Ay 122 - Fall 2006

Imaging and Photometry

(Many slides today c/o

Mike Bolte, UCSC)

Imaging and Photometry

• Now essentially always done with imaging arrays (e.g.,CCDs); it used to be with single-channel instruments

• Two basic purposes:

1. Flux measurements (photometry)• Aperture photometry or S/N-like weighting• For unresolved sources: PSF fitting• Could be time-resolved (e.g., for variability)• Could involve polarimetry• Panoramic imaging especially useful if the surface

density of sources is high

2. Morphology and structures

• Surface photometry or other parametrizations

What Properties of Electromagnetic

Radiation Can We Measure?

• Specific flux = Intensity (in ergs or photons) per unitarea (or solid angle), time, wavelength (or frequency),e.g., f! = 10-15 erg/cm2/s/Å - a good spectroscopic unit

• It is usually integrated over some finite bandpass (as inphotometry) or a spectral resolution element or a line

• It can be distributed on the sky (surface photometry,e.g., galaxies), or changing in time (variable sources)

• You can also measure the polarization parameters(photometry ! polarimetry, spectroscopy ! spectro-polarimetry); common in radio astronomy

(From P. Armitage)

Real detectors are sensitive over a finite range of ! (or ").

Fluxes are always measured over some finite bandpass.

!

F = F" (")d"#Total energy flux: Integral of f" over

all frequencies

Units: erg s-1 cm-2 Hz-1

A standard unit for specific flux (initially in radio, but now

more common):

!

1 Jansky (Jy) =10"23 erg s-1 cm-2 Hz-1

f" is often called the flux density - to get the power, one

integrates it over the bandwith, and multiplies by the area

Measuring Flux = Energy/(unit time)/(unit area)

Fluxes and Magnitudes

(From P. Armitage)

For historical reasons, fluxes in the optical and IR are

measured in magnitudes:

!

m = "2.5log10 F + constant

If F is the total flux, then m is the bolometric magnitude.

Usually instead consider a finite bandpass, e.g., V band.

f!

e.g. in the V band (! ~ 550 nm):

!

mV

= "2.5log10 F + constant

flux integrated over the range

of wavelengths for this band!

Some Common

Photometric

Systems

(in the visible)

Johnson !

Gunn/SDSS!

There areway, waytoo manyphotometricsystems outthere …

(Bandpass curves

from Fukugita et

al. 1995, PASP,

107, 945)

… and more …… and more …… and more …

(From P. Armitage)

Using Magnitudes

Consider two stars, one of which is a hundred times

fainter than the other in some waveband (say V).

!

m1 = "2.5logF1 + constant

m2 = "2.5log(0.01F1) + constant

= "2.5log(0.01) " 2.5logF1 + constant

= 5 " 2.5logF1 + constant

= 5 + m1

Source that is 100 times fainter in flux is five magnitudes

fainter (larger number).

Faintest objects detectable with HST have magnitudes of

~ 28 in R/I bands. The sun has mV = -26.75 mag

Magnitude Zero Points

f!

!

Vega =# Lyrae

Alas, for the standard UBVRIJKL…system (and many others) the

magnitude zero-point in any bandis determined by the spectrum

of Vega ! const!

U B V R I

Vega calibration (m = 0): at ! = 5556: f! = 3.39 !10 -9 erg/cm2/s/Å

f" = 3.50 !10 -20 erg/cm2/s/Hz

N! = 948 photons/cm2/s/ÅA more logical system is AB"

magnitudes: AB" = -2.5 log f" [cgs] - 48.60

Photometric Zero-Points (Visible)

(From Fukugita et al. 1995)

Magnitudes, A Formal Definition

e.g., Because Vega(= # Lyrae) isdeclared to be

the zero-point!(at least for the

UBV… system)

Defining

effective

wavelengths

(and the

corresponding

bandpass

averaged

fluxes)

The Infrared Photometric Bands… where the atmospheric transmission windows are

Infrared Bandpasses Infrared Bandpasses

IR Sky Backgrounds

I

108

105

102

1µ 10µ 100µ2µ

!-4 Rayleigh scattered sunlight

273K blackbody

thermal IR

(atmosphere, telescopes)

OH can be resolved

Zodiacal lightatmosphere

IR Sky Backgrounds

Colors From Magnitudes

(From P. Armitage)

The color of an object is defined as the difference in the

magnitude in each of two bandpasses: e.g. the (B-V)

color is: B-V = mB-mV

Stars radiate roughly as

blackbodies, so the color

reflects surface temperature.

Vega has T = 9500 K, by

definition color is zero.

Apparent vs. Absolute Magnitudes

(From P. Armitage)

The absolute magnitude is defined as the apparent mag.

a source would have if it were at a distance of 10 pc:

It is a measure of the luminosity in some waveband.

For Sun: M"B = 5.47, M"V = 4.82, M"bol = 4.74

Difference between the

apparent magnitude m and

the absolute magnitude M

(any band) is a measure of

the distance to the source

!

m "M = 5log10

d

10 pc

#

$ %

&

' (

Distance modulus

M = m + 5 - 5 log d/pc

Why Do We Need

This Mess?

An example: the Color-

Magnitude DiagramThe quantitativeoperational frameworkfor studies of stellarphysics, evolution,populations, distances …

Relative measurements

are generally easier and

more robust than the

absolute ones; and often

that is enough.

The Concept of Signal-to-Noise (S/N)or: How good is that measurement really?

• S/N = signal/error (If the noise is Gaussian, wespeak of 3-$, 5- $, … detections. This translates into aprobability that the detection is spurious.)

• For a counting process (e.g., photons), error = " n, andthus S/N = n / " n = " n (“Poissonian noise”). This isthe minimum possible error; there may be other sourcesof error (e.g., from the detector itself)

• If a source is seen against some back(fore)ground, then

$2 total = $2 signal + $2 background + $2 other

Signal-to-Noise (S/N)

• Signal=R*• t time

detected rate in e-/second

• Consider the case where we count all the detected e- in a circular aperture with radius r.

I

sky

r

r

• Noise Sources:

!

R* " t # shot noise from source

Rsky " t " $r2 # shot noise from sky in aperture

RN2" $r2 # readout noise in aperture

RN2 + (0.5 % gain)2[ ] " $r2 # more general RN

Dark " t " $r2 # shot noise in dark current in aperture

R* = e&/sec from the source

Rsky = e&/sec/pixel from the sky

RN = read noise (as if RN2 e& had been detected)

Dark = e_ /second/pixel

S/N for object measured in aperture with radius r: npix=# ofpixels in the aperture= #r2

!

R*t

R* " t + Rsky " t " npix + RN +gain

2

#

$ %

&

' ( 2

" npix +Dark " t " npix)

* +

,

- .

1

2

Signal

Noise

!

(R*" t)

2

All the noise terms added in quadratureNote: always calculate in e-

Noise from sky e- in aperture

Noise from the darkcurrent in aperture

Readnoise in aperture

Side Issue: S/N %mag

!

m ± "(m) = co# 2.5log(S ± N)

= co# 2.5log S 1± N

S( )[ ]= c

o# 2.5log(S) # 2.5log(1± N

S)

%mm

!

"(m) # 2.5log(1+ 1S /N)

=2.5

2.3N

S$ 12(NS)2 + 1

3(NS)3 $ ...[ ]

#1.087(NS)

Note:in log +/- notsymetric

This is the basis of people referring to +/- 0.02mag error as “2%”

Fractional error

Sky BackgroundSignal from the sky background ispresent in every pixel of theaperture. Because each instrumentgenerally has a different pixel scale,the sky brightness is usuallytabulated for a site in units ofmag/arcsecond2.

˝(mag/ )

19.219.920.019.517.014

19.520.320.720.718.510

19.720.621.421.619.97

19.920.821.722.421.53

19.920.921.822.722.00

IRVBULunarage

(days)

NaD OH

Hg

!

Scale""/pix (LRIS - R : 0.218"/pix)

Area of 1 pixel = (Scale)2 (LRIS#R : 0.0475"2

this is the ratio of flux/pix to flux/"

In magnitudes :

Ipix = I"Scale2 I" Intensity (e-/sec)

#2.5log(Ipix ) = #2.5[log(I") + log(Scale2)]

mpix = m"# 2.5log(Scale2) (for LRIS - R : add 3.303mag)

and

Rsky (mpix ) = R(m = 20) $10(0.4#m pix )

Example, LRIS in the R - band :

Rsky =1890 $100.4(20#24.21)= 39.1 e- /pix /sec

R sky = 6.35e- /pix /sec % RN in just 1 second

S/N - some limiting cases. Let’s assume CCD with Dark=0,well sampled read noise.

!

R*t

R* " t + Rsky " t " npix + RN( )2" npix[ ]

1

2

Bright Sources: (R*t)1/2 dominates noise term

!

S/N "R*t

R*t

= R*t # t

1

2

Sky Limited

!

( Rskyt > 3"RN) : S/N#R*t

npixRskyt# t

Note: seeing comes in with npix term

What is ignored in this S/N eqn?

• Bias level/structure correction

• Flat-fielding errors

• Charge Transfer Efficiency (CTE)0.99999/pixel transfer

• Non-linearity when approaching full well

• Scale changes in focal plane

• A zillion other potential problems

Photometry With An Imaging Array

• Aperture Photometry: some modern ref’s– DaCosta, 1992, ASP Conf Ser 23– Stetson, 1987, PASP, 99, 191– Stetson, 1990, PASP, 102, 932

Sum counts in allpixels in aperture

Determine skyin annulus,subtract offsky/pixel incentral aperture

Aperture Photometry

!

I = Iijij

" # npix $ sky/pixel

Total counts inaperture from source

Counts in each pixel in aperture

Number of pixels in aperture

m = c0 - 2.5log(I)

Aperture Photometry

• What do you need?– Source center

– Sky value

– Aperture radius

Centers

• The usual approach is to use ``marginal sums’’.

!

"x i = Iijj

# : Sum along columns

j

i

Marginal Sums

• With noise and multiple sources you have todecide what is a source and to isolate sources.

• Find peaks: use $ &x/$ x zeros

• Isolate peaks: use ``symmetry cleaning’’1. Find peak

2. compare pairs of points equidistant fromcenter

3. if Ileft >> Iright, set Ileft = Iright

• Finding centers: Intensity-weightedcentroid

!

xcenter

=

"x i

xi

i

#

"x i

i

#; $ 2

=

"ix

i

2

i

#

"i

i

#% x

i

2

• Alternative for centers: Gaussian fit to &:

• DAOPHOT FIND algorithm uses marginalsums in subrasters, symmetry cleaning,reraster and Gaussian fit.

!

"i = background+ h•e# (x i #x c )/$( )

2/ 2[ ]

Height of peakSolving for center

Sky

• To determine the sky, typically use a localannulus, evaluate the distribution of countsin pixels in a way to reject the bias towardhigher-than-background values.

• Remember the 3 Ms.

Counts

#of

pixe

ls

Mode (peak of this histogram)

Arithmetic mean

Median (1/2 above, 1/2 below)

Because essentially all deviations from the sky arepositive counts (stars and galaxies), the mode is thebest approximation to the sky.

Sky From Minmax Rejection

Pixel value: 1000 200 180 180

Average with minmax rejection, reject 2 highest value averaginglowest two will give the sky value. NOTE! Must normalizeframes to common mean or mode before combining! Sometimesit is necessary to pre-clean the frames before combining.

Aperture size and growth curves

• First, it is VERY hard to measure the total light assome light is scattered to very large radius.

Radial intensity distributionfor a bright, isolated star.

Radius from center in pixels

Inner/outer sky radii

Perhaps you have most ofthe light within this radius

• Radial intensity distribution for a faint star:

Same frames as previous example

Bright star aperture

The wings of afaint star are lostto sky noise at adifferent radiusthan the wings ofa bright star.

Radial profile with neighbors

Neighbors OK insky annulus(mode), troublein star apertures

One Approach Is To Use “Growth Curves”

• Idea is to use a small aperture (highest S againstbackground and smaller chance of contamination) foreverything and determine a correction to larger radiibased on several relatively isolates, relatively brightstars in a frame.

• Note! This assumes a linear response so that all pointsources have the same fraction of light within a givenradius.

• Howell, 1989, PASP, 101, 616

• Stetson, 1990, PASP, 102, 932

Growth Curves

0.000.020.0570.0940.1840.3240.430cMean

0.120.300.050.100.190.330.41Star#6

0.190.210.190.090.180.320.42Star#5

0.140.120.140.220.180.330.44Star#4

-0.010.020.060.100.180.320.43Star#3

0.040.110.210.080.190.330.42Star#2

0.000.020.050.090.170.310.43Star#1

8765432Aperture

$mag forapertures n-1, n

Sum of these is the total aperture correction to beadded to magnitude measured in aperture 1

Aperture Radius

$m

ag[a

per(

n+1)

- a

per(

n)]

5 10 15 20 25 30

-0.15

-0.10

-0.05

0

0.05Aperture Photometry Summary

1. Identify brightness peaks

!

Ixy

xy

" # (sky•aperture area)

!

2.

Use small aperture

3. Add in ``aperture correction’’determined from bright, isolated stars

Easy, fast, works well except for the case of overlapping images

Crowded-field Photometry

• As was assumed for aperture corrections, all pointsources have the same PSF (linear detector)

• PSF fitting allows for an optimal S/N weighting• Various codes have been written that do:

1. Automatic star finding2. Construction of PSF3. Fitting of PSF to (multiple) stars

• Many programs exist: DAOPHOT, ROMAPHOT,DOPHOT, STARMAN, …

• DAOPHOT is perhaps the most useful one: Stetson,1987, PASP, 99, 191

• To construct a PSF:

1. Choose unsaturated, relatively isolated stars

2. If PSF varies over the frame, sample the fullfield

3. Make 1st iteration of the PSF

4. Subtract psf-star neighbors

5. Make another PSF

• PSF can be represented either as a table ofnumbers, or as a fitting function (e.g., aGaussian + power-law or exponentialwings, etc.), or as a combination

Photometric Calibration

• The photometric standard systems have tended to be zero-pointed arbitrarily. Vega is the most widely used and wasoriginal defined with V= 0 and all colors = 0.

• Hayes & Latham (1975, ApJ, 197, 587) put the Vega scaleon an absolute scale.

• The AB scale (Oke, 1974, ApJS, 27, 21) is a physical-unit-based scale with:

m(AB) = -2.5log(f) - 48.60 where f is monochromatic flux is in units of

erg/sec/cm2/Hz. Objects with constant flux/unit frequencyinterval have zero color on this scale

Photometric calibration

1. Instrumental magnitudes

!

m = c0 " 2.5log(I • t)

= c0 " 2.5log(I) " 2.5log(t)

Counts/sec

minstrumental

Photometric Calibration

• To convert to a standard magnitude youneed to observe some standard stars andsolve for the constants in an equation like:

!

minst =M + c0 + c1X + c2(color)+ c3(UT) +c4 (color)2

+ ...

Stnd mag

Instr mag

zpt

Extinction coeff(mag/airmass)

airmass Color term

• Extinction coefficients:– Increase with decreasing wavelength

– Can vary by 50% over time and by some amount duringa night

– Are measured by observing standards at a range ofairmass during the night

Slope of thisline is c1

Photometric Standards

• Landolt (1992, AJ, 104, 336)• Stetson (2000, PASP, 112, 995)• Fields containing several well measured stars of

similar brightness and a big range in color. Theblue stars are the hard ones to find and severalfields are center on PG sources.

• Measure the fields over at least the the airmassrange of your program objects and interspersestandard field observations throughout the night.

Photometric Calibration: Standard Stars

Magnitudes of Vega (or othersystems primary flux standards) aretransferred to many other, secondarystandards. They are observed alongwith your main science targets, andprocessed in the same way.

(Bedin, Anderson,

King, Piotto, 2001)

NGC 6397

M4

(King, Anderson,

Cool, Piotto, 1999)

We often need to compare

observations with models,

on the same photometric

system

Low

metallicity

Intermediate

metallicity

Always, always transform

models to observational

system, e.g., by integrating

model spectra through your

bandpasses

Alas, Even The “Same” Photometric

Systems Are Seldom Really The Same … … From mismatches between the effective bandpasses ofyour filter system and those of the standard system.Objects with different spectral shapes have differentoffsets:

This Generates Color Terms …

A photometric system is thus effectively (operationally)defined by a set of standard stars - since the actualbandpasses may not be well known.

Surface Photometry

Simple approach of aperturephotometry works OK for somepurposes.

mag=c0 - 2.5(cntsaper- #r2sky)

Typically working with muchlarger apertures - prone to contamination - sky determination even morecritical - often want to know more thantotal brightness

• The way to quantify the 2-dimensional distribution oflight, e.g., in galaxies

• Many references, e.g.,– Davis et al., AJ, 90, 1985

– Jedrzejewski, MNRAS, 226, 747, 1987

• Could fit (or find) isophotes, and the most commonprocedure is to fit elliptical isophotes.

• Isophotal parameters are: surface brightness itself (µ),xcenter, ycenter, ellipticity ()), position angle (PA), theenclosed magnitude (m), and sometimes higher ordershape terms, all as functions of radius (r) or semi-majoraxis (a).

Surface Photometry

Start with guesses for xc, yc,R, ) and p.a., then comparethe ellipse with real data allalong the ellipse (all ' values)

Iellipse- Iimage

'

0 Good isophote

'

fitted isophotetrue isophote

(I

'

Fit the (I - ' plot and iterate on xc, yc, p.a., and ) to minimizethe coefficients in an expression like:

I(')= I0 + A1sin(') + B1cos(') + A2sin(2') + B2cos(2')

Changes to xc and yc mostly affect A1, B1, p.a. “ “ A2 ) “ “ B2

More specifically,

iterate the following:

!

"(major axis center) =#B1

$ I

"(minor axis center) =#A1(1#%)

$ I

"(%) =#2B2(1#%)

a0$ I

"(p.a) =2A2(1#%)

a0$ I [(1#%)2

#1]

where :

$ I =&I

&R a0

• After finding the best-fitting ellipticalisophotes, the residuals are ofteninteresting. Fit:

I = I0 + Ansin(n') + Bncos(n')

already minimized n=1 and n=2, n=3is usually not significant, but:

B4 is negative for “boxy” isophotes

B4 positive for “disky” isophotes

! Major axis surf. brightness profiles

Isophotal shape and orientation prof’s"

Examples of Surface

Photometry of Ellipticals

One can also measure the deviationsfrom the elliptical shape (boxy/disky)

Disky and Boxy Elliptical Isophotes Calculate mean and RMS pixel intensity forannulus, toss any values above mean + nRMS

From your isophotal fits, you can then construct the best2-dimensional elliptical model for the light distribution

… And subtract it from the image to reveal anydeviations from the assumed elliptical symmetry

Saturated core

Hidden disk

Non elliptical structuresBad job of clipping

Panoramic Imaging

• Generally used to study populations of sources (e.g.,faint galaxy counts; star clusters; etc.)

• Commonly done in (wide-area) surveys

• Image (pixels) ! catalogs of objects and theirmeasured properties

• If done properly, essentially all information isextracted into a more useful form; but not always…

• Key steps:1. Object finding (there is always some spatial “filter”)2. Object measurements / parametrization3. Object classification (e.g., star/galaxy)

Thresholding is an alternative to peak finding. Look forcontiguous pixels above a threshold value.

• User sets area, threshold value.• Sometimes combine with a smoothing filter

Deblending based on multiple-pass thresholding

Image (De-)Blending and Segmentation

How are the individual sources -or source components - defined?

Easy Not so easy

• One very common program for panoramic photometry

is Sextractor– Bertin & Arnouts, 1996, A&AS, 117, 393

• Not for good a detailed surface photometry, but goodfor classification and rough photometric and structuralparameter derivation for large fields.

Steps:

1. Background map (skydetermination)

2. Identification of objects(thresholding)

3. Deblending

4. Photometry

5. Shape analysis

SextractorFlowchart

Star-Galaxy separation

• Galaxies are resolved, stars are not

• All methods use various approaches to comparingthe amount of light at large and small radii.

I

Pixel position

Star

galaxy

m1-

m2

m1

stars

galaxies too noisy

Star-Galaxy Separation

• Important, since for most studies you want either stars(or quasars), or galaxies; and then the depth to which areliable classiffication can be done is the effectivelimiting depth of your catalog - not the detection depth– There is generally more to measure for a non-PSF object

• You’d like to have an automated and objective process,with some estimate of the accuracy as a f (mag)

– Generally classification fails at the faint end

• Most methods use some measures of light concentrationvs. magnitude (perhaps more than one), and/or somemeasure of the PSF fit quality (e.g., *2)

• For more advanced approaches, use some machinelearning method, e.g., neural nets or decision trees

Sextractor Star/GalaxySeparation

• Lots of talk about neural-net algorithms, but in the end it isa moment analysis.

• “Stellarity”. Typically test it with artificial stars and find itis very good to some limiting magnitude.

s-g going bad at R~22

Typical Parameter Space for S/G Classif.

Stellar locus#

Galaxies

(From DPOSS)

More S/G Classification Parameter

Spaces: Normalized By The Stellar Locus

Then a set of such parameters can be fed into an automated classifier(ANN, DT, …) which can be trained with a “ground truth” sample

Automated Star-Galaxy Classification:

Artificial Neural Nets (ANN)

Input:

variousimageshapeparameters.

Output:

Star, p(s)

Galaxy, p(g)

Other, p(o)

(Odewahn et al. 1992)

Automated Star-Galaxy Classification:

Decision Trees (DTs)

(Weir et al. 1995)

Automated Star-Galaxy Classification:

Unsupervised Classifiers

No training data set - the program decides on the number of classespresent in the data, and partitions the data set accordingly.

Star

Star+fuzz

Gal1 (E?)

Gal2 (Sp?)

An example:AutoClass(Cheeseman et al.)

Uses Bayesianapproach inmachinelearning (ML).

This applicationfrom DPOSS(Weir et al. 1995)

Seeing - A Key Issue• The “image size”, given by a convolution of the atmospheric

turbulence, and instrument optics• Affects two important aspects of imaging photometry:

1. S/N: because a larger image spreads the available signalover more noise-contributing pixels

• Defines the detection limit• Should have pixel scale matched to the expected image quality,

to at least Nyquist sampling

2. Morphological resolution: as high spatial frequencyinformation is lost

• Defines the star-galaxy classification limit• Can be recovered only partly by image deconvolutions, which

require a good S/N, sampling, and a well measured PSF

• A tricky issue is how to combine data obtained in differentseeing conditions

Exampleof seeingvariationsin theground-based data(from thePQ survey)

Good seeing

Mediocre seeing

Summary of the Key Points

• Photometry = flux measurement over a finite bandpass,could be integral (the entire object) or resolved (surfacephotometry)

• The arcana of the magnitudes and many differentphotometric systems … !

• The S/N computation - many sources of noise• Issues in the photometry with an imaging array: object

finding and centering, sky determination, aperturephotometry, PSF fitting, calibrations …

• Surface photometry and isophote fitting lore …• Star-galaxy classification in panoramic imaging• The effects of the seeing: flux measurements and

morphology