwarf


cepheids known to date that are brighter than visual magnitude 11.5 atmaximum. Coordinates can be found in Eggen's references or by usingthe BD or HD catalogs.

No exclusive catalog exists for RR Lyrae variables. Tsesevich35 pre-sents older visual observations for several hundred RR Lyrae stars andthis work should be reviewed by the interested reader. A complete cat-alog of the several thousand Magellanic Cloud RR Lyrae variables hasbeen published by the Gaposchkins,37-38 but these stars are too faint andin overcrowded fields for photoelectric work by all but the large profes-sional telescopes. Sturch39 lists photoelectric observations on 100 fieldRR Lyrae stars.

Table 10.2 lists the magnitudes and coordinates for selected veryshort-period (less than 1 day) variables. Table 10.2 is by no means com-plete, but can be used as a stepping stone to more difficult objects.

Because of their relative rarity and larger luminosity, cepheid vari-ables have been cataloged more carefully than the other variables.Schaltenbrand and Tammann40 give a complete list of those cepheidswith photoelectric data. Henden41'42 gives an extensive list of short-period (less than 5 days) cepheids that are visible from the NorthernHemisphere. Table 10.3 lists a representative sample of cepheid vari-ables with periods less than 10 days.

10.4b Medium-period Variables

These stars are classified primarily as RV Tauri stars or long-periodcepheids. Eggen43 gives a review of these stars whose periods range from30 to 100 days or so. Most of the long-period cepheids are found in theSmall Magellanic or Large Magellanic Cloud. Very little data exist onthese variables because of their rarity and long periods. These and thelonger period Miras are prime candidates for dedicated amateurs astheir periods are too long for good coverage from national observatories.RV Tauri light curves are unstable and show irregularities in bothshape and period. Like RR Lyrae stars, they belong to the old galactichalo star population type. The long-period cepheids have not been stud-ied extensively, and in some cases, such as RU Cam, they have quitpulsating for extended periods of time. Representative members of bothof these classes with periods between 10 to 100 days are shown in Table10.4.

TABLE 10.2. Intrinsic Variables, 0.1 < P < 1.0 Days

t 1950.0)NAME

BS AQRON AQRX ARIVZ CNCAD CMIV743 CENRZ CEPRR GETXZ CETXZ CYGDX DELSU DRAXZ DRARX ERISV ERICS ERISS FORRS GRUV 1NDTT LYNRR LVRTY MENV429 ORIDH PEGRU PSCV440 S6RV703 SCOMT TELTU UMAAI VEL

2323387132211

192011194322

21218

1954

221

191718118

R,461653850253729573145359

479

355

3985923405312112939582712

,A. DEC.11.628.548.09.5

11.717. 727.934.052.627.45.06.824.428.527.010.936.148.211 .049.052. 138. 143.055.042.020.00.63# . 89.726.2

- 8-2410Iff1

-51641

-1656126764

-15-11-43-27-48-454442

-81- 3624

-23-32-4630

-44

2533150431

355

35161636464932106

251647413836349

5730432025

24.28,23.10.39.58.42.12.15.47.42.27.33.35.36.47.5.6.42.6.

11.19.30.12.6.36.0.27.37.21 .

57841560

2209130

3955a910000037e

23233e7

132211

192011194322

212191954221

191719118

i: 1985.0}R.A.471874052273831593246379

491136742102

24375514133141I

2913

591941302640213310446

373730115309

599274036261761

35

DEC.- 8-241091

-51641

-1656126764

-15-11-43-26-48-454442

-81- 3624-23-32-4630-44

1321235238124615252124245045241

5615a3845373344205331409

31

44592541125039595202649259424183064724141239126026246

TYPERRSRRABRRABRRSRRSRRSRRRRABRRRRABRRABRRABRRABRRABRRABRRCRRABRRSRRABRRABRRABRRSRRABRRCRRCRRABRRSRRCRRABRRS

91097989989999998979977991097a96

MAG..4-10.,0-10,.2-10..5- 8..1- 9..3- 8..5-10..3-10..5- 9.. 1-10..5-10..2-10..6-10..2-10..6-10..7- 9..5-10..9- 8..1-10..5-10..2- 8.,7- 7..0-10..3- 9..0-10..8-11..8- 8..7- 9..3-10..4- 7.

EPOCH0B5PSB2f>4B5P3B3P2P5B3V2V6VIV2V2V6V5V5V2V6B9P0v8V4V3B5B3V2VIV

28095.28425.37583.33631.36601.

-38207.17501 .

-36933.30950.20605.27985.21692.28398.-

3866B.34325.27993.36651.38241 .38314.28876.38251 .24057.37526.37186.38479.38510.

-

33028456884618228

9384421

9815067569648479200

9512931567358460493413872945324365332756

PERIOD0.197822780.634640.6511390.178363760. 1229740.1040.3086450.55302530.4510.4665790.472616730.660419260.47649440.587246220.71375900.3113310.4954320.147011470.4795910.59743790.56680540.187470.50170.2555100.39031740.4774740.115217890.3168990.5576590.11157396

TABLE 10,3. Intrinsic Variables, 1.0 < P < 10.0 Days

(1950.0)NAME

FF AQLV1162 AQLETA AQLV CARGI CARTU CASV381 CENV4I9 CENV553 CENV659 CENDELTA CEPAX CIRV1334 CVGTX DELBETA DORW GEMGH LUPV5Z6 MONAU PEG16 PSAAP PUPS SGEV636 SCOST TAUSZ TAUU TRAALFA UMIAH VELBG VELT VULa VUL

R.A.1819198

110131114132214212056156

21237

191754161892019

56494927112347284328274817473332205921056S31942342481064934

1.235.255. S42. S48.036.722.534.232.212.818,529.922.442.011.35.656. S21. 140.444.11.0

44.95.413.320.251.248.825.639.120.626.5

17-110

-59-5751-57-56-31-6158-63383

-6215

-52- 1IB

-35-3916

-451318-6289-46-512820

DEC.17295257380193757199361

27312240331

5930343326461

29143

13

32.446. Z33.217.918.113.558.422.142.029.731.817.831.6S4.820.216.438.929.949.412.714. 64.01.1

23.635.236.643.736.80.043.712.3

1819198

1101311M132214212056157

21237191754

162892019

(1985.0)R.A.57 3451 3151 4228 2513 2025 3049 4330 1245 3830 3228 3651 2118 4649 2733 2934 523 311 7

23 IS2 3957 U55 2021 3944 1236 225 58

16 1611 317 4650 4935 58

DEC.17

-110

-60-5751-57-56-32-6158-63383

-6215

-52- 118

-34-4016

-451318

-6289-46-512820

20245744911304863020441035292048612494

35363430521135221117

24205920445122573018175525455835432515356401

IS4B154856313755

TYPECDELCEPCDELCDELCEPCWcwCEPCEPCEPCDELCEPCEPCWCDELCDELCEPCEPCWCEPCDELCDELCEPCWCDELCEPCWCEPCDELCDELCDEL

MAG.5.8-8,6-4.1-7.8-8.8-7.4-7.9-8.6-8.7-7.1-3.9-5.6-5.8-B.8-4.0-7.3-7.8-9.0-9.0-5.2-7.1-5.9-7.2-8.5-7.1-7.7-1.9-5,5-7.4-5.4-6.8-

6.4B9.3P5.4BB.8B9. 288.9B9.0B9.2B9.3P7.4P5.2B6.1V6.0V9.5V5. IB8.5B8.2P9.4P9.4V5.5P7.BV7.0B8.0B9.6B7.7B9. IB2.1V5.9V7.9V6.1V7.5V

EPOCH41576.25803.32926.35612,34521.36792.34932.34906.34235.30049.42756.38199.41760.42947.40905.42755.38202.40286.41739.38260.40689.42678.34906.41761.41659.19722.39253,40742.41053.4 1 705 .42526.

428400749164094294365637458325900033261721452904392502t7834796319428423143012132B

PERIOD4.4709165.37617.1766416.696384.430612.1392925.078785.507462.061195.6216055.3662705.27343.3330206.1659079.842007.9137799.2852.6749852.405257.9755.06431028.3820866.796634.0342293.1483802.5684383.969784.227136.923574.4354627.990629

TABLE 10.4. Intrinsic Variables with 10 < P < 100 Days

( 1950.0)NAME

DS AQRV341 ARARU CAMTW CAMTW CAPU CARIW CARL CARV420 CENRS COLX CYGSS GEMZETA GEMAC HERAP HERFW LUPT MONU MON¥ OPHEl PEGRS PUPST PUPAR PUPLS PUPLX PUPR SGEAL VIRU VIRV VULSV VUL

22IB74

201099

1152067

18181567

1723a667B

2$14132019

R505316161155254337134151

28481922284919U471

566

118

233449

A.36.02.020.439.040.045.642.952.424.033.026.633.58.69.013,07.233.224.357.815.09.012.010.058.06.0

46.726.826.924.027.8

DEC.-18-636957

-13-59-63-62-47-283522202115-407

- 9- 612

-34-37-36-29-1616

-13- 32627

51745195927241641482437384952446407

1925132710183447

2519

30.054.053.418.030.050.843.336.46.024.023.932.843.453. 142.053.351.015.459.018.035.66.018.018.048.026.232.98.848.052.8

221674201099

115206718181567

172386B78

2014132019

(1985.0)R.A.52 2856 IB20 719 3213 3757 1126 3244 5039 614 5542 487 403 13

29 3749 4721 2524 2630 451 5021 012 2948 242 2758 227 4113 2210 2025 1535 5350 53

DEC.-18-636957

-13-59-63-62-47-283522202115

-407

- 9- 612

-34-37-36-29-1616

-13- 32627

40114124633933265246313735515552544B3031153316244014183325

191188196552IB4445912352110233941264857311325649253717

TVPERVCEPCWRVACWCOELRVBCDELCWCEPCDELRVCDELRVACWCEPCDELRVBCEPSRACDELCWRVBCEPCEPRVBCWCWRVACDEL

MAG.10.3-1110.9-119.3-1010.4-1110.3-126.7- 87.9- 94.3- 59.9-119.0- 96.6- 89.3-103.7-47.4- 910.4-119.2- 95.6- 66.1- 87.1- 710.0-116.5-79.7-118.7-109.8-109.5-109.5-119.1- 99.5-108.1- 96.7- 7

EPOCH-6P-3V.48.5P.05.58.6P.SB.66.4P.48.7P.2V.76-2V.6P-0V. IP. 9B.5P.6V.SB.9P. SP,0P.5E.9V.7V.3V.8V

3097234237373562864735664411182940140736253502780925739343653441635052

-38197408383034734921309613573435617

-38376373142362737462326971487138268

.6

.6

.9

.

,8.2

.44

.67

.90

.78

.

.3

.59

.49

.0

.426

.45

.500

.25

.0

.5

.783

. 1

.9

PERIOD78.3011.9522.05585.628.557838.768167.535.533024.767B14.6616.386689.3110.1508275.461910.40816.7327.020592.2617.1232661.1541.387618.886475.14.146413.8870,59410.304017,273675.7245.035


10.4c Long-period Variables

The Mira type of variable has regular periods of 150 to 450 days. Theydiffer from the semiregular (SR) variables primarily in their amplitudeof pulsation. Miras vary by more than 2.5 magnitudes in the visual,whereas SR variables are defined to have smaller amplitudes than this.Miras are red giants, and are named after their prototype Mira Ceti,the first intrinsic variable star ever discovered. Most long-period vari-ables show bright hydrogen emission lines and titanium oxide bands intheir spectra. Table 10.5 lists the red long-period variables whoseperiods range from 100 to 1000 days.

Landolt has one of the longest series of UBV observations on long-period variables. The papers in his series are listed below.

I: 1966, Pub. A.S.P. 78,531.II: 1967, Pub. A.S.P. 79, 336.Ill: 1968, Pub. A.S.P. 80, 228.IV: 1968, Pub. A.S.P. 80, 450.V: 1968, Pub.A.S.P. 80,680.VI: 1969, Pub.AS.P. 81, 134.VII: 1969, Pub. A.S.P. 81, 381.VIII: 1973, Pub. A.S.P. 85, 625.

From these papers, it is immediately obvious that one of the problemsof observing this class of variables is: they are very red, with typical(B — V) indices of 1.5 to 4.0 magnitudes. Therefore, when a longperiod variable is near minimum at V equals 10, it may be fourteenthmagnitude at B\

10.4d The Eggen Paper Series

Eggen was one of the pioneers in observing variable stars photoelectri-cally, and has been classifying variables systematically for many years.His series of papers should be consulted for information concerning anyvariable class, and contain many observations that are useful in settingup your own program. The papers are somewhat technical and may bedifficult for the amateur to read, but are listed below.

I: "The Red Variables of Type N," 1972. Ap. J. 174, 45.II: "The Red Variables of S and Related Types," 1972. Ap. J. 177,

489.

TABLE 10.5. Intrinsic Variables, 100 < P < 1000 Days

{ 1950.0)NAME

VX ANDTHET APSV AQRRV BOORW BOORV CAMV CVNRU CEPT GETOMI GETW CYGRS CYGR DORUX DRAAH DRATV GEMUW HERAK HYAT INDT MICRY MONX OPHTW PEGL2 PUPU PYXBM SCOCE TAURY UMAVW UMASS VIRSW VI R

N01

0142014144

13102

21204

19166178

21207

182278175

12101213

R17044373926171419163411362347a

1237162443511227372918552211

.A.15.023.317.79.36.2

31.917.123.714.549.08.2

34.510.422.424.050.939.035.752.252.531.057.643.20.749.142.816.84.038.040.029.7

DEC.44

-762

3231574584-20- 34538

-627657Zl36

-17-45-28- 7

622-44-30-3ZIB61701

- 2

2633154547184752Z01293410275352257

142528476339

11333515232

0.24.11.15.6.12.22.10.6.

13.0.36.31.41.59.51.26.23.3.39.48.19.20.26.25.21.3Z.13.25.48.32.

079Z674124448726746207449103506

0142014144

13102

21204

19166178

Zl207182278175

12101213

I; 1985.0)R.A.193

46384029181921183512362248101339192763731329393119582413

6503

37352648503527B0357057521210012372141451194402717

DEC.44

-762

3231574585-20- 34538-6276572136-17-45-28- 7822-44-30-32IB61700

- 2

3743223638223638218406

31505223145

1832491637161235234

5143

392954138462112273425592149212145010425

1131429262

34101039

TYPESRASRBSRBSRBSRBSRBSRASRDSRBMSRBSRASRBSRASRBSRCSRBSRBSRBSRBSRAMSRSRBSRSRDSRCSRASRMSRB

MAG.7677886862665588877775728668868

.8-

.4-

.6-

.9-

.0-

.2-

.8-

.2-

.6-

989999897

.0-10

.8-

.5-

.9-

.9-

.5-

.7-

.6-

.8-

.7-

.7-

.7-,3-.0-.6-,6-.8-.1-,1-.4-.0-.2-

896699989999969869999

.3V

.6P-4V.6P,5P.0P-8V.4V,7P.IV.9P-3V.9V.5V.3P.5P.5P.2P.4P.6P.2V.2V-2V.0V.4P.7?.5P.3P-IP.6V.4P

EPOCH25558.28625.34275.

--

28861 .34930,-

36460.38457.30684.37930.

--30520.

----

30100.23743.38475.3*37*.36446.

---

34670.

-38890.-

PERIOD369.119.244 .137.209.101.191.68109.159.331.65130.85418.03 3 8 .168.158.182.100.112.320.347.466.334 .22956.4140.83345.8S0.165.311.2125.354.66150.


III: "Calibration of the Luminosities of Small-Amplitude Red Vari-ables of the Old Disk Population," 1973. Ap. J. 180, 857.

IV: "Very-Small-Amplitude, Very Short-Period Red Variables,"1973. Ap.J, 184,793.

V: "The Large-Amplitude Red Variables," 1975. Ap. J. 195, 661.VI: "The Long-Period Cepheids," 1977. Ap. J. Supl Ser. 34, 1.VII: "The Medium-Amplitude Red Variables," 1977. Ap. J. 213,

767.VIII: "Ultrashort-Period Cepheids," 1979. Ap. J. Supl. Ser. 41, 413.IX: "The Very Short-Period Cepheids," Ap. J. In press.

10.5 ECLIPSING BINARIES

At least 50 percent of all stars are members of systems in which two ormore stars orbit around their common center of mass. Many of the starsthat appear single are in fact unresolved binaries. Some of these systemshave orbital planes that lie nearly along our line of sight. As a result,the two stars take turns blocking each other from our sight during eachorbital period. The apparent single point of light seen on earth fadesand recovers as a star goes through eclipse. There are two eclipses eachorbital period. The amount of light lost depends on the temperature ofthe two stars. The greatest light loss, called primary eclipse, occurswhen the hotter star is blocked from view. The shallower, secondaryeclipse occurs when the cooler star is blocked from view. These stellarsystems are referred to as eclipsing binary systems. Their light curvescontain valuable information about the stellar sizes, shapes, limb dark-ening, mass exchange, and surface spots, to name but a few items. Asa result, the study of these systems forms an important part of stellarresearch.

It has been traditional to classify an eclipsing binary system as oneof three types based on the shape of its light curve. When binaries areclassified in this way, many systems of diverse physical structure andevolutionary state are grouped together. Furthermore, there are systemsfor which light curves defy classification in this scheme. It has becomeclear in recent years that this classification system is now obsolete.However, there is no new classification scheme that is universallyaccepted. For this and historical reasons, we describe the three tradi-tional classification types.


Algol or |8 Persei is a naked-eye star that was discovered to be a var-iable star by ancient Arab astronomers. This was the first known vari-able star. It is now the prototype of the Algol class of eclipsing binaries,which now numbers several hundred. Typically, these systems containan early-type (B to A) star that is brighter and more massive than itslate-type (G to K) companion. The primary eclipse is deep because ofthe loss of light from the hot early-type star. On the other hand, thesecondary eclipse, because of the light loss of the cooler star, is veryshallow and difficult to detect. The orbital periods of Algol systemsrange from about 1 day to more than a month. The light curve in Figure10.2a is from a typical Algol system. Table 10.6 lists a few of thebrighter systems.

The light curve of Figure 10.2b is that of a typical j8 Lyrae typeeclipsing binary. In these systems, the apparent magnitude changes con-

AN ALGOL SYSTEM

A 0LYRAESYSTEM

i I I

A WUMa SYSTEM

0.75 0.25

PHASE

0.50 0.75

Figure 10.2. Light curves of eclipsing binaries.

M(7100

Name

XZAndV889 Aql

RZCasTVCas

V636 CenUCepUCrB

SWCygA[ Dra

TW DraS Equ

CDEriTTHyaAU Mon

8 Per (Algol)IZPer

RW PerYPsc

BH PupUSgeA Tau

RWTauXTri

ACUMaTXUMa

TABLE 10.6. Algol-Type Binary Systems

(1950.0)RA Dec.

01*53-49'19 16 3402 44 2300 16 3614 13 4000 57 4615 16 092005 2416 55 0915 33 0720 54 4403 45 2111 10 4606 52 220304 5501 28 5604 16 4823 3! 5308 06 3419 16 3703 57 540400 4901 57 4308 51 3610 42 24

41"51'26"16 093069 25 3658 51 40

-49 424881 362531 494246 092152 46 3064 04 2304 53 10

-08 4604-26 11 36-01 1842

40 45 5053 454242 11 4107 38 52

-41 525519 310612 210427 592427 384865 094545 4947

(1985.0)RA Dec.

01 "5 5-57'19 18 0902 47 3300 18 3014 15 5801 00 5615 17 3520 06 301655 571533 3820 56 2903 47 0311 1 2 2906 54 090307 1201 31 0804 19 1423 33 4008 07 4619 18 0903 59 500402 5801 59 4308 54 371044 27

42'01'41*16 132269 342159 03 19

-49 523281 474331 420446 152752 43 1 563 572405 01 16

-08 3937-26 2302-01 21 24

40 535353 563042 164307 5028

-41 590519 345812 265828 05 1027 485765 01 4445 3844

Mag.

10.02- 12.99/78.7-9.3 p6.38-7.89 p7.3-8.39 p8.7-9.2 v6.80-9.10 V7.04-8.35 p9.24-11.83 V7.05-8.09 V8.2-10.5/78.0-10.08 V9.5-10.2 p7.5-9.5 p8-2-9.5 v2.12-3.40 V7.8-9.0 p9.7-11.45 V9.0-12.0 v8.4-9. 1 p6.58-9.18 V3-3-3.80 p8.02-11.59 V8.9-11. 5 p9. 2-10.2 p7.06-8.76 V

Epoch

2441954.34027210.59639025.302536483.809134540.34042327.769716747.96438602.600937544.509538539.445737968.34529910.56724615.38832888.55440953.465725571.36039063.68441225.47326100.89140774.463835089.20440160.377140299.29642521.5839193.310

Period (Days)

1.35729311.120711.19524991.81261304.283982.4930833.452204164.5731161.198815202.80683523.4360722.8767666.9534124

11.113062.86730753.687661

13.198913.7658591.9158543.38062603.9529552.76884250.97153306.854933.063243


tinuously even outside of eclipse because of the distorted shapes of thestars. The stars in these systems are nearly touching, and their mutualgravitational attraction has distorted them into elongated, egg-likeshapes whose long dimensions always point toward each other. As thestars move in their orbits, their projected surface area as seen by theobserver varies. The light curve then peaks when both stars are seen"broadside" and then fade as they turn. These systems usually containearly-type stars and often show complications in their spectra indicatinggas streams and shells. Their orbital periods usually exceed 1 day. Table10.7 is a list of some bright /3 Lyrae-type systems.

The curve in Figure 10.2c belongs to a typical member of the WUrsae Majoris {W UMa) class. These systems too are highly distortedby gravitational fields. The two component stars have nearly identicalsurface temperatures, which results in near equality in the depths of thetwo eclipses. The similarity in temperature is believed to be the resultof a common atmosphere that surrounds both stars. These stars reallyare touching! Unlike the /3 Lyrae systems, these stars belong to spectralclasses F, G, and K. The orbital periods are less than I day. Table 10.8is a list of some brighter W UMa-type systems.

There is a newly recognized type of eclipsing binary, RS Canun Ven-aticorum (RS CVn) systems, which shows one of the shortcomings ofthe old classification system. In the past, members of this group hadbeen classified with the Algols, yet they clearly contain stars that arevery different from those found in Algol systems. The hotter star is typeF or G and the cooler is late G to early K. They have orbital periods ofa few days. The curious feature of their light curve is a broad depressionof about OT1 which slowly migrates in phase. Several of these systemsare now known to flare in the radio and x-ray regions of the spectrum.The depression in the light curve is believed to be the result of largestarspots that migrate slowly in longitude on the surface of the coolerstar. These spots are like sunspots, only they cover a much higher per-centage (about 10 percent) of the stellar surface. The x-ray and radioemissions are believed to be related to strong coronal activity aroundthe spotted star. A more complete review can be found in an article byZeilik et al.44 Table 10.9 contains a partial list of some RS CVn-typesystems. Anyone seriously considering observing these systems woulddo well to contact Douglas Hall,45 who is coordinating the photometricwork of amateur and professional observers. When monitoring thesesystems, it is important to follow the light curve distortions and to cor-

TABLE 10.7. 0 Lyrae-Type Binary Systems

Name

AN Ando Agl

LR AraTT AurSZCamCXCMa

X CarAOCasLWCenLZCen

AHCepV366 CygV548 CygV836 Cyg

uHer0Lyrj?Ori A

VVOriAUPup

VPupV525 SgrV453 ScoV499 Sco

AC VelAY Vei

(1950.0)RA Dec

&MTW19 36 4316 49 070506 150403 2407 19 5708 30 1100 15 0411 35 121 1 48 0522 46 042043 0619 55 4721 19 2117 15 29IS 48 140521 5805 30 5908 15 5607 56 481904 0217 53 0017 25 451044 1808 18 37

4r29'59"05 1658

-61 300839 31 2462 1 1 59

-25 4644-59 0327

51 0922-63 0418-60 3059

64 474953 55 1054 395135 31 2533 09 1333 18 13

-02 2627-01 11 24-41 3305-49 06 30-30 1424-32 2808-32 5754-56 3359-43 4327

(1985.0)RA Dec.

23*17*41'1938 271652 1705 08 400406 290721 2308 30 5700 16 5611 36 501 1 49 4922 47 2020 44 0519 56 3721 20 4717 16 4718 49 3205 23 4405 32 4608 17 0907 57 481906 161755 1717 28 0310 45 4308 19 48

41°4]'28"05 2148

-61 333739 340362 1737

-25 5046-59 1035

51 2102-63 1556-60 4240

64 585554 02 4954 45 3235 402233 065933 2041

-02 24 34-01 0958-41 3939-49 12 14-30 1107-32 2826-32 5935-56 4503-43 5007

Mag.

6. 0-6.16 p5.18-5.36 B10.0-10.6/78.3-9.2 p7.0-7.29 B9.9-10.6 p8.0-8.7 p5.96-6.11 B9.3-9.6 p8.0-8.6 p6.9-7.12 p10.0-10.46 p8. 9-9. 72 p8.59-9.30 B4.6-5.28 p3.34-4.34 V3.14-3.35 B5.14-5.51 p8.50-9.40 V4.74-5.25 p7.9-8.8 p6.36-6.73 V8.8-9.36 p8.5-9.0 p9.1-9.8 p

Epoch

2421060.32622486.79728004.43021242.256427533.519128095.60115021.11424002.57924824.46226096.38434989.370234489.59338972.170626547.522427640.65436379.53233420.21540545.89939237.98528648.304829662.459341762.5828340.40523936.28526308.903

Period (Days)

3.2195651 .950261.5193041.332733652.69843780.9546081.08263103.5234871.00256742.7577171.77472741.09601831.8052440.653410902.0510272

12.930167.989261.485377691.1264111.45448670.70512200

12.00612.33329774.56224261.617653

TABLE 10.8. W UMa- Type Binary System

Name

AB AndBXAnd

S AntOOAgl

iBooXYBooTXCnCRRCen

VWCepRZCom

« CrAYYEriAKHerSWLacAM LeoUZLeoXYLeo

V502 OphV566 OphV839 Oph

UPegAW UMa

WUMaAG VirAH Vir

(1950.0)RA Dec.

23h09m08>

02 05 5809 30 071945 471502 0813 46 4808 37 1114 13 2520 38 0312 32 351855 2104 09 4617 11 432251 2211 59 351037 5309 58 5516 38 481754 261806 5923 55 251 1 27 260940 161 1 58 2912 11 48

36°37'22"40 3330

-28 242509 110347 505120 260219 1037

-57 37 1975 245723 3651

-37 1025-10 3541

16 243237 402010 095913 494217 390700 360704 592909 082615 403030 143556 01 5213 171212 0600

(1985.0)RA Dec.

23MO-48'02 08 070931 3919 47 281503 191348 2808 39 1114 15 5420 37 311234 2018 57 4304 11 2617 13 1722 52 5911 01 251039 4510 00 501640 351756 091808 392357 1211 29 1709 42 431200 1712 1335

36°48'47"40 4326

-28 334309 16 1847 424120 153719 03 10

-57 470375 322323 25 17

-37 0734-10 3019

16 220837 51 3109 584213 384417 290000 320604 591509 08 5015 52 1130 030056 01 1513 053111 5420

Mag.

10.4-11.27/78.9-9.57 v6.7-7.22 B9.2-10.0 v6.5-7.10 v10.0-10.36 p10.45-10.78 p7.46-8.1 B7.8-8.21 p10.96-11. 66 B4.96-5.22 p8.8-9.50 B8.83-9.32 B10.2-11.23/J8.2-8.65 v9.58-10.15 V10.43-10.93 B8.34-8.84 V7.60-8.09 p9.4-9.99 p9.23-9.80 V6.84-7.10 V7.9-8.63 V8.4-8.98 V9.6-10.31 p

Epoch

2440128.794543809.887335139.92940522,29439370.422239953.962138011.390929036.032 J41880.802734837.419839707.661933617.519838176.509243459.747639936.833740673.666641005.535141174.228841835.861836361.731736511.668838044.781541004.397739946.747235245.6522

Period (Days)

0.331893050.610115080.6483450.50678870.26781600.370546630.3828815370.605691210.27831610.338506040.59142640.3214962120.421523680.32072160.365797200.61804290.284110.453393450.409643990.40899460.37478190.43873180.333636960.642647870.40752189

TABLE 10.9. Eclipsing Binaries of the RS CVn Type

Name

CQAurSSBooSSCam

RUCncRSCVnAD CapUXComRTCrB

WW DraZHer

AW HerMM HerPW HerGKHyaRTLacARLacRVLibVVMonLXPerSZPscTY Pyx

RWUMa

(1950.0)RA Dec.

06*00"39»15 11 3907 10 1908 34 341308 1821 37 031259 0715 35 591638 211755 521823 2717 56 321808 3508 28 1321 59 292206 391433 020700 5103 09 5223 10 508 57 36

11 38 05

31' 19*52*38 45 1 573 25 1623 441536 1200

-16 140028 53 5029 390160 474515 083118 155022 08 5833 223402 265543 385545 2946

-17 4905-05 3942

47 550502 2406

-27 371452 1629

(1985.0)RA Dec.

06*02-55'15 12 5907 14 3608 36 381309 5521 38 581300 4815 37 2516 38 5017 57 2718 24 591758 011809 5208 30 022200 542208 0414 34 5907 02 3403 12 1823 1 2 3708 59 0611 39 58

31'19'37*38 372673 21 3823 365536 0050

-16 042928 423229 32 1060 434115 082118 170422 08 5033 230202 195043 490345 4004

-17 58 14-05 4249

48 025602 35 32

-27 452652 04 50

Mag.

9.6-10.6 p10.2- 11, 2 p10.1-10,7 v9.9-11.5 v8.4-9.92 p9.3-9.9 p10.91-11.8 B10.2-10.7 v8.29-9.49 V7.3-8.1 p9.5-10.9 v9.8-10.8 p10.7-11.8 p9.1-9-Sp8.84-9.89 V6.11-6.77 V9. 8-10.4 p9.6-10.4 v8.4-9.4 p8.02-8.69 B6.87-7.47 V10.3-11.9 p

Epoch

2429558.7820707.37535223.2822650.72038889.330030603.5525798.37028273.28028020.369313086.34827717.215131302.45128248.56426411.46039073.802039376.495530887.23626037.52927033.12036114.56527154.32533006.308

Period (Days)

10.621487.6062154.8241

10.1729884.7978556.118263.6423865.117124.6295833.99280128.800867.960372.88100163.5870355.0740121.9831987

10.7221646.050798.0380443.966371.5992927.328251


relate any optical changes that occur simultaneously with radio and x-ray outbursts.

Finally, there is a group of binary systems that display dramaticallythe results of stellar evolution. The cataclysmic systems contain a faintred dwarf star and a white dwarf companion. The red dwarf star isslowly expanding, as stars are known to do when their evolution carriesthem from the main sequence to the red giant stage. In this case, how-ever, as the star expands, its atmosphere is drawn by the gravitationalattraction of the white dwarf. This transferred matter forms a diskaround the white dwarf as it spirals into the star. This process of masstransfer occurs in many binary systems, including Algols, but nowhereare the effects as visibly dramatic as in cataclysmic systems. For thesesystems, the mass transfer is so rapid that a thick, extensive accretiondisk forms. The continuous emission from the disk becomes so brightthat it can outshine the stars. Matter streams in, striking the disk andproducing a flickering hot spot. High-speed photometry (with time res-olution of a fraction of a second) shows this flickering clearly. It dis-appears when the spot is eclipsed. There are several subclasses withinthe cataclysmic group, one of which is novae. It is believed that theexplosive events in these systems are a direct result of the mass accre-tion process. A very interesting review of cataclysmic systems has beengiven by Trimble.46 A review of high-speed photometry of these systemshas been published by Warner and Nather.12 Unfortunately, nearly allthese systems are faint and beyond the grasp of small telescopes. Forthis reason, we have not included a list of these objects. However, onecan be found in a well-written review article by Robinson.47

In general, a photometric study of an eclipsing system can have oneor both of the following goals. The first is a set of observations to deter-mine the time of mideclipse, which is often called the time of minimumlight. This is done in order to determine the orbital period. In manyeclipsing systems, the orbital periods are not constant. Mass transfer ormass ejected from the system may cause a slight shift in the position ofthe center of mass of the system. As a result, there is a small change inthe orbital period. With observations of the time of minimum light, wecan check to see if the eclipses are occurring "on time." Eclipses thatoccur earlier or later than predicted indicate a period change. Varia-tions in the orbital period give us indirect information about the masstransfer or mass loss processes. A second reason for observing the timesof minimum light is so that spectroscopists can determine accurately


the orbital phases at which their spectrograms are taken. This is essen-tial for the proper interpretation of the spectroscopic data.

The second observational goal mentioned above is the observation ofthe entire light curve. For short-period systems like the W UMa bin-aries, it is possible to observe a sizable portion of the light curve in onenight. The data from several nights, covering portions of differentorbital periods, can be combined into a composite light curve similar toFigure 1.1. Then you can calculate the orbital phase of each observa-tion. This is explained below.

The analysis of a light curve can yield many physical parameters ofthe stars. However, analysis is a rather advanced topic and the readeris referred to Irwin,48 Binnendijk,49 and Tsesevich50 for an introduction.A review of the most modern synthetic light curve analysis can be foundin Binnendijk.51 By no means should an observer be discouraged fromobtaining a complete light curve even if the analysis appears too diffi-cult. There are many professional astronomers who specialize in thesestudies and would welcome the data. Many binary systems show dis-torted light curves whose shapes can change because of circumstellarmatter or starspots. Such systems should have their entire light curvesmonitored frequently. A comparison of light curves over many yearscan often provide clues as to the location and nature of the matterresponsible for the distortions. See Bookmyer and Kaitchuck,52 forexample.

The determination of the time of minimum light is a good programfor the newcomer to photometry. This involves simple differential pho-tometry without the need to transform to the standard system. It pro-vides valuable practice in observing techniques and data reduction. Inaddition, it provides valuable information. After the binary has beenchosen and a comparison star has been found, the next step is to cal-culate the predicted time of minimum light. Tables 10.2 through 10.5contain two numbers for each binary labeled epoch and period. Thesenumbers are referred to as the light elements. They allow for calculationof binary orbital phase for any given date and time. The epoch is aheliocentric Julian date of a primary eclipse. The second number is theorbital period in days. You may be surprised by the accuracy to whichthe orbital periods are quoted. Although the time of minimum light canbe determined only to an accuracy of a minute or so, we can measurethe orbital period to an accuracy of a few seconds or even less. This isbecause for short-period systems a small error in the orbital period can


accumulate in a few years to a large discrepancy between the predictedand observed times of eclipse.

Orbital phase is defined to be zero at mideclipse and 0.50 at one-halfan orbit later, etc., as seen in Figures 1.1 and 10.2. The first step inpredicting the time of minimum light is to compute the heliocentricJulian date (HJD) at 0 hour UT, as explained in Sections 5.3d and 5.3efor the day and time in question. We then compute a quantity called H,by

- - - epoch \H = fractional part of

\ period /

if HJD is greater than the epoch, or

, /HJD-epodAH = 1 — fractional part of

\ period /

if HJD is less than the epoch.The heliocentric phase is then given by

UT(in hours)/24 hoursPhase (Hel.) = H + - — .

period

To predict the time of minimum light, we need only to find a day andtime for which phase 0.0 or equivalently 1.0, occurs after dark and withour star above the horizon. Because the orbital periods change, the lightelements in Tables 10.2 through 10.5 give only an approximate time ofminimum light. This is especially true for light elements with smallepoch values where the error could be as much as an hour. You shouldbegin observing early to avoid the possibility of missing part of theeclipse. The most recent references to published light elements can beobtained from the astronomy department at the University of Florida.53

They maintain a file of index cards containing literature references toresearch papers, times of minimum light, and the most recent light ele-ments for hundreds of eclipsing-binary systems. A xerox of these cardscan be obtained by writing to them and specifying which binary systemsyou are studying.

The actual observing procedure to follow is outlined in Section 9.3cand the data reduction is explained in Section 4.8. For short-period sys-


terns, you can observe most or all of the eclipse in a few hours. For long-period systems, the eclipse can last days and you have to spread yourobserving over several nights. It is important to observe as much of thedescending and ascending branches of the curve as possible. Thisimproves the accuracy of your determination of the time of minimumlight greatly. Because of observational error, the time of minimum lightdoes not simply refer to the time of your lowest measurement. Thereare several methods for determining the time of minimum light fromthe observations.

The bisection of chords technique can be applied if the eclipse curveis symmetric about mideciipse. First, a plot is made of Am versus HJD.A smooth freehand curve is drawn through the data points and severalhorizontal lines are drawn connecting points of equal brightness on theascending and descending sides of the curve. The midpoint of each lineis measured and the average is taken to give an estimate of the time ofmideciipse. The standard deviation from the mean gives an errorestimate.

The tracing-paper method is still simpler. On the data plot, a verticalline is drawn to indicate your best estimate of the time of minimumlight. The data points, the vertical line, and the horizontal axis are thentransferred to a piece of tracing paper. The tracing paper is then turnedover, so the time axis is reversed, and laid back on the original plot. Thetracing paper is then moved horizontally so that data points and theirtracing-paper counterparts give the best fit with a minimum of scatter.In general, the vertical lines on the plot and tracing paper do not align.The time of minimum light is midway between these two lines and canbe read directly from the horizontal axis. Shifting the tracing paper tothe right and left until the fit obviously worsens gives an error estimate.

A more analytical technique has been devised by Hertzsprung.54

Because this reference is difficult to find in many libraries, we illustrateit in detail. This method is more time-consuming but has the advantageof eliminating most individual bias. It can also be implemented easilyon a computer or programmable calculator. A plot of the observations,Am versus HJD, is made on a large sheet of graph paper, say 50 X 100centimeters. The plotting scales should be chosen so that the eclipsecurve has a slope of about 45 °. Each data point is then connected bystraight line segments. Figure 10.3 shows this plot (greatly reduced insize) for an eclipse of V566 Oph observed by Bookmyer.55 First, esti-


-0.16 -

-0.12 -

-0.08 -

0.120 -

0.1600.530 0.550 0.570 0.590

HJD (+2,441,866.0)

Figure 10.3. Eclipse observations.

0.610 0.630

mate the time of minimum light, called r0, by simply looking at the plot.In this case, t0 was estimated to be 0.5850, the Julian decimal. Place anarrow on the plot at 10 or more time intervals Ar to either side of t0. Inthis case, Af was chosen as 0.003 day. Read Am from the plot at eacharrow, using the straight line segments. These values are recorded inpairs that correspond to the same time interval on either side of tQ. Theabsolute value of the differences between the numbers in each pair isthen found. The middle three columns of Table 10.10 shows these mea-surements and their differences. Ideally, if we had guessed tQ exactlyright, and the light was symmetric with no observational error, the col-umn of differences would only contain zeros.

This process is then repeated, taking the first arrow on the right to bea new estimate of t0 (the rightmost three columns of Table 10.10). Thisreally does not involve much work; it requires moving some numbers inthe table to different rows. The process is repeated a third time usingthe first arrow on the left as the new tQ (the leftmost three columns inTable 10.10). We now let Y~, Y°, and K+ be the sum of the squares


TABLE 10.1O. Time of Minimum Light Data

(0 - Al = 0.5820Am

Derending Ascending |Diff.

t0 = 0.5850Am

ending Ascending |Diff.|

t0 + At = 0.5880Am

Derending Ascending jDiff.

-0.141-0.117-0.101-0.083-0.058-0.033-0.007+0.018+ 0.048+0.076+0 107+0.119+0.132+0.142

-0.108-0.083-0,057-0,028-0.004+0.022+0.053+0.078+0.104+0.132+0.139+0.140+0.148+0.137y~ =

=

0.0330.0340.0440.0550.0540.0550.0600.0600.0560.0560.0320.0210.0160.005

EIDiff . l 2

0.02836

-0.117-0.101-0.083-0.058-0.033-11.007+0.018+ 0.048+0.076+0.107+0.119+0.132+0.142+0.137

-0.130-0.108-0.083-0.057-0.028-0.004+0.022+0.053+0.078+0.104+0.132+0.139+0.140+0.148yO _

"

0.0130.0070.0000.0010.0050.0030.0040.0050.0020.0030.0130.0070.0020.011

C|Diff.|2

0.00065

-0.101-0.083-0.058-0.033-0.007+0.018+0.048+0.076+0.107+0.119+0.132+0.142+0.137+0.148

-0.152-0.130-0.108-0,083-0.057-0.028-0.004+0.022+0.053+0.078+0.104+0.132+0.139+0.140y+ =

=

0.0510.0470.0500.0500.0500.0460.0520.0540.0540.0410.0280.0100.0020.008

E|DLffv | 2

0.02560

of the values in columns 3, 6, and 9, respectively. The time of minimumlight, tmim is then computed by

tmin = to +

In this particular example,

Y~ - 2Y°Ar.

tmin = 0.5850 +

= 0.5851

0.02836 - 0.025600.02836 - 2(0.000650) + 0.02560

0.003

For even higher accuracy, the entire procedure can be iterated usingthe above value as a new estimate of (0- However, this is probably notjustified unless the observational data is of very high quality. An esti-mate of the error in the time of minimum light can be obtained bychanging Af and recomputing tmin.

When reporting a time of minimum light, it is customary to alsoinclude a second number called the (O ~ C) value. This is the differ-ence in decimal day between the observed and computed time of min-imum light. Because this computed time of minimum light depends on


the set of light elements used, they should also be reported. The com-puted time of minimum light can be calculated by

Tmin = T0 + P • E,

where T0 and P are the initial epoch and period, respectively, and E isan integer. With a little trial and error, a value of E can be found, whichgives a value of Tmin as close as possible to the observed time of mini-mum light. In our example above, the observed time of minimum lightis calculated from a set of light elements published by Bookmyer56 manyyears before, namely

TV, = 2,436,744.4200 + 0.40964091 E.

If E is chosen as 12,504, then the calculated time of minimum light is2,441,866.5699. The observed time of minimum, tmin, minus the com-puted time of minimum light, Tmim is

(O- C) = 0.5851 - 0.5699= 0.0152 day

or about 22 minutes. This large difference is certainly easy to measure.When (O — C) values over intervals of a few years are collected and

plotted versus Julian date, the story of period variations is told. Forinstance, if the (O — C) values scatter about zero, resulting in a hori-zontal line on the plot, the period is constant. If the (O — C) pointstrace out a curved path on the plot, this indicates that the period iscontinuously changing. The (O — C) plot for V566 Oph shows a longinterval of a constant period and then abruptly at about Julian date2,440,000, or the year 1968, the (O - C) values begin to rise followinga straight line.57 Such a straight, sloping line on an (O — C) plot indi-cates that a single abrupt period change occurred. From this date(2,440,000) until the time of minimum light measured in our exampleabove, the binary system completed a number of orbits given by

2,441,866.6 - 2.440,000period


or

1866.60.409644

= 4556.

If it took 4556 orbits to accumulate an (O — C) value as large as 23minutes, then the change in orbital period must have been

23 minutes4556

= 0.005 minute

or 0.3 second! This example demonstrates that times of minimum lightdetermined to an accuracy of only a few minutes can often detect periodchanges of just a few tenths of a second.

10.6 SOLAR SYSTEM OBJECTS

In the past, less photoelectric photometry has been done on solar systemobjects than on stellar or galactic objects. There are many reasons forthis, among them being, that there are fewer objects to study and anincorrect impression that little new information could be learned. Thereare three areas that amateurs can contribute most readily: comets,asteroids, and satellites.

Very little comet photometry has been performed. Most astronomersconcentrate on spectra when comets appear, trying to decipher the com-position of comets. However, recent research at GSFC by Niedner58 hasshown that magnetic sector boundaries emanating from the sun causetail disconnections and brightness flareups. This latter phenomenonmay be related to the brightness variations shown by periodic cometssuch as Schwassmann-Wachmann and therefore observing these com-ets photoelectrically may help in our understanding of the interplane-tary magnetic field. Observing comets as they approach from beyondJupiter can provide a probe at great distances from the sun. One prob-lem with comets is that most of the light from the coma/nucleus regionis from emission lines, making interpretation of wide-band colors asused in the UBV system difficult. We have the following suggestions forcomet observations:

1. Use UBV photometry on new comets as soon as they are brightenough to be detected by your telescope. Be careful in measuring


a comet with a tail and coma, as your diaphragm will not containthe entire comet. Center on the nucleus and keep accurate recordsof what diaphragms you use. Sky determinations are also difficultbecause of contamination by the comet.

2. On large, bright comets, you can obtain intensity isophotes (con-tours of equal intensity) by centering on the comet nucleus andmeasuring its brightness in diaphragms from as small to as largeas possible.

3. On bright comets, use narrow-band or interference filters to isolateparticular emission features. Borra and Wehlau59 used 75 A widebandpasses centered at 3878 A (CN bands), 4870 A (reflectedsunlight continuum), and 5117 A (Swan C2 bands) for their pho-tographic photometry.

There are several hundred asteroids brighter than eleventh magni-tude. All are within range of amateur telescopes but only a few havebeen observed photoelectrically. Not only can photoelectric measure-ments provide information about their albedo and size, but most aster-oids are nonspherical and show brightness variations during their rota-tions. Obtaining light curves can give clues as to their shape and theorientation of their spin axis. A study of 43 Ariadne by Burchi andMilano60 gives examples of parameters obtainable from asteroid pho-tometry. The current hypothesis that many asteroids have satellites alsomeans that eclipsing asteroids should be visible, but have not yet beendetected. Because asteroids are constantly moving with respect to stars,you need to obtain ephemerides from the Almanac24 or another similarsource. Comparison stars are difficult to obtain because of the asteroids'movement. Stars from the VBV catalog61 should probably be used.

Planetary satellites are very difficult to measure. They are alwaysmuch fainter than their parent body and lie very close in angular dis-tance. You must be very careful to avoid contamination from thebrighter planet. Eclipses behind the shadow of the planet can yield threemain pieces of information: the diameter of the satellite, its orbit, andthe structure of the planetary upper atmosphere. The Galilean satellitesalso have intervals of mutual eclipses twice per Jovian year. Theseeclipses of one satellite by another can give better information abouttheir diameter and orbits, and can occur at a greater angular separation


from the planet. In all eclipse observations, record the eclipse time tothe nearest tenth of a second, preferably from inspection of the stripchart or microcomputer time series. Williamon62 gives examples of thephotometric appearance of this eclipse.

Observations of the planets have been carried out at major observa-tories. Uranus and Neptune have been observed for many years at Low-ell Observatory to check the constancy of their solar luminosity. Thesetwo planets are faint enough that nearby comparison stars can be found.The rotation period of Pluto may be best detected photoelectri-cally63 because of the low amplitude of its light variations, about 0.2magnitude in V. The variation may be partially a result of the newlydiscovered satellite. You too can participate in such determinations ifyou have access to a meter-sized telescope.

10.7 EXTRAGALACTIC PHOTOMETRY

Extragalactic photometry is a very active area of research for manyprofessional astronomers. This type of photometry can take many dif-ferent forms. For example, drift scan photometry is a way of obtainingbrightness and color profiles by drifting the photometer aperture acrossthe face of a galaxy. Such data yield information about the distributionof dust and stellar populations within the galaxy. Aperture photometryis another technique that yields much the same information. Measure-ments of a galaxy are made through photometer diaphragms of varioussizes centered on the galactic nucleus. Quasars, BL Lac objects, com-pact galaxies, and many radio galaxies appear starlike in a telescope.Photometry of these objects is done much as it is for ordinary stars.

The problem with all these projects is that we are dealing with verydistant objects that appear very faint. Because much of this book hasbeen aimed at small telescope users, it would be out of place to discussin detail projects that exceed the capabilities of their equipment. Weinstead mention one particular project which is within the reach of a30- to 40-centimeter (12- to 16-inch) telescope and is of great scientificvalue. Many quasars, BL Lac objects, and radio galaxies are variablein the optical region of the spectrum. For many such objects, there is aserious shortage of observations over the long term, necessary for anunderstanding of the nature of these variations. The reason for theshortage of data is simply that these objects greatly outnumber thenumber of astronomers working in this field.


There are very few quasars brighter than thirteenth visual magni-tude. The quasar 3C273 is about visual magnitude 12.5 and its varia-tions have been well documented. The AAVSO3 can supply a finderchart for this object at a modest cost. There are several radio galaxiesand BL Lac objects brighter than thirteenth visual magnitude whichcould be monitored profitably. Burbidge and Crowne64 have compiledan optical catalog of radio galaxies. A similar catalog for quasistellarobjects has been published by Hewitt and Burbidge.65 Both referencesrepresent a good starting place for finding potential objects to study andfor further references to published papers and finding charts. It shouldbe emphasized that these objects are faint and are difficult to measureaccurately with a small telescope. However, even observations with anerror of 0.1 magnitude are valuable, particularly if they are made on aroutine basis.

10.8 PUBLICATION OF DATA

The acquisition of photoelectric data is enjoyable in itself. As you gainexperience and observational data, you may reach the conclusion thatyou would like to publish your data. Publication serves three purposes:it provides a means of rapid dissemination of important data or results,it gives permanence to results so that they may be used decades in thefuture, and it gets your name in print.

There are several methods to publish your data. These are listedbelow.

1. The American Association of Variable Star Observers (AAVSO)publishes newsletters and its own journal. They are the centraldepository for all visual observations and also most amateur pho-toelectric data. Check with them at their headquarters.3 They areoften consulted by professional observatories for information onlong-period variables.

2. The International Astronomical Union (IAU) has a depository ofphotoelectric data.66 This is a good alternative if you do not wishto publish your results in meeting abstracts or a journal. It is usedheavily for extensive sets of data on a particular star such as RWTau or SS 433.

3. Societies such as the American Astronomical Society, the Astro-nomical Society of the Pacific, and the International Astronomical


Union sponsor meetings and publish the proceedings. Generally,only the abstract of your paper or talk is published, but interestedparties then know who to contact for further details.

4. Among the small journals are the Information Bulletin of Varia-ble Stars, the Journal of the AA VSO, the Journal of the RoyalAstronomical Society of Canada, and the Monthly Notices of theAstronomical Society of South Africa. In many cases, these jour-nals do not have page charges and may not have your paper refer-eed formally.

5. Major astronomical journals include the Publications of theAstronomical Society of the Pacific, Astronomy and Astrophysics,the Astronomical Journal, the Astrophysical Journal, and theMonthly Notices of the Royal Astronomical Society. These jour-nals send submitted papers to independent referees who assess thequality of the paper and who can ask for changes in addition torecommending acceptance or rejection. In addition, most majorjournals have page charges because of the technical nature of thepapers, the cost of typesetting, and the large volume of papers,The cost may range from around $40 to $100 per page or more.

The large number of astronomers and the use of modern techniquesyield more data than in the past, and the emphasis on professional pub-lication forces more papers to be written, contributing to the necessityof these page charges. At the same time, avenues have been opened toamateur observers, as few professionals can devote time to long-termprojects or projects that contain classical astronomy that do not yieldimmediate results.

When you decide that you would like to publish results, we suggestthat you find a professional astronomer with whom to collaborate. Theastronomer probably knows more about photometry theory and publish-ing procedures than you do, and in addition may be able to give youguidance and access to a library or computer center. Other reasons forcollaboration are that the astronomer may have an ongoing project towhich you can contribute, the astronomer may be a good reference ifyou are contemplating a career in astronomy, and the astronomer'sinstitution may be able to pay page charges on a joint publication.Check with local universities or read journals to find someone whoappears to be interested in the same field that you are.

There is not enough room in this text to give complete details of the


techniques of technical writing. The IAU Style Book67 and the AIPStyle Manual68 give the grammatical aspects for astronomical papers.There are several texts on scientific writing, and each journal mentionsthe format that they wish submitted papers to obey. The best methodis to read the journals and see how others write and learn by example.

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1. Landis, H. J. 1977. /. AAVSO6, 4.2. Stanton, R. H. 1978. J. AAVSO1, 14.3. American Association of Variable Star Observers, 187 Concord Avenue, Cam-

bridge, MA 02138.4. Argue, A. N., Bok, B. J., and Miller, P. W. 1973. A Catalog of Photometric

Sequences. Tucson: Univ. of Arizona Press.5. Hertzsprung, E. 1924, Bui. Ast. Inst. Neth. 2, 87.6. Luyton, W. J. 1949. Ap. J. 109, 532.7. Gordon, K., and Kron, G. 1949. Pub. A. S. P. 61, 210.8. Moffett, T, J., 1974. Sky and Tel. 48, 94.9. Love!!, Sir Bernard 1971. Quart. J.R.A.S. 12, 98.

10. Gershberg, R, E. 1971. Flares and Red Dwarf Stars. D. J. Mullan, trans.Armagh, Ireland: Armagh Observatory.

11. Gurzadyan, G. A. 1980. Flare Stars. Elmsford, NY: Pergamon.12. Warner, B., and Nather, R. E. 1972. Sky and Tel. 43, 82.13. Nather, R. E. 1973. Vistas in Astr. 15, 91.14. McGraw, J. T., Wells, D. C., and Wiant, J. R. 1973. Rev. Sci. Inst. 44, 748.15. Povenmire, H. R. 1979. Graze Observer's Handbook. Indian Harbor Beach, Flor-

ida: JSB Enterprises.16. Elliot, J. L. 1978. AJ. 83, 90.17. Walker, A. R. 1980. I.A.U. Or. 3466.18. Evans, D. S. 1977. Sky and Tel. 56, 164.19. Evans, D. S. 1977. Sky and Tel. 56, 289.20. Nather, R. E., and Evans, D. S. 1970. AJ. 75, 575.21. Nather, R. E. 1970. AJ. 75, 583.22. Evans, D. S. 1970. A.J. 75, 589.23. Nather, R. E., and McCants, M. M. 1970. A.J. 57, 963.24. The Astronomical Almanac. Washington, D.C.: Government Printing Office.

Issued annually.25. J. R. Percy, ed. The Observer's Handbook. Royal Astronomical Society of Can-

ada. Toronto: Univ. of Toronto Press. Issued annually.26. Smart, W. M. 1971. Text-Book on Spherical Astronomy. New York: Cambridge

Univ. Press.27. International Occultation Timing Association, P.O. Box 596, Tinley Park, IL

60477.28. Glasby, J. S. 1969. Variable Stars. Cambridge: Harvard Univ. Press.


29. Strohmeier, W. 1972. Variable Stars. Edited by A. J. Meadows. Elmsford, NewYork: Pergamon.

30. Kukarkin, B. V. 1975. Pulsating Stars. New York: Halsted (translation).31. Kukarkin, B. V., Kholopur, P. N., Efremuv, Yu. N., and Kurochkin, N. E. 1965.

The Second Catalogue of Suspected Variable Stars. Moscow: U.S.S.R. Academyof Sciences.

32. Code 680, Goddard Space Flight Center, Greenbelt, MD 20771.33. Becvar, A. 1964. Atlas of the Heavens—// Catalogue. Cambridge: Sky Publish-

ing Co.34. Eggen, O. J. 1979. Ap. J. Supl. Ser. 41, 413.35. Tsesevich, V. P. 1969. RR Lyrae Stars. Springfield, Virginia: National Technical

Information Service, U.S. Department of Commerce.36. Percy, J. R., Dick, R., Meier, R., and Welch, D. 1978. / AAVSO1, 19.37. Payne-Gaposchkin, C. H. 1971. Smithsonian Contributions to Astrophysics. No.

13 (LMC).38. Payne-Gaposchkin, C. H., and Gaposchkin, S. 1966. Smithsonian Contributions

to Astrophysics. No. 9 (SMC).39. Sturch, C. 1966. Ap. J. 143, 774,40. Schaltenbrand, R., and Tammann, G. A. 1971. Ast. and Ap. Supl. 4, 265.41. Henden, A. A. 1979. MNRAS 189, 149.42. Henden, A. A. 1980. MNRAS 192, 621.43. Eggen, O. J. 1977. Ap. J. Supl. Ser. 34, 1.44. Zeilik, M., Hall, D. S., Feldman, P. A., and Walter, F. 1979. Sky and Tel, 57,

132.45. Dr. Douglas S. Hall, Dyer Observatory, Vanderbilt University, Nashville, TN

37235.46. Trimble, V. 1980. Mercury 9, 8.47. Robinson, E. L. 1-976. Ann. Rev. Astr. and Ap. 14, 119.48. Irwin, J. B. 1962. Astronomical Techniques. Edited by W. A. Hiltner. Chicago:

Univ. of Chicago Press, p. 584.49. Binnendijk, L. 1960. Properties of Double Stars. Philadelphia: Univ. of Pennsyl-

vania Press, p. 258.50. Tsesevich, V. P., ed. Eclipsing Variable Stars. New York: Halsted Press.51. Binnendijk, L. 1977. Vistas in Astr. 21, 359.52. Bookmyer, B. B., and Kaitchuck, R. H. 1979. Pub. A. S. P. 91, 234.53. Curator, Card Catalog of Eclipsing Variables, Department of Physics and Astron-

omy, University of Florida, Gainesville, FL 32611.54. Hertzsprung, E. 1928. Bui Astr. Inst. Neth. 4, 179.55. Bookmyer, B. B. 1976. Pub. A. S. P. 88, 473.56. Bookmyer, B. B. 1969. A. J. 74, 1197.57. Kaitchuck, R. H. 1974. J. AAVSO 3, 1.58. Niedner, M. B. 1980. Ap. J. 241, 820.59. Borra, E. F., and Wehlau, W. H. 1971. Pub. A. S. P. 83, 184.60. Burchi, R., and Milano, L. 1974. Ast. and Ap. Supl. 15, 173.61. Blanco, V. M., Demers, S., Douglass, G. G., and Fitzgerald, M. P. 1968. Photo-

electric Catalog. Washington, D.C.: U.S. Naval Observatory. XXI, second series.


62. Williamon, R. M., 1976. Pub. A. S. P. 88, 73.63. Neff, J. S., Lane, W. A., and Fix, J. D. 1974. Pub. A. S. P. 86, 225.64. Burbidge, G., and Crowne, A. H. 1979. Ap. J. Supl. Ser. 40, 583.65. Hewitt, A., and Burbidge, G. 1980. Ap. J. Supl. Ser. 43, 57.66. Breger, M. 1979. Information Bulletin on Variable Stars, No. 1659.67. Anon., 1971. International Astronomical Union Style Book. Transactions IAU,

XIVB, 261.68. Hathwell, D., and Metzner, A. W. K., for the AIP Publications Board, American

Institute of Physics Style Manual. New York: AIP. Revised periodically.

APPENDIX AFIRST-ORDER EXTINCTION STARS

This appendix lists those stars that are particularly useful in determining thefirst-order extinction coefficients. Several sources were consulted with the fol-lowing criteria used for star selection:

1. The star must be fainter than 4m.O (V).2. -0.15 < B - K < +0.15.3. -0.15 < U - B < +0.15.4. No star is a common close visual or spectroscopic binary.5. There are no peculiar spectral types.6. No star is a known variable.

The following tables indicate the SAO 1950.0 coordinates along with pre-cessed 1985 coordinates (with no proper motion corrections). Those stars withno HR number or Flamsteed number are designated by their HD or BD num-ber in an obvious manner.

Table A. 1 lists the northern declination stars, while Table A.2 lists the south-ern declination stars. The observer is indicated in the rightmost column of eachtable.

AZ-TNT. = Iriarteetal.1

COUSINS = Cousins*JOHNSON = Johnson'

For those who need fainter standards, the list of equatorial standards (10 -13m) by Landolt* is extremely useful.

279


REFERENCES

1. Iriarte, B., Johnson, H. L., Mitchell, R. 1., and Wisniewski, W. K. 1965. Sky andTel. 30, 25.

2. Cousins, A. W. J. 1971. Roy. Obs. Annals. No. 7.3. Johnson, H. L. 1963. In Basic Astronomical Data. Edited by K. AA Strand. Chi-

cago: Univ. of Chicago Press, p. 208.4. Landolt, A. U. 1973. A. J. 78, 959.

TABLE A.I. Northern First-Order Extinction Stars

( 1950.0)HR0063006803430378033305900620066407180879093209721002114812611324138713891448154415701724180718722034210321742Z09223824042543258425352629265427102751

R.A.0 140 151 91 151 161 592 52 142 252 553 63 123 183 454 24 144 224 224 314 474 525 145 235 315 505 566 66 136 156 326 496 526 536 597 17 97 14

28.342.52.613.042.77.227.720.029.833.327.71 .35.02.950.923.423.135.629.553.08.45.2

57. 138.811.015.221.220.412.73.82.1

51 .458.32.244.811.344.3

+ 38+ 36+ 54+ 3+ 27+ 72+ 37+ 33+ 8+ 39+ 74+ 20+ 43+ 71+ 50+ 50+ 22+ 17+ 5+ 8+ 10+ 1+ 6+ 3+ 27+ 0+ 2+ 69+ 59+ 7+ 3+ 8+ 45+ 4+ 1+ 5+ 49

DEC.243053210103737142712519

101310104827484

534944363230201

366

239

53334433

14.29,4.6.6.50.22.1.

13.50.22.37.2.

51.3.28.51.55.54.57.22.36.38.3.8.59.32.27.54.46.10.23.40.26.50.20.22.

9732888517190639929655755781286172671

0011122222333344444445555566666667777

( 1985.0)R.A.161710171827

1627571014204851724243349541525335258817183350545603

1117

183291

388

342421471622645276283620473

5450292231011175752453153333

23

DEC.3B36553277237338

3974204371505022175810163

27026959738

45415

49

35424

32112047462336205916171815155332527

SB5145363330191

35320650304029

549

1498

57204436131925341643373841163144532527358104026

38395324414933

444543444444444444544665455446664666S

V.61.50.36.15.76.98.83.01.28.70.87.89.94.66.29.65.23.29.68.35.66.42.42.35.61.21.73.79.47.44.38.28.90.63.56.ye.04

B-V0.060.060.170.070.03-0.010. 120.03-0.060.060.03-0.020.060.030.020.050. 120-040.050.010.08-0.02-0.020.04-0.020.000.060.030.01-0.010.040.040.040.060.01-2.020.08

U-B0.070.070.130.100.100.030.140.01-0.130.120.04-0.010.050.05-0.040.050.150.080.100.030. 080.02-0.050.060.010.010.04-0.01£.03-0.030.090.060.030.09-0.07-0.050.09

SPEC.A2VA2VA7VA3VA3VA1VA4VA0VB9IIIA2VA0VA0IVA3VA3IVB9VA2A7VA3VA3AtfVA0VB8VB9A2B9.5VAlA0.5IA0VA2VA0A0A0A2VA0B9A0A3III

NAMETHE ANDSIG ANDTHE CAS89 PSCUPS PSC50 CAS58 ANDGAM TRI

CHI2 GETPI PER

ZET ARI32 PERGAM CAMLAM PERB PERKAP TAU68 TAU

PI2 ORIPI1 ORI

38 ORI136 TAU

60 ORI

22 CAM2 LYN14 MON

16 LYN

DBS.JOHNSONAZ-TNT.JOHNSONCOUSINSAZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.JOHNSONAZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.COUSINSAZ-TNT.AZ-TNT.COUSINSCOUSINSCOUSINSAZ-TNT.COUSINSCOUSINSAZ-TNT.AZ-TNT.COUSINSCOUSINSCOUSINSAZ-TNT.COUSINSCOUSINSCOUSINSAZ-TNT.

M03M

HRZ81S2946306731363173341034123449349Z357336513799390639744248430043564386438045284505459947894805482650215037506251125264585959726161616863556436

NUK 1HEK*

( 1950.0)R.A.

77778a88a89991010101111111111111212121313131313151616161717

223850SB435354045559

31494

515911181645575832353916192332594202832315

56.847.226.435.242.30.622.723.747.133.236.224.637.429.26.539.811.933. 524.720.823.218.621.631.421.219.09.016. &24.85.855.*8.44.229.33.456.7

+ 49+ 58+ 26* 5+ 5159

21614

+ 52+ 2+ 35+ 43+ 20+ 0+ 6+ 38+ 8+ 3+ 6+ 22+ 3+ 10+ 3+ 2+ 55+ 49+ 1+ S+ 22+ 68+ 42+ 12+ 37

DEC.1849631

395246381

444164129Z7261218273156535433305720141647365652324820

46.747.048.77.39.9

45.62.458.825.18.523.230.217.321.424.054.29.913.136.425.31.0

35.115.426.539.21.6

57.853.015.88.510.031 .034.821.129.134.1

777a8888a89991010111111111111IZ121212131313U14151616161717

h I K b l URUtK EXTINCTION SIAKi

( 1985.0)R.A.2541520736374247571133516

531

12201847590343741182024330441

2833417

35443426205116253821254826326

3159211981066

1875553950523838037409

DEC.4958264

5159

21513

5223543200638a36

2231032

5549152268421237

14444855334537315336557

3119161506161944414221194593537295048284518

3449201732440243804482451236434264419534153a595857310374201

3821

V4. S34.994.995.644.844.176.524.664.376.596.144.506.024.494.714.415.424.064.795.315.364.674.816.324.886.625.704.034.704.275.584.835.014.204.914.66

B-V-0.020.080.090.000.050.00-0.020.02-0.050.060.000.00-0.040. 19-0.050.04-0.02-0.070.120.020.000.130.000.010.090.060.050.150.120.090.040.07-0.06-0.020.120.05

U-B-0.020.090.120.010.000.02-0.040.01-0.050.080.000.04-0.080.07-0.060.04-0.05-0.120.030.040.000.10

-0.010.010.060.020.020.090.110.140.030.05-0.10-a. 100.06-0.03

SPEC.A1IVA3IIIA4VA0A2VA0VA0A1VA0VA0A0A2VA1VA7VA1VA1VA0B9VA2VAlAfA4VA3IIIA0A1VA1IVA0A5VA4VA3 I 11A0VA3VB9IVB9VA3IVA2V

NAME21 LVN24 LVNPHI GEM

27 LVNDEL HYA37 CNCGAM CNCRHO HVA

26 UMA7 SEX21 LMIOMI UMA60 LEO69 LEOSIG LEO55 UMA4 V1R7 VIRPI VIR23 COM

RHO VIR

80 UMA24 CVNTAU VIR

PI SER15 DRASIG HER60 HER69 HER

OBS.AZ-TNT.AZ-TNT.AZ-TNT.COUSINSAZ-TNT.AZ-TNT.COUSINSAZ-TNT.AZ-TNT.COUSINSCOUSINSAZ-TNT.COUSINSJOHNSONAZ-TNT.AZ-TNT.COUSINSAZ-TNT.AZ-TNT.COUSINSCOUSINSAZ-TNT.AZ-TNT.COUSINSAZ-TNT.COUSINSCOUSINSJOHNSONAZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-THT.

6723678969237Bfi9708573137371754675927724773077367740785778717891809B826583283491864187178738904 Z

171718181819191919202020202020202121212222222223

59482344471520465111111Z12313236835441339&25660

12.918.310.748.74.216.624.945.420.157.743.639.614.129.158.217.25.414.141.830.324.342. S11.231.0

+ 1+ 86+ 58+ 18* 0* t+ 65+ 19+ 23+ 15+ 46+ 36+ 56+ 9+ 14+ 21* 9+ 6+ 2+ 8+ 29+ a+ 7* 1

183646746563705623939245330150232718232448

17.334.816.528.040.626.75.255.552.838.448.97.850.915.02.228.738.433.914.61.6

46.155.921.745.3

181718181819191919202020202020202121212222222223

0372346481720485213121313333437936461541545752

69041215123518493449582103650475828142285718

1865818B2

65192415463656101421962829a72

1835479490

416294645310378593336281344150

17412846615612222

113314271652131

58294583626

4.424.354.984.366.246.184.595.004. 584.954.824.994.306.S64.694.826.076.205.646.204.794.916.336.Z8

0.040.010.080. 130.030.010.020.10-0.060.090. 100.120.110.080.11-0.020.020.020.000.02-0.010.000.060.00

0.020.*40.050.07-0.010.010.060.06-0.130.010. 140.000.080.050.11-0.070.040.01-0.01-0.040.00-0.010.01-0.01

A1VAWA1VA3VA0A0A2IVA3VB9.5IA2VA 3 I I IA2II1A3IVA0A3VB9.6VA2VA1VA0A0A1VAWA0A2

68 OPHDEL UHI39 DRA111 HER

PI DRAZET S6E13 VULRHO AOL30 CVG29 CYG33 CYG

ZET DEL29 VUL6 EOU3 PEG11 PEG

OM PEGRHO PEG

25 PSC

AZ-TNT.AZ-THT.AZ-TNT.AZ-TNT.COUSINSCOUSINSAZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.AZ-TNT.COUSINSAZ-TNT.AZ-TNT.COUSINSCOUSINSCOUSINSCOUSINSAZ-TNT.AZ-TNT.COUSINSCOUSINS

TABLE A.2. Southern First-Order Extinction Stars

£ ( 1950.0)KB

0125019104440558060706?30634070507JJ80789080608370875089212721383152215961 62 117621826193720392071£15521952210229523282414271431313383342634373*523556

R.A.0 290 411 301 522 02 132 172 202 232 372 382 442 542 564 34 214 434 554 595 185 265 365 495 536 36 a6 106 206 2A6 327 97 578 318 358 388 398 53

0.66.8

33.917.637.60.89.9

51.232.053.648.845.96.910.731.911 .353.739.615.318.726.927.844.57.3

53.535.413.854.419.357.618.637.529.953.136. 134.622.8

-49-57- 9-42- 0- 9- 4-68-12-43-68-b7- 3- 2- 8- 3- 3- 2-20-21- 3- 7- 9- 4-14- 6- 2- 4- 7-22- 0-18- i-42- 8-47-27

DEC.4

4416446

413453606

2849545859512

177

1720143

475544293928552415584352829

47.13.16.30,42.52,28.11.54.19.50.35.45.51.23.34.35.18.24.19.47.21.11 .41.45.31 .26.37.49.26.30.39.46.47.23.16.18.

3232379837940298120254355654614289827

0011222222222244445555556666667788888

< 1985.0)R.A.3042325321418212539394555575

224557019283851545101122263411593337404054

414018442543.552913132*1852561355382546481192450281759381

255

11166174352

DEC.-48-57- 9-420

- 9- 4-68-12-42-68-67- 3- 2- 8- 3- 2- 2-20-21- 3- 7- 9- 4-14- 6- 2- 4- 7-22- 0-18- 2-42- 8-47-27

53325

343

322443515719404650534658144

IS19132

4755453040305728215

56591537

1143291222750392719514818284444486

22147

124223590043681

255910534721

4465566444445B65664464564666644454644

V.77.36.59.11.42.54.50.08.89,74.11.83.16.23.26. 18.33,34.91.70.38.81.95.28.67.14.62,66.26.54. 15.61.80.14.62.77.88

B-V0.010.00-0.04-0.050. 14-0.010.070.03-0.020.06-0.060.050.080.000.060.070.040.09-0.05-0.05-0.010.140. 100.050.050.010.040.06-0.03-0.040.00

0.070.000. 10-0.020. 120.12

LI0-0-0-0

0-080-00-0

0

*00000

~ei-8-a$&08a0

0-0-00000000

-B.04.03.10.14.13,07.08.05.04.06.13.08.06.04.08.08.06.11.15.11.06.09.08.06.00.03.08.00. \SS.02.03.08.00. 12.02.12.15

SPEC.A0VA0VA0B9A5A0A2A2VB9VA2VB9IIIA2AWA2A3VA1VA2A0B9A0VB9A4IVA0A0AWA0A0B9AJJA0VA0IVA3VA0A9IIA0A5IIA3V

NAMELAM1PHEETA

PHI60

DELRHO

EPSZET

XSI

49

THE

PHE

PHECET

HYICET

HYIHYI

ERI

OKI

LEP

CHI2CMADEL

DEL

MON

PYX

OBS.COUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSAZ-TNT.COUSINSCOUSINSCOUSINSJOHNSONCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSAZ-TNT.COUSINSCOUSINSAZ-TNT.COUSINSCOUSINSAZ-TNT.COUSINSCOUSINSCOUSINSCOUSINSAZ-TNT.AZ-TNT.COUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINS

3615 93787 93832 93981 103989 104109 104138 104293 104343 115163 138342 145367 145489 145670 155724 155959 15603 1 1 66070 166446 176519 176581 176930 186963 187029 197254 197440 197773 208075 218431 228451 228573 223840 238959 238988 239016 239XT98 0

M00or

1293557

2629579

41131741132258915182838263006

3317357

2812354346

I

39.826.024.322.738.411.94.4

51.311.717.954.530.454.734.95.15.415.812.00.721.736.120.842.216.54.340.053.58.328.345.10.159.79.97.819.510.8

-66- 0- 9- 0- 8- 3-71-41-22- 5- 2-37-34-58-38- 8- 9-28-12-23-12-14- 5- 3-37-24-12-17-33- 4-10- 3-45-!«-28-17

11571179

2944573314573958363316562947555135564659595525148564646492436

46. 248.056.035.342.89.77.0

26.49.052.052.623.152.658.528.617.110.229.352. 133.01.056.059.940.03.544.24.567.80.323.94.19.29. 117.424.751.4

99910101010101113141414ISIS15161617171718181919192021222222232323230

2313779272959104315194416245911171930402832283519579

29143741482

121371022585727557

43383202158102258293420347264749530345147256858

-66- 1- 9- 0- 8- 3-71-42-22- 5- 3-37-35-58-38- 8-10-28-12-23-12-14- 5- 3-37-24-12-17-33- 3-10- 3-45-14-28-17

207

2117203954844257497

4440221

3449575234554355554817358453434371225

6524522545443332436044415110333656633423344022432433

17413138449

4.004.566.404.495.906.044.734.384.486.526.144.044.924.064.605.554.924.774.314.814.244.716.365.424.124.594.764.074.506.274.825.554.734.514.574.56

0.150.11-0.04-SI. 040.820.040.030.110.030.040.00-0.030.020.090.000.040.090.020.03B.0B0.080.070.020.000.04-0.06-0.04-0.010.050.00-0.060.060.08-0.040.00-0.04

00-0-0-000000-0-0-00-0-00

-00-000-0-00-0-00

0

-0-003-0f)-0

.14

.09

.09

.07

.06

.05

.07

.13

.07

.02

.01

.11

.04

.08

.07

.04

.10

.01

.03

.07

.10

.04

.05

.07

.07

.12

.11

.01

.05

.07

.12

.05

.08

.13

.00

.12

A5VA3IIIA0A0HIA0A0A2VA2IVA2IIIA0A3A01VAlA3VA0IVAlA2VA0VA1VAlA2VA3VA0A0A2VB9B9VA0VA2VA0A0IVA2A2VB9.5VA0VB9IV

ALF VOLTAU2HVA34 HYAALP SEX17 SEX

BET CRT

PSl CEN

BET CIR

50 LIBPSl SCOD SCOMJ SER51 OPHOMI SERGAM SCT

14 AQLALF CRA52 S6RNU CAPTHE CAPMU PSA

THE AQR

OM2 AQRDEL SCL2 CET

COUSINSAZ-THT.COUSINSAZ-TNT.COUSINSCOUSINSCOUSINSCOUSINSAZ-TNT.COUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSAZ-TNT.AZ-TNT.AZ-TNT.COUSINSAZ-TNT.AZ-TNT.COUSINSCOUSINSCOUSINSAZ-TNT.AZ-TNT.AZ-TNT.COUSINSCOUSINSAZ-TNT.COUSINSCOUSINSAZ-TNT.AZ-TNT.AZ-TNT.

APPENDIX BSECOND-ORDER EXTINCTION PAIRS

As mentioned in Section 4.4, second-order extinction can be determined byobserving a close optical pair of widely differing colors as they pass throughvarying air mass. A list of bright pairs in the equatorial plane was publishedby Crawford, Golson, and Landolt.1 Barnes and Moffett2 extended this list toinclude R and I magnitudes and one additional star. Their paper is extremelyuseful as it gives finding charts for all 37 stars of their extinction network.

An examination of the Johnson, Tonantzintla, and Cousins lists of standardstars yielded 23 more stars useful in the extinction determination. Table B.Ilists all 60 stars.

Extinction examples using stars from this list are presented in Appendix G.

REFERENCES

1. Crawford, D. L., Golson, J. C., and Landolt, A. U. 1971. Pub. A. S. P. 83, 652.2. Barnes, T. G., Ill, and Moffett, T. J. 1979. Pub. A. S. P. 91, 289.

286

TABLE B.1. Second-Order Extinction Pairs

Star

HR0607HR0610

HR0718HR0725

HD 16581HD 16608

HR1373HR1389

HD30544HD30545

HR1826HR1830

HR2071HR2070

HD40983HD41029

HD50279HD50167

h

22

22

22

44

44

55

55

56

66

EPOCHRAm s

0 381 14

25 3026 55

37 037 11

20 322 36

46 246 7

26 2726 54

53 753 2

59 470 5

SO 149 29

1950DEC

d m s

-0-0

89

11

1717

33

-3-3

-4-4

11

11

6 4234 45

14 1320 37

9 1454 57

25 3748 55

33 4530 7

20 4729 5

47 4137 24

5 266 55

9 518 45

EPOCH 1985RA DEC

h m s d m s

2 2 262 3 1

2 27 222 28 47

2 38 4B2 38 60

4 22 44 24 37

4 47 524 47 57

5 28 125 28 39

5 54 515 54 46

6 1 356 1 53

6 51 496 51 18

0 3 23-0 24 41

8 23 369 29 58

1 18 162 3 58

17 30 3117 53 41

3 37 253 33 47

-3 19 7-3 27 27

-4 47 23-4 37 5

1 5 241 6 52

1 6 301 16 12

V

5.425.92

4.286.07

8.198.31

3.764.29

7.326.01

6.385.80

6.285.87

8. 568.17

8.177.85

B-V

0.140.88

-0.061.02

-0.061.51

0.990.04

-0.061.21

-0.011.15

0.051.17

0.000.98

-0.061.55

U-B

0.130.51

-0.130.86

-0.271.79

0.830.08

-0.311.15

-0.061.07

0.061.22

-0.040.76

-0.281.72

SPEC. NAME

A5 60 GETG5II 61 CET

B9III CH12 CETK2III

B9K4

Kill I DEL TAUA3V 68 TAU

B9KG

B9G8

AOK2

B9KO

B8K5

OBS.

COUSINSCOUSINS

JOHNSONSCOUSINS

KITT PK.KITT PK.

AZ-TNT.AZ-TNT.

KITT PK.KITT PK.

COUSINSCOUSINS

COUSINSCOUSINS

KITT PK.KITT PK.

KITT PK.KITT PK.

Star

SECOND ORDER EXTINCTION PAIRS (CONT)EPOCH 1950 EPOCH 1985

RA DEC RA DECm s d m s h m s d m s V B-V U-B SPEC. NAME DBS.

HR2710HR2713

HD63390HD63368

HD75012HD75138

HOB4971HD84916

HR3989HR3996

HD97991HD98007

HD111133HD111165

HD 11 8246HD118129

HD 129956HD 129975

7 97 9

7 467 46

8 458 45

9 469 46

10 710 8

11 1311 13

12 4412 44

13 3313 32

14 4214 43

1128

104

144

1251

3827

3944

3043

727

574

5 445 33

-0 8-0 37

0 150 44

-2 28-4 10

-8 9-8 10

-3 11-3 29

6 137 16

-5 54-6 42

0 55-0 6

2133

07

4425

5025

4316

5719

2751

444

3817

7 11 37 11 20

7 47 577 47 51

8 46 498 47 32

9 47 589 47 37

10 9 2210 10 11

11 15 2611 15 31

12 46 1712 46 29

13 34 5713 34 17

14 44 4414 44 52

5 40 505 30 1

-0 13 16-0 42 23

0 7 590 36 39

-2 38 35-4 20 11

-8 20 3-8 20 37

-3 23 25-3 40 47

6 1 597 5 23

-6 4 47-6 53 28

0 46 48-0 15 7

6.086.15

8.748.43

7.837.24

8.658.66

5.905.64

7.418.94

6.368.46

8.078.18

5.708.37

-0.021.15

0.050.95

0.081.48

-0.171.16

0.021.31

-0.240.71

-0.061.16

-0.161.07

-0.031.50

-0.050.97

0.050.57

0.091.78

-0.761.12

-0.061.41

-0.930.32

-0.051.21

-0.630.95

-0.061.86

AOKG

B9KO

B9K2

B5K5

AOK2

B3KO

B9KO

88K2

B9K5

COUSINSCOUSINS

KITT PKKITT PK

KITT PKKITT PK

KITT PKKITT PK

17 SEX COUSINS18 SEX COUSINS

KITT PKKITT PK

KITT PKKITT PK

KITT PKKITT PK

KITT PKKITT PK

SECOND ORDER EXTINCTION PARIS (CONT)

Star

HD 140873HD1408SO

MD161261HD 161 242

HD171732HD171731

HR7313HR7319

HD 184 790HD 18491 4

HD 196426HD 196395

HD205556HD205584

HD209905HD209796

HR8451HR8453

HR9042HR9033

h

1515

1717

1818

1919

1919

2020

2121

2222

2222

2323

EPOCH

RA

m s

43 3043 23

41 4941 45

33 5633 56

15 1716 0

33 3634 07

34 4534 33

33 2633 44

4 63 29

7 457 57

50 3149 24

1950DEC

d m s

-1 38 56-1 17 26

5 44 75 16 17

-3 7 13-2 31 36

1 56 270 59 34

-2 55 07-4 24 44

-0-04 41-0-41 34

5 15 75 54 46

2 11 431 0 48

-4 8 24-4 30 49

1 48 452 39 09

EPOCH 1985RA DEC

h m s d m s

15 4515 45

17 4317 43

18 3518 35

19 1719 17

19 3519 35

20 3620 36

21 3521 35

22 522 5

22 922 9

23 5223 51

1911

3228

4646

347

2658

3321

1129

5316

3446

1811

-1 45-1 23

5 435 15

-3 5-2 29

2 01 3

-2 50-4 20

0 2-0 34

5 246 4

2 211 11

-3 58-4 20

2 02 50

2758

1424

2750

1625

261

4014

3211

582

428

2650

V

5.408.80

8.317.80

9.129.06

6.185.09

8.128.16

6.238.72

8.327.72

6.528.94

6.276.00

6.285.55

8-V

-0.031.66

0.051.28

0.281.13

0.011.15

0.171.20

-0.091.66

-0.061.26

-0.061.21

0.000.98

0.001.53

U-B

-0.422.02

-0.141.10

-0.101.05

0.011.01

-0.320.93

-0.392.04

-0.351.32

-0.231.17

-0.070.84

-0.011.86

SPEC

68K5

69K2

B9K2

AOK2II

68K5

B8K5

69K2

B9K2

AOKO

NAME

23 AQL

OBS.

KITT PK.KITT PK.

KITT PK.

KITT PK.

KITT PK.

KITT PK.

COUSINSCOUSINS

KITT PK.

KITT PK.

KITT PK.

KITT PK.

KITT PK.

KITT PK.

KITT PK.KITT PK.

COUSINS

COUSINS

APPENDIX CUBV STANDARD FIELD STARS

Having a photometric system without standard stars is like measuring the dis-tance from New York to Paris in meters without defining the length of themeter. The standard stars are an integral part of a photometric system. Theyare as important as the filter responses themselves.

After the definition of the UBV system by Johnson and Morgan,1 Johnsonand Harris2 published a list of 108 stars intended for use as photometric stan-dards for the system. There were 10 primary standards, stars that were mea-sured every possible night, and 98 additional stars that were measured fromtwo to 17 times, averaging 7.3 measurements each. Because of the few mea-surements of some of these stars, internal accuracy of the system is on the orderof 0!"03 in K That is, if you select a large subset of these stars and measurethem, your mean probable error of a measurement should be about this size.

The Johnson standard list has no stars south of —20° declination. Tradi-tionally, this has made Southern Hemisphere transformations difficult. For thisreason, Cousins3 has proposed a second list of standards between +10" and— 10° declination, and secondary standards in the E and F regions of thesouthern sky.4 The equatorial standards are presented by Cousins with theirHR numbers but without coordinates, making them difficult for the amateurto use. The E and F region stars have the coordinates listed, but the amateurmay have difficulty finding the reference in the local library. For this reason,the E and F region stars are included in this appendix and the list is separatedinto stars north and south of the celestial equator. Table C.I lists the northernstandard field stars, while Table C.2 lists the southern standard field stars.

Another problem with the standard stars listed in references 2 through 4 isthe preponderance of bright stars. Transformations using bright stars and largetelescopes are very difficult because of dead-time corrections and photomulti-plier tube fatigue. For instance, with the Goethe Link 40-centimeter (16-inch)telescope, you cannot easily look at stars brighter than 4 !"0 in V, and seldomuse stars brighter than 6TO as standards. Fainter stars that can be used as

290

UBV STANDARD FIELD STARS 291

secondary standards can be found in the list by Landolt.s These equatorial starsrange from seventh to fourteenth magnitude, with most about twelfth magni-tude in V. Another list of brighter UBV secondary standards is readily avail-able from Sky Publishing6 and is recommended for purchase.

To use the stars from this appendix for transforming to the standard system,pick 20 or more from the list, distributed over the sky but greater than 30"above the horizon. This removes effects caused by a small number of standardsand minimizes systematic regional errors.

REFERENCES

1. Johnson, H. L. and Morgan, W. W. 1953. Ap. J. 117, 313.2. Johnson, H. L. and Harris, D. L., Ill 1954. Ap. J. 120, 196.3. Cousins, A. W. J. 1971. Roy. Obs. Annals. No. 7.4. Cousins, A. W. J. 1973. Mem. Roy. Astr. Soc. 77, 223.5. Landolt, A. U. 1973. A. J. 78, 959.6. Iriarte, B., Johnson, H. L., Mitchell, R. I., and Wisniewski, W. K. 1965. Sky and

Tel. 30, 25. (Available as a reprint.)

TABLE C.I. Northern UBV Standard Field StarsMtoN

HR303900S 30ZZ6034304030437049305530617

071807b30753099610301048134613731409154315521641179017912010

Z4Z12763

2852Z98S?2493454356936653S153974

( 1990.0)R.A.

0 10 39.40 M 28.30 47 2.81 8 2.61 ZZ 31 .51 28 48.21 39 46.61 51 52.32 4 20.92 5 5.32 2b 29.82 33 20. 12 33 20. 13 16 44.13 22 7.13 26 10.5

16 56.720 2.825 41,647 7.448 32.4

9 3 0.25 ZZ 26.85 Z3 7.75 40 44.36 28 51 .78 34 49.47 15 13.27 19 35.07 25 53.87 41 25.98 13 48.38 40 36.78 55 47.69 11 45.89 32 40.010 4 29.2

DEC.14 5438 2440 4854 b351 5815 520 120 3323 13Z 578 146 386 383 118 51

55 16IS 3017 Z519 46 525 31

41 106 IB28 3412 3817 4016 26Ifi 375 39

31 5324 319 203 34

4B 14Z 31

36 235 29

20.614.925.24.3

34.419.434.35Z.037.054.013.157.857.817.315.250.830.636.816.432.316.38.4

21.61.7

13.612.037.356. 142.08.310.527.745.721.834.614.721.4

(1985.0)R.A.

0 12 270 16 180 48 5tf1 10 91 24 481 30 401 41 411 53 482 6 182 6 542 27 212 35 112 35 113 18 333 24 03 28 514 18 554 22 34 27 444 49 04 50 245 5 275 24 195 25 205 48 426 30 546 36 507 17 147 21 277 28 87 43 320 15 428 42 268 58 129 13 349 34 4710 fi 32

DEC.15 6 138 35 5440 59 5155 4 1460 9 2916 16 720 12 920 44 1023 23 3E3 7 528 23 366 48 86 48 83 18 538 58 38

55 24 5IS 35 3217 30 3019 8 546 56 85 34 48

41 12 576 20 13

Z8 35 5012 38 5117 38 4016 24 4816 34 65 35 39

31 48 4824 26 79 13 583 27 11

48 6 122 22 51

35 52 5136 19 5

V2.834.614.534.332.683.625.232.652.0010.034.285.82

11.654.8Z3.595.083.653.763.543.193.693.171.641.654.909.631.933.589.824.163.573.524.303.153.88S.414.48

B-V-0.230.06-0.150.170.130.970.830.131.151.44

-0.060.971.610.680.890.050.990.981.020.45-0. 17-0.18-0.23-0.13-0.071.500.000.111.560.320.931.48

-0.190.1B-0.060.770.18

U-B-0.870.04-0.580.110.120.760.500.101.121.08

-0.130.791.120.180.620.030.820.820.88-0.01-0.80-0.67-0.87-0.49-0.181 .180.030.101.12

-0.030.61)1.78

-0.740.07-0.130.450.08

SPEC.B2IVA2VB5VA7VA5VG81IIK1VA5VK2III

B9IIIK3V

G5VG B I I IA 1 VK0IIIK0IIIK01IIF8VB2I1IB3VB2II1B7UIB91V

A0IVA3V

F0VG81IIM i l lB3VA7VAffP

caiv-A7V

NAMEGAM PEGTHE ANDNU ANDTHE CASDEL CASETA PSC107 PSCBET A R IALF ARI2 348CHI2 GETABKAP CETOM1 TAU

GAM TAUDEL TAUEPS TAUPI3 OR IP14 OR IETA AURGAM OR IBET TAU134 TAU17 1320GAM GEMLAM GEM5 1668RHO GEMKAP GEMBET CNCETA HYAIOT UMATHE HVA11 LMI21 LMI

OBS.JOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONPRIMARYJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONPRIMARYPRIMARYJOHNSONJQHNSONJOHNSONJOHNSON

39B2 1»4033 10

104133 104456 114534 114540 114550 114554 114660 124931 124983 135062 13btfi'Z 135235 135511 14*854 155867 155868 155933 155947 15&J09Z IS

176556 176603 176629 17

177001 197178 187235 19

1919

7557 197602 1979»6 208622 228781 238832 23

K 3969 2323

5 42.614 5.421 33.430 10. e32 6.446 30.64S 5.450 6.251 12.612 57.653 35.49 32.421 55. 825 59.1152 18. 243 43.141 48.243 5Z.744 0.854 8.5Sb 39.918 14.123 15.932 36.741 0.045 23.0515 42.93b U.757 4.33 6.69 51.8

31 7.248 20.652 51.437 18.937 0.82 16.110 51.937 22.646 35.6

12 1243 91 379 3317 414 512 238 453 5857 1856 3828 755 1014 218 3D2 66 3415 347 3015 4927 146 252 1012 30<f 35Z 434 1638 4432 3713 474 573 398 446 1618 4438 4714 5656 535 212 8

44.553.554.052.224.65.847.639.222.036.97.852.056.642.851.39,053,937,430,724.817.453,614.241.911 .828,354.09.611.315.954,07.85.8

49,84,322,49.2

31,318.611.7

If 7 3410 16 1210 23 2110 32 111 33 5511 48 1811 49 S311 51 5511 53 212 14 401 3 0 413 11 1213 ?3 1913 27 4213 53 5814 45 2915 43 3115 45 2915 45 4315 55 4415 56 5816 19 1717 25 117 34 1317 42 4317 47 817 52 2618 36 2518 58 2219 4 4319 11 3519 32 5219 50 119 54 3420 38 5422 38 3523 4 B23 12 2423 39 923 48 23

12 2 2642 59 241 27 159 23 216 52 4714 39 251 51 4

37 52 58S3 46 4067 6 5656 26 4927 66 4354 59 5713 51 5018 28 331 57 206 28 1815 28 67 24 0IS 43 2026 55 1646 20 532 8 2412 34 204 34 162 42 464 16 28

38 45 5832 40 613 50 295 1 273 43 428 49 276 22 2318 51 3038 58 1915 7 2987 4 675 32 672 19 52

1.363. 459.633.855.952.143.616.452.443.314.934.284.014.982.693.742.653.674.433.854.153.897.542.082.773.759.540.043.252.999.136.820.773.713.774.882.495.574.138.98

-IMI0.031.52

-0.14-0.160.090.550.750.000.080.360.570.160.710.580.001.170,060.600.481.23

-0.151.360.151.160.041.740.00-0.050.001.490.020.220.86-0.06-0.20-0.051.010.511.48

-0.360.061.19

-0.95-0.640.070.100.170.010.070.010.070.080.260.19-0.031.240.070.10-0.031.28-0.561.260.101.240.041.29

-0.01-0.09-0.011.16

-0.830.080.48-0.22-1.04-0.060.89

«.«*1.09

BBVA2IV

BUBB3VA3VF8VG8VPA0VA3VF2VG0VA5VG5VG0IVA0VK2IIIA2IVG0VF6VK3IIIB5IVK7VA5IIIK2II1A0VM5VA0VB9IIIA0VM3.5VBBA71V-G8IVB9V

09VB9VK3VF7VM2V

ALF LEOLAH UHA1 2447RHO LEO90AB LEOBET LEOBET VIR

GAM UKADEL UHA78 UKABET COM80 UMA70 VIRETA BOO109 VIRALF SERBET SERLAM SERGAM SEREPS CRBTAU HER157881ALF OPHBET OPHGAM OPH4 3561ALF LYRGAM LVRZET AQL4 4048184279ALF AQLBET AQLALF DEL10 LACALF PEG

IOT PSC1 4774

JOHNSONJOHNSONJOHNSONJOHNSONJOHHSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONPRIMARYJOHNSONJOHNSONJOHNSONPRIMARYPRIMARYJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONPRIMARYJOHNSONPRIMARYJOHNSONJOHNSON

TABLE C.2. Southern UBV Standard Field Stars( 1950. f)

HR0331937*

04 It

0509

0875

1M4

12911316

16661781

19551861169919031998

2462

294633143654

3730

R.A.51214223Z

1 411 582 543 253 303 363 483 494 84 74 IV4 154 164 355 55 21S 265 Z95 30S 325 335 446 296 306 436 466 438 239 99 179 20

31.055.946.831.713.744.7e.z6.947,034.447.6Z6.13.4

48. 20.756.055.227.99.123.4S.8

55.330.69.5

59. 140.541 .334.217.647.547.230.29.715.41.1

34.3

DEC.-41 45-45 47-42 47-44 47-45 55-16 12-18 18- 3 54-76 55- 9 37-74 8-45 32-74 50-44 48-45 59-44 29-75 55-45 46-73 IS- 5 8- 0 12- J 41- 7 20- 1 37- S 5f- 1 13-14 50-43 40-40 10-47 10-43 44-45 23- 3 44-44 39-44 47-45 50

14.253.143.918.162.20.530.040.09.9

34. B17. aa. e43.93.9

46.342.250.92W.439.558.518.74.1

12.935.728.1F6.121. Z51.428.88.9

37,924.031.545.246.90.5

( 1985.0)R.A.

1 71 14

1624334359

2 553 243 323 363 493 48

18121417

4 345 75 ZZ5 305 315 315 345 355 466 306 3)6 446 476 496 249 109 189 21

6282024226405258152233264752

55323165639125541271637134550315431IB51

DEC.-41-45-42-44-45-16-18- 3-76- 9-74-45-74-44-45-44-75-45-73- S- 8- 3- 7- 1- 5- 1-14-43-48-47-43-45- 3-44-44-45

34 136 4736 4036 2246 71 28B 20

46 1847 5130 311 26

25 4944 2342 1654 1524 2250 4141 1514 246 IS10 2339 3118 4236 755 812 3849 3642 2312 212 2347 125 5251 2348 2256 3959 0

V5.214.967.866.276.923.5010.185.176.803.737.616.947.128.206.586.717.237.546.812.805.707.974.635.352.771.703.556.694.947.227.426.543.904.987.205.74

B-V0.160.57-0.081.140.900.721.530.080.200.891.140.940.390.130.381.481.640.640.960.13-0.221.47

-0.26-0.20-0.25-0. 190. 101.000.88-0.160.161.51

-0.020.231.110.92

U-B0.080.10-0.371.080.480.201.160.050.100.571.010.70-0.050.150.001.801.970.170.660.10-0.871.21

-1.07-0.94-1.08-1.040.060.740.61-0.740.1Z1 .82

-0.02-0.570.940.65

SPEC.A3G089K0C5GBVP

A1VA0K2VK0G5F2A2F0K0K0G0K0A3IIIB5VKZB0VB1V091 1 1B0IAA3VK0K083A3KMA0VB5K0C5

NAMEUPS PHENU PHEHD7795

H09733TAU CET-IB 0359

HD21940EPS ERIHD231Z8HD24Z91HD24636HDZ5653

HDZ77Z8HD27471HD29751BET ERI3529936395UPS ORI36591EPS ORIEPS ORIZET LEPH046415

HD49260HD49850

HD80527

OBS.COUSINSCOUSINSCOUSINSCOUSINSCOUSINSJOHNSONJOHNSONPRIMARYCOUSINSJOHNSONCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSJOHNSONCOUSINSCOUSINSCOUSINS

4519

46014624

4662

4B04

50195056b40l544455»955305531bbze55805685

6175

637164276460

74467498

9 219 249 2711 4011 4311 bl11 5712 112 612 1112 1312 1612 1912 3S12 4813 1513 2214 Z'J14 3314 4414 4714 4814 4814 5615 1415 1)16 2216 3417 216 5717 717 1517 ZS17 2217 3418 219 3119 4319 45

39-115.727.21.3

15.04.21 .98.5

18.125.313.851.421.859.89.7

47.133.358- 11.9

16.455.06.4

21.99.718.752.713.924. 127.039.64.6

46.935.31.7

13.328.312.141.327.3

-48-12-42-74-45-4B-46-73-44-45-17-76-73-75- 0-18-10-45-46-43-15-15-43-42- 9- 7-12-10- 4- 4-44-44-44-44-42- 3- 7-72-72

4515856242721562261531135

292

54541

2047502257111118295841291074432183735

19.142.013.058.544.450.216.27.849.945.352.0Zl .535.642.726.31.33.4

33.041 .654.025.96.6

11.139.658.918.06.02.8S8.40.042.717.90.18.40.152.624.742.416.9

999

111111111212121212121212131314141414141414IS1516161716171717171710191919

22 S325 5628 4741 3444 5952 5058 432 57B 713 1515 218 5421 2338 1549 5717 3924 2426 1435 1946 3349 5150 250 3958 2716 1113 4424 1036 204 18

59 309 36

IB 1923 724 3536 484 18

36 447 4249 30

-48-13-43-75-45-46-46-74-44-45-17-76-73-75- 0-!8-11-46-46-43-15-15-43-43- 9- 7-12-10- S- 4-44-44-44-44-42- 3- 7-72-72

13 200 497 268 37

36 2439 3132 577 49

14 3138 2527 3243 025 1417 1540 5113 34 583 5810 5029 4056 558 4530 496 119 4019 522 5332 161 48

44 532 1912 288 5646 033 141 423 4132 3049 59

6.2710.066.606.475.297.226.666.455.755.312.606.846.786.488.494.750.96S.825.546.305.162.754.326.102.6110.5610.132.S67.7310.075.076.64S.ll7.657.179.384.965.397.30

-0.141.530.480.52-0.121.040.561.220.241.43

-0. 110.440. 160.091.410.71-0.230.311.491.080.410.15-0.160.60-0,111.611.600,021.161.430.8B0.20-0.051.250.651,52

-0,010.240.46

-0.611.150.060.05-0.550.830.090.980.151.56

-0.35-0.010. 14-0.231 .260.25-0.940.141.710.88-0.040.08-0.620.28-0.371.201.18-0.861.051 .090.560.05-0.401.140.171.21-0.870.10-0.02

B5

FSG0B85SF8K0A2K0B8IIIF5A2B9M0.5VG6VB1VA3K2G5F51VAMB5F8BBV

09.5VK5VM3.5VG5A0B8K0G5K5B0.5IA3FB

HD81347-12 2918H082224HD101805

HD103281HD104138

HD106321GAM CRVH0107145HD107547

0 298961 VIRALF VIR

ALF1 LIBALF2 LIBONI LUP

BET LIB-7 4003-12 4523ZET OPH154363-4 4226

HD157487HD159656-3 4233KAP AOL

HD186502

COUSINSJOHNSONCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSJOHNSONCOUSINSCOUSINSCOUSINSJOHNSONJOHNSONJOHNSONCOUSINSCOUSINSCOUSINSJOHNSONJOHNSONCOUSINSCOUSINSPRIMARYJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONCOUSINSCOUSINSCOUSINSCOUSINSCOUSINSJOHNSONJOHNSONCOUSINSCOUSINS

SOUTHERN UBV FIELD STANDARD STARS

( 1950.0)HR7590

7950

86570062

8704

R.A.1919192020Z*2020202222222Z2Z»12

54SB597

1Z\Z14154427294243444550

50.839.64.0

48.051.256.850.314.258.218. 113.844.146.731.927.550.8

-73-44-45-43-73-72-43,-42- 9-42-43-46-47-45-15-11

DEC.236?.a488

57046403231481213g52

43.821.410.240.33.750.355.030.848.251 .514.238.112.344.842.058.3

19202020202020202022222222222222

< 1905.0)R.A.5811

10161617174629314445464752

SI73213465013375122184750341841

DEC.-72-44-45-43-73-72-42-42- 9-42-43-46-47-45-14-11

573014421

515439332220371Z4941

1301825342022573525346393647

V3.957.916.576.546.566.937.017.453.776.926.915.516.567.2210.175.81

B-V-0.03-0.041.220.881.411.030.330.590.01-0.050.981.320.300.601.60-0.08

U-B-0.06-0.251. 220,571.670.870.020.160.04

-0.080.771.430.100.071.15-*.3Z

SPEC.A0A0K069K2K0F0G0A1VA0C5KBASG0

B9

NAME OBS.EPS PAV COUSINSH0189502 COUSINSHD1B9563 COUSINSHD191349 COUSINSHD191937 COUSINSKD191973 COUSINSHD192758 COUSINSH01928Z6 COUSINSEPS AOR JOHNSONHD213155 COUSINSHD213457 COUSINS

COUSINSCOUSINS

HD215657 COUSINS~15 629* JOHNSON74 AQR JOHNSON

APPENDIX DJOHNSON UBV STANDARD CLUSTERS

When Johnson and Morgan created the UBV system, they included a list ofstandard stars. This list covered the entire sky visible from the United Statesand is used by some observers as the primary method of calibrating theirinstrumental magnitudes. However, at the same time, Johnson observed starsin or near three open clusters. These standard clusters were also to be used indetermining the transforming coefficients.

The advantages of using star clusters to obtain coefficients include:

1. First-order extinction corrections are very small because all stars arewithin 1 ° of each other.

2. It is easy to observe many stars quickly because they can be located ona single finding chart and only small movements are required to changefrom one star to another.

The disadvantages of using one of these standard clusters include:

1. Red stars are almost always fainter than blue ones, because most starsare on the main sequence and are at the same distance.

2. A clear spot for obtaining a sky reading can be hard to find.3. The density of stars makes the use of a large diaphragm difficult because

"companions" become included.4. Transformation difficulties can exist because each cluster is in an isolated

region in the sky and was placed on the standard system differently thanthe whole-sky standard list.

Still, for many observers the cluster method of calculating color coefficientsis highly useful. We recommend that you use both the whole-sky and the clus-ter methods sometime during your observing season.

297


In observing these clusters, Johnson picked from one to three regional stan-dard stars, and measured all cluster stars with respect to these standards. Hethen tied the regional standards to the UBV system by using the whole-skystandards. Therefore, the clusters are placed on the UBV system in a two-stepprocess, though with exactly the same equipment which was used to define theUBV system.

Each cluster is discussed in a similar manner. There is a brief description ofthe cluster and of the pertinent references. This is followed by a list of 26 starsfor each cluster. These stars were picked from the much larger lists by Johnsonto include stars over a wide magnitude and spectral range, and with as manyJohnson observations as possible.

Each star's coordinates are taken from the SAO catalog, or from the catalogreference if the star is fainter than tenth magnitude. The stars are named onthe right by a letter of the alphabet and the zone BD number where available.The last item for each cluster is a finding chart with 26 stars identified.

D.1 THE PLEIADES

This famous naked-eye cluster is often called the Seven Sisters. It is a veryyoung cluster, containing many bright blue stars and nebulosity. There areabout 600 stars brighter than fourteenth magnitude in the central 1 ° field.

Because of the youth of the cluster, only three red stars were observed thatare brighter than tenth magnitude. This places a large weight on their mea-surement, especially star D, and this should be recognized by the observer.Table D.I lists the information for this cluster.

Approximately 250 stars were observed with the 53- and 103-centimeter(21- and 42-inch) Lowell reflectors. All stars in the region were observed dif-ferentially with respect to the regional standard stars E, 1, and Alcyone.

Catalog:Hertzsprung, E., 1947. Ann. Leiden Obs. 19, 1A.

Photometry:Johnson, H. L., and Morgan, W. W. 1953. Ap. J. 117, 313.Johnson, H. L., and Mitchell, R, I. 1958. Ap. J. 128, 31.

Chart:Red Palomar chart MLP 357 (E-641).

D. 2 THE PRAESEPE

The Praesepe, M44, or the Beehive, is one of the largest and nearest open clus-ters. It is visible to the naked eye, though not as easily as the Pleiades, and hasseveral hundred members.

TABLE D.1. Pleiades Cluster Standards

(1950.0)HR116511451172

R.A.333333333333333333333333

444245454445474443424446444245434643434144434644

30.413.622.97.00.331.128.838.736.235.644.255.811.141.027.555.725.542.38.35.019.07.7

26. 139.0

232423242424242323242324242423232323242424252324

DEC.571816502211202728182347

28532541204251404934

7.642.98.89.10.436.442.123.112,230.426.62.5

26.022.048.050.019.040.047,017.018.051.058.052.0

333333333333333333333333

{ 1985.0)R.A.454447474647494645444649464447464845454346454846

351827125

36344340404801646320304613924133044

DEC.242423242424242323242324242424232323242424252324

325225628182733342529101334032472749112075641

351734353022

51423

5424555513194210185446222120

V2.874.315.456.466.826.957.427.728.118.608.799.169.469.7010.0210.5211.3512.0212.0512.5112.6114.3615.7216.42

B-V-0.09-0.11-0.071.700.030. 120. 131.230.35J0.351.150. 160.470.550.560.640.780.991 .010.811.181 .011.150.60

U-B-0.34-0.46-0.322.07

-0.070.090.121.120.290. 110.810.150.020.050.090.160,380.540.840.301.000.470.980.25

SPEC.B7IIB6VBSVK5B9VA0A2KfA0A2K0A0F8F8

F9G2G8G5GlG9V

ABCDEFGHIKLMN0PQRSTUVXYz

NAME OBS.JOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSON


w'PLEIADES

-T- - .'

;30'

Figure D.I. The Pleiades cluster standards. Copyright by the National Geographic Society—Palomar Observatory Sky Survey. Reproduced by permission.

Johnson made observations of 150 stars in the region of the Praesepe cluster.The 26 stars listed in this table are therefore a small subset of the total. Tworegional standards were used in this cluster, stars B and F. All stars in thecluster were then observed differentially with respect to these two standards.Table D.2 lists the information for this cluster.

Catalog:Klein-Wassink, W. J. 1927. Pub. of Kapelyn Astronomical Laboratory at Gr<Snin-gen. No. 41, 5.Vanderlinden, H. L. 1933. Etude de I'amas de Praesepe. Gembloux: Joules Duculot.

Photometry:Johnson, H. L. 1952. Ap. J. 116, 640.

Chart:Blue Palomar chart MLP 426 (O-1311).

TABLE D.2. Praesepe Cluster Standards

HR

3429

34283428

EPOCH

RAh m s

8 36 598 37 148 37 308 32 278 37 358 37 198 38 48 37 518 38 588 37 268 36 178 35 558 37 98 35 408 37 38 38 348 37 488 36 228 38 428 36 148 36 378 38 248 37 138 36 23

1950

d

192019191920191920191919201920201920201919191920

DECm

43115045438

4553

34246401838147

50238

2857555614

s

78

53482357325213361039493034225317545551371251

EPOCH

RAh m s

8 38 598 39 158 39 318 34 288 39 358 39 208 40 48 39 528 40 598 39 268 38 188 37 556 39 108 37 408 39 48 40 358 39 488 38 238 40 438 38 148 38 388 40 258 39 148 38 24

1985

d

192019191920191919191919201920191920201919191920

DECm

353

433835

138465535383311317

594315

1215048487

s

414126335530

324429

45152278

5225522431267

4526

V

6.5906.3906.4406.5806.3006.6106.7806.8506.9007.5408.5009.0009.670

10.01010.11010.72010.87011.31011.71012.37012.64013.70014.61014.940

B-V

0.9600.9801.0200.6700.1700.0100.1700.2000.9600.1600.2500.3200.4401.0100.4900.6000.6800.7000.7800.4601.0000.8100.9501.480

U-B

0.7200.8300.9000.2500.1600.0200.1400.1500.7400.1300.0700.030

-0.0200.7800.0000.1000.1900.2400.380

-0.0500.7600.4000.6901.100

SPEC.

KOI IKOI IKOI IGDI IA2AOA6VA9IIKOI IFOIIA9VA5F6VG5

NAME

A.2150B,2158C.21660,2118E,2171F,21596,2175H,2172J.2185K.21631,2144N.,2139N,2156P,2056QR.2181STUVUXYZ

OBS.

JOHNSON

JOHNSONJOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSON

JOHNSONJOHNSON

JOHNSON


Figure D.2. The Praesepe cluster standards. Copyright by the National Geographic Society—Palomar Observatory Sky Survey. Reproduced by permission.

D.3 1C 4665

The standard region near the open cluster 1C 4665 was observed by Johnsonin 1954. The cluster itself contains a few blue stars, some of which were mea-sured, but most of the standard stars in this region are field stars within 1 ° ofthe cluster center.

All stars in the region were measured differentially with respect to theregional standard, star A. Some of these stars were observed at McDonald andsome at Mount Wilson observatories, but most were measured at Lowell obser-vatory. Table D.3 lists the information for this cluster.

This region is interesting because several reasonably bright red stars areincluded, thereby reducing the error in the color transformation that can betroublesome on the other two standard regions.

Catalog:Kopff. 1943. Mitt. Hamburg Sternw. 8, 93.

JOHNSON UBV STANDARD CLUSTERS 303

IC4665. *N|".'. . :' , / " ' - . . ' . '.'

J

20'-

. .-. . - •• .' - • .*rx %•. / , •" '.." •: •'- . • . *' . ^"\'. . -• ' ; • . - * • . - • "

W: ^ ' / ' ' X^ ' : • - " "* ' ' ' ' ' *• ' ' • • • • • • • • ••-• ' • '

Figure D.3. The 1C 4665 cluster standards. Copyright by the National Geographic Society-Palomar Observatory Sky Survey. Reproduced by permission.

Photometry:Johnson, H. L. 1954. Ap. J. 119, 181.

Chart:Red Palomar chart MLP 569 (E-780).

TABLE D.3. 1C 4665 Regional Standards

HR( 1950.0)

P.. A.17 4317 4417 4317 4117 4617 4317 4417 4117 4417 4417 4117 4417 4117 4517 4517 4417 4517 4317 4217 4217 4417 4217 4217 4317 4417 43

40.114.043.736.916. 329.99,845.42.10.824.819.448.81.7

21.030.355.37.433.619.841.055.034.052.041.045.0

55565565655555555555555556

DEC.32474044242S

IE15245134443722261926321331355257136

54.531.035.251.459.446.418.116.614.254.857.257.46.714.643.335.132.044.133.614.039.01.0

46.021.015.044.0

1717171717171717171717171717171717171717171717171717

( 1935.0)R.A.4545454347454543454543464346474647444444464444454645

2356261959125228444372

31444133850162243716342427

DEC.55565565655555555&55S55556

324639342417

1514245134433622251825311230345156125

7454857205832232782

1213311

5051554222541155343056

67777777778B8888899991010101011

V.85.12.34.43.49.59.74.83.89.94.05.22.31.33.40.89.96.10.39.68.81. 10.21.61.75.33

B-V0.010.020.0Z0.330.020. 00-0.011.281.030.450.070.110.061.731.23».n1 .250.260.311.270.680.121.290.450.370.53

U-B-0.54-0.48-0.460.15-0.41-0.49-0.551.100.77-0.01-0. 17-0.30-0.162.081.04

-0.271.070.100. 171.040.230.011.270.150.22-0.05

SPEC.B4VB8B9A2B9B6VB9K2K0F0A0B9B9K5K2B9V

A2V

A0

A2

ABCDEFGHIJKLMN0PQRSTUVwXYz

NAME.3483,3490,3484,3514.3504,3432,3525,3469,3524,3488.3466,3491,3471,3498.3500,3493,3503,3479,3473,3472,3496,3477,3474,3485,3495,3523

DBS.JOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONJOHNSONOOHNSONJOHNSON

APPENDIX ENORTH POLAR SEQUENCE STARS

The North Polar Sequence (NPS) was developed at Harvard College Obser-vatory in 1906 to provide stars with standard photographic magnitudes thatcould be used to derive magnitudes for other stars. The sequence began with10 and rapidly expanded to 96 stars. Three separate sequences evolved: red (r),blue (no suffix), and supplementary (s, mostly yellow). Mount Wilson collab-orated in the latter stages, and the sequences formed the basis of the Interna-tional System of magnitudes adopted by the IAU in 1922. See Pickering,1

Leavitt,2 and Stebbins et al.3 for more detail.The NPS has only historical significance now because it has been superseded

by sequences in clusters and in the equatorial plane, accessible by major tele-scopes in both hemispheres. In addition, only photographic magnitudes andcolors have been derived for the majority of the stars, and even these are inerror for the fainter members of the NPS.

One modern application for which the NPS is well suited is to determinetemporal consistency of the atmosphere. This is because none of the stars liesmore than 3 ° from the north celestial pole, and therefore remain at the sameair mass with time. It is a good idea to observe one of these stars several timesduring the night to be sure that the atmosphere is remaining constant.

Table E.I lists the sequence stars that are brighter than eleventh magnitudeon photovisual plates. The V magnitudes of those stars in the table with no(U — B) color indicated are photovisual magnitudes, and the color index isphotographic-photovisual. The accompanying finding chart identifies the tablestars.

For amateurs in the Southern Hemisphere, a short sequence near the southcelestial pole has been set up by Soonthornthum and Tritton.4 Their photoelec-tric sequence includes nine stars ranging from V = 6.53 to V = 12.66. Theyinclude a finding chart in their article.

305


NORTH POLAR SEQUENCE12"

• 7,

3s

-5r

\3r

K,12 .

10 -14.*/

18'

Figure E.I. The North Polar Region. Lick Observatory photograph.

REFERENCES

1. Pickering, E. C. 1912. Harvard College Circular. No. 170.2. Leavitt, H. C. 1917. Ann. Harvard College Obs. 71, 47.3. Stebbins, J., Whitford, A. E., and Johnson, H. L. 1950. Ap. J. 112, 469.4. Soonthornthum, B., and Tritton, K. P. 1980. Observatory 100, 4.

TABLE E.1. North Polar Sequence Stars

< 1950.0)NP589101R6R6457

1411zs3S6S5R3R5S127R13Bft142R4R23

R,A.12377a89

111111121212121213131313141717ISIS2223

104918121722454913234514153043S3

115465247484921581727

57.344.347.531.250.639.753.627.559.615.752.745.024.54.427.122.140.61 .8

56.820.5IB. 518.312.221.829.533.634.2

888888698789888987898987868988878989898989868689888587

DEC.455545447

214619545112584225375538133325293659346511

18.916.056.052.034.530.614.844.943.354.024.937.649.316.945.65.7

11 .928.926.31 .4

12.734.831.93.5

55.627.054.4

133979910111 11112121212121213131314171717182223

(1985.0)R.A,25235116331112191946501516243955411

16264

373636231427

4439122654397

45361416101744474042612120142

38329

DEC.888988898789888987398987868988878989898989863689838687

56 203 18

52 5138 93 1713 4038 09 29

43 1440 150 4446 5731 913 4026 1543 4526 472 17

22 3814 2319 4535 4153 392 59

49 11 57

13 28

V6.448.088.899.055.049.247.099.897.5510.579.616.226.3310.728.637.4710.069.789.8710.2710.414.355.746.348.225.245.56

B-V0. 110.410.200.121 .571 .230.060.47-0. 170.450.220.260.330.671 .531 .42i .080.351. 120.241 .020,010. 141.561.02

-0.J320.16

U-B0.14-0.060.000.006,008.000.008.000.000.000,000.000.000.000.00I .510.000.000.000.000.000.040.000.000.00-0.220.00

SPEC,A2F0AA5M0G8A0S0EBF2F2F0F2G8K0K2G5A3G8A5G8A1VA3M3K0A0F0

NAMEeoesBD88BD88BD89B087BD89BD89BD89S088BD89BD89BD38B087BD89BD89BD88BD89BD89BD89BDB9BD89BD86BD86BD88BD88BD85BD86

49133519131264118711072622763725352931269272112114383344

OBS.JOHNSONJOHNSONNPSNFSNPSNPSNPSNPSNPSNPSNPSNPSNPSNPSNPSJOHNSONNPSNPSNPSNPSNPSJOHNSONNPSNPSNPSJOHNSONNPS

APPENDIX FDEAD-TIME EXAMPLE

This appendix gives an example of how to calculate the dead-time coefficientfor a photomultiplier pulse-counting system. Section 4.2 presents the basis forthis example and should be consulted for more detail.

The main requirement for determining the dead-time coefficient is a set ofstars, some of which should have negligible correction and some with a largecorrection. For our system, this means stars with rates around 100,000 countsper second and 800,000 counts per second to bracket the changeover point. Oursample of eight stars should be considered the minimum necessary to deter-mine the coefficient.

Several stars were measured with the neutral density filter used at IndianaUniversity, which has the v, — v0 relation:

v, - v0 = -0.008(B - V) + 3.934 (F.I)

The data are listed in Table F.I. The steps required to obtain the coefficientfollow.

1. Calculate v, — v0 for each star from Equation F.I, using the known colorindex given in column 2.

2. Using Equation 4.7, calculate the intensity ratio b (= /0//,).3. Using the observed count rate obtained with the filter in the light path

(nL) and the rate without the filter («„), calculate the true count ratewithout the filter using Equation 4.3a:

NH = bnL

308

DEAD-TIME EXAMPLE 309

TABLE F.1. Dead-Time Coefficient Data

Collected Data

Star

HR8334HR8465HR8469HR8494HR8498HR8585HR8622HR8694

B - V

0.511.550.230.271.460.01

-0.201.05

nL

10,67025,170

519011,72012,40017,190

617021,430

*374,200824,700190,100407,300427,400582,700224,200708,100

v , - v 0

3.9303.9223.9323.9323.9223.9343.9363.926

Calculated Results

b

37.3237.0437.4037.3937.0637.4637.5237.17

ff*

398,200932,200194,100438,200459,600643,900231,500796,600

A

0.06220.12250.02080.07310.07260.09990.03200.1178

4. Calculate the quantity

A-*(£*\ Iff

for each star (where In is the natural logarithm).

0.12

0.10

0.08

0.06

0.04

0.02

0.00 i I I I i I i I10 30 50 70 90

NH (UNITS OF 10,000)

Figure F.I. A versus NH, where A = (NH/nH).


5. Now plot A versus NH from Equation 4.4,

A = tNH

The slope of the resultant line is I, the dead-time coefficient. The plot forthis example is given in Figure F.I.

For our example using eight stars, the slope (using linear least squares) is t= 1.40 X 10~7 seconds. An example of the use of the dead-time coefficientcan be seen in Appendix H.

APPENDIX GEXTINCTION EXAMPLE

The approach used to determine the extinction coefficients depends, to someextent, on the type of observing program. In a program of differential photom-etry the atmospheric extinction corrections can usually be ignored. Occasion-ally, however, an extinction correction is advisable if the two stars are widelyseparated. In this case, the comparison star measurements themselves yield theextinction coefficients. When the observing program calls for measuring manydifferent stars at various locations in the sky, it is necessary to follow anotherapproach. In this case, a separate set of standard stars must be observed todetermine the extinction coefficients. No matter which kind of observing pro-gram is conducted, another set of observations must be made to determine thesecond-order extinction coefficients.

We now take you through each of these procedures using actual photometricdata.

G.1 EXTINCTION CORRECTION FOR DIFFERENTIAL PHOTOMETRY

Because the second-order extinction corrections applied to differential photom-etry are small enough to ignore, we concentrate on the principal extinctioncoefficients. As stated above, it is the comparison star measurements that allowus to find the principal extinction coefficients. The success of this methoddepends on the fact that the comparison star is usually observed through alarge range of air mass. Second, the method assumes that the transmission ofthe atmosphere has remained constant and that the electronics remain freefrom gain drift throughout the night. These conditions are usually satisfied ona clear night and with well-designed electronics.

As the comparison star rises or sets, the amount of light detected changes asmore or less light is absorbed by the earth's atmosphere. Using Equations 2.1through 2.3 we can calculate the instrumental magnitude of the comparison

311


star through each filter at each air mass. These magnitudes can be correctedfor atmospheric absorption by using Equation 4.20 for each filter, that is

(G.I)(G.2)(G.3)

v0 = v -b0= b~u = u ~ k'X

where v, b, and u are the instrumental magnitudes, VQ, b0, and u0 are the instru-mental magnitudes corrected for extinction and k'v, k'b, and k'u are the principalextinction coefficients. These coefficients are, of course, unknown. However, aplot of v versus X yields a straight line slope k'v. Similarly, plots of b versus Xand u versus X yield the slopes k'b and k'u, respectively.

1. The comparison star used in this example has a right ascension of 2h07m

and a declination of 40°23'. The observatory has a latitude of 39°33'north. In Table G. I, column 1 contains the hour angle of each comparisonstar observation. Column 2 contains the air mass, X, calculated by Equa-tions 4.17 and 4.18. Try calculating X to check a few entries in column2. If you have difficulties, refer to the example in Section 4.4.

2. Columns 3, 4, and 5 of Table G.I contain the counts per second recordedthrough each filter with the sky background subtracted. These have notbeen corrected for dead time because the count rates are rather low. Ifthese observations had been made with a DC system, these three columnswould contain the deflection (usually in percent of full scale) and the

TABLE G.1. Comparison Star Data

Star a = 2k07IB, 5 = 40° 23' Observatory: 39°33' north latitude

HACounts per Second

v Filter b Filter u FilterInstrumental Magnitudes

v b u

2-A2E 1.1651:53 1.0761:14 1.0320:30 1.0050:09W 1.0010:14 1.0011:001:432:243:154:014:31

.021

.062

.127

.252

.423

.579

566058545878588758975883583856555568541551875042

755079488110814380888083800678217511719568196450

141315301596163816111617158815281424129711881033

-9.382-9.419-9.423-9.425-9.427-9.424-9.416-9.381-9.364-9.334-9.287-9.256

-9.695-9.751-9.772-9.777-9.770-9.769-9.759-9.733-9.689-9.643-9.584-9.524

-7.876-7.962-8.008-8.036-8.018-8.022-8.002-7.961-7.884-7.782-7.687-7.535

EXTINCTION EXAMPLE 313

amplifier gain settings. Columns 6, 7, amd 8 are the instrumental mag-nitudes calculated by Equations 2.1, 2.2,, and 2.3. Check some of theseentries. If DC measurements had been made, we would substitute thedeflection for the count rate and add the amplifier gain (in magnitudes).That is,

v = -2.5 log K) + Gv

b = -2.5 log (rft) + Gb

u = -2.5log(</H) + Ga.

3. We now plot v versus X, b versus X, and u versus X. These are shown inFigure G.I. A linear least-squares analysis yields

*; = 0.306k'b = 0.441

*: = 0.840.

(See Section 3.5 for an explanation of linear least-squares analysis.)4. The magnitude difference between the variable and the comparison star

can now be corrected for extinction by using Equations 2.35 through 2.37(or 2.38 through 2.40) and 2.43 through 2.45. The color indices are com-puted by Equations 2.41 and 2.42. These colors can be corrected forextinction by noting that

If' — 1r> — If'"•bf Kft **•»

Kub ~ k'u — tCb

and applying Equations 2.46 and 2.47. The third term on the right sideof Equation 2.46 can usually be dropped because k"v is small. Further-more, if the variable and comparison star are nearly the same color, A(6- v) is nearly zero. However, if you wish, this term can be included by

finding k"bv as outlined in Section G.3.

G.2. EXTINCTION CORRECTION FOR "ALL-SKY" PHOTOMETRY

The procedure described in this section is applied to the situation in which starshave been measured at various positions in the sky. Each star is observedbriefly through a limited range of air mass and hence the procedure of SectionG.I cannot be used. There are two approaches that can be followed, dependingon whether or not the transformation coefficients to the standard system areknown.


-7.6 -

-7.51.00 1.10 1.20 1.30

X

1.40 1.50 1.60

Figure G. 1 . Instrumental magnitudes versus air mass.

Unknown Transformation Coefficients

The observing procedure is quite simple. At various times during the night, oneobserves an early A-type standard star from the list in Appendix A or C. Thereason for choosing this type of star becomes apparent in the following. If wesubstitute Equation 4.20 into 4.32, we obtain

- V)

or

K - v = -I&C + <(B - K) + fv. (G.4)

For early A stars, (B — K) is very small. Furthermore, e is a small number,

EXTINCTION EXAMPLE 31 5

so their product is extremely small. Thus, to a good approximation the aboveequation becomes

V - v = f¥ - k'JC (G.5)

A plot of V — v versus X for the early A stars yields a straight line with slopek'v. Substituting equation 4.29 into 4.33 yields

(B - V) = rib - v) - Mb ~ v)k'bvX - nk'bvX + f*. (G.6)

Because /x is usually very nearly equal to one for most photometers and k"bv isvery small, we can make the following approximation.

(B - V} - (b - v) ^ -k>hyX + ffc (G.7)

A plot of (B — V) — (b — v) versus X yields a straight line with slope— k'bv. A similar procedure with substitution of Equation 4.22 into 4.34 yields

(U - B) = ftu - b) - tk'ttbX + U- (G.8)

We again note that ^ — 1 for most photometers. We thus obtain our finalequation

(U- B)-(u-b)=* -kfubX + ^. (G.9)

A plot of (U - B) - (u - b) versus X yields a straight line with slope k'ub.

1. Table G.2 contains the data for a number of extinction stars obtainedduring one night. Some stars appear more than once because they wereobserved later in the night at an appreciably different air mass. Most ofthe observing time was spent on program stars that do not appear in thetable. Columns 2, 3, and 4 contain data taken from Appendices A and C.Column 5 contains the air mass calculated by Equations 4.17 and 4.18.Columns 6, 7, and 8 contain the count rates through each filter (after skysubtraction and a dead-time correction was made). Check some of theentries in column 9, using Equation 4.14 for v.

2. Check some of the entries in columns 10 and 11 using Equations 4.15and 4.16.

3. Figure G.2 shows the plots of (K - v) versus X, (B - V) - (b - v)versus X and (U — B) — (u — b) versus X. A linear regression analysis


TABLE G.2. Extinction Star Measurements

Counts per Second

Star

SOUMa109 VirBSer ArHerTOphT Ser68Oph68OphxSer109 VirTOph57Cyg

V

4.033.743.673.893.754.834.424.424.833.743.754.77

B - V

0.150.000.06

-0.150.040.070.040.040.070.000.04

-0.14

U - B X

0.09 1.042-0.03

0.07-0.56

0.040.05

.269

.161

.080

.868

.0970.02 1.9310.020.05

-0.030.04

-0.58

.340

.082

.673

.245

.006

V

399,726524,038559,793452,832454,939194,252234,468260,411192,633458,829501,527197,291

b

799,2961,152,5291,209,1841,204,795

887,150407,561456,535559,110402,426974,150

1,102,259515,202

u

140,792201,823195,872380,022122,53871,05965,53794,77371,980

148,048180,078175,144

(B- V) -V - v (b - v)

18.01418.03818.04018.03017,89518.05117.84517.95918.04217.89418.00118.008

0.9120.8560.8960.9100.7650.8750.7640.8700.8700.8170.8950.902

(U- B)-( M - />)

-1.805-1.922-1.906-1.813-2.109-1.846-2.088-1.907-1.819-2.076-1.927-1.752

I I I I

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Figure G.2. Extinction plots when transformation coefficients are unknown.


yields

k'v = 0.200k'bv = 0.153k'ab = 0.353.

Known Transformation Coefficients

When the transformation coefficients are known, the procedure of finding theextinction coefficients is simplified. Now all that is required are observations offive or more standard stars, of any color, at various air masses. These can bethe same standards you would observe to determine your nightly zero-pointconstants (fv, fftv, i"ufr). If Equation G.4 is rearranged, we obtain

- K) = -k'v (G.10)

Because < is known, we do not need to use A stars in order to approximate thee(fi — K) term as zero. A plot of K — v — t(B — K) versus X yields a straightline with slope k'v. Likewise, Equation G.6 becomes (assuming k"bv ~~ 0)

(B - K) - + (G.ll)

A plot of (B — K) — n(bEquation, G.8 becomes

(V- B)

Again, a plot of (t/ — B) —

— v) versus X yields a slope nk'bv. Rearranging

— b) versus A" yields a slope

1. Table G.3 contains data on 10 standard stars observed on one night. Thefirst four columns contain values taken from Appendix C. Column 5 con-tains the air mass and columns 6, 7, and 8 contain the instrumental mag-nitudes and colors calculated by Equations 4.14, 4.15, and 4.16. Thetransformation coefficients were determined on a previous night to be

( = -0.084n = 1.083^ = 1.006.

Compute the left-hand side of Equation G.10 for a few stars to check theentries in column 9. Likewise, compute the left-hand side of EquationG.I i and G.12 to check the entries in columns 10 and 11.

TABLE G.3. Observations of Standard Stars

Star

i PscHR883210 Lacf Aqra DelBAql7 OphrHertCrb(3SerA

V

4.135.574.883.773.773.713.753.894.153.67

(B- V)

0.511.01

-0.200.01

-0.060.860.04

-0.151.230.06

(U- B)

0.000.89

-1.040.04

-0.220.480.04

-0.561.280.07

X

2.1831.2621.1831.5621.1031 . 1 9 1

1.5141.3611.7542.340

v

-13.721-12.502-13.206-14.199-14.312-14.344-14.293-14.179-13.862-14.177

(b - v)

-0.0010.334

-0.794-0.564-0.694

0.190-0.514-0.721

0.618-0.352

(u- b)

2.1832.7410.7092.0331.6122.2692.0021.2883.3422.286

V - v -t(B~ V)

17.85118.15718.06917.97018.07718.12618.04618.0561 8 . 1 1 517.852

n(b — v)

0.5110.6480.6600.6210.6920.6540.5970.6310.5610.441

(V - B} -$(u - b)

-2.196-

-—-—-

.867

.753..005.842.803.974.856

-2.082-2.230


18.5

18.0

17.5

17.0 J L J L

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4

Figure G.3. Extinction plots when transformation coefficients are known.

2. Plots of V - v - ((B - V) versus X, (B - V) - n(b - v) versus X,and (U — B) — $(u — b) versus X appears in Figure G.3. A linearregression analysis of each plot yields the following slopes.

k'v = 0.212k'bv = 0.163k'ub = 0.373.

Note that this same linear regression analysis gives the intercepts thatare the zero-point constants. These turn out to be

f, = 18.359

f* = 0.875= -1-381.


G.3 SECOND-ORDER EXTINCTION COEFFICIENTS

Because k" is very small and k"ub is defined as zero, we confine this example tok"bv. Experience has shown that k"bv is both small and fairly stable. Therefore,k"bv need be determined only once or twice a year. The procedure is to observea closely spaced pair of stars, of very different colors, at various air masses. Ifwe let subscripts 1 and 2 refer to each star and use Equation 4.29 to form thedifferences in color indices, we obtain

(b - v)01 - (b - v)02 = (6 - v), - k"bvX,(b - v),

l -(b- v)2 + k"bvX2(b ~ v)2

- v).

Because Xt =* X2, this reduces to

- v)0 = A(6 - v) - k'b (G.I 3)

Because A(fc - v)0 is constant, a plot of A(A - v) versus Xk(b — v) givesa straight line with slope k"bv.

TABLE G.4. Observations of HD30544 andHD30545

HD30544 HD30545(b - v) (b - v) A(6 - v) X A(6 - v)X

-0.628-0.676-0.718-0.756-0.778-0.783

0.4580.3960.3760.3440.3390.335

-1.086-1.072-1.094-1.100-1.117-1.118

2.0791.7861.5511.3671.2791.231

-2.258-1.915-1.697-1.504-1.429-1.376

1.15

1.10

1.05

-1.00-1.2 -1.4 -1.6 -1.8

X A(b-v)

-2.0 -2.2 -2.4

Figure G.4. Second order extinction plot.


1. The pair of stars used in this example is HD30544 and HD30545. Theseand other pairs can be found in Appendix B. Columns 1 and 2 of TableG.4 contain the color indices of each star calculated by Equation 4.15and 4.16. Column 3 is just column 1 minus column 2. Column 4 is themean air mass at the time of observation. Finally, column 5 is the productof columns 3 and 4.

2. Figure G.4 shows a plot of A(fc — v)X versus A(6 — v). A linear regres-sion analysis yields a slope

k"hy = -0.042.

APPENDIX HTRANSFORMATION COEFFICIENTS

EXAMPLE

H.1 DC EXAMPLE

The transformation coefficients are often determined from standard starswithin a cluster that is near the zenith. This procedure reduces the impact ofthe extinction corrections greatly. Appendix D contains finder charts and listsof standard stars for three clusters. In this particular example, however, a clus-ter was not used. Standard stars near the meridian were selected from the listin Appendix C. While this technique works, it usually gives inferior resultsunder variable sky conditions to the cluster method. A 60-centimeter (24-inch)telescope was used. A completely different telescope and photometer were usedfor the pulse-counting example to follow.

1. The first step is to measure each star through each filter. The sky back-ground should be measured in each filter using the same amplifier gain anddiaphragm as used for each star. In fact, the same diaphragm should be usedfor all measurements. Record the gain setting of each gain switch, not theirtotal. This is necessary because there is more than one combination of gainswitch positions which gives the same apparent total gain. However, each com-bination has a different set of gain corrections as determined by the gain cor-rection table.

Table H.I lists the coordinates, magnitudes, and colors of the stars observed.For now, ignore the three columns on the right. While in this particular exam-ple, only nine stars were used, it is recommended that you observe as many asis practically possible. Table H.2 lists the actual observations.

2. The next step is to subtract the sky background to produce the net deflec-tion shown in column 5 of Table H.2. The settings of the two gain switches areshown in column 6, separated by a slash. For the particular amplifier used inthis example, the coarse switch requires no gain corrections. The corrections

322

TRANSFORMATION COEFFICIENTS EXAMPLE 323

TABLE H.I. Standard Stars

Star

11 LMi21 LMiA UMaa Leoe Leov UMa

31 Leo72 Leo/?Leo

Star11 LMi

21 LMi

A UMa

a Leo

e Leo

v UMa

31 Leo

72 Leo

/3 Leo

RAh m

9 34100610 161007944948

100611 1311 48

FiltervbuV

buV

buV

buV

buV

buV

buV

buV

bu

Dec.V

35 56 5.4135 22 4.4843 03 3.4512 05 1.3623 58 2.9859 14 3.8210 12 4.3623 18 4.6014 43 2.14

TABLE H.2.

Deflection41.036.237.836.351.536.635.557.736.637.567.855.752.344.135.940.651.935.546.034.627.552.744.645.053.378.069.2

StandardB - V U - B

0.77 0.450.18 0.080.03 0.06

-0.11 -0.360.81 0.460.29 0.101.45 1.751.66 1.850.09 0.07

CalculatedV B-V

5-43 0.784.52 0.183.47 0.051.35 -0.132.95 0.79

U -B

0.380.130.11

-0.370.43

3.77 0.29 0.094.33 1.434.64 1.682.13 0.09

1.821.800.05

Stellar Measurements

Sky10.910.910.97-67.6

14.15-85.88.76.46.410.76.06.0

11.56.76.7

12.013.713.010.213.010-314.113.813.814.1

Net Gain v30.1 5/225.3 5/2

3.306b-v0.189

u — b2.433

26.9 7.5/228.7 5/143.9 5/1

2.378 -0.462 2.199

22.5 5/2.529.7 5/051.9 5/0

1.318 -0.606 2.184

27.9 5/1.531.1 2.5/061.4 2.5/0

.5 -0.7385

-0.739 1.847

45.0 2.5/246.3 5/038.1 5/024.4 5/2

0.836

33.9 5/0.5 1.668

0.212

-0.312

2.487

2.22045.2 5/0.523.5 5/232.3 5/1 2.250 0.925 3.72721.6 5/1.517.3 7.5/2.539.7 5/1.5 2.513 1.145 3.64034.3 5/2.530.9 10/139.5 2.5/164.2 2.5/155.1 5/1

.5 0.0195

-0.527 2.178


for the fine gain switch were determined by the procedure outlined in Section8.6. The table below gives the actual fine gain values. The amplifier gain isthen the total of the coarse switch position and the actual fine gain read fromthe table.

Fine GainSwitch Position Actual Gain in Magnitudes

00.51.01.52.02.5

00.4941.0231.5102.0032.496

The atmospheric, instrumental magnitudes and colors can be computed byEquations 4.11, 4.12, and 4.13. Check some of the results in the last threecolumns of Table H.2.

3. The magnitudes and colors determined above must now be corrected foratmospheric extinction. It is necessary to determine the air mass, X, for eachstar. Because the three filter measurements were made within a few minutesof each other, a single value of X is used for each star. Column 2 of Table H.3lists the hour angle at the time of observation of each star. In this case, thetelescope had an hour angle read-out device. If your telescope is not soequipped, you need to calculate hour angle as described in Section 5.3c. Theeast-west notation in the table is really unnecessary because it makes no dif-

TABLE H.3. Extinction Correction and Transformation CoefficientDetermination

Star HA(B- V) - (V-B)(b-v)D ( u - 6 )

11 LMi21 LMi\ U M aa Leo< LeovUMa31 Leo72 Leo0Leo

1:23 W0:47W0:33W1:24W1:18W1:18W1:06W0:11W0.58E

1.0441.0141.0141.1601.0691.1181.1521.0261.102

2.9091.9930.940

-1.1790.4301.2431.8122.123

-0.400

-0.124-0.766-0.917-0.991-0.109-0.647

0.5790.859

-0.858

1.8981.6951.6651.2531.9401.6483.1373.1251.614

2.5012.4872.5102.5392.5502.5772.5512.4762.540

0.8950.9460.9480.8810.9190.9380.8710.8010.948

-1.449-1.615-1.605-1.613-1.479-1.547-1.387-1.275-1.544


ference for the calculation of X. As an example, consider 21 LMi.

hour angle = H = 0:4747 minutes

X (15°/hour)

(4.18)

60 minutes/hour= 11.75°

5 = 35.37* (declination from Table H.I )# = 35.92° (latitude of observatory)X = (sin 5 sin <t> + cos 5 cos 0 cos H)~l

X = 1.014

The more precise method described in Section 4.4a (Equation 4.19) is notneeded because measurements were made near the zenith.

The extinction coefficients were determined by the method outlined inAppendix G. They were found to be

k'v = 0.380kL = 0.300k'* = 0.512k"bv 0

The extinction corrections were then made by calculating for each star, assum-ing that k"ub = 0:

v0 = v - k'v X

(b- v)0 = (b- v)-k'bvX

(II- *)„- ( « - * ) - faX.

The results of these calculations appear in columns 4 through 6 of Table H.3.These are the instrumental magnitudes and colors. Check some of these results.

4. To determine e and {"„ a plot of K — v0 versus (B — F) is needed. Thevalues of V and (B — K) are found in Table H.I. The computed values ofK — v0 appear in Table H.3. The graph appears in Figure H.I. A linear least-squares fit yields a slope of —0.012 and an intercept of 2.532. From Equation4.38,

e = -0.012fv = 2.532.

5. To determine n and f^, a plot of [(B — V) — (b — v)0] versus (B — V)is required. Again, (B -- K) is found in Table H.I and [(B -• V) -


2.7

o 2.61

> 2.5

74

—

r

_ * • •

•

1 1-0.1 0.1 0.3 0.5 0.7 0.9

B-V

1 1 1.3 1.5

Figure H.I. Plot to determine e and fv.

1.7

1 0

A 0.9I> 0.8

0 7-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

(B-V)

Figure H.2. Plot to determine M and £&•

(b — v)0] is calculated and shown in Table H.3. The graph appears in FigureH.2. A linear least-squares fit yields a slope of —0.059 and an intercept of0.940. From Equation 4.40,

and

1 - - = -0.059MM = 0.944

— = 0.940M

fh, = 0.887.

6. The procedure in step 5 is repeated for [(U — B) — (u — fc)0] versus(U — B). The graph appears in Figure H.3. A linear least-squares fit yields aslope of 0.139 and an intercept of —1.570. From Equation 4.41,

1 - -^ = 0.139

& = 1.161

and

%= -1.570Vfu6 = -1.823.


-0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-1.7

Figure H.3. Plot to determine ^ and fub.

Now the standardized magnitudes and colors of any star measured thatsame night can be found by the following expressions:

(B - V) = 0.944 (b ~ v)0 + 0.887V = v0 - 0.012 (B - K) + 2.532

(U - B) = 1.161 (u - b)0- 1.823.

As a check on the quality of these transformation equations, the magnitudesand colors of the standard stars were computed and compared to the knownvalues. The results appear in the rightmost three columns of Table H.I. Com-pare Figures H.I, H.2, and H.3 to Figures H.4, H.5, and H.6. This last set offigures, from the pulse-counting example, represents a very good set of obser-vations and hence, a good set of transformation equations. The results of theDC example show that the observations are of marginal quality and the trans-formation equations are not very dependable. This is primarily because of poorsky conditions. The cluster method tends to work better when the sky condi-tions are less than excellent. Note that, because different telescopes and pho-tometers were used, the color coefficients from the two examples are not thesame.

As explained in Chapter 4, the coefficients f, n, and ^ are constant over fairlylong time periods. However, $„ f ^ , and fut must be determined nightly.

H.2 PULSE-COUNTING EXAMPLE

The transformation coefficients can be determined from standard stars withina cluster that is near the zenith. This procedure reduces the impact of theextinction corrections greatly. Appendix D contains finder charts and lists ofstandard stars for three clusters. In this case, the cluster 1C 4665 was observed.


A 40-centimeter (16-inch) telescope was used. A completely different telescopeand photometer were used for the DC example.

1. The first step is to measure each star through each filter. The sky back-ground should be measured several times during this process. The same dia-phragm should be used for all measurements. Table H.4 lists the coordinates,magnitudes, and colors of the stars observed. For now, ignore the three columnson the right. Table H.5 lists the actual observations. The third column containsthe number of counts per second, which was found by averaging the countsobtained in several 10-second intervals and dividing by 10.

2. The next step is to correct the observations for dead time. The dead-timecoefficient, /, was found by the technique outlined in Chapter 4 with a workedexample in Appendix F. For the particular photomultiplier used in this exam-ple, t = 1.54 X 10~7 seconds per count. The observed count rate n is relatedto the true count rate N by

n = Ne~*'.

A FORTRAN subroutine to solve this equation in an iterative fashion can befound in Section I.I. To illustrate the essential idea behind the solution, anumerical example is given using the first observation in Table H.5. Theobserved count rate is 28,925. As an initial guess, suppose that /V is equal tothis observed rate. Then

n = 28,925 e-(28-92S)1-54x10"7

n = 28,796

TABLE H.4. The Standard Stars

Star

AFGIJN0PSuVw

RA

17"1717171717171717171717

45"4545454546474644464444

Dec.

5'56655555555

32'4208152437222632313451

V

6.857.597.747.897.948.338.408.899.399.81

10.1010.21

StandardB - V

0.0110.002

-0.0091.0280.4491.7281.2320.1060.3140.6760.1221.292

U - B V

-0.54-0.49-0.55

0.77-0.01

2.081.04

-0.270.170.230.011.27

6.847.587.767.887.958.348.408.889.399.80

10.1110.21

CalculatedB - V U - B

0.0180.003

-0.0071.0340.4331.7111.2430.0990.3060.7010.1111.296

-0.554-0.491-0.544

0.735-0.019

2.0980.992

-0.2640.2210.2010.0361.301

TABLE H.5. Stellar Measurements

Star

A

F

Sky

G

I

Sky

J

N

O

P

Sky

S

U

V

W

Sky

Filter

V

bitV

buV

buV

buV

buV

buV

buV

buV

buV

buV

buV

buV

buV

buV

buV

bu

Counts perSecond

28,92561,71818,57514,81932,152

913654.193.228.6

12,53927,547

828710,70810,139

91151.884.628.2

10,38715,994

291768103756

11566135310

387452491652129

51.482.826.6

28264856

72319262416

37315083028

54212961034

7650.275.825.2

Dead-timeCorrected

29,05562,31318,62814,85332,312

9149

12,56327,664

829810,72610,155

911

10,40416,033

291868173758

11566205314

387452791782129

28274860723

19262416

37315083029542

12961034

76

Net

29,00162,22018,59914,79932,219

9120

12,51027,575

827010,67310,066

883

10,35215,949

289167653674

8765685230360

447590942102

27764781

69818752337

34714572950516

1245955

50

v b — v u — b

-11.156-0.829

1.311-10.426

-0.8451.370

-10.243-0.858

1.308-10.071

0.0642.642

-10.038-0.469

1.854-9.576

0.6634.062

-9.5440.247

2.906-9.127

-0.7701.590

-8.609-0.590

2.090-8.183

-0.2392.070

-7.909-0.766

1.893-7.738

0.2883.211

329


Because this value is lower than the actual observed rate, we increase the nextguess of N by the difference between these two numbers, i.e.

N (second guess) = 28,925 + (28,925 - 28,796)= 29,054

We can now recompute n by

/i = 29,054= 28,924.

This differs from the observed rate by only 1. Thus as a third guess,

N (third guess) = 29,054 + 1

and

« = 29,055 e-<29-0»)154x|0~7

n = 28,925.

Because this value agrees with the observed rate, we know that the true countrate is 29,055. This number appears in column 4 of Table H.5 along with theother corrected count rates.

3. The sky background is now subtracted from each value in column 4.When the stellar measurement is sandwiched between two sky measurements,an average sky measurement is subtracted.

The atmospheric instrumental magnitudes and colors can be computed byEquations 4.14 through 4.16. The results appear in columns 6, 7, and 8 ofTable H.5. Check some of the results with your calculator.

4. The magnitudes and colors determined in step 3 must be corrected foratmospheric extinction. It is necessary to determine the air mass, X, for eachstar. Because the three filter measurements were made within a few minutesof each other, a single value of X is used for each star. Column 2 in Table H.6lists the hour angle at the time of observation of each star. In this case, thetelescope had an hour angle read-out device. If your telescope is not soequipped, you need to calculate the hour angle as described in Section 5.3c.The east-west notation in the table is really unnecessary because it makes nodifference for the calculation of X. As an example, consider star A.

hour angle = H = 1:54

H= 7\ X(15 ' /hour )


H = 28.50°

5 = 5.53° (declination from Table H.4)0 = 39.17° (latitude of observatory)X = (sin 5 sin <f> + cos 6 cos (f> cos //)"'X = 1.353

The more precise method of finding X described in Section 4.4 (Equation 4.19)is not needed since the measurements were made near the zenith.

The extinction coefficients were determined by the method outlined inAppendix G. They were found to be

k'v = 0.209k'bv = 0.162k'ub = 0.337k"bv •• 0.088

The extinction corrections were then made by calculating, for each star, assum-ing k»b = 0,

VQ = v -(b - v)0 = (* - v) (1 - k"hvX) - k'bvX(u- b). = (u- b) - k'uhX.

The results of these calculations appear in columns 4 through 6 of Table H.6.These are instrumental magnitudes and colors. Check some of these results.

TABLE H.6. Extinction Correction and Transformation CoefficientDetermination

Star HA

A 1:54EF 1:50EG 1:46EI 1:43EJ 1:38EN 1:34EOPSuVw

:31E:27E:19E:18E:13E:08E

X

1.3531.3391.3201.3111.3131.3001.2971.2881.2701.2691.2591.247

*t

-11.439-10.703-10.519-10.345-10.312-9.847-9.815-9.396-8.874-8.448-8.172-7.999

(b - v)0

-1.146-1.161-1.171-0.141-0.736

0.5280.065

-1.066-0.862-0.471-1.054

0.118

(« - H,

0.8550.9190.8632.2001.4123.6242.4681.1561.6621.6421.4692.791

y- vn

18.28918.29518.25918.23518.25218.17718.21518.28618.26418.25818.27218.209

(B- V} -(b - v)0

1.1571.1631.1621.1691.1851.2001.1671.1721.1761.1471.1761.174

(U- B)-(« - b)0

-1.395-1.409-1.413-1.430-1.422-1.544-1.428-1.426-1.492-1.412-1.459-1.521


5. To determine e and £„ a plot of V — v0 versus (B — V) is needed. Thevalues of Kand (B — K) are found in Table H.4. The computed values ofV — v0 appear in Table H.6. The graph appears in Figure H.4. A linear least-squares fit yields a slope of —0.057 and an intercept of 18.284. From Equation4.38,

( = -0.057fc, = 18.284.

6. To determine fj. and iV, a plot of [(B -• V) — (b -- v)0] versus(B - V) is required. Again (B • V} is found in Table H.4 and[(B — V) — (b — v)0] was calculated and appears in Table H.6. The graphappears in Figure H.5. A linear least-squares fit yields a slope of 0.011 and anintercept of 1.164. From equation 4.40,

1 - - • 0.011Mp = 1.011

18.3

18.2

18.1-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7

(B-V)

Figure H.4. Plot to determine t and fv.

T 1.20 -

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

(B-V)

Figure H.5. Plot 10 determine ^ and &„.

and


= 1.164^f* = 1-177

7. The procedure in step 6 is repeated for [(U — B) — (u — b)0] versus(U — B). The graph appears in Figure H.6. A linear least-squares fit yieldsa slope of —0.045 and an intercept of — 1.432. From Equation 4.41,

and

1 - -r - -0.045

* = 0.957

% = -1.432V

= -1.370.

Now the standardized magnitudes and colors of any "unknown" starobserved that same night can be found by the following expressions.

(B- K) = 1.011 (b- v ) 0 + 1.177y = v0 - 0.057 (B - V) + 18.284

(U - B) = 0.957 (u - b)0 ~ 1.370

As a check on the quality of these transformation equations, the magnitudesand colors of the standard stars were computed and compared to the knownvalues. The results appear in the rightmost three columns of Table H.4. This

-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 22

Figure H.6. Plot to determine 0 and f,b.


comparison shows that the transformations are fairly good. Compare FiguresH.4, H.5, and H.6 to H.I, H.2, and H.3. This last set of Figures, from the DCexample, represents a rather poor set of observations and the resulting trans-formation equations are of questionable quality. Note that, because differenttelescopes and photometers were used, the color coefficients from the twoexamples are not the same.

As explained in Chapter 4, the coefficients e, p, and ty are constant over fairlylong time periods. However, f,, f^ and £ub must be determined nightly.

APPENDIX IUSEFUL FORTRAN SUBROUTINES

These routines were written by one of the authors (AAH) for the FORTRAN ivcompiler of Control Data Corporation computers. The programming is func-tional but not elegant. Most of the routines are in constant use at IndianaUniversity and have not been found in error.

Further information on programming and numerical analysis is available inmany texts. Three books the authors have found most useful are:

• Bevington, A. R. 1969. Data Reduction and Error Analysis for the PhysicalSciences. New York: McGraw-Hill.

• Carnahan, B., Luther, H. A., and Wilkes, J. O. 1969. Applied NumericalMethods. New York: Wiley and Sons.

• Murril, P. W., and Smith, C. L. 1969. FORTRAN IV Programming forEngineers and Scientists. Scranton: International Textbook Co.

335


ccccccccccccc

c

c

cc

c510

90020

1.1 DEAD-TIME CORRECTION FOR PULSE-COUNTING METHOD

FUNCTION CNTCR (OBSD.TAU)

*"* DEAD-TIME CORRECTION FOR PULSE COUNTING ***

INPUTS -TAU DEAD-TIME COEF IN SECONDSOBSD OBSERVED COUNT RATE IN INVERSE SECONDS

OUTPUTS -CNTCR CORRECTED COUNT RATE IN INVERSE SECONDS

*** COMMENTED VERSION BV AftNF A. HENDEN 1978 ****** FROM ORIGINAL FROM UBVLSQ AT I.U. ***

CALCULATE INITIAL COUNT VALUE FOR ITERATIONA=OBSD*EXP(OBSD*TAUlITERATE FOR 20 TRIES. IF NO CONVERGENCE,GET HELP

DO 10 0=1,20CALCULATE NEXT GUESS FOR RATE

B*OBSD*EXP(A"TAU>IS THIS GUESS WITHIN 0.001 COUNTS?(ACTUALLY, PERCENTAGE ERROR WOULD BE BETTER)IF <ABSIA-B)-0.001> 20,20,5NO...RESET OLD RATE VALUE TO NEW VALUE AND TRY AGAIN

A-BCONTINUEWRITE (6,900)FORMAT* *NO CONVERGENCE AFTER 20 TRIES*)CNTCR-BRETURNEND

1.2 CALCULATING JULIAN DATE FROM UT DATE

SUBROUTINE JDAY <ID,IM,IY,UTHR .DATE )

"** CALCULATE JULIAN DATE FROM UT DATE ***

INPUTS -IDIMIYUTHR

OUTPUTS -DATE (JULIAN DATE - 2400000>

INTEGER UT DAYINTEGER UT MONTHINTEGER UT YEARF.P. UT HOUR (Z4HR CLOCK)

*** PROGRAMMED BY ARNE A. HENDEN 1977 ***

MONTH IS NUMBER OF ELAPSED DAYS IN NORMAL YEARDIMENSION MONTH!12)DATA MONTH/0,31,59,90,120,151, 181,212,243,273,304,334/LEAP IS NUMBER OF LEAP DAYS SINCE 1900

LEAP«IY/4CHECK TO SEE IF THIS YEAR IS LEAP AND MONTH-JAN,FEBIF «4«LEAP-IY>.EQ.0.AND.IM.LT.3> LEAP-LEAP-1CALCULATE INTEGRAL NUMBER OF JULIAN DAYS

OD0=1502tf+IY*365tLF_AP+MONTH( IM1 + IDADD IN UTHR AND SUBTRACT A HALF-DAYDATE*FLOAT( JD0) + UTHR/24" .-0.5RETURNEND

USEFUL FORTRAN SUBROUTINES 337

1.3 GENERAL METHOD FOR COORDINATE PRECESSION

SUBROUTINE PRCS f R.DD, VR*,VR, IH, IN, IS. ID, IOM, IDS )

*«* RIGOROUS COORDINATE PRECESSION U/0 P.M. CORRECTIONS ***

INPUTS-R R.A. IN DECIMAL HOURSUD DEC. IN DECIMAL DEGREESYR0 COORDINATE EPOCHYR YEAR TO BE PRECESSED TO

OUTPUTS-IH.IM.IS R.A. H,M,S IN INTEGER FORMID.IDM.IDS PRECESSED DEC. IN INTEGER FORM

*** WRITTEN 8V A, HENDEN 1978 **«

, 1 .9E-3/DATA A , B , C , D , E / 1 . 6001 7 , 5 . 25E-4 , 7 . SE-5 , 1 .DATA F , G , H , P / 3 . 3 E - 5 , 1 . 3 9 2 1 4 , 7 . 3 9 E - 4 , 1 . 8 E - 4 /DATA PICON/0. 0174532925/

CONVERT INPUTS TO RADIANSRA-R*f»ICONM8,OEC=DD*PICOHYAU»l Y R - Y R £ T > / 2 5 0 .

T A U * f A * T A U * { B + TAU*O) "PICONU * { D ^ T A U * ( E + T A U * F »*PICONTAU«(G-TAU'I(H + TAU*P) ) "PICON

AMU-ZET>ZBET=RA+ZETQ-SIN(THET)*(TAN(DEC)+C05(DET)*TAN(THET*0.5»GAM"0.5*ATAN(Q*SIN(BET)/( 1 . -Q*COS( BET > ) )DEE-2.*ATAN(TAN(THET*0.5)*COS(BET+GAM)/COS<GAM)>DR-GAM+GAM+AMURA=(RA+DR)/<PICON*15. )DEC=(DEC+DEE I/PICONWE NOW HAVE DECIMAL FORMS. .. CONVERT TO INTEGER FORMIH-INT(RA)X"(RA-FLOAT( IH»*60.IM-INT(X)IS-INT({X-FLOAT( IM) )*6«. )ID-INT(OEC)RM-< DEC-FLOAT) ID))"60.IF {RM.LT.0. ) RM--RMIDM-INT(RM)IDS-INT((RM-FLOAT( 10M)>«60. )RETURNEND


1.4 LINEAR REGRESSION (LEAST-SQUARES) METHOD

Subroutine SMOOTH (X,Y,H,A,B)

c *** Linear least squares routine from HieLson ***c Simple solution with no weighting factorscc Inputs:c x,y data arraysc m number of points in arraysc Outputs:c a,b where y=ax+bcc *** written by A. Henden 1973 ***c

dimension x(m),y(m)c initialize sunning parameters

a2=0.a3=0.c1=0.c2=0.a1=m

c Loop to set up matrix coefficientsdo 10 i=1,ma2=a2+x(i)a3=a3+x(i)*x(i)c1=c1+y(i)c2=c2*y(i)*x(i)

10 continuec solve matrix - simple since only 2x2

det=1./(al*a3-a2*a2)b=-(a2*c2-d*a3)*deta=Ca1*c2-c1*a2)*detreturnend


1.5 LINEAR REGRESSION (LEAST-SQUARES! METHOD USING THE UBV

TRANSFORMATION EQUATIONS

SUBROUTINE SOLVE ( N , X , SMAG , URMAG , SEXT , FEXT , COEF , ZERO >

LEAST SQUARES SOLUTION OF THE UBV TRANSFORMATION EQUATIONSINPUT -

SMAGIN.3) STANDARD MAGNITUDES AND COLORS IN THE FORM:SMAGIN.1) - VSMAG(N,2> - B-VSMAG(N.3> - U-B3) YOUR CORRESPONDING INSTRUMENTAL MAGNITUDES & COLORS

AIR MASS VALUESNUMBER OF OBSERVATIONS TO REDUCESECOND ORDER EXTINCTION

URMAGINX(N)NSEXTO)

OUTPUT -FEXTO)COEF<3)ZERO<3>

FIRST ORDER EXTINCTIONTRANSFORMATION COLOR COEFFICIENTSZERO POINTS

TO HAVE FEXT, COEF AND ZERO TO ALL BE VALID, YOUMUST USE STANDARDS WITH BOTH A WIDE RANGE OF COLORSAND A WIDE RANGE OF AIRMASSES. WITH A CLUSTER, YOUWILL GET VALID COEF PARAMETERS, BUT FEXT AND ZERO MAYBE INVALID.. .SO BEWARE I

WRITTEN BY A. HENOEN 1980

DIMENSION X(N>.SMAG(N.3),URMAGtN,3),SEXT(3)DIMENSION FEXT(3>.COEF(3>,ZERO(3)

LOOP OVER COLORSDO 20 K-1,3

ZERO MATRIX ELEMENTSAl-0 S A2=0. $ A3=0. $ A4-0. S A5-0. S A6-0.A7-0. S A8-0. S A9=0. $ A10-0. $ All-0. $ A12-0.

LOOP OVER NUMBER OF STANDARDSDO 10 I-l.N

CALCULATE MATRIX ELEMENT SUMSTEMP-URMAGfI ,K>«( 1 .-SEXT<K>*X< I »STD-SMAG<I,K>IF (K.NE.1) GO TO 5

NOTE: THE V EQUATION HAS SLIGHTLY DIFFERENT FORMTEMP-SMAGl1,2)STD-STO-URMAG<I,K>CONTINUE

THE MATRIX LOOKS LIKE:( At A2 A3 . A4 )( AS A6 A7 . A8 )( A9 A10 All . A12 )

NOTE: A2-A5, A3-A9, AND A7-A1*. BUT EXPLICIT HERE FOR CLARITY

A1-A1+TEMP*TEMPA2=A2 + TEMP"X(I )A3=A3+TEMPA4-A4+STD*TEMP

A6-A6*X(I )*X( I )A7-A7+X(I)A8-A8 + STD*X( I)A9-A9+TEMPA10-A10+X<I)A11-A11+1A12-A12+STD

(Continued on p. 340)


1.5 LINEAR REGRESSION (LEAST-SQUARES) METHOD USING THE UBVTRANSFORMATION EQUATIONS (Continued)

If CONTINUEC CALCULATE MINORS

AA-A7*A10-A6*A11BB-A5*A11-A7«A9CC-A6«A9-A5*A10DO-A8*A11-A7"A12EE-A6*A12-A8*A10FF-A5*A12-A8*A9

C CALCULATE DETERMINANTDET-A1*AA+A2*BB+A3*CC

C SOLVE FOR WANTED VALUESCOEF(K)-(A4«AA+A2-DD+A3*EE>/DETZERO<K)-<A2*FF-A1*EE+A4*CC)/DETFEXTIK).(A1*DD-A4*BB+A3*FF>/OETIF (K.NE.l) FEXT(K)-FEXT<K)/COEF(K)

28 CONTINUERETURNEND

1.6 CALCULATING SIDEREAL TIME

SUBROUTINE STIME (DATE.UT.ALON.ST.1ST)CC *** CALCULATE S IDEREAL TIME ***CC INPUTS -C DATE JULIAN DATE - 2400000.C UT UT IN DECIMAL HOURSC ALOH OBSERVER LONGITUDE IN HOURSC { LONGITUDE IN DEG / 15)C OUTPUTS -C ST SIDEREAL TIME IN DECIMAL HOURSC 1ST S.T. IN FORMAT HHMM (INTEGER)CC *** PROGRAMMED BY ARNE A. HENDEN 1977 ***CC STHR IS # SIDEREAL HRS SINCE OAN 1, 1900

STHR-6.646*556 + 2400.0512617«(DATE-15020. )/365Z5.+S UTM.00273-AION

C GET S.T. FROM STHR - * WHOLE S.T. DAYS SINCE JAN 1,1900ST-STHR-INT(STHR/24.)-24.

C NOW COMBINE TO GET 1STIST-INT(ST)IST=IST*10tf+INT«ST-FLOAT< 1ST! )*60. >RETURNEND


I.7 CALCULATING CARTESIAN COORDINATES FOR 1950.0

SUBROUTINE XYZ <DATE,X,V,Z)CC CALCULATE HELIOCENTRIC X.Y.Z COORDINATES FOR 1950.8CC INPUTS-C DATE JULIAN DAV - 2400000C OUTPUTS-C X.Y.Z HELIOCENTRIC RECTANGULAR COORDINATESC IN A.U.CC EQUATIONS FROM ALMANAC FOR COMPUTERS, DOGGETT. ET. AL.C USNO 1978C *** PROGRAMMED BY ARNE A. HENDEN 1978 ***C

DATA PICON/0.01745329/C T IS A RELATIVE JULIAN CENTURY

T-lDATE-15020.>/36525.C EL IS THE MEAN SOLAR LONGITUDE, PRECESSED BACK TO 1950.0

EL-279.696678+36000.76892*T+0.000303*T*T-$ ( 1.396041+0.000308MT+0.5) )*{T-0.499998)

C G IS THE MEAN SOLAR ANOMALYG-358.475833+35999.04975«T-0.00015*T*T

C AJ IS THE MEAN JUPITER ANOMALYAJ-22S.444651+2880.0*T+154.906654*T

C CONVERT DEGREES TO RADIANS FOR TRIG FUNCTIONSEL»EL*PICONG-G*PICONAJ=AJ*PICON

C CALCULATE X.Y.Z USING TRIGONOMETRIC SERIESX-0.99986*COS(EL J-0.025127*COS( G-EL )+0. 008374*COS< G+ED +

S 0.000105*COS<G+G + EL >+0.000063*T*COS<G-EL) +C 0.000035*COSCG+G-EL)-0.000026*5IN<G-EL-AJ)-S 0.000021*T*COS(G+EL)

Y«0.917308-SIN(EL)+0.023053*51N(G-EL)*0.007683*5IN(G + EL> +S 0.000097*51N(G+G + EL)-0.000057*T*SIN(G-EL>-$ 0.000032*5INIG+G-EL >-0.000024*COS(G-EL-AJ)-S 0.000019*T*SIN<G+EL>

Z-0.397825-SIN(EL 1+0.009998*5 IN(G-EL)+0.003332*51N(G + EL) +S 0.000042*51NIG+G + EL)-0.000025*T*SIN( G-EL>-S 0.000014*S1N(G+G-CL)-0.000010*COS(G-EL-AJ)

RETURNEND

APPENDIX JTHE LIGHT RADIATION FROM STARS

J.1 INTENSITY, FLUX, AND LUMINOSITY

The concepts of intensity, flux, and luminosity are very important and yet theyare often confused. These terms cannot be used interchangeably as they havevery different meanings. The following discussion is based on treatments byAller1 and Gray.2 Figure J.I shows a small area AA on the surface of a star.What we wish to consider is the amount of energy emitted by this small areain the direction 8 from the normal (a line perpendicular to the surface).Throughout this discussion, we assume symmetry about the normal. That is tosay, the results are the same if 0 is drawn to the left of the normal in FigureJ.I instead of to the right. In order to measure the energy, some detector mustbe used to collect the energy falling on its surface. In practice, you cannotmeasure energy at a "point" because a point has no area. In Figure J.I, AX'represents the energy-collecting area, and Ao) is the solid angle subtended bythis area as seen from the star. Strictly speaking, the only point on the surfacethat can radiate energy into the cone, Aw, is the point at the vertex of this cone.For the moment, assume that AX is very small compared to this cone, so thatAX looks almost like a point.

The amount of energy emitted each second into the cone depends on Aw.Obviously, if the cone is made larger, it will contain a larger fraction of thetotal energy emitted by AX. A second factor is the range of wavelength orfrequency to be measured. No detector can be made to measure light in aninfinitely narrow wavelength interval. Likewise, no single detector can measureevery portion of the electromagnetic spectrum. It is for this reason we build x-ray, 7-ray, radio, and infrared as well as optical telescopes. A large bandpasscontains more energy than a narrow one. Finally, the projected size of AXaffects the energy in the cone Aw. To understand this, imagine you are viewingAX by looking down the axis of the cone. If 0 equals zero, you are viewing AXperpendicular to its surface and it appears its actual size. As 0 increases, the

342

THE LIGHT RADIATION FROM STARS 343

NORMAL

Figure J. 1. Geometry used to define flux and intensity.

area appears smaller and smaller until at Q = 90° the apparent area is zero.Likewise, as B increases, the apparent brightness decreases until at 6 - 90° itreaches zero. The projected area is &A cos B.

Combining all of the above ideas, it can be said that the energy emitted(A£) in a time interval (Ar) into the cone is proportional to the size of thecone (Ato), the wavelength interval (AX), and the projected surface area (A/4cos 6). Symbolically,

— oc (Ay4 cos 8) AXAo).A/ (J.I)

The "constant of proportionality" must contain the information that describesthe radiation emerging from the star through its surface. Its value is set by thephysical conditions in the star's atmosphere such as temperature, pressure, andgravity. This quantity, called the specific intensity, /x, is defined by rearrangingEquation J.I and taking the limit as Af, Ao?, AX, and &A go to zero. Then

7X = limAf—0

A--0

AX-0

A£\AtA/1 cos 0 Aw AX

(J.2)

or

A =dt dA cos 0

(J-3)


The subscript X has been added to remind us that intensity is a function ofwavelength. Specific intensity is not really a constant because it can be a func-tion of direction, wavelength, and perhaps time. The units of specific intensity(usually just called intensity) are ergs per second, per Angstrom, per squarecentimeter, per square radian.

An important point to note is that intensity can be defined at any point inspace, not just on the surface of the star. The area &A could be an imaginarysurface at any distance from the star, and the definition of intensity could pro-ceed exactly as before. A second point to note is that intensity is independentof the distance from the source. Intensity is a measure of the energy flowing ina solid angle and this does not depend on distance. As an example, supposethat a star radiates evenly in all directions, that is, isotropically. A solid angleof one steradian (one square radian) then contains l/4x times the total energyflow from the star. One steradian always contains this much energy no matterhow far one is from the star.

Flux is defined as the net energy flow across an element of area per secondper wavelength interval. For any given surface of area A/t, we must sum allthe energy flowing in and out of this surface from every angle. Figure J.2 illus-trates this idea. Because we want the net energy crossing AX, inward and out-ward energy flows have opposite algebraic signs. Flux is then defined as

AX A / A X(J.4)

or

Solving Equations J.3 for dEx and substituting into Equation J.5 results in

FX = /A(0) cos 9 dw = 27T /A(0) sin 8 cos 6 d8. (J.6)all inglei

At the surface of a star, the integration can be broken into two pans, the fluxleaving the star and the flux directed inward, or

f «/2 r»7X(0) sin 6 cos 0 d6 + 2* /x(0) sin 8 cos 6 d8. (S.I)

.\ -J~n


+AE

Figure J.2. Energy flow across AA.

In the second term, /x(0) must be zero because at the surface of a star all ofthe energy is directed outward. Therefore,

F = 2-B- r•J nsin 0 cos 8 d8. (J-8)

If /x is independent of direction, then

= 2irh I sin 6 cos 6 dQ (J.9)

F, = IT/,. (J.10)

Keep in mind that this result applies to a rather special case and in generalflux and intensity are not related so simply.

To illustrate the difference between intensity and flux, consider the followingexperiment (based partly on Gray,2 page 103). Imagine that a telescope lensfocuses an image of the sun onto a projection screen. Assume that the solardisk is illuminated uniformly, that is, no limb darkening. The screen has asmall hole of area AD that allows light to reach a photomultiplier tube. FigureJ.3 illustrates the experiment. As is the area on the sun's surface that corre-


Figure J.3. Illustration to clarify difference between flux and intensity. See text.

spends to AB on the screen. In other words, the light entering the hole, AD, wasemitted from area A, on the sun. Light emitted elsewhere on the sun is imagedto another portion of the screen. The only energy reaching the photomultipliertube is that contained in the solid angle a? defined by

w = A,ji (J.l l)

where r is the distance between the lens and the sun. Because the output of thePM tube depends on this solid angle, this must be a measure of intensity. (Fluxdoes not depend on solid angle.) If the distance to the sun could be slowlyincreased, the intensity measured by the photomultiplier tube would remainthe same as long as the solar image is larger than AD. To see this, note that

/oc (J.I 2)

where E, is the energy emitted each second by the surface element A, into thelens. However, E, is proportional to A, for a uniformly emitting solar disk.Therefore,

,/ o c - = —a; A,

(J.I 3)

If we assume that the light rays pass through the center of the lens, and henceare undeviated, then

= 0) (J.14)


where /is the focal length of the lens. This last equation says that <a is deter-mined by the size of our detector, A& and the focal length of our telescope.The projected area of the detector on the source is A,. According to EquationJ.I4, o> is constant and Equation J.I3 shows that intensity is constant.

Another way of obtaining the same result is to note that as the distance tothe sun increases, A, must increase because the angle at which the conediverges from the lens is unchanged. If we assume, for the sake of simplicity,that A. is circular, then

As = irr]

However, r, increases with distance so that

rs= re

(J.I 5)

(J.16)

where 6 is the angle subtended by r,. Combining Equations J.15 and J.16, onefinds

A, oc r2

Combining this with Equation J.I3,

/ oc constant.

(J.I 7)

(J.I 8)

In other words, as the distance to the source increases, the area of the sourceimaged on the detector, A,, increases in a compensating manner to keep theintensity constant. However, a transition occurs when the distance increases tothe point where A, equals the area of the disk of the sun. At this point, A, hasreached its maximum value and it no longer increases as the distance to thesun increases. The sun is now unresolved because the projected image is smallerthan the PM tube opening. It is at this point that the photometer is no longermeasuring intensity, but instead is measuring flux. The output of the PM tubeis independent of o>, as long as w contains the entire light source. As the dis-tance between the telescope and the sun continues to increase, the output ofthe PM tube decreases because flux does depend on distance to the light source.

We can distinguish between two types of photometry. The surface photom-etry of galaxies is in fact a measurement of intensity. The procedure is to movethe image of the galaxy across a small diaphragm opening. The data from thistype of research specify a magnitude per square arc second at points across thevisible surface of the galaxy. Stellar photometry, on the other hand, involves


Figure J.4. Two concentric spheres.

unresolved point sources* and hence there is no specification of a solid angle,i.e., stars just have a "magnitude." We should note also that the use of a tele-scope or optical system can in no way increase the intensity reaching the detec-tor. However, the total energy collected by the telescope depends on the areaof the primary lens or mirror. This results in an increase in the flux passingthrough the focal point, which is a primary function of the telescope. Whenattempting to measure a faint star, the largest possible telescope should beused. When measuring an extended source (and therefore measuring inten-sity), the advantage of the large telescope is increased resolution. A small tele-scope can measure the intensity across the surface of a galaxy as well as a largetelescope. However, in a small telescope, large regions may appear as an unre-solved "blob," whereas in a large telescope, fine detail may be resolved andmeasured.

Flux obeys the familiar inverse square law of light. Consider Figure J.4,which shows a point light source surrounded by two transparent spheres ofradii r, and r2. If the light in the center is the only light source, then the sametotal energy that crosses the inner sphere per second must cross the outersphere. The flux at each sphere is just the total energy radiated per seconddivided by the area of each sphere, that is,

Thus the flux is inversely proportional to the square of the distance from thesource:

'Distant galaxies can also appear as unresolved point sources.


The luminosity of a star is the total energy emitted per second at all wave-lengths in all directions from the entire surface. We must therefore sum theenergy contribution from each small element of stellar surface over all solidangles and all wavelengths. The geometry is the same as in Figure J.I. As inthe previous discussion of intensity, the projected area, dA cos 9, must be used.The luminosity, L, is given by

L = \ /X(cos0 dA)

If /, is constant over the surface of the star, then

L = 4irR••II'- cos 6 dd) d\

(J.21)

(J.22)

(J.23)

where R. is the radius of the star. By Equation J.6,

L-**K$Fla

where F[ is the flux at the surface of the star. We demonstrate later that, toa good approximation, the above integral can be replaced by Stefan's law.

J.2 BLACKBODY RADIATORS

The description of the radiation leaving a star is an enormously complex prob-lem, and there is no simple mathematical expression that accurately describes/* for a star. It would be extremely helpful to have a simple expression that atleast approximates /x for a star. The blackbody radiator, a highly idealizedradiation source described by theoretical physics, fulfills this need. A blackbodyis an object that absorbs all radiation falling upon it. This object also emits asmuch energy as it receives and is therefore in an equilibrium state at sometemperature. Note that a blackbody radiates and therefore need not appearblack. For a blackbody, the intensity of the radiation it emits, 7X, depends onlyon its temperature T and in the wavelength interval d\:

(J-24)

This expression is known as Planck's law. In this expression, T is the temper-ature in degrees Kelvin, h is Planck's constant (6.63 X 10~27 erg-seconds), cis the speed of light in a vacuum (3.00 X 10'° centimeters/second), k is theBoltzmann constant (1.38 X 10~16 erg/°K), and the wavelength is in centi-meters. The units of Ix are ergs per square centimeter per second per square


radian per wavelength interval of one centimeter. Note that these are the unitsof intensity. By substituting all the constants, Equation J.24 takes on a moreusable form of

1 19 x 10"(J-25>

where X is still in centimeters.The continuum spectrum of most stars at least roughly approximates a black

body spectrum. Figure J.5 shows the shape of the blackbody spectrum (a plotof Equation J.25) for two different temperatures. Note that the curves for dif-ferent temperatures do not cross; a hot blackbody is brighter than a cool oneat all wavelengths provided they are viewed from the same distance. The wave-length at which the curve reaches its maximum height, XmM, is a function oftemperature. To find this wavelength, it is necessary to solve

—* = 0 (J.26)

for X, which then equals X^. The result, when T is in degrees Kelvin, is

0.290\na* = —^~ centimeter. (J.27)

This result is referred to as Wien's displacement law. It is important to notethat the constant in this equation applies to wavelength intervals only. That is,this equation cannot be used to find the frequency of maximum emission bysubstituting c divided by the frequency for X,, . The constant must also bechanged. That is, when T is in degrees Kelvin,

v^ = 5.88 X 10'° T (Hertz). (J.28)

where i%M is the frequency of maximum emission. Wien's displacement law isvery useful for determining the temperature of a blackbody. It is simply a mat-ter of measuring Xm<1 or vmn to find the temperature. This technique can alsobe applied to stars because they radiate approximately as blackbodies.

Stefan's law is an expression that gives the total energy radiated per secondper square centimeter per square radian at all wavelengths. Stefan's law isobtained by integrating Planck's law over all wavelengths. The result is

I ( T ) = - T4 (J.29)1T


Figure J.5. Blackbody curves.

where a — 5.67 X 10 s ergs per square centimeter per Kelvin degree to thefourth power. I(T) is simply the area under the curve in Figure J.5. For ablackbody radiator because / is independent of direction, Equation J.29 can besubstituted into Equation J.10 to give the flux at its surface,

F = aT4. (J.30)

This says that the total flux at all wavelengths depends solely on temperature.In this equation, F is related to Fx by

•£•J n F>.d\. (J.31)

Equation J.30 can be substituted into Equation J.23 to yield an approximationfor the luminosity of a star, that is

L = 4irffR2. T4.. (J.32)

J.3 ATMOSPHERIC EXTINCTION CORRECTIONS

Figure J.6 shows the earth's atmosphere as a plane-parallel slab, i.e., the cur-vature of the earth is ignored. This is a valid approximation for stars that aremore than 30° from the horizon. The relative loss of light flux t/Fx in travelinga distance ds in the earth's atmosphere is shown in Figure J.7. This loss mustbe proportional to Fx (the more flux, the greater the absorption, in an absolutesense), to the absorption coefficient, ax (the fraction of flux lost per unit dis-tance, in units of cm"1), and the distance traveled through the atmosphere, ds.Stated mathematically,

= — Fv a-, ds. (J-33)


ZENITH

Figure J.6. Path Length is a function of Zenith angle.

els

Figure J.7. Absorption of light.

The minus sign indicates that Fx is decreasing, or being absorbed, with distancetraveled. Equation J.33 can be rewritten as

— = -ak ds (J.34)FI

and integrated over the path length, s, traveled in the atmosphere to yield

ln(Fx/Fxo) = - '^ds (J.35)J n

or

= exp f r( -\ •* Q

(J-36)

where F^, is the flux at the top of the atmosphere and Fx is the flux reachingthe ground. Astronomers often define the optical depth, T, by

«x ds,J n

(J.37)


as this term is dependent only on the absorbing material and the geometricorientation, not on the source of radiation. Then we can rewrite Equation J.36as

= e (J.38)

Note that if rx = 1, then the flux reaching the ground is l/e of the incidentflux on the top of the atmosphere.

To convert this flux ratio to a magnitude difference, we apply Equation 1.3to yield

= -2.5 log (FJFM)= -2.5 log (*-**)

U-39)

where mx and mxo are the apparent magnitude of the star at the earth's surfaceand above the atmosphere, respectively. Equation J.39 becomes

- mxo = 2.5 (log (J.40)

or

~ 1.086rx. (J.41)

This equation can be placed in a more useful form if we show the variation ofTX with location in the sky. By Figure J.6

cos 2 = y/s (J-42)

or

and

s = y sec z

ds = dy sec 2

(J-43)

(J.44)

where z is the zenith angle, y is the thickness of the atmosphere at the zenith,and s is the path length of the light. By Equations J.37 and J.44, the opticaldepth can be expressed as

rx = sec z I ax dy.'oJ. (J-45)


The integral is simply the optical depth at the zenith, a constant factor. Thus,Equation J.41 becomes

mxo = >"x — 1 086 sec z r ay dy-*

or

wxo = mx — &xsec z (J.46)

where & x is called the principal extinction coefficient and sec z represents theair mass, or the relative amount of atmosphere traversed. Air mass is oftendesignated as X.

There are two different kinds of particles in our atmosphere that causeextinction. Each has different wavelength dependences. The major constituentare the molecules, where k\ varies as X~4. These particles are roughly the samesize as the wavelength of light. Larger particles, like dust, are called aerosolsand cause k'K to vary as X ~ ' or X°. The relative fraction of each type dependson the atmospheric conditions and the zenith distance.

We have taken a simple approach to extinction. Our hypothetical caseaccounts for the wavelength dependence of the absorption, but we assume aplane-parallel atmosphere and niters with infinitely sharp bandpasses in orderto obtain a magnitude at a specific wavelength. The actual case of a sphericalearth is covered in Chapter 4. The bandwidth effect is discussed here and givesrise to a correction to the extinction known as second-order extinction.

Within a filter's bandpass, some wavelengths suffer more extinction thanothers. In general, blue wavelengths are absorbed and scattered more readily.An average extinction in the bandpass could be used, but then stars with risingflux toward the blue end of the bandpass (in general, hot stars) would system-atically be given less extinction than is the real case, and those rising towardthe red (cool stars) would be given more extinction and a fainter magnitude.In other words, using an average extinction introduces a systematic error in themagnitude determination, an error that is dependent on the color of the starand the air mass through which it is observed. Correcting Equation J.46 forthis color-dependent term, we obtain

wxo = m\ — fc* sec z — k{c sec z (J-47)

or

sec z (J.48)


where fcx is called the second-order extinction coefficient and c is the instru-mental color index of the star.

Because atmospheric extinction is wavelength-dependent, the apparent colorindex of a star is also affected. Equation J.47 can be written with a subscript1 or 2 for the two wavelengths. That is,

^xoi = wx, — £X 1 sec z — k^ c sec z

>«xo2 = wX2 — k'X2 sec z — &x2 c sec z.

Subtracting, we then obtain the color index

= (mxl - mX2) - (k'u - *xz) sec z

(J.49)

(J.50)

If we let the color indices c and c0 be defined as

c = (mx, - mX2)

c0 = (wXOi — mx02) = (mx, —

and let

then

kc =

c0 = c — k'c sec z — A:^ (sec z) c. (J.52)

In practice, the extinction coefficients can be determined by a few measure-ments of stars at various altitudes without any knowledge of ax or the actualphysical processes of absorption. There are pitfalls in the determination ofthese coefficients because of their temporal and spatial variability. The prac-tical details are presented in Chapter 4.

J.4 TRANSFORMING TO THE STANDARD SYSTEM

For observers at different observatories to be able to compare observations,observations must be transformed from the instrumental systems (which areall different) to the standard system. The main reason for this differencebetween photometer systems is that the equivalent wavelengths of observation


are slightly different. The equivalent wavelength (X^) is an average wavelengthof observation weighted by the response function of the equipment, and isdefined by

f

J n

J

(J.53)

where (f>(\) = * as defined in Chapter 1. Small changesin the spectral response of the measuring equipment change X^. This meansthat a magnitude determined by one instrumental system differs slightly fromthat found by a second system. For example, suppose you are measuring a hotstar whose flux is rising very rapidly toward the blue. If your equivalent wave-length is slightly blueward of the standard system, your measured magnitudewill be brighter than the accepted value. Because the difference in X,,, is usuallysmall, the magnitude on the standard system, M, can be approximated by aTaylor expansion in X.,, about the instrumental magnitude, m0. That is,

+ higher-order terms. (J.54)

Figure J.8 illustrates this situation. Note that in this discussion the magnitudesare assumed to have already been corrected for atmospheric extinction. Theterm in parentheses is the slope of the star's continuum, which is proportionalto the star's color or color index in that region. Any instrumental magnitude,mxo, can be converted to the standard magnitude, M, by an expression of the

STELLARCONTINUUM

WAVELENGTH

Figure J.8. Difference between the standard and instrumental equivalent wavelength.


form

(J.55)

0X and 7X are constants which are unique to the photometer in use and C is thestandard color index of the star, near the wavelength in question.

The transformation of a color index (that has already been corrected forextinction) is found by applying Equation J.55 to each spectral region andforming the difference. That is,

- /3X2)C

or

C = 7, (J.56)

where C is the color index on the standard system, c is the instrumental colorindex and 0, 7, 0r, and 7,. are constants. If we define a constant, 6, by

1

then

1 -

C = 8c0 (J.57)

For both the magnitude and color index transformations, we have assumedthat the higher-order terms beyond a linear relation to the color index are neg-ligible. This is not always the case. An example is the use of a "neutral" densityfilter that adds additional color dependence to the instrumental response. Thesesituations are apparent when the coefficient determination is attempted, asdepartures from a linear dependence are evident. For these situations, the nexthigher-order term (quadratic in color) must be added to the procedure. Asthese conditions rarely arise, further treatment is not presented in this text, butyou should be aware of the possibility.

REFERENCES

1. Alter, L. H. 1963. The Atmospheres of the Sun and Stars. New York: RonaldPress.

2. Gray, D. F. 1976. The Observation and Analysis of Stellar Photospheres. NewYork: John Wiley and Sons.

APPENDIX KADVANCED STATISTICS

Appendix K explains some of the finer points in statistics and their applicationto astronomy. The subject matter treated here is not essential for basic pho-tometry, but is not covered elsewhere in this text and is given here to stimulatethe interested to read further.

Much of the presented material has been drawn from elementary statisticstexts1-2-3 or from the astronomical literature.4-5'6 You should know the meaningof the terms matrix, determinant, derivative, and partial derivative before youcan make full use of this appendix.

K.I STATISTICAL DISTRIBUTIONS

To introduce the idea of a probability distribution, assume that we are execut-ing a coin toss. It is relatively simple to calculate the probability of tossingthree heads and seven tails out of 1 0 tosses, but what is the probability of toss-ing n heads and ( m — n ) tails out of m tosses? These numbers can be thoughtof as describing the function p(n), where p is the probability that n of thetosses will be heads. Such a function is called a probability distribution.

If the index n varies between 1 and m, then we have included all possibleevents and the sum of these probabilities must be unity. That is,

(K.I)

The two types of probability distributions extensively used in experimentaldata in photometry are the Gaussian, for experimental values, and the Poisson,for photon arrivals. You need not know the functional forms of these distri-butions to use the results that mathmaticians have derived from them, such as

358

ADVANCED STATISTICS 359

the standard deviation; but it is important to at least look at the shapes of thedistributions.

The Gaussian (or normal) distribution is defined by Equation 3.10, rewrittenhere as

p ( x ) =1

exp -i x — x\

J (K.2)

This distribution is shown in Figure 3.1 and repeated in Figure K.I for com-pleteness. It has been shown to approximate errors of measurement very closelyand therefore is the most widely known and used of all distributions.

Often, Equation K.2 is standardized when given in tables in the back ofstatistics texts. That is, the function is given with respect to the standardizednormal variable, z, where

x — xz = (K.3)

-* .08 -

.04-

00-3

Figure K. I . Gaussian distribution for x = 5.0, a = 2.3.


then p'(z) is tabulated where

-z'/2 (K.4)

This allows tables of the function to be given without having to list functionpoints for every combination of a and x.

For approximating arrival rates of photons, the Poisson distribution is themost commonly used function. It is defined by

• \ \ e A rv occurrences in \ , for .-. I — i v A .> Ua given time unity

y = 0, 1,2, . . .

(K.5)0, otherwise

where p is the probability of occurrence and X is the mean arrival rate, thenumber of photons per second. This function looks much like the Gaussian,except:

1. The variance is given by a2 = \.2. It is a discrete distribution, not continuous like the Gaussian, because a

photon either arrives or it does not arrive.3. It is skewed to the right because there are no negative occurrences, that

is, y is never negative.4. As \ -* co, the Poisson distribution approaches the Gaussian distribution

in form.

An example of the Poisson distribution for X = 5.0 is given in Figure K.2.Just as the standard deviation was defined for a Gaussian distribution, a

similar "standard deviation" can be derived for the Poisson distribution. IfEquation K.5 is substituted in a generalized form of Equation 3.6 (a messyproposition!), then the standard deviation of Poisson-distributed data is givenby

op = VN (K.6)

where N is the total number of observations. Note that this value is not thesame as for the Gaussian distribution. In fact, no two distributions have thesame value for the standard deviation. This means that if there is a bias in yoursample so that it does not fit a Gaussian distribution curve, the standard devia-tion that you derive using Equation 3.6 is not the correct one for your sample!You must be very careful to remove any chance of bias from your calculations.


0.18

0.17

0.16

0.15

0.14

0.13

0.12

0.11

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

------------

I .1 2 3 4 5 6 7 8 9 10 11 12 13

Figure K.2. Poisson distribution for A = 5.0.

K.2 PROPAGATION OF ERRORS

If you use various experimental observations to calculate a result, and theseobservations each have uncertainties associated with them, then the error inthe result will be a function of the errors of the individual observations. Forexample, if you want to calculate the density, or weight per unit volume, of aliquid in a container, you need to both find the volume of the container and itsweight. It is not entirely obvious what the density error is if you know thevolume to ± 0.1 percent and the weight to ± 1.0 percent. To find the resultanterror, you need to propagate each individual error through the calculation andsee how it affects the final result.

In a general form, assume that the function f ( x ) is similar to that shown inFigure K.3. We can see that an error in the measurement of jt, <;„ causes acorresponding error, e7, in our evaluation of the function. Now

dx


Figure K.3. Error in f(x) due

or

(K.7)

where/'(x) is a shorthand way of expressing the derivative of f ( x ) , i.e., howfast it varies with a change in x.

We can extend Equation K.7 to functions of more than one variable:

dfdz

(K.8)

nrwhere — is the partial derivative of/with respect to x, which means simply

dxthat we recognize / is a function of many variables, but we are evaluating thederivative with respect to x and hold the variations due to the other variablesconstant during the process.

Now

N

df

(K.9)

(K.10)


which reduces upon expansion and removal of cross terms to:

dx(K.ll)

that is, errors add in quadrature.We seldom know the mathematical form of the function / There are some

examples in photometry, however, where Equation K. 11 can be applied. TheUBV transformation equations depend on the values of the zero points, thecolor terms, the extinction coefficients, and the air mass. By knowing the stan-dard deviation in each of these parameters, you can calculate the expecteddeviation in the derived standard magnitudes and also see which terms arerelatively unimportant.

K.3 MULTIVARIATE LEAST SQUARES

In Chapter 3, we considered the case of fitting a least-squares line to thefunction

f(x,) = a + bxt. (K.I 2)

That is, by knowing the approximate f ( x t ) for several values of x,, we couldsolve for the parameters a and b. This is simple linear regression in that thef unction f ( x ) is dependent only in a linear fashion on one variable, x.

There are two ways that this result could be generalized: (1) f ( x ) could havea nonlinear dependence on x, such as

/U) = a + bx*

or (2) the function /could be dependent on more than one variable, such as

Equation K.13 is known as multiple linear regression and is covered in thissection. The nonlinear case can be found in Young,1 Harnell,2 and Bevington.3

Multiple linear regression is very important in photometry in solving thetransformation equations. For example, substituting Equation 4.28 into 4.32yields

V = c(B - V} + (v - k»v(b - v)X) + (K.14)


which shows that Kis a function of the star, the instrument, and the air massthrough which the star is observed.

A mathematical representation of the regression equation is

y = a + V, + b2z2 + • • • + bmzm (K.15)

where z,, z2, . . . are not the individual values of one variable (z), but areindependent variables. For example, a comparison with Equation K.I3 givesz, = x, z: = y. Then we minimize the sum of the squares of the deviations,D:

- E <* - *>'fiT (K.16)

where n is the number of observations. The procedure for minimizing D is thesame as in the simple linear case, except that partial derivatives must be takenfor m + 1 variables instead of just two. Setting these m + 1 partial derivativesto zero and solving, we obtain the following set of normal equations:

an + fc,£z1( + b2I.z2l + ---- h bj^z^ = Eyf

+ 6,Ezf ; + 62Ezuz2, + ---- h 6fflZzuzmi- = Ezuyt

+ b&ZuZu + b2Lz22i + ---- h bmEz2izmt = Ez2/y,

(K.17)

In matrix form,

Ajl( = T.ZjZk where j = 0,« and k = 0,/i

Nj = Xify,Bj = bj where B(0) = a

then


Removing subscripts,

AB = NB = A - ' N (K.18)

Equation K.18 gives the coefficient vector fi, providing that matrix A can beinverted. There are more elegant methods of solving the normal equations, forinstance, by assuming they are separable, solving for bltb2 ... ,bm and thenback-substituting for the constant term 60.

As an example, we solve Equation 2.10 for the zero-point, first-order extinc-tion, and the transformation coefficient, that is a "full-blown" solution. If werewrite Equation 2.10 by substituting Equation 2.7 to obtain

We can see by correspondence that we have

y, - a + 6,2! + b2z2

y, = B - V

b, = Mz, = (b- v)[l - k"bvX]b2 = —fik'by

z2 = X.

Then, writing the normal equations,

an + 2>iEzu + 6iEz,( = Ev,

we obtain

(K.20)

a matrix equation, which can be solved by back-substitution or the use ofminors. See Arfken7 for more detail.


The transformation equations for Kand (U — B) can be solved in a similarmanner. A subroutine to calculate the full treatment UBV least-squares solu-tion can be found in Section 1.5. You should be very wary of solving for boththe color and extinction coefficients at once, as they interact with each otherand bad data points are not obvious. We suggest using simple linear leastsquares for the color coefficients and a separate solution for extinction, as pre-sented in Chapter 4.

K.4 SIGNAL-TO-NOISE RATIO

Each time a star is measured, there is an uncertainty in the value obtained. Ifsystematic errors are ignored, the uncertainty can be defined as the noise, orthe standard deviation of a single measurement from the mean of all the mea-sures made on the star. The ratio of the value derived for the observed countrate to the uncertainty in that number is called the output signal-to-noise ratio(S/N).

If an idealized detector is considered, where no background is present andthe only source of noise is from statistical fluctuations from the star, then thearriving signal is given by

S-m = Ctt (K.22)

where Sin is the number of input photons to the detector in time t, C is therate of photon arrival in photons per second, and t is the integration time, orthe time required for the measurement (seconds). For a weak beam of photons,Poisson statistics are a good approximation to the statistical fluctuations of thebeam. These statistics state that the noise, or standard deviation of the signalfrom the mean, is given by Equation K.6, or in another form,

^Vin = Si'/2 = (CO"2- (K.23)

The S/N calculated using Equations K.22 and K.23 tells us, in a sense, whatfraction of the arriving signal is contributed by the noise. This ratio is given by

(K.24)

or

=(CO"2- (K.25)


One hundred total source counts would yield an S/N of 10. Because the noisevaries as the square root of the signal, 10,000 total counts would be requiredto reduce the noise to one percent of the signal or equivalently, to have anS/N of 100.

K.4a Detective Quantum Efficiency

Very few photomultiplier tubes are perfectly efficient. In general, only a frac-tion, Q, of the incident photons are detected. The parameter Q can be thoughtof as the efficiency of detection of quanta or photons, or the quantum efficiency.Realizing this, the S/N (Equation K.24) for the output signal can be writtenas:

(K.26)

(K.27)

_

, (CO)"2

Q is generally not known, but must be measured experimentally. Rearranging,

N

(K.28)

Theoretically, Equation K.28 holds only for the case where the number ofdetected photons is just a simple linear fraction of the number of incident pho-tons. We can generalize to the more common case by defining

DQE = (K.29)


where DQE is the detective quantum efficiency, so called because it is a mea-sure of how badly the actual detector deviates from a perfect detector. Theactual detector or system performs as if it were an ideal detector with the inputsignal decreased by the factor DQE (:<!). The DQE of a system is always lessthan the quantum efficiency of the photocathode, as some electrons are lost inthe dynodes and different detection systems weight the anode pulses differ-ently. A typical photomultiplier system has a DQE of 2 to 4 percent.

So far we have discussed only shot or Poisson noise, the fluctuations inherentin the source itself. There are several other possible sources of noise, both inthe sky and the actual detection system. All do not affect the output S/N inthe same manner, and at a given source intensity, one noise component may bedominant or several may contribute. For faint sources, the source signal maybe much smaller than the noise.

Listed below are the three major types of noise sources and how they affectthe signal-to-noise ratio when they are dominant. Discussions of these sourcescan be found in other chapters.

1. Noise dependent on the square root of the signal,a. Photon shot noise, A7,, in the signal.

N, = C(CCO"2

Here Equation K.26 becomes

Q ** _ / x W - y / j (K3())

where G is the internal gain of the detector. Note that the S/N is pro-portional to the square root of the number of counts; increasing counts bya factor of 100 increases the S/N by only a factor of 10. By taking log-arithms of both sides,

log - = & log «?O) = K log (counts). (K.31)

So plotting several readings of log (S/N) versus log (counts) gives astraight line slope %..

2. Noise linearly dependent on the signal.a. Scintillation noise from the atmosphere.

This noise source, A^ is a fractional modulation, m, of the incidentbeam, that is

Nsc= mQ Cst. (K.32)


In other words, stars of all magnitudes vary by the same percentage.If a 1 Om star is varying by 10 percent, then so is an 18m star. EquationsK.26 and K.31 then become

_1_m

^S\log ( — I = constant

\™ / out

where the slope of the plot is zero.3. Noise independent of the signal,

a. Amplifier noise.Equations K.26 and K.31 become

(K.33)

(K.34)

GQC,t

\log I - \ = log G - log Nam + log (QC,t)

/ out

(K.35)

(K.36)

where Nam is the amplifier noise in net equivalent photons, that is, thenumber of incident photons that would be required to produce a noisesignal the size of the amplifier noise. In this example, the slope of theplot equals 1.

b. Background shot noise, N& such as from the sky.

Nb =

Our equations become:

S \ GQC,t

log = log ~ * log

(K.37)

(K.38)

where Cb is the background count rate. Again, the slope is equal to 1.Each of these sources becomes dominant in different count regimes. Forinstance, bright sources have negligible background, sources near the horizonexhibit much more scintillation, etc. For count rates not dominated by onenoise component, all contributors must be accounted for by adding the noisecontributions in quadrature, i.e.,

,1/2 (K.39)


so that

GQC,t(N} + N]c + N

(K.40)

A derivation of why uncorrelated noises (or standard deviations) add in quad-rature is beyond the scope of this section, but can be described as taking thepartial derivative of the total noise count function with respect to each of thecontributors (source, sky, etc.) and noting that cross terms vanish (N, is notdependent on Nb etc.).

A plot of the theoretical signal-to-noise ratio obtainable with t = 1 second,Q = G = 1, and Cb = 30 counts per second for various sources (i.e., varyingC) is shown in Figure K.4. Note that for high count rates, the slopeapproaches & and for count rates near Cb the slope is approximately 1.

K.4b Regimes of Noise Dominance

This section examines three noise regimes as examples of signal-to-noise ratioconsiderations. These are:

1. Background. So far, only those cases where the background and sourceare uniquely known have been discussed. This is not normally the case,as the background is generally unknown and must be measured during

100 t-

10,000

Figure K.4. Signal-to-noise ratio comparisons for differing sources.


part of the total observing time. For this case of first measuring sourcewith sky background and then sky by itself, the noise or standard devia-tion from the mean is a combination of Poisson noise and backgroundnoise:

N =

where / is the amount of time spent measuring source or background andCx is the count rate for a measurement, with subscript s referring to thesource and subscript b the background. Also, the data are represented by:

Total source counts = S ± N (K.42)S N

Count rate for source (per unit time) = — ± - (K.43)

2. Scintillation. For brighter stars, the dominant noise source is scintillation.But if scintillation and background noises are both important for a source,one should realize that scintillation affects only the source counts and notthe sky counts. The S/N for this case is;

As long as the source counts are low or the scintillation modulation, m,is small, the S/N takes on its previous appearance, Equation K.41.

3. Amplifier noise. In many cases, such as astronomical TV systems, bothamplifier and photon shot noise are important. The S/N for this case is:

GQC,t

GQC,t(G2QC,t

__N N* Y/2 (K'45>

Note that the effect of the amplifier noise is decreased by the internalgain of the detector. The ratio is then added, much as a source of photons.


This noise-to-gain ratio can be thought of as the number of equivalentincident photons on the detector that would give shot noise of the sameamount. In other words,

N- = amplifier noise referred to the input.

G

We then see why amplifier noise, DC amplifier or pulse preamp, is notimportant for photomultiplier systems, because the noise-free gain in thephotomultiplier tube is 106, which reduces the effect of amplifier noise tonegligible levels.

K.5 THEORETICAL DIFFERENCES BETWEEN DC AND PULSE-COUNTINGTECHNIQUES

There are two basic methods of detecting the output from a photomultipliertube. DC techniques measure the feeble current generated by the incident pho-tons while pulse-counting techniques measure the number of photons or pulsesdirectly. Proponents of both sides have been arguing for decades over whichtechnique, if either, is superior. The major area in which one may have anadvantage over the other is in the signal-to-noise ratio. For this reason, thecomparison between DC and pulse-counting methods is included in thischapter.

K.5a Pulse Height Distribution

Each dynode of a 1P21 photomultiplier tube has an average gain of about 5.However, statistically one dynode might have a gain of 4 one time, and 6 oreven 10 the next. This means that the number of electrons collected at theanode is not constant, depending on which dynode deviates from the mean.This was shown earlier in Figure 7.2 for a typical photomultiplier tube output.The noise is given by NamjG and is small for a photomultiplier tube but notnegligible on other devices nor for larger dark currents. Typically the distri-bution of pulse sizes is called the pulse height distribution, shown for an ideal-ized source in Figure 7.3.

The general features of such a distribution are readily seen. As x approacheszero, all pulses are a result of dark current or amplifier noise and the numberof such pulses approaches infinity. After this initial peak, the next larger peakis a result of a photon liberating one photoelectron at the cathode. Other peaksoccur at larger heights because more than one photoelectron is released by thephoton or by cosmic ray induced electrons. Many photoelectrons may be pro-duced if the cosmic ray strikes the photocathode at near grazing incidence. The


—H f— A>

Figure K.5. Schematic pulse height distribution.

probability of more than one photoelectron being liberated by a photon is verysmall, and most of these higher energy events are caused by the extraneousnoise sources such as cosmic rays at a rate of about two per square centimeterof cathode per minute. The shape of the distribution is not constant, but variesaccording to dynode voltages, temperature, wavelength, and cathode area.

10K.5b Effect of Weighting Events on the DQE

The primary theoretical difference between the two detection methods is intheir weighting of photomultiplier pulses. The DC method gives the averagecurrent from the photomultiplier tube; therefore, a large pulse gives more cur-rent and is weighted more heavily, even though it signifies the arrival of onlyone photon just as a smaller pulse does. Pulse-counting techniques treat allpulses equally.

A schematic form of a pulse height distribution is shown in Figure K.5,where n(x) is the number of detected incident photons which produce a pulseof height x.

S = Q n(x,) Ax, + Q n(x2) Ax2 +

or, because Ax, = Ax2 = Axj, and so forth,

(K.46)

s = ;r<?*(xr.)Ax./-i (K.47)


In integral form (Ax -» 0),

f OO

S = Q n(x) dx. (K.48)^o

Incorporating weighting factors, the equations become

S = H<X,) Q «(*,) Ax + w(x2) Q n(x2) Ax + - - - (K.49)N

S = £ w(*,) e«(x,) Ax (K.50)1-1

S = w(x) Qn(x) dx. (K.51)Jo

The noise for unweighted signals is just

N = VS. (K.52)

With weights, the noise equation is much more complicated. Because each binin Figure K.5 can be thought of as a separate event, the total noise is given byadding in quadrature:

/i(x,) Ax]1/2}2 + (w(X2)[Q «(*2)Ax]"2}2 + • • -]"2 (K.53)

In integral form,

N = >v(x)2 Qn(x) dx . (K.54)L -^o J

Therefore, the output S/N for weighted events is given by

f OO

\ Lr rc

IL Jo

(K.55)

From our definition of DQE, we have


DQE = = Qn(x) dx]

[J n ( x ) dx] Un(x)dx](K.56)

or

where

/ =

DQE = Qf

n(x) dx]n(x) dx] [$n(

(K.57)

(K.58)

So/is the factor that the unweighted DQE is modified by when weights areincluded. Two cases are involved:

1. Weights equal to one, as in pulse counting (w(x) = 1). Then

DQE = Q( I n ( x ) d x ]

Un(x)dx] [ $ n ( x ) d x ]

This reduces to

DQE = Q (K.59)

for an ideal photomultiplier.2. Weights proportional to the size of the pulse, as in DC photometry (w(x)

Then

DQE =Q[$xn(x)dx]

[ $ n ( x ) d x ](K.60)

Some assumptions need to be made about the pulse height distributionbefore this equation can be solved. Typical trials include n(x) — constantover a window, n(x) — e~x, or n(x) = Poisson distribution. These alllead to complicated reductions. Rather than show all three, we use the


window function as an example. The pulse height distribution can beapproximated by a box, i.e.,

n(x) = HO, a < x < bn(x) ~ 0, x < a or x > b,

as shown in Figure K.6. Then

DQE =

r rQ U,-J „

x dx

r r«o x2 dx «o dx

Qnl

= (2

b2- a-

b3 - a(b- a)

(b + a)2(b - a)2

+ ab + a2) (b - a) (b - a)

b2 + 2ab + a

If the lower limit a is set to zero, then

DQE = %Q

(K.61)

(K.62)

or the DQE for DC photometry is about 25 percent less than that forpulse counting.

Other functions and experimental results give

0.5 Q DQE < 0.8 Q


nn — A

/X

s

^T/ \/ \

/ \/ N

\\

Figure K.6. Box window function.

for DC work. That is, pulse counting gives a better DQE in all cases, by asmuch as a factor of two. This advantage is greatest for weak signals andbecomes much less for strong signals. In general, if the signal is larger than thenoise, DC and pulse-counting methods are roughly equivalent, with perhaps 20percent difference between the two techniques under good conditions. Becauseof a relatively narrow pulse height distribution, the 1P21 photomultiplier tubeis slightly better for DC work than other types, such as the Venetian blindtubes.

K.6 PRACTICAL PULSE-DC COMPARISON

In addition to the theoretical signal-to-noise comparison shown above, thereare several practical comparisons that must be made when deciding whichtechnique to use. It seems appropriate that, because we have discussed the the-oretical differences, we list these observational considerations.

1. Pulse counting is relatively insensitive to amplifier drifts.2. When high precision (less than 1 percent) is needed, pulse counting is

better unless voltage-to-frequency conversion techniques are used withthe DC amplifier.

3. When several measurements are to be added or a large number of obser-vations are to be reduced, the data are much easier to handle when indigital form, obtained directly with pulse counting.

4. A digital system has a symmetric and sharp filter function, whereas DCmethods have an RC filter, which tends to partially correlate readings,if a long time constant is used.


5. An analog system gives good real-time indication of sky conditions, dif-ficult to obtain from digital methods.

6. Pulse counting provides discrimination against dark current. However,at the same time, some primary photoelectrons must be rejected. A bet-ter method to reduce dark current is to cool the photomultiplier tube.

7. Pulse counting is insensitive to leakage currents, but more sensitive toRF fields, such as generated by motors and relays in the telescope'senvironment.

8. Pulse counting can be performed at higher speed, DC is limited by thepen or meter movement. This advantage is negated if the DC signal isrecorded on an instrumentation tape recorder and played back at aslower speed.

9. Pulse counting requires dead-time corrections that can be very signifi-cant for bright stars.

10. Complexity of instrumentation for the two methods is approximately thesame.

K.7 THEORETICAL S/N COMPARISON OF A PHOTODIODE AND APHOTOMULTIPLIER TUBE

This section is designed as an example of how two detectors can be comparedon paper by their S/N characteristics. This calculation is also intended to bea fairly realistic comparison to help potential photometer builders decidebetween a photodiode or a photomultiplier as a detector. We have adopted thenoise and quantum efficiency characteristics of the 1P21, and typical photo-diode characteristics from the EG & G Electro-Optics catalog.8

Until now, the S/N discussion has assumed a pulse-counting photometer.However, a photodiode cannot be used for pulse counting because it lacksinternal gain. The output of a photomultiplier consists of pulses, as seen inFigure 7.2, because each electron released at the photocathode is amplified bythe dynode string by a factor of 106. Each detected photon produces a pulsethat can be readily counted. Many of the noise sources within the photomul-tiplier produce smaller pulses at the output because they result from amplifi-cation by only part of the dynode string. It is easy to discriminate against muchof the noise.

This is not the case for the photodiode because each detected photon con-tributes just one electron to the output current. There is no strong burst ofelectrons at the output when a photon is detected. The current produced by thephotons looks identical to that produced by noise within the photodiode. Thiscomparison assumes that both detectors are used with excellent quality DCamplifiers.

There are some minor modifications to be made to the S/N treatment forthe DC case. Shot noise, whether for photons or electrical current, is a "white"


noise. This means it is equally strong at all frequencies. The amount of noisedepends on the bandpass of the amplifier. An amplifier that measures a fre-quency range from 1000 to 10,000 Hz detects much more noise than one thatspans 1000 to 2000 Hz. Any equation for shot noise must include the width ofthe amplifier bandpass. For instance, photon shot noise is written as

JV, = G(2QC.B)l/2 t (K.63)

where B is the bandwidth (Hz). For pulse-counting photometry, the bandwidthhas a very simple relationship with the integration time, t, namely,

» - £ . (K.64)

For DC photometry, this relation is

B = J- (K.65)4r

where T is the RC time constant of the amplifier. Note that if Equation K.64is substituted into Equation K.63, we get the same expression for photon shotnoise used earlier. Also note that Equations K.64 and K.65 imply that

t = 2r.

Photon shot noise for the DC case is given by

^. (K.66)

The noise sources that must be considered are the photon shot noise, shotnoise from the sky background, amplifier noise, and detector noise. We ignorenoise due to atmospheric scintillation because this is highly variable and a func-tion of observatory location. Then by Equation K.40, we have

(K.67)

where Ndel is the detector noise. For a photomultiplier, this is given by the shotnoise of the dark current. For the 1P2I at room temperature, Nda is about 2X 10~" B l /2ampsor 1.3 X 10" B1/2 electrons per second. If the tube is cooledto dry-ice temperature, Ndrl drops to 1.3 X 106 electrons per second. Persha9


has recommended that photodiodes be used in a photovoltaic mode for astro-nomical photometry. In this case, there is no dark current but there is a thermalnoise generated at the p-n junction given by

, -

where k is Boltzmann's constant, T is the temperature (Kelvin), and R is theshunt resistance (ohms) of the photodiode. This resistance is also a function oftemperature. According to the EG & G catalog, R approximately doubles foreach 5 Celsius degree temperature drop. The largest shunt resistance nowavailable is about 10* ohms at room temperature. The noise current for a pho-todiode is given by

/„ = 2.5 X 104\/B electrons/second (K.69)

at room temperature and

/„ = 41 3 \I~B electrons/second (K.70)

at dry-ice temperature.The amplifier noise for both detectors, referred to the amplifier input, is

assumed to be 2500 B]/1 electrons per second. This is about the best that canbe expected with present FET input amplifiers. We adopt an amplifier timeconstant of 1 second which fixes B at 0.25 Hz. The internal gain, G, for thephotomultiplier is about 10* while for the photodiode, which lacks internalamplification, U is 1.0.

Using these various parameters. Equation K.67 becomes for thephotomultiplier,

and for the photodiode,

^ 2gC

4r

(K.72)


The equations have been left in this form to illustrate how a high gain detectorminimizes the effects of Nam and Ndtt. This advantage over the photodiodebecomes evident in the calculations to follow.

The photon arrival rates, C, and Cb, are calculated for the B filter for variousapparent magnitudes using Equation 2.33 with approximate corrections for thetransmission of the atmosphere, telescope optics, and filter. A 20-centimeter(8-inch) diameter telescope is assumed. A sky background of twelfth magni-tude was established from the known background at a dark site and scaled toa 0.5-millimeter diaphragm. This is a typical size for an active area of a pho-todiode. The quantum efficiencies at 4400 A for the photomultiplier and pho-todiode are 0.10 and 0.60, respectively.

Figure K.7 shows the S/N, calculated from Equations K.71 and K.72 versusB magnitude for uncooled detectors. For very bright stars, the two detectorsgive comparable results. The reason for this can be seen in Equations K.71 andK.72. For very high photon rates, the S/N approaches a value of ((?O'/2-Because C, is the same for both detectors, the difference in S/N is set by theirquantum efficiencies. In the high photon rate limit, the S/N of the photodiodeshould exceed that of a photomultiplier by a ratio of (0.6/0. l) l / 2 or 2.5. For allbut the very brightest stars, the photomultiplier is clearly superior. This is aresult of the amplifier and detector noise terms becoming important at lowphoton rates. These terms are not nearly as important for the photomultiplierbecause of its high internal gain. This more than compensates for the lowerquantum efficiency of the photomultiplier.

The S/N curves become a little more meaningful if we consider an example.An 8.4 magnitude star produces a S/N of 100 with a photomultiplier tube andan amplifier time constant of 1 second. With this same time constant, this starwould only produce a S/N of 4.5 with a photodiode detector. Because theS/N improves with the square root of the observing time, the observation withthe photodiode would have to be 500 times the duration of the observationmade with the photomultiplier to obtain the same accuracy. Finally, if we con-sider a S/N of 1 to be the detection limit. Figure K.7 shows that the photo-multiplier can go almost four magnitudes fainter than the photodiode.

The situation improves if the photodiode is cooled to dry-ice temperature.The calculation proceeds as before except that the detector noise terms arereduced as discussed earlier. Figure K.8 shows an uncooled photomultipliercompared to a cooled photodiode. A cooled photodiode comes much closer tomatching the performance of an uncooled photomultiplier tube. While the pho-todiode still appears to be inferior the uncertainties in the calculation are suchthat it is safest to say that they are roughly comparable. Figure K.9 comparesdetectors when both are cooled to dry-ice temperature. The photomultiplier hasa superior S/N for stars fainter than fifth magnitude while the photodiode is


10000

1000

Uncooled detectors20 cm telescope

12m sky1 sec time constant

100 -

14 16

B-mag.

Figure K.7. S/N of an uncooled photodiode and an S-4 photomultiplier tube.


10000

1000

100

10

Uncooled S-4photomultiplier

10 12 14 16

Figure K.8. S/N of a cooled photodiode and an uncooled S-4 photomultiplier tube.


10000

1000

TOO

10

1 I I T \ I T

Cooled detectors

S-4photomultjplier

10 12 14 16

B-mag.

Figure K.9. S/N of a cooled photodiode and a cooled S-4 photomultiplier tube.


10000

1000 -

100 -

10 12 14 16

R-mag.

Figure K.IO. S/N of a cooled photodiode and a cooled S-1 photomultiplier tube.


better for stars brighter than this. The photomultiplier has a detection limitabout 3.5 magnitudes fainter than the photodiode.

Unlike the S-4 response of the 1P21, the photodiode has very high quantumefficiency in the near-infrared. The above calculation was repeated for the Rbandpass (7000 A) comparing the photodiode to an S-l photomultiplier. Bothdetectors were assumed to be cooled. The superior quantum efficiency of thephotodiode makes a significant difference in the infrared. An S-l surface hasa quantum efficiency of only 0.4 percent compared to the photodiode's 83 per-cent at 7000 A. Figure K.10 shows that this difference makes the photodiodea superior detector down to the eighth magnitude, despite the higher internalgain of the photomultiplier. The curves would look much the same in the Ibandpass (9000 A) because the quantum efficiencies are nearly the same.

The above calculations are intended only as a rough comparison of these twodetectors. The S/N curve for the photomultiplier has been checked at a fewpoints with observational data and has compared well. Unfortunately, theauthors have not had access to comparable empirical data for a photodiodephotometer. However, we suspect that the theoretical calculation is not in largeerror. Figures K.7, K.8, K.9, and K.10 can be used to draw some general con-clusions about detector selection. First, if you intend to use an uncooled detec-tor, the photomultiplier tube is clearly the better detector. If you can designand build a cooling system, the photodiode will be only slightly inferior to anuncooled photomultiplier. A cooled photodiode offers the convenience of a sin-gle detector with sensitivity from the ultraviolet to the infrared. However, ingeneral, the high internal gain of the photomultiplier makes it a superior detec-tor overall. Finally, it should be noted that the photomultiplier has a potentialS/N gain over the photodiode, which has not been included in these calcula-tions. Because of its high internal gain, the photomultiplier can be used to pho-ton count. By the results of Section K.5, this means an additional increase ofas much as a factor of \/2 in the S/N compared to the photodiode, which mustuse DC measuring techniques.

REFERENCES

1. Young, H. D. 1962. Statistical Treatment of Experimental Data. New York:McGraw-Hill.

2. Harnell, D. L. 1975. Introduction to Statistical Methods. 2d ed. Reading, Mass.:Addison-Wesley.

3. Bevington, P. R. 1969. Data Reduction and Error Analysis for the Physical Sci-ences. New York: McGraw-Hill.

4. Young, A. T. 1974. In Methods of Experimental Physics: Astrophysics, Edited byN. Carleton. New York: Academic Press, vol. 12A.


5. Golay, M. 1974. Introduction to Astronomical Photometry. Boston: D. Reidel.6. Meaburn, J. 1976. Detection and Spectrometry of Faint Light. Boston: D. Reidel.7. Arflten, G. 1970. Mathematical Methods for Physicists. New York: Academic

Press.8. EG & G Electro-Optics Division, 35 Congress St., Salem, Mass. 01970.9. Persha, G. 1980. IAPPP Com. 2, 11.

10. Meaburn, J. 1976. Detection and Spectrometry of Faint Light. Boston: D. Reidel.

INDEX

AAVSO 31, 32, 239, 273Aerosols 354Airglow 230Air mass 29, 53, 86-88, 324, 330Altitude 119-123American Astronomical Society (The)

273Amplifier (See Preamp)

DC 16, 188-193, 196-201meter 193noise 369, 371-372operational 185pulse 167-172

Aperture stops 83Asteroids 271Astronomical Almanac (The) 111,

113, 114, 119Astronomical Journal (The) 274Astronomical Society of the Pacific

(The) 274Astronomy and Astrophysics 274Astrophysical Journal (The) 274Atlas

photographic 204Atlas Stellarum 1950.0 204,

220Canterbury Sky Atlas 204ESO/SRC Atlas of the

Southern Sky 204Lick Observatory Sky Atlas

204National Geographic-Palomar

Observatory Sky Survey 204Photographic Star Atlas

(Falkau Atlas) 204True Visual Magnitude

Photographic Star Atlas 204positional 203

Atlas Australis 203Atlas Borealis 203

Atlas Eclipticalis 203Banner Durchmustemng (BD)

117, 203, 208Cordoba Durchmusterung

(CD) 203, 208Smithsonian Astrophysical

Observatory Atlas 203, 208,298

Atmosphere 28Atmospheric refraction 104-108Attenuation, methods of 83-85Aurora 231Azimuth 119-123

BBackground

noise 369-371sky 10

Balmerdiscontinuity 36, 40, 45, 56limit 40, 56lines 45

Batteries 149Binary stars (See Eclipsing binary

systems)Blackbody 45, 46, 349

radiator 349-351Bond, W.C- 6Brown, F.C- 7

Calculators 101Calibration

absolute 50-52DC amplifier 197diaphragm 227process 26

Catalog of Suspected Variables 248

389

390 IND6X

Cellsphotoconductive 7photoelectric 7

Cepheids (See Variable Stars)Chart Recorders 8, 9, 194Check Star 24, 209Chromatic aberration 12CHU 109Clock, sideral rate 112Cold box 133-134

suppliers 134Color-color diagram 45Color excess 46Color index 27-28, 57Colors, instrumental 25, 85

standard 92Comparison stars 23, 53

advantages 207-208selection 208use 209

Computers, 101-104, 178, 247programs 335-341

cartesian coordinates 341coordinate precession 337dead-time correction 336Julian date 336linear regression method 338,

339-340sideral time 340

Construction, electronic 147Coordinates, precession 116-119, 337Current

constant sources 196input bias 190leakage 132

nDark current 15-16, 133Date

Heliocentric Julian 113-116Julian 112,336

Daysideral 110solar 108

Dead time 81-85, 308, 328, 336calculation of 308-310correction of 82

De Veny, J.B. 161Depletion region 20Detent positioning 146Detective quantum efficiency 367-370

Diaphragm 9, 84, 138-141background removal 226calibration 227selection 220-228

Diode, PIN 18-28, 378-386Direct current (DC) methods 16

measurements 213-216pulse counting comparison

372-378Discriminator (See preamp)Dispersion 64Distribution

Gaussian 66, 358Normal 66Poisson 360probability 358

Dwarf Cepheids (See Variable Stars)Dynode 14

EEclipse, primary 256Eclipsing binary systems 2

Algol 2579 Lyrae 257-258Cataclysmic Systems 263observing 256-270RS CVn 259W UMa 259

Eggen, O.J. 254Eggen Paper Series 254, 256Einstein, A. 13Electron multiplier tube (See

photomultiplier tube)Electronics

DC 184-201pulse 167-183suppliers 148

Emissionfield 14secondary 14thermionic 14

Epoch 264Equation

normal 70of condition 69

Errorillegitimate 60probable 66progration 361-363random 61standard 71

INDEX 391

Error (cont.)standard deviation 64systematic 61

Extinctionatmospheric 28, 228correction 23, 86-91, 325, 331first order 28, 88-90, 311-319

stars 279-285second order 28, 90-91, 320-321

stars 286-289theory 351-355

Extrapolation 77

H

Hall, D. 259Harvard spectral classification 41Henry Draper Catalog 38, 164, 208Hertz, H. 13Hertzsprung-Russell (H-R) diagram 2,

41, 45main sequence matching 44-45

High voltage power supply 149-155Hiitner, W. 157Hipparchus 5H/3 photometry 57

Fabry lens 10, 23, 125-127Faint sources 218-220Filters

neutral density 84slide 136UBV 35, 134-136wheel 137

Finding charts 202-207preparation 205published 206

AAVSO Variable Star Atlas206

Atlas of Finding Charts ofVariable Stars 206

Atlas Stellarum Variabilium206

Charts for Southern Variables206

Odessa Charts (The) 206Sonneberg Charts (The) 206

Field emission 14Flare Stars 240-244Flip mirror 127, 141Flux 25, 344-349F-ratio 12Frequency counter 167

Gain table 199, 322-324Galaxies 272, 347Galvanometer 7, 8General Catalog of Variable Stars

(GCVS) 206, 207, 248Goodness of fit 71Guthnick, p. 8

IAPPP 31IC4665 302-304Infrared 55International System 7, 34, 36Interpolation 73-76

exact 74smoothed 76

Intensity 342-348Interstellar

absorption 46-50reddening 46-50

Johnson, H.L. 34, 36, 290-296,297-307

Johnson noise 190Johnson standard list 94, 290-304Journal of the AAVSO 274Journal of the Royal Astronomical

Society of Canada 274Julian Date 112, 336

Heliocentric 113-116

KKeenan, P.O. 38, 41Kellrnan, E. 41Kephart, J.E. 178Kron, G.E. 8

Least squareslinear 68-73, 338, 339muitivariate 363-366

392 INDEX

Lightnight sky 229-231zodiacal 229

Light elements 264Light radiation

flux 5, 344-348intensity 342-348luminosity 349

Line blanketing 55Luminosity 349

criteria 41

M

Magnitudedifference 5,24extra atmospheric 86instrumental 27, 85, 356photographic 6photovisual 6standard 29, 92

Mean, sample 61Median, sample 63Metals 40Meter, amplifier 193Minchin, G.M. 7Minimum light 263, 266Mira (See variable stars)Mirrors, coating 12M-K spectral classification 34, 38—42

luminosity classes 41Moffat, A.F.J. 99Monthly Notices of the Astronomical

Society of South Africa 274Monthly Notices of the Royal

Astronomical Society 274Morgan, W.W. 34, 36, 41, 44, 290Mount Wilson Observatory 7, 41Multiple Linear Regression 363-366

N

National Geographic Palomar SkySurvey 204, 240

Night airglow 230Noise 366

amplifier 369, 371-372background 369, 371-372scintillation 368, 371shot 368sources 368-370thermal 190

North Polar Sequence 7, 8, 305-307Novae 263

oObserving

application 238-275diaphragm selection 224faint sources 218-220first night 231-236optimizing time 212

Occultation photometry 245-247

Period 264Photocathode 14

materials 16-18opaque 17semitransparent 17

Photodiodes PIN 18-23compared to photomultiplier

tube 378-386Photodiode Photometer 18-23

advantages 21-22Photoelectric cell 7-8Photoelectric effect 13—14Photographic magnitudes (See

magnitude)Photography 6Photometer Head 9, 124-148, 157-164

Steps for constructingAdjustable High-Voltage

Supplies 149-155Diaphragms 138-141Electronic Circuitry 147-148Optical Layout 124-128

Specialized designschopping 159-161dual-beam 161-162multifilter 163-164single beam 157-159

Photometric sequences 238-240Photometric Systems 33—59Photometry

all sky 94cluster 94differential 23, 52-54, 95-98,

216-218extragalactic 272occultation 245photographic 6-7

INDEX 393

Photometry (cont.)references 30visual 6

Photomultiplier tube (PMT) 8, 11,13-18, 128-134

compared to a photodiode378-386

1P21 8, 14-15, 16, 35, 36,128-133, 372

931A 8, 129cathode types 16-18dynode 14EMI 6256 128EMI 9789 129FW-118 129FW-130 129end-on 17fatigue 132R869 129RCA 7102 129housing 133PIN comparison 378-386selection 128-129suppliers 128, 129

Photon counting (See Pulse Counting)Photon 13Planck's law 349-350Planets 272Pleiades 298-300Pogson, N.R. 5-6Pogson scale 5, 7Point source 220Praesepe 298, 301-302Preamp 167-172

suppliers 170Precession 116-119, 337Probability distribution

Gaussian 358-360poisson 358, 360

Ptolemy, Claudius 5Publication of data 275Publications of the Astronomical

Society of the Pacific 274Pulse counter (See also Electronics,

Pulse)design 173-181microprocessor 178-181suppliers 172-173

Pulse countingDC comparison 215, 372-378measurements 210-213operation 16, 182

Pulse generator 181-182

Pulse height distribution 169, 372

Quantum efficiency 14, 18, 21, 367detective 367-368

R

Red leak 17Reddening 46Reference light sources 155-157Refraction

atmospheric 104calculation 104differential 107-108effect on air mass 106

Regression analysis 68Rejection of data 66-68RF oscillator power supply 153Roberts, G.L. 163Rosenberg, H. 8RR Lyrae variables (See variable stars)RV Tauri stars (See variable stars)

SAO (See Atlas, positional)Satellites 271Scaliger, J.J. 112Schultz, W.F. 8Secondary emission 14Selenium photoconductive cells 7Semiconductor 19Seven Sisters (See Pleiades)Signal-to-noise (S/N) ratio 77, 210,

366-372Solar System objects, observing

270-272Standard deviation 64Standard stars 33, 290-304Standard system

transforming 29, 92, 355-357Star

DC measurements 213profile 223pulse measurements 210

Statistics 60-78, 358-386sources 78

Stebbins, J. 7, 8Stefan's law 350-351

394 INDEX

Strip-chart recorder (See Chartrecorders)

Stromgren 4-color system 55-57Subroutines, FORTRAN 335-341Sun, rectangular coordinates 114, 341

Taylor, D.J. 170, 172Telescope

alignment 11Cassegrain 12focal point 12F-ratio 12Newtonian 12optical system 12refracting 12Schmidt-Cassegrain 12selection 11

Thermionic emission 14, 16Time

sideral 110-112, 340solar 108of minimum light 263-270universal 109

Transformation coefficients 93DC example 322-327pulse example 327-334theory 355-357use of star clusters 297

Twilight emission lines 230

Variable starscataclysmic 263cepheids 3, 249, 250dwarf cepheids 249intrinsic 248-256Mira 254RR Lyrae 249RV Tauri 250semiregular 254UV Ceti 2416 Scuti 249

Variance 65Vogt, N. 99Voltage-to-frequency converters 195

wWhitford, A.E. 8Wien's displacement law 350

Zero point constants 38, 91

uU-B problem 98-101UBV

filters 35IR extension 55secondary standards list 291standard cluster stars 297 -304standard field stars 290-296system 34transformation coefficients 38, 93transformation equations 37-38.

93zero point constants 38, 91

Software for AstronomicalPhotometry

The authors of this book have written software intended to aid in the re-duction of astronomical photoelectric photometry data from either a pulsecounting or DC photometer system. This software is composed of a suite of7 major programs, 4 utility programs and 6 standard star files. The majorprograms are:

1. DATA provides a means for entering the raw observational data andcreating an output file for use by other programs.

2. STARLIST creates a file containing information about the observedstars which is used by several other programs.

3. INSTRU uses the files created by DATA and STARLIST to make an outputfile containing the heliocentric Julian date, universal time, air massand instrumental magnitude for each filter of each star.

4. DIFF is for reducing differential photometry using an instrumentalmagnitude file created by INSTRU.

5. EXTINC a program that allows the user to select from four differentmethods of extinction coefficient determination.

6. COEFF computes the transformation coefficients to the standard pho-tometric system.

7. CONVERT uses the coefficients in the TRAN.DAT file to convert the in-strumental magnitudes found in the input file to standard magnitudesand colors.

The utility programs are:

1. JULDATE computes the Julian date from a given UT date or optionallycomputes the heliocentric time correction if the object's coordinatesare entered

2. PRECESS will precess the coordinates of a star between any two epochs.

3. DTIME computes the dead time correction to the count rate enteredat the keyboard. The dead time coefficient can be read from thePARAM.DAT file or it can be entered from the keyboard.

4. SIDTIME computes the local sidereal time from the UT date and time.The observer's longitude can be read from either the PARAM.DAT fileor the keyboard.

The standard star files, from the appendices of Astronomical Photometry,in the STARLIST format are:

1. PLEIADES.LSI contains the UBV data for the Pleiades cluster.

2. M44.LST contains the UBV data for the Praesepe cluster (M44).

3. IC4665. LSI contains the UBV data for the cluster IC4665.

4. 1DRDER.LST is the northern hemisphere first-order extinction tablefrom Appendix A

5. 20RDER.LST is the second-order extinction table from Appendix B

6. JOHNSON. LSI is a list of all of the Johnson UBV standards

7. LANDDLT.LST contains the 223 Landolt UBVRI Equatorial Standards.

All of these programs are designed to run on an IBM PC or compatiblesystem, with or without a math co-processor chip. If a co-processor ispresent the computations will be greatly accelerated. The programs requireno more than 128K of memory to run.

The output from most of the programs is in the form of an ASCII diskfile. These files are read by the other programs and can be typed to thescreen or printed by the user to record the various stages of the reductionprocess. Because these files are ASCII in form, they can be edited with atext editor if so desired. This is sometimes helpful to correct an input errorthat affected only one star in the file.

The programs are written in FORTRAN 77 and both source code andexecutable (compiled) programs are provided.

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