This dissertation has been

microfUmed exactly as received 69-16,660

NAJITA, Kazutoshi, 1925-ENHANCED BLACK BODY RADIATION AS AGENERATING MECHANISM FOR WI-IITE LIGHTSOLAR FLARES.

University of Hawaii, Ph.D., 1969Astronomy'

University Microfilms, Inc., Ann Arbor, Michigan

ENHANCED BLACK BODY RADIATION

AS A GENERATING MECHANISM FOR WHITE LIGHT SOLAR FLARES

A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THEUNIVERSITY OF HAWAII IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN PHYSICS

JANUARY 1969

By

Kazutoshi Najita

Dissertation Committee:

Frank Q. Orrall, ChairmanJohn T. Jefferies

John R. HolmesHoward C. McAllister

Victor J. Stenger

ACKNOWLEDGMENT

The author acknowledges with pleasure the special

assistance of the following individuals:

J. A. Van Allen, University of Iowa;

T. L. Cline, S. S. Holt, and E. W. Hones, Jr., NASA;

R. G. Mann, Haleakala Observatory;

Mount Wilson and Palomar Observatories;

A. Maxwell, Harvard Radio Astronomy Observatory;

J. W. Evans, Sacramento Peak Observatory;

P. E. Tallant, Sacramento Peak Observatory;

H. L. DeMastus and R. R. Stover, Sacramento Peak

Observatory; and

T. Takakura, Tokyo Astronomical Observatory.

ABSTRACT

A white light flare is a rare impulsive event observed

in the visible continuum radiating from localized regions

during the early explosive stage of a few solar flares. A

self-consistent model is developed which attempts to explain

the white light flare in terms of an enhanced local black

body radiation due to a temperature perturbation at about

optical depth unity in the photosphere.

Solar cosmic ray and radio observations indicate that

energetic protons and electrons are generated during the

early phase of a solar proton flare. The model assumes a

blast of energetic electrons and protons in equal numbers in

the 10 to 1000 MeV range, incident on the photosphere from

above, releasing most of its energy to the ambient gas at

about the depth one sees the normal photospheric continuum

radiation. This is interpreted by the study in terms of a

temperature perturbation of the layer and a reradiation of

the energy from an optically thin medium with a radiative

relaxation time of several seconds and a radiation temperature

of several hundred degrees above the normal.

For the purpose of analysis, an inverse power law energy

distribution for the energetic particles is assumed, and the

analysis is applied to the May 23, 1967 white light flare

event. The number of energetic particles required to account

for the enhanced continuum radiation by this mechanism is set

between 10 31 and 10 32• No other process seems to account as

v

efficiently for the observed emission except possibly the

synchrotron process, but then only with the most favorable

geometry. Since the Planck function at 6000 0 K predicts

negligible radio and x-radiation, the model allows for the

generation of these radiations by other mechanisms, for

example, the synchrotron and bremsstrahlung mechanisms,

respectively. Assuming that the bremsstrahlung mechanism is

responsible for the hard x-radiation, the inverse power law

distribution is truncated at the low energy end of the spec-

trum. The final result is a differential energy spectrum,

dNdE = 4 x 10 32 ,

= 4 X 1032E-s/3,

0.50 MeV < E < 1 MeV

1 MeV < E < 1 BeV,

which would account for the radiations during the impulsive

phase of the May 23, 1967 event. Although the predictions of

the model agree well with the observations, new observations,

particularly of white light polarization, are needed to test

the model.

TABLE OF CONTENTS

ACKNOWLEDGMENT iii

ABSTRACT iv

LIST OF TABLES viii

LIST OF ILLUSTRATIONS ix

CHAPTER I. INTRODUCTION 1

CHAPTER II. WHITE LIGHT FLARE OBSERVATIONSClassification of Visible Continuum

Radiations 8Visual Descriptions and Occurrences

of White Light Flares 11Observations of the White Light Flare

o f May 23, 1967 15Summary on Observations and Conditions

Imposed on a White Light Flare Model 18

CHAPTER III. COMPARISON OF WHITE LIGHT FLAREGENERATING MECHANISMS

Introduction 20Review of Selected Mechanisms 20Synchrotron Radiation 24

74

6468

CHAPTER V.

CHAPTER IV. PROPOSED MODEL FOR WHITE LIGHT FLARESIntroduction 36Heating of the Photosphere by Beam

of Energetic Particles 38Temperature Perturbation of Finite

Photospheric Layer 48Enhanced Radiation from Thermally

Perturbed Finite Atmosphere 52Example: The White Light Flare of

May 23, 1967 58

OTHER MANIFESTATIONSX-radiationMicrowave BurstEnergy Distribution for Energetic

Particles

CHAPTER VI. SUMMARY AND CONCLUSIONS 76

APPENDIX A. FLARES WITH CONTINUUM OPTICALRADIATION AND ASSOCIATED EVENTS 78

B. OBSERVATIONS ASSOCIATED WITH THEMAY 23, 1967 WHITE LIGHT FLARE 83

BIBLIOGRAPHY

vii

C. RADIATION DUE TO SELECTED MECHANISMS1. Radiation Energy Loss Rates

Per Electron in SolarMagnetop1asma 102

2. Inverse Compton Effect 1033. Br~msstrah1ung Radiation 1044. Synchrotron Radiation 112

D. DIVERGENCE OF ENERGY FOR ENERGETICPARTICLES1. Model Atmosphere 1332. Divergence of Energy for

Energetic Protons in SolarAtmosphere 134

3. Divergence of Energy forEnergetic Electrons in SolarAtmosphere 139

144

Properties of Microwave Bursts 69

Required Energetic Particle Flux 73

Flares with Continuum Optical Radiationand Associated Events 78

TABLE 4.1.

TABLE 4.2.

TABLE 5.1.

TABLE 5.2.

TABLE 5.3

TABLE A-1.

TABLE C-3.1.

TABLE C-4.1.

TABLE C-4.2.

TABLE D-1.1.

LIST OF TABLES

Relationships in the Inverse Power LawDistributions

Requirements for the May 23, 1967 WhiteLight Flare Event

Comparison of the 1835 and the 1936 UTEvents

Values for L(x,V) and J(X,v)

The Functions F (x) and Fp (x)

The Functions G(x) and Gp (x) for X = 5/3

Model Atmosphere

48

63

65

110

121

128

133

FIGURE 2.1.

FIGURE 2.2.

FIGURE 2.3.

FIGURE 3.1.

FIGURE 3.2.

FIGURE 4.1.

FIGURE 4.2.

FIGURE 4.3.

FIGURE 4.4.

FIGURE 4.5.

FIGURE 4.6.

FIGURE 5.1.

FIGURE B-1.

FIGURE B-2.

FIGURE B-3.

FIGURE B-3a.

LIST OF ILLUSTRATIONS

Sketch of White Light Flare ofSept. 1, 1859

White Light Flare of March 23, 1958

Occurrences of White Light Flares inRelation to the Sunspot Cycle

Peak Flux for Radio Bursts During theMay 23, 1967 Solar Flares

Differential Energy Spectra of SelectedCosmic Ray Events

Energy Loss for Protons with InversePower Law Distributions

Energy Loss for Electrons with InversePower Law Distributions

Combined Energy Loss by Equal Numbersof Electrons and Protons

Radiative Relaxation Times for theLower Chromosphere and the UpperPhotosphere

Square Wave Temperature Perturbation

Combined Energy Loss by Equal Numbersof Electrons and Protons

A Truncated Inverse Power LawDistribution for Energetic Electronsin the May 23, 1967 Event

Calcium Plage Reports for McMath­Hulbert Plage No. 8818

Longitudinal Sunspot Magnetic Fieldfor McMath Plage No. 8818 (1967)

The Sunspot Groups in Plage 8818Obtained by Mount Wilson Observatory,May 22, 1967

The Sunspot Groups in Plage 8818Obtained by Mount Wilson Observatory,May 24, 1967

12

12

14

29

33

45

46

47

55

57

62

75

84

86

87

88

x

FIGURE B-4. Sunspot Configuration and MagneticDensity for May 23, 1967 89

FIGURE B-5. Sequence of Ha Flare Patrol Photographsfor May 23, 1967 90

FIGURE B-6. Ha Flare Patrol Photographs for theMay 23, 1967 White Light Flare 91

FIGURE B-7. Ha Flare Patrol Photographs for theMay 23, 1967 White Light Flare 92

oFIGURE B-8. Sunspot Patrol Centered at 5800A for

the May 23, 1967 White Light Flare 93

FIGURE B-9. Videometer Tracings for the May 23,1967 Flare 94

FIGURE B-lO. Sketch of Intensity Curve for WhiteLight Flare of May 23, 1967 95

FIGURE B-ll. Solar X-rays> 80 keV for May 23, 1967 95o

FIGURE B-12. Soft X-rays 2-l2A for May 23, 1967 96

FIGURE B-13. 8800 MHz Burst for May 23, 1967 97

FIGURE B-14. The Radio Burst Associated with theSolar Proton Flare of May 23, 1967 98

FIGURE B-15. Details of the 2800 MHz Solar RadioBursts of May 23, 1967 99

FIGURE B-16. Dynamic Spectral Record of the May 23,1967 White Light Flare 100

FIGURE C-l. Radiation Energy Loss Rates perElectron in Solar Magnetoplasma 102

FIGURE C-2.l. Inverse Compton Radiation by Mono-Energetic Electrons 103

FIGURE C-2.2. Inverse Compton Radiation by Electronswith Inverse Power Law Distribution 103

FIGURE C-3.l. Non-Thermal Bremsstrahlung Due toSingle Electron 111

FIGURE C-3.2. Non-Thermal Bremsstrahlung Due toElectrons with Inverse Power LawDistribution 111

FIGURE C-4.la. Angular Relationship for ElectronFollowing a Helical Orbit in aUniform Magnetic Field H

FIGURE C-4.lb. The Reference Frames for theRelativistic Electrons and theStationary Observer

FIGURE C-4.2. The Functions F(x), Fp(X) and theDegree of Polarization, TI

FIGURE C-4.3. The Functions G(x) and Gp(x) forX = 5/3

FIGURE C-4.4. Volume Emissivity nv(~), Calculatedfor 10 2 Gauss

FIGURE C-4.5. Volume Emissivity nv(~)' Calculatedfor 10 3 Gauss

FIGURE C-4.6. Volume Emissivity nv(~)' Calculatedfor 10 4 Gauss

FIGURE C-4.7. Volume Emissivity nv(~)' Calculatedfor 10 3 Gauss

FIGURE D-2.l. Energy Loss for Energetic Protons

FIGURE D-2.2. Energy Loss for Energetic Protons

FIGURE D-3.l. Energy Loss for Energetic Electrons

FIGURE D-3.2. Energy Loss for Energetic Electrons

xi

115

116

122

128

129

129

130

130

137

138

142

143

CHAPTER I

INTRODUCTION

A white light flare is a sudden brightening of a local­

ized region in the optical continuum brighter than the disk

radiation during the very early stages of a few solar flares.

It is usually described as having a bluish-white or white

color (as in A or 0 type stars). It is usually observed

within several minutes after the onset of the Ha flare and

lasts for about 2 to 10 minutes. It is my objective to de-

velop a self-consistent model which can account for the ob­

servations of this phenomenon and is consistent with the

various associated solar flare manifestations.

Although the first solar flare was observed independently

in integrated light by Carrington (1859) and Hodgson (1859)

in 1859, there has been until recently very little interest

in white light flares. Interest in this rather rare, short­

lived phenomenon, believed to be observed in the solar chro­

mosphere, has gained momentum as it is thought that these

flares may provide valuable clues to the solar flare problem

inasmuch as they occur during the very early stages of the

solar flare event.

Several observations of white light flares were reported

during the last solar cycle maximum (Angle 1961; Cragg 1959;

McNarry 1960; Nagasawa, et al. 1961; Notuki, et al. 1956;

Waldmeier 1958). Although this type of flare is, generally

associated with large proton flares (importance 3 or 3+),

2

there have been cases of its association with flares esti-

mated to be only of importance 1 (Cragg 1959).

The solar activity is again on its rising phase, and in-

creased efforts are being made to obtain observational infor-

mation on white light flares. Instruments for photographing

these events have been developed at Sacramento Peak Observa-

tory by Dr. Frank Q. Orra11 (1968) and flare events are

regularly photographed at 5800 A, in a region of the solar

spectrum where there are no lines. Dr. Orra11 has also de-

ve10ped and put into operation on Ha1eaka1a Observatory

similar instruments for photographing these events. His

method affords a possible route in determining the po1ariza-

tion as well as the total intensity. The first event for

this solar maximum period was recorded on May 23, 1967 by

observers at Sacramento Peak Observatory (DeMastus and Stover

1968). Observations on this flare will be presented in some

detail in Chapter II.

In 1963 Stein and Ney,(1963) considered the various

possible mechanisms for the generation of continuous emission

in the visible spectrum. These included (a) scattering by

free electrons, (b) free-bound (recombination) transitions

and electrons captured by hydrogen (H- ions), (c) Cerenkov

radiation, (d) bremsstrahlung or free-free transitions,

(e) synchrotron radiation, (f) black body radiation.

They conclude that synchrotron radiation could simu1ta-

neous1y explain the continuum radiations in the visible and

3

the centimeter radio ranges if one assumed an exponential

rigidity spectrum with rigidity Ro ~ 200 MV for the electrons

orbiting in a magnetic field of about 500 gauss for conditions

at the peak of the synchrotron radiation. Such an assumption

is probably not correct inasmuch as the radio spectrum and

the optical continuum may originate at quite different heights

in the solar atmosphere. For example, if the visible and the

microwave continuous radiation were generated simultaneously

in the same location in the low chromosphere, the index of

refraction for the centimeter radio waves may be imaginary

and the atmosphere above would be opaque to the microwave

radiation, resulting in attenuation or even complete reflec­

tion. On the other hand, the atmosphere above is transparent

to the white light. The authors have not accounted for this

opacity of the atmosphere to the radio wave in their normal­

ization process. There are several other observational objec­

tions to the synchrotron hypothesis which will be pointed out

later.

Stein and Ney (1963) reject all of the other mechanisms,

in some cases using rather inadequate arguments. In partic­

ular, they reject black body radiation as a possible mechanism

for the white light flares on the basis that this type of

flare occurs in the chromosphere where the number density of

emitting and absorbing atoms and ions is too low. Although

some limb sprays have been seen in the optical continuum ex­

tending thousands of kilometers above the limb at about the

4

brightness of the disk (Kiepenheuer and Kuenzer 1958), the

evidence has not been persuasive to indicate that the contin­

uum radiations associated with the sprays are the same as

those for white light flares observed on the disk.

There are evidences of energetic particles being re­

leased at some altitude above the photosphere. For example,

Wild, et ale (1963), have hypothesized the radio radiation

over the entire frequency range and types of radiations,

x-rays, and cosmic rays by supposing some form of explosive

release of energetic particles in the lower corona. They

reason that if there were high energy particles already

trapped in the magnetic field in the corona, synchrotron

radiation in the radio range would be observed long before

the flash phase. However, this precludes neither the exist­

ence of moderately energetic electrons trapped high in the

corona where the magnetic field is small, nor moderately

energetic protons at relatively low altitudes and strong

magnetic fields. During the flash phase, some kind of accel­

eration process takes place to increase the energy of the low

to moderately energetic particles to very high energies. This

acceleration process may be associated with the opening of

the magnetic bottle. If the magnetic bottle partially opens

in the direction of the photosphere, these high energy par­

ticles could be dumped into the lower chromosphere, and

possibly even into the photosphere, generating the continuum

radiation there by such processes as bremsstrahlung and local

5

black body radiation after thermalization of the high energy

particles. This process could explain a white light flare

for even a moderately small flare in Ha .

On the other hand, if the magnetic bottle does not open

in the direction of the photosphere, then the particles may

be further accelerated to very high energies and generate

white light by the synchrotron mechanism, according 'to the

proposal presented by Gordon (1954) and by Stein and Ney (1963).

A decade ago Mustel (1957) suggested that raising the

temperature of the photosphere by about 200 0 could account

for the sudden brightening in the continuum. His idea that

the flare penetrates into the photosphere from above was re­

jected by Svestka (1966) because there are no observational

evidences for (a) any correlated motions at the photospheric

surface and (b) any measurable changes in the profiles of the

photospheric metal lines (Severny and Hoklova 1959). If the

radiation indeed were to corne from the photosphere, then it

would be necessary to assume that a temperature increase

occurs inside the photosphere without any direct penetration

of parts of the flaring chromospheric region in the form of a

well into the photosphere.

This study proposes a detailed model of a white light

flare self-consistent with the observations of (1) the mag­

netic field structure of the region of the flare; (2) the

continuous radiation in the x-ray, the visible, and the

radio spectra; (3) temperature; (4) electron density;

6

(5) particle energy spectrum; and (6) other parameters deter­

mined by established methods. It is suggested that the white

light flare is due to an enhanced black body radiation from

the upper layers of the photosphere. Energetic particles are

assumed to be released during the flash phase either in the

corona or the chromosphere. They penetrate into the photo­

sphere where they give up most of their kinetic energies.

This energy increases the internal energy of the gas, which

can be described in terms of a temperature perturbation of

certain regions of the photosphere. The perturbed gas relaxes

by radiation according to the Planck law at a local black body

temperature. For this enhanced continuum radiation to be

visible on the disk, the temperature perturbation should

occur about the T = 1.0 layer.

For analytical purposes, electrons and protons with in­

verse power law distribution in energy will be assumed. The

height distribution of the energy loss will be determined and

the resulting enhanced emergent radiation will be compared

with the data for the May 23, 1967 flare event. An estimate

of the most efficient power law distribution for generating

the white light flare will be given. It will be shown that

the calculated particle distribution and the energetic parti­

cle flux required by our model do predict peak intensities

consistent with those of observed microwave and x-ray bursts

during the flash phase.

Since the white light is observed during the very early

7

stages of the flare, it is believed that the results of this

study would establish better estimates of conditions that

exist immediately before the flash phase. Hopefully, such

results may prove useful in the ultimate solution of the

solar flare problem, particularly, the determination of the

source and mechanism for the release of energy in a solar

flare. This latter aspect, however, is not considered within

the scope of this study and will not be discussed further.

This study is organized in six chapters with reference

matter presented in four appendices. Chapter II sets the

background in providing a summary and classification of the

observations of visible continuum radiation. Data on the

visible and non-optical observations for the May 23, 1967

white light flare event are relegated to Appendix B. The

possible mechanisms suggested in the past for generating a

white light flare are compared in Chapter III and the details

of selected theories are given in Appendix C. The enhanced

photospheric radiation model proposed in this thesis is

developed in Chapter IV. Associated non-thermal x-ray and

microwave bursts are discussed in Chapter V. Finally, an

evaluation of the model, along with a discussion of additional

aspects of the problem, are presented in Chapter VI.

CHAPTER II

WHITE LIGHT FLARE OBSERVATIONS

Classification of Visible Con"tinuum Obs'erva"tions

While there may be many physical causes for the observed

continuous spectra of flares, from the observational point of

view, Svestka (1966) suggests that they may be classified

into three basic types. A brief description of the classi­

fication will be given below.

Type I: This type of continuous emission is associated

with the brightest regions of large intense solar flares ob-

served in an emission line such as H~. It has a very intense,

short-lived white or bluish white color, similar to the normal

photospheric light. It is highly impulsive, rising to its

maximum very rapidly and decaying rapidly.

Type II: This type of continuous emission, occurring in

grain-like areas is visible in the flare spectrum for a rea-

sonable length of time during the flare development. The

pattern of the continuous emission grains need not coincide

with that of the line emission. These emission grains are

not limited to the great flares but may occur in active

plages without any line emission. Although inconclusive, theo

intensity is estimated to attain a maximum at about 4000 A

and decreases with increasing wavelength. The excess inten-

sity is similar to that in an A-type star with a temperature

of about 10~oK (Svestka 1966). These continuous emission

threads are sometimes observed over the limb. Since these

9

emissions show no center-to-limb variation, they are believed

to be optically thin in the visible spectral range. These

continuous emission threads are sometimes observed over the

limb.

Type III: This type of continuous emission occurs in

flares observed above the limb. It has a weak continuum

below the Balmer limit. Such emissions with absolute inten­

sity of about 0.2 percent of the continuum are of the same

intensity as the scattered photospheric light and would not

be visible if projected on the disk.

Most solar flares associated with continuum radiation in

the visible range reported are listed in Table A-I. In this

table, those events indicated by an * are considered to

belong to the Type I continuous emission. They are, generally

associated with large flares of importance approximately 3 or

greater. For the more recent cases where solar cosmic ray

proton and radio data have been recorded, these flares are

usually associated with energetic protons that produce

ground-level events (GLE) and Type IV radio emission. The

other events listed were either measured in an optical region

of narrow width and hence are not conclusive, or information

reported was insufficient for classification.

One of the most extensive searches for white light flares

was carried out by J. M. Beckers (1962) from July to October

1959. Observations were carried out through the use of a

green filter with a band width of 500 Acentered at 4900 A.

10

The lower detection limit was set at an increased emission

of 0.5 percent of the normal solar background emission.

Beckers studied four limb flares, 41 1- flares, 20 1 flares,

and six 2 flares. He could account for only one continuous

emission coincident with and one not coincident with an Ha

flare. Thus, he reasoned, the continuous emission of solar

flares of class 2 or smaller at 4900 Ais generally smaller

than 10- 22 ergs/ster-cm2 -ang-sec.

Intensive photographic observations of loop prominences,

even at the early stages, have never disclosed continuous

radiations as bright as those seen in white light disk

flares (Jefferies and Orrall 1961). Although Menzel (1961)

proposed that the white light flare could be a bright loop

prominence, to the author's knowledge, no one has analyzed

this model.

After an extensive study of the line and continuum of

60 different flares, Michard (1959) concluded (1) that the

center to limb variation for the continuum is similar to

those of faculae, (2) that flares seen outside the limb are

generally related to the phase following the flash of disk

flares and (3) their emission comes from higher layers and

has different characteristics.

Although it is not clear how a white light flare ob­

served on the solar disk would appear if it were seen at the

limb, we will assume that the various co~tinuous emissions

reported may be separated into the three classifications

11

indicated above. Furthermore, for the purpose of this study,

attention will be restricted to those continuous radiations

classified as Type I and designate them white light flares.

Visual Description"s" and Occurre"n'c'esof White Light Flares

Permanent photographic records have been made only in

the most recent white light flare events. Previous observa­

tions were limited usually to sketches and descriptions of

the observer's impressions of the events. As a consequence,

our intuition of what to expect of these events has been

guided to a great extent by these personal sketches and de-

scriptions. This seems inevitable since these events are so

rare and transitory that one would have to carry on a patient

and extended observational study in order to make any kind of

average or statistical interpretation.

Koyama and Murayama (Nagasawa, et ale 1961) at the

National Science Museum of Tokyo describe (in rather familiar

language) the naked eye account of the white light flare of

November 15, 1960 as follows:

"A part of a large sunspot of type F suddenlybrightened in pearly white colour at 02h2lm andagain at 02h23.5m at an adjacent part. The dura­tions were both less than a minute. II

Although the outline of the continuum radiation region

has no fixed form, it is (a) coincident with structures sep-

arating sunspots of possibly opposite polarity and (b) located

primarily outside the penumb~a of bipolar or complex sunspots.

These concepts are illustrated in Figures 2.1 and 2.2 and

Figure B-8 in Appendix B. Becker (1958) locates the white

12

FIGURE 2.1. SKETCH OF WHITE LIGHT FLARE OF SEPT. 1, 1859.A and B were observed at 1118 G.T. C and D were observed at1123 G.T. Distance between the two pairs of spots is ~ 56,000km (from R. C. Carrington, 1859).

(ll)

FIGUlill 2.2. mUTE LIGHT FLARE OF HARCH 23, 1958.The white light flare, which is shown in cross-hatch, is for.1004 U.T. (a) Anacapri-Beobachtungen and (b) Schauins1and­Beobachtungen (from U. Becker, 1958).

13

spots outside the penumbra as exactly below the strongest

intensity in Ha . A similar impression is given by comparing

the Ha flare patrol and sunspot patrol photographs of the

May 23, 1967 flare (see Appendix B, Figures B-6, 7 and 8).

The color of these flares is, as a rule, described as

white, bluish-white or intense white. However, there has

been an exception where it was reported as magenta (duMartheray

1948). It is not clear at this point whether this observation

should be discarded or whether any model must indeed account

for this variation.

As white light flares are primarily associated with

solar flares of large importance, their occurrence correlate

very well with the sunspot cycle and the number of proton

events. It is noted that there have been no white light

flares reported during the sunspot minimum years. Indeed,

the major events have tended to appear during the rising and

declining phases of the sunspot cycle. These relationships

are presented in Figure 2.3.

Dobbie (Dobbie, et a1.1938) compared the ionospheric

effect (radio fadeout) of the March 31, 1938 white light

flare event with that of the Ha flare event of April 13, 1938.

The April 13 event was more extended than the former but had

less ionospheric effect. He remarks that "the long wave

records reveal the highly significant fact that the eruption

of March 31, of higher peak intensity, had the greater effect

on the ionosphere. Thus in this instance peak intensity of a

200 ----------.----

14

-------_._----------...,

'"d 0 150<lJZ

..c:

.jJ.jJ 100o 0o P.lE; UlU) ~ 50

:JU)

o1850 1860 1870 1880 1890 1900 1910 1920 1930 1940

(a) Early

. 2500z.jJ 2000C-l.lUl~ 150:JU)

'"d 100<lJ

..c:

.jJ500

0S

(J)0

observations indicated by vertical bars.

-----------_._--

Year 1942 44 46 .48\ 50\52 t 56 58 60 l62 64 66 68 70

-Sea-level

o ---IF 0Neutron 2 0 1 2 0 6 2 0 1!1onitorEvents * ,--. --- 1---- --PCA Events* 10 4 2 6 27 20 10 5 5

- -W.L. Events 1 0 2 1 1 1 0 2 6 10 0 0 0 2

--* The sunspot cycle and proton events are from J!'ichtel and

McDonald, 1967.

(b) Recent observations compared with sunspot cycle andsolar proton events.

FIGURE 2.3. OCCURRENCES 0::<' \'iI-IITE LIGH'l' lcLAHESIN RELATION TO THE SUNSPOT CYCLE.

15

solar eruption appears to be a more important characteristic

than great area. II

Aside from Dobbie's study of the ionospheric effect,

there appears not to have been any specific study correlating

x-ray with white light flare observations, although radio

fadeouts have been known to occur during solar flares. It is

essential to establish some understanding of the nature of

x-rays associated with the various types of flares in order

to obtain a self-consistent picture of a solar flare event.

Such studies of x-radiation and white light flares have prob­

ably not been carried out since x-ray observations have been

systematically undertaken only in recent years.

Radio frequency radiations from the sun have been exten­

sively studied and are reported by Kundu in IISo l ar Radio

Astronomy" (1965). He indicates that there is strong corre­

lation between microwave impulsive bursts with x-rays present

in the very early stages of some solar flares.

Observations of the White Light Flare of May 23, 1967

The May 23, 1967 event is probably the first white light

flare event for which there have been rather complete non­

optical observations. These data are presented in Appendix B

in form suited to the analysis undertaken here. Some of the

crucial points concerning various manifestations are pointed

out in this section.

(a) Calcium plage. There is a close geometrical rela­

tionship between the Ca II emission and the magnetic field

16

(Leighton 1959). There is also close relationship among the

photospheric facula and appearance of sunspots, and the radio

and x-ray plages (Christiansen, et ale 1960). Thus, the cal­

cium report shows the centers of activity and the evolution

of these centers. The evolution of the McMath plage number

8818, in which the white light flare appeared, is given in

Appendix B, Figure B-1. It is believed that the differentially

rotating plasma below the photospheric level drags the magnetic

field along. The logjam effect in the calcium plages may

imply that the magnetic field in the solar atmosphere is

strained and distorted.

(b) Sunspot magnetic field. During the time that the

McMath plage number 8818 made its transit across the solar

disk, there were several great flares on May 21, 23, and 28.

After each major flare, there were noticeable increase,

grouping, and dispersion of the longitudinal sunspot magnetic

field. The Mount Wilson sunspot magnetic field map before

and after the May 23 flare shows a large amount of regrouping

and extension of the entire spot group.

For this solar cycle, the leading pole in the northern

hemisphere is of south polarity. Hence, for investigative

purposes, the sunspots were grouped into possible bipolar

pairs based on the history of the calcium plage groups from

which number 8818 was formed and the leading polarity of the

magnetic field.

(c) Ha and white light flare observations. According

17

to the sequence of photographs of the solar flare in Ha taken

at Haleakala, the flare tends to occur in pre-existing plage

areas along parallel ribbons on either side of the line of

polarity reversal. Even photographs made much earlier than

1815 UT also show these same structures. The Sacramento Peak

Observatory photographs of the event in white light and the

Ha flare in the lower chromosphere show that these points

again coincide with the bay-like area outside the penumbra.

The white light appears most intense in the bay-like region

and does not appear to penetrate into the penumbra.

(d) Electromagnetic radiations. The white light flareo

was photographed with a 500 A wide filter centered at

A5800 1 by H. L. DeMastus and R. R. Stover (1968) at

Sacramento Peak Observatory. The right white spot in Figure

B-8 increased its intensity from a pre-flare value of 0.90 to

a flare maximum of 1.06 of the center of the solar disk. The

left white spot increased from 0.91 to 1.05 of the solar disk.

The beginning, maximum and ending times for the electro-

magnetic radiations are accurately known. Although accurate

intensity measurements as functions of time have not been

determined, using the crude information available, output

curves have been plotted in Figure B-lO and B-9 for the white

light and Ha flares, respectively.

The soft and hard x-rays and the various radio bursts

are given in Figures B-ll through B-15. A photograph of the

swept frequency observations in the decameter to decimeter

18

range is included in Figure B-16. This photograph indicates

that a very strong Type III radio burst occurred about 1838

UTe The rapid rise in the white light, the Ha , the x-ray and

~~he centimeter bursts are strongly time-correlated with this

Type III burst at 1838 UTe

Summary on Observation's 'and COhdi'ti'oh'S' 'Imposed 'oha White

Light Flare Model

Listed below are some of the observational conditions

imposed on a white light flare model.

(a) The increased radiation in white light are very

restricted and localized.

(b) The increased radiation in white light are found

just outside the penumbra, in bipolar or complex sunspots.

(c) White light flares appear to be rare events. Evi­

dently rather special circumstances are needed to produce

them, otherwise we should observe them more frequently.

(d) Some large proton events and impUlsive x-ray flares

do not correlate with white light events, while some small

flares (importance 1) are visible in white light.

(e) The flares are usually described as white or bluish­

white, or as the color of A or 0 type stars. Thus, the mech­

anism should give a spectrum with intensity increasing with

decreasing wavelength in the optical range.

(f) The duration of the continuum radiation is usually

between 2 to 10 minutes. The intensity rises to its peak

value very rapidly and seems to decay more slowly.

19

(g) The peak intensity may vary from several percent to

20 or 30 percent of the photospheric intensity at the center

of the disk.

(h) The white light flare starts about 3 minutes after

the onset of the Ha flare.

(i) The peak of the white light flare does not coincide

with the peak of the Ha flare. The peak of the Ha intensity

lags behind the peak intensity for the white light.

(j) The white light and the centimeter burst starting

and peak times coincide. The decay is also similar.

(k) The x-ray, white light and microwave radiations are

time-correlated with the group of Type III decimeter, meter

and decameter bursts which started shortly after 1837 UT as

shown by the swept spectral radio data from Fort Davis.

During the period from 1838 to 1840 UT when the white light

flare is increasing to its peak intensity, there are no Type

IV bursts.

While the observational conditions (a) to (h) listed

above apply to white light flares in general, the conditions

(i) to (k) are observations associated particularly with the

May 23, 1967 event.

CHAPTER III

COMPARISON OF WHITE LIGHT FLARE GENERATING MECHANISMS

Introducti'on

Although numerous studies on the various mechanisms for

generating continuous electromagnetic radiation over the en­

tire spectral range from the radio to the x-rays have been

reported, there have been relatively few attempts at explain­

ing the visible continuum radiation of the white light flares.

Following the last solar flare maximum, there have been sev­

eral attempts and proposals explaining the source of this

visible continuum radiation. A brief discussion and compari­

son of these mechanisms will be presented in this chapter.

The first systematic study of the continuum radiation

was carried out by Stein and Ney (1963). In it they list

the possible continuum radiation mechanisms (see Chapter I).

They conclude that synchrotron radiation is the only promising

mechanism from the entire list. Since this mechanism is one

of the foremost hypotheses explaining the white light flare,

it will be discussed in detail in the last section of this

chapter. A general review of a selected list of possible

mechanisms is presented in the next section. Those processes

not included here are given in sufficient detail by Stein and

Ney (1963) and Svestka (1966).

Review of Sele'c't'e'd Me'chani'sms

Korchak (1965a) takes issue with Stein and Ney (1963)

for eliminating the inverse Compton scattering by relativistic

21

electrons as a possible mechanism for explaining the white

light flare. He compares the rate of energy losses by syn­

chrotron and bremsstrahlung radiation with the energy losses

by inverse Compton scattering. Korchak reasons that the

mechanism should not be ignored if the process were to take

place in regions of very low magnetic field and low particle

densities. This is illustrated in Figure C-l.

The power loss per Hz for the inverse Compton scattering

for relativistic electrons with mono-energetic and power law

distributions are derived by Korchak (1965a, 1967). The

results are plotted in Figures C-2.l and 2.2, respectively.

The radiation output is several orders of magnitude too low

in the visible and the white light flare does not seem to be

a coronal phenomenon because the region over which the intense

light is observed is much less than the region of the intense

chromospheric Ha flare. Thus this mechanism appears to be

better suited to explain the x-radiation. However, let us

suppose that the radiation were indeed due to this process.

Figure C-2.l indicates that electrons with kinetic energy in

the 100 to 200 MeV range contribute most to the radiation in

the visible range. The electron energy distribution may be

approximated by an inverse power law at least over a short

energy interval in this range. Inverse Compton radiation

from such an electron distribution is, given in Figure C-2.2,

which shows that for any reasonable inverse power law distri­

bution, the predicted x-radiation would be of the same order

22

as that in the visible. This, however, is contrary to

reported observations.

The nontherma1 bremsstrahlung calculations for the mono­

energetic electrons and the electrons with power law spectral

distribution are illustrated in Figures C-3.1 and 3.2, respec­

tively (bremsstrahlung radiation is discussed in Appendix

C-3). Although the total energy radiated by this mechanism

increases greatly as the energy of the incident particles

increases, the radiation of photons with energies much lower

than the electron energy is not strongly dependent on the

electron energy. This mechanism depends primarily on the

number of scattering nuclei of the ambient gas. The result,

as presented in Figure C-3.2, may not be correct because the

integration is carried out to Ek = hv. It is not known

whether the power law distribution does apply to low energy

particles, while bremsstrahlung may still be efficient at low

energies and would contribute considerably to the low frequency

radiation. Since the spectrum is shaded to the red and does

not decrease too rapidly, this mechanism may have considerable

influence in the x-ray region, but not in the visible.

Recently, Svestka (1966) reviewed the various observa­

tions in the visible range during solar flares and the mech­

anisms for generating continuous radiation in the visible

range. He shows that the hydrogen free-bound transition

could explain the weak continuum observed above the limb dur­

ing some flares. This type has been studied by Jefferies and

23

Orrall (1961) and belongs to the Type III model presented in

Chapter II.

Svestka (1966) considers the H- formation in the flare

region as one of the most promising of all the possible

mechanisms. He shows that if the electron density ne~1013/cm3

and if the electron temperature is about 6500 oK, then the H-

process would generate radiation an order of magnitude larger

than the Balmer continuum. However, since the H- process

cannot produce a visible continuum if the electron tempera-

ture becomes greater than 7500 oK, one could reason that the

radiation must originate very low in the chromosphere. This

is borne out in observations, which note that the emission

°increases toward the blue with a maximum at about 4500 A for

flares close to the limb, but for flares near the center of

the disk the maximum shifts to longer wavelengths. There is

some contradiction as to the electron density in the chromo-

sphere. Jefferies and Orrall estimated densities for the

March 7, 1959 limb flare at least an order of magnitude lower

than Svestka's estimate, which is at least 1013/cm 3 in certain

regions of the chromosphere (Jefferies and Orrall 1961;

Svestka 1966).

According to Wien's law, the black body Planck function

is maximum for TAmax=O.290 em degree. At T=6000oK, the peak

°is at 4840 A. The black body radiation mechanism is attrac-

tive in that a slight increase in the photospheric temperature

could easily account for the often described "intense white"

24

or II shaded to the blue ll phenomenon of this type of flare.

It also contributes only negligibly to the radiation in the

radio and the x-ray, which may best be explained by possibly

more efficient processes in those frequency ranges. Stein

and Ney rejected this as a possible mechanism because they

assumed that the source of the white light flare was some­

where high in the chromosphere or low corona, and they could

not account for the high particle densities and opacities

required for observation of black body radiation (Svestka

1966). A simple energy consideration would also show that a

temperature increase of only about 200 0 K is required in order

to account for the observed increase in the continuum radia­

tion. The black body radiation model of the white light

flare will be considered in detail in the following chapter.

Synchrotron Radiat'i'on

Synchrotron radiation from relativistic electrons in

magnetic fields has been studied and detailed derivations

have been carried out by many investigators. The subject has

been reviewed many times within different contexts (Oort and

Walraven 1956; Thorne 1963; Westfold 1959; Ginzburg 1959;

Chang 1962; Takakura 1960, 1960a, 1967). Therefore, the

detailed analysis will not be repeated here. However, it

should be noted that there has been some confusion as to what

radiation a distant observer would receive of the radiation

emitted from electrons moving along helical orbits in a uni­

form magnetic field. Westfold (1959) derived the expression

25

for the power radiated by a single electron. This derivation

has been criticized (Takakura and Uchida 1968; Epstein and

Feldman 1967) based on an erroneous interpretation that this

expression represented the power received by a distant ob-

server. However, the power emitted in a time interval t is

received by the observer in a shorter time interval t sin 2¢

(see notations in Appendix C-4). Thus, the power received by

the distant observer is Prec. = Pemitt./sin2¢ (Scheuer 1968).

This expression turns out to be equivalent to that given in

Appendix C-4, if Pemitt. is the corresponding expression for

a single particle as given by Westfo1d (1959).

The spectral power distribution from a single electron

received by the distant observer is given by

p(V) = (C-4.24b)

Since vc~ sin¢ and F(V/Vc ) decreases more rapidly than sin¢,

p(v) goes to zero with decrease in angle ¢. In general for

O<¢<TI/2, the fundamental gyrofrequency and its harmonics are

shifted to higher frequencies. However, the observer will

lie within the cone of radiation for a longer time as ¢ be-

comes smaller, the radiation pulses wo~ld not be as discon­

tinuous and there will be less higher harmonics required to

reconstruct the pulses of radiation. As a consequence, the

peak of the radiation frequency will decrease and eventually

all the radiated energy will lie in the Doppler shifted gyro­

frequency along with a few of its immediate higher harmonics

accompanied by a drastic decrease in output at the higher

26

harmonics. Thus, maximum power output will occur at the

higher harmonics for pitch angle ~ = rr/2.

There are two ways of calculating the power received by

an observer at a distance from an aggregate of relativistic

electrons moving in helical paths toward the observer. The

Westfold method is to "multiply the power radiated per elec-

tron by the mean number of electrons in the source region

with the appropriate pitch angle required to radiate toward

the observer." The Takakura or Epstein and Feldman method

is to "multiply the power received at the observer by the

mean number of electrons contributing to the radiation ob-

served at a particular instant." (Scheuer 1968) The latter

nV(~) =

approach has been used in Appendix C-4. As expected, both

approaches predict the same volume emissivity, which for an

inverse power law distribution within the energy limits E1

and E2 is given by

Ke 2 vHo8rr12 (3/2)X/2 (v/vHo ) (1-X)/2 (sin~) (X-d/2 x

(C-4.37b)

This expression has been evaluated and plotted as a function

of v for X = 5/3 and for representative values of magnetic

fields, energy intervals, and helix pitch angles in Figures

C-4.4 to 4.7. These curves have the characteristic V{1-X)/2

relationship at the higher frequencies and a relatively sharp

cutoff at the lower frequencies due to the fact that the

27

spectral representation has been cut off at E1 • The Doppler

shift to higher frequency is overcompensated by the decrease

in higher harmonics, resulting in (1) a net shift of the peak

output to lower frequencies and (2) a decrease in output as

the pitch angle ~ is decreased to zero.

In the energy range of 10 MeV to 1 BeV, the rigidity is

quite different from the energy for the proton. However, the

rigidity spectrum and the kinetic energy spectrum for elec­

trons are almost indistinguishable in this energy range. The

exponential spectrum for the electrons tends to de-emphasize

the number of low and very high energy particles and empha­

size the number of intermediate energy particles in compar­

ison with the inverse power law spectrum. Thus, the

exponential rigidity spectrum may be approximated by a

constant energy spectrum at the lower energies up to El ~ 200

MeV and an inverse energy power law at the higher energies.

This would shift the maximum output to higher frequencies,

as has been shown by direct calculations by Stein and

Ney (1963). Although these calculations do not predict the

radiation behavior over the entire electromagnetic spectrum

(since there has been no attempt made to represent the entire

energy range), they do give a clear indication of the radia­

tion behavior in the optical region about v ~ 6 X 10 14 Hz.

Stein and Ney (1963) assumed that the microwave burst

and the visible radiation were generated by the same elec­

trons. At magnetic fields of the order of 500 gauss, as they

28

assumed, the gyrofrequency is 1.4 x 10 9 Hz. Therefore, in

the microwave region it is not clear whether the continuum

formulation of the synchrotron radiation is valid at all.

Takakura (1960, 1960a, 1967) has proposed and made several

calculations and predictions of the microwave bursts on the

basis of discrete frequencies for quasi-relativistic electrons.

One of the key points presented by Stein and Ney to give

plausible weight to their proposed model was that the micro­

wave burst at about 10GHzwas continually increasing in some

solar flares. However, recent radio measurements made beyond

10 GHzindicate that the trend is contrary to their assumption.

Figure 3.1 shows the peak flux increases for the May 23, 1967

white light flare event. The peak radiation in the radio

range occurred during the time interval 1839.2 to 1840.7 UT

for frequency ranges 606 to 8800 MHz (see Appendix B, Figure

B-14), and the peak value of 3200 flux units for the 35 GHz

burst occurred at 1839.16 UT (Castelli, et al. 1967). The

peak values are plotted in Figure 3.1. Since the white light

peak was predicted at about 1840 UT, these radiations seem to

reach their p~ak values at essentially the same time. However,

the resultant curve shows a maximum at about 20 GHz and does

not continue to increase as assumed by Stein and Ney. It is

possible that the 35 GHz point is low because of the in­

creased attenuation by the earth's atmosphere at the higher

frequencies. The radiation at 35 GHz falls in a window of

the absorption coefficient spectrum for the earth's atmosphere.

29

10 7r- _

o,\\\\\\\\\'b\\\\\III _ 1946 VT

~- .... , -;;:I ,,"'...... I

\ /0 ',II I 0

I /I ,

\ / 18 110 VTI /0I /I I'0./

UlQ)UlrclQ)l-lUI=:

•.-1

~~~

4-l

10 2 I I I I 1.-.--L1--'--'---'-1.....1....1.....1...1 -'-_.L--_'---'-''-'--'-'-'

100 1000 10,000 100,000:E'requency

FIGURE 3.1. PEAK FLUX FOR P~DIO BURSTS DURING'THE HAY 23, 1967 SOLAR FLARES

Peak Flux Increases (lO-22w/ m2-Hz )

30

At this frequency the attenuation has been calculated to be

about 0.07 db/km at sea level (Evans and Hagfors 1968). The

estimated depression of the maximum flux increase at 35 GHz

is indicated on the curve in Figure 3.1. The absorption

coefficient curve also shows a peak of 1.8 db/km at about

22 GHz, which means that the maximum flux increase curve at

this frequency should be increased. Thus, it is not possible

to arrive at any form of increasing curve as assumed by Stein

and Ney. Furthermore, the 1936 UT event had a similar curve

but with greater flux than the 1835 UT event, although there

was no associated white light flare. Thus, it is not obvious

that one may assume that the microwave and the visible radia­

tion come from the same source.

The structure of the magnetic field in a sunspot region

is tubular and differs from the general dipolar magnetic

field of the sun. The spot field decreases from spot center

to the edge following the relationship H(r) = Hm(l - r 2 /b 2)

and usually has a value of about 50 gauss at the outer edges

of the penumbra. In most magnetic field intensity measure­

ments reported, only the umbra field is given. This value

has an upper limit of about 4000 gauss for the very large

sunspots (Kiepenheuer 1953). According to the magnetic field

intensity measurements for the May 23, 1967 flare (see Figure

B-4), the magnetic field density in the umbra adjacent to the

white light flare positions was about 2500 gauss.

Assuming a tubular magnetic field structure, there are

31

two extreme positions from where one might observe synchro­

tron radiation: (a) at the foot of the magnetic loop where

the electrons reverse their direction of motion, and (b) at

the top of the loop. At position (a), the electrons will have

very flat orbits with pitch angles nearly 90 degrees with re­

spect to the magnetic fields. Since the position of the sun­

spot group was near the center of the disk, radiation from

relativistic electrons with such flat orbits could not be re­

ceived by an observer in a direction perpendicular to the

solar disk. Since the white light appears on the outer edge

of the penumbra, the magnetic field is small and the synchro­

tron mechanism becomes very inefficient. Moreover, near the

optical range the efficiency of radiation decreases for par­

ticles with smaller pitch angles (see Figures C-4.4 to 4.7).

On the other hand, one might conceive of an extremely tilted

magnetic field such that the observer will lie in the plane

of the flat orbits. This configuration might account for the

apparent displacement of the white spots. Although it cannot

be excluded, this extreme tilting of the magnetic loop would

seem inconceivable even in the wedged region of the bipolar

fields.

At position (b), one would be observing from a point in

a plane which is orthogonal to the magnetic field. At this

height, the magnetic field density would be much lower than

near the lower chromospheric level. Also it would be highly

inconceivable to find all the electrons with very flat orbits

32

at this position. Those particles with pitch angles of 90

degrees at point (b) will not move along the magnetic field.

Therefore, it is reasonable to assume a magnetic field of

about 1000 gauss and a spatially isotropic distribution of

pitch angles for the electrons.

Solar cosmic ray spectral observations and studies have

been carried out for a number of years. However, there ap-

pears to be no accepted theory relating the cosmic ray spec­

trum observed at the earth's radius to the spectrum at the

source. This difficulty arises from the strong east-west

anisotropy of the magnetic field which is being dragged along

in the rotational process of the sun at the center of the

solar system. However, if we approximate the spectrum by an

inverse power law in the energy range about 100 MeV, one

finds that those cosmic ray spectra associated with white

light flare events have flatter spectra than those not asso-

ciated with white light flares (see Figure 3.2). Thus, it is

reasonable to consider an inverse power law of between

x = 3/2 to 5/3.

For the inverse power law spectrum, the range over which

the power law holds must be of a factor of 100 for X ~ 1.5.

Therefore, it would be of some interest to compare the effi-

ciency of synchrotron radiation with other competitive models

for the white light flare for these conditions. Since the

integral

(C-4.38d)

33

10 35r----.----

10000

(a)Feb. 23, 1956

(c) Nov. 15, 1960

100 1000Kinetic energy (~eV)

o

~\\ 0\

0\' \\ \ \ \\\ \ \\ 0\ \\ \ \\\ \ \\\ \ \\ \ \ \\ \ \ \ 0\ \ \ \\ \ \ \

0\ \ \'P\ \ \\\ \ \\ \ \,\ \~ \\\ \ \ \\ \ \\\ \ \ \\ \ \ \\ \ \ \

\ \ \\ \ \ \\ b \ \ ~\ \ \ \\ \ \ \(f) July 18, 1961\ \ \\ \ \\ \ \r \ \\ \ 0\ (e) J u 1 y 1. 6, 1 9 5 9

\ ~(g) Sept. 28, 1961

(d) &.Aug. 22, 1958 (b) Sept. 3, 1960

__..J...---L-.J._L_L.LLLL L_L_l_L.LLLLl_-l_l-J.-l-LLU

UlCl)

r-Io

.r-!-l-lH

~ 10 33

lHoHQ)

~Z

FIGURE 3.2. DIFFERENTIAL ENERGY SPECTFJ\. OFSELECTED COSMIC R~Y EVENTS

The events a, b, and c were associated with white light flares,while the others were not (from W. R. Webber, 1964).

34

can be evaluated exactly and has been tabulated for the case

x = 5/3, it would be convenient to consider this case for

comparison purposes.

The weighted volume emissivity, nv(~)/A, corresponding

to the conditions applicable to our case is plotted in Figure

C-4.5, where A = K/4~ and K is the coefficient defined in the

inverse power law energy distribution. In the visible range

The monochromatic intensity in the normal direction is

given by

1L [n v (0) ] [AC]= - dsd;\

A ;\ 2o(3.1 )

= [nv (0 ) 1[Ac 1

A ;\2 LdL (3.2)

oFor the above conditions and at 5800 A,

(3.3)

~ 4.48 X 10- 3 ~~ ergs/cm 3 -sec-ster. (3.4)

The monochromatic intensity in the normal direction oI;\(O) of

the continuum background at 5800 Ais given as 34.7 x 10 13

ergs/cm2 -sec-ster-cm (Minnaert 1953). For the May 23 flare,

assuming a 15 percent enhancement of the continuum radiation

due to the white light flare and equating this to the esti­

mated oI;\(O), we have (0.15) (34.7 x 10 13) = 4.48 X 10- 3 KL/4~.

4~ (5."21 x '10 1 3)Therefore, KL = ~ 1.46 x 10 17

•4.48 x 10- 3

35

The area of the white light disk is about 2 x 10 17cm2 • Thus,

the constant for the power law for the total number of parti­

cles per second becomes 3 x 10 34 /sec. This would correspond

to ~ 4.5 X 10 33 electrons per second with energies greater

than 10 MeV. This estimate of the number of energetic parti­

cles required to predict the white light flare radiation is

at least an order of magnitude greater than that calculated

by Stein and Ney's method.

CHAPTER IV

PROPOSED MODEL FOR WHITE LIGHT FLARES

Introduction

We will assume that energetic particles (protons and

possibly electrons) which have been accelerated to high ener­

gies and possibly stored in the magnetic bottle in the corona

and chromosphere are released and form a beam incident on the

photosphere. The particles are thermalized through release

of energy resulting from collision and ionization of the am­

bient neutral atoms. Depending on the initial energy of the

particles, a temperature perturbation of different layers of

the photosphere will result. There will also be enhanced

black body radiation from this thermally perturbed layer of

the photosphere. If this layer should lie in the region

where the optical depth of the medium is about one, we will

see this enhanced radiation over the undisturbed photospheric

radiation. The local magnetic field plays only a passive

role of guiding the beam of particles during its transit from

the release to the thermalization points.

Stein and Ney (1963) reject the black body radiation

process by stating that "the emission of white light from

flares is a chromospheric phenomenon." In order for absorp­

tion and emission in the visible wavelength to occur, they

reason, the number density of the absorbing and emitting

atoms and ions must be equal to that of the photosphere. There

are, however, no observations which account for such high

37

particle densities in the chromosphere.

Although numerous observations have indicated that the

Ha flare occurs in the chromosphere, there is no evidence

that the white light flare occurs in the chromosphere. As

Stein and Ney state, to sight an optical continuum radiation

as intense as it is claimed to be would require high particle

densities or media with large optical depth. Hence, it is

reasonable to hypothesize the origin of the white light radia­

tion to be in the photosphere. On the other hand, there is

some evidence to suggest that it may occur very low in the

chromosphere or indeed in the upper layers of the photosphere

as indicated by the following characteristics. The white

light flares (1) have distinct shapes which do not change (a

brightening of a pre-existing structure); (2) are very impul­

sive (characteristic of radiation from a thin layer in a dense

medium); (3) satisfy the black body spectrum (tend to be

bluish white at temperatures slightly higher than the normal

photosphere).

Furthermore, for even moderately energetic particles re­

leased in the chromosphere or even in the corona, their mean

free path would be so long that they would strike the photo­

sphere before losing an appreciable fraction of its initial

energy (Schatzman 1963). The total energy supplied to the

photosphere results in (1) raising the temperature (i.e.,

increasing the internal energy of the photospheric gas) and

(2) re-radiating in the continuum. Estimates for these calcu­

lations are within reasonable limits. Finally, the relaxation

38

time for a thermally perturbed photospheric layer due to

radiation compares favorably with the observed relaxation

time for the white light flares.

Heating of the Photosphere by Beam of Energetic Particles

In this section calculations will be made of the energy

loss by energetic particles as they penetrate into the lower

chromosphere and the photosphere. Ultimately a determination

will be made of the number of energetic particles required to

raise the temperature by some predetermined amount in a layer

of the photosphere of optical depth one in the continuum. In

carrying out this program, all energy loss processes in the

path of the incident particles will be considered and the

total energy loss at a given height will be determined. It

is assumed that this energy loss at a given height in the

atmosphere goes into raising the local temperature of the gas

in the region.

For the purpose of these calculations it will be assumed

that the magnetic lines of force in the region of interest

are essentially perpendicular to the surface of the photo­

sphere and the energetic particles are moving parallel to the

lines of force. Hence there is only negligible loss due to

synchrotron radiation. We will also assume (1) a plasma of

essentially electrons, protons and neutral hydrogen and

(2) electrons and protons as the initial energetic particles.

The loss mechanisms for these particles include elastic, ex­

citation and ionization collisions, plasma oscillations,

39

Cherenkov and bremsstrahlung radiations, capture of free

electrons and charge exchange with neutral hydrogen by pro­

tons, and possible nuclear reactions.

For the protons as the incident particles with kinetic

energies between 1 MeV to 1 BeV, the various mechanisms may

be separated into those playing major and secondary roles as

follows:

(a) Major roles:

(1) Inelastic ionization collisions

These are inelastic collisions with the bound

electrons of the retarding medium which lead to

excitation and ionization of the atoms.

(2) Elastic collisions

These are collisions with the free electrons,

nuclei and atoms of the stopping gas by which

part of the kinetic energy is transferred to the

recoil particles.

(b) Secondary roles:

(1) Inelastic collisions with nuclei

These collisions lead to bremsstrahlung, excita­

tion of nuclear levels, and nuclear reactions.

(2) Elastic collisions with bound electrons

(3) Cherenkov effect

Emission of light by particles passing through

matter with velocity exceeding the phase velocity

of light waves in the medium.

The inelastic collisions with nuclei are relatively

40

unimportant in reducing the velocity of a beam of heavy

charged particles as compared with inelastic ionization col­

lisions with bound atomic electrons. Similarly, the elastic

collisions with free electrons and with nuclei (atoms) have

more influence in reducing the velocity of a beam of highly

charged particles than elastic collisions with bound elec­

trons. For fast particles of small charge, v ~ cZ/137, the

energy is lost mainly in small amounts during inelastic col­

lisions with bound electrons. The effective charge of the

particles remains unchanged along a large segment of its path,

at least as long as the velocity of the particle exceeds the

velocity of the electrons in its K-shell by a considerable

amount. When the velocity of the charged particles becomes

comparable to that of the orbiting electrons, v ~ cZ/137, the

inelastic collisions with electrons become less effective in

retarding the particle. This is usually taken into account

by introducing corrections for the electron binding effect.

As the particle velocity decreases further, charge ex­

change (capture and loss of electrons by particles) begins to

take on importance. At this point, the effective charge of

the particle decreases at a continuing rate. The rate of

energy loss due to inelastic collisions with electrons be­

comes smaller and at the same time elastic collisions with

atoms become more important. For protons of about 25 keV

(passing through hydrogen), charge exchange losses amount to

15 to 17 percent of the total losses. At present there is no

41

satisfactory theory explaining charge exchange and energy

loss of retarded charged particles (Sturodubtsev and Romanov

1962) •

In the very high velocity range there are possible in­

elastic nuclear reactions and Cherenkov radiations. At

proton energies ~ 10 9 eV, inelastic nuclear reactions become

important~ however, if we consider a particle distribution

spectrum with a very small number of high energy particles,

then the over-all contribution from the highly energetic

particles from this mechanism may be negligible. Cherenkov

radiation is also ineffective in the chromosphere and upper

photosphere because the index of refraction of the medium is

not essentially different from 1.00 at these densities.

The relative importance of these mechanisms may further

be modified by the nature of the medium. For example, in the

high chromosphere where most of the hydrogen is ionized, in­

elastic ionization collisions will be obviously, less impor­

tant than elastic collisions.

For the incident energetic electrons with kinetic energy

greater than 0.5 MeV, passing through matter, the various

loss mechanisms may be separated into those playing major and

secondary roles as follows:

(a) Major roles:

(1) Inelastic ionization collisions

These are inelastic collisions with the bound

electrons of the retarding medium in which the

42

energy lost by the incident electron is re­

flected in the excitation and ionization of

atoms.

(2) Elastic collisions

These are collisions with free electrons, free

protons, and nuclei of atoms and ions, in which

part of the kinetic energy of the incident elec­

tron is transferred to the recoil particles.

(3) Bremsstrahlung

This is radiation by the incident electron as it

is decelerated upon encounter with protons and

nuclei of atoms and ions.

(b) Secondary roles:

(1) Inelastic collisions with nuclei

These are collisions which lead to excitation of

nuclear levels and to nuclear reactions.

(2) Elastic collisions with bound electrons

(3) Cherenkov radiation

The collision of an incident electron with an electron in the

atom of the stopping medium leads to relatively large energy

loss and to considerable change in its direction of motion.

Thus the intensity of a beam of electrons in a heavy medium

decreases almost exponentially with distance. For electron

kinetic energies greater than 0.5 MeV, radiation losses due

to electrons being decelerated in the nuclear field become

increasingly important with rise in energy. The principal

mechanisms for energy loss in this range are inelastic

43

scattering on the bound electrons of the stopping medium 1ead-

ing to the ionization and excitation of the atoms, and elastic

scattering with the nuclei of the stopping medium.

The ionization energy losses per unit length for e1ec-

trons and protons of the same velocity are almost equal. The

probability of elastic scattering by nuclei is much higher

for the incident electrons than for the incident protons.

The importance of scattering increases with decreasing e1ec-

tron energy (Sturodubtsev and Romanov 1962). Again the re1a-

tive importance of these mechanisms are modified by the na-

ture of the ambient stopping medium.

In Appendices D-2 and D-3, the expressions for the mean

divergence of the proton and electron energies due to the re­

spective important competing processes in the energy range of

interest are derived as functions of the various types of

field particle densities. Their results are respectively

given by equations

_~ = 27fe~ 1 { (2 {ffiec2eV).2{2+y)y2j+(me c 2) (ffipc 2) y ne 1n 13.6 eV

dYe- __ tV

dxand

(4mec2y )}

2n o1n I

27fe~ [(l+Y) {meC2 ).2 Ye 2me c 2 S2 no 1n 2I 2

(D-2.6)

(D-3.4)

44

where y is the ratio of the kinetic energy to the rest energy

of the incident particle.

Simple numerical integrations immediately show that for

electrons and protons of about 20 MeV or greater energy inci­

dent from any height in the chromosphere, most of their

energy is deposited in the photosphere. Furthermore, for

energies below about 500 MeV and greater than 10 MeV, most of

the energy is transferred to the photospheric gas near the

top of the photosphere between optical depth 0.004 and 1.

The very energetic protons penetrate to much greater depths

and lose their energies there. Thus, if there are sufficient

numbers of energetic particles penetrating the" photospheric

surface, the interior layer could be heated without visibly

disturbing the surface. The results of the relative energy

divergence in the different heights for inverse power law in

energy, N = KE-X, are given in Figures 4.1 and 4.2 for the

protons and the electrons, respectively, and for the combined

proton and electron energy divergence in Figure 4.3. These

curves for the different exponent X are calculated for the

same number of electrons and protons of 5 x 1015jcm2 each.

The curves of Figure 4.3 indicate that for the same number of

particles, more energy will be associated with the smallest

exponent X and consequently show greater losses. Furthermore,

the particles with smaller X will have greater energy loss

between -200 and -300 km than for particles with larger X.

In order to relate these curves to different number of

~

U1

.'------- / "- - -.-; -;- ;;-.-."'= -::-:.-.;-:-. "."X=3

.\"

\"

" "\, ., .\, ., .

\, .\, .

\, '.\, ."\,

\,

'\'\'\

" ............, ...

""

'''', ~' ..... , ......." .

:-.. ......

~ .... '". ..............."\...."', ..

'~:...~ .

..c:

-~..!:L:......

FIGURE 4.1.There are 5x10 15 protonsjcm2 withsphere.

~

x = 1.5, 2.0, and 3.0

.t:::­0"1

I , ! I, .., " t , , , •

II

I I!! I 1111 I! I I I II I ----- ~ !

900

800

700I

600 ~500r400 I"

..-.. 300 rS~~. 200 I..c: r

I

100 ~I

0 1"

-100 I-200 r.-300

-4001 10 100 1000 10000

/:"W ergs/cm 3

FIGURE 4.2. ENERGY LOSS FOR ELECTRONS WITH INVERSE POWER LAW DISTRIBUTIONS. There are5x10 1S electrons/cm2 with energy greater than 10 MeV incident on the photosphere.

X = 1. 5, 2. 0,· and 3. 0

700

600

500

400

300,.....

EO: 200~.....-..c:

100

0

-100

-200

-300

-400

-5001 10 100 1000 10,000

~W(ergs/cm3)

FIGURE 4.3. COMBINED ENERGY LOSS BY EQUAL NUMBERS OF ELECTRONS AND PROTONSThe particles have the same inverse power distribution and equal number of particles,5 X10ls /cm 2

, electrons and protons incident on the photosphere. X = 1.5, 2.0, and 3.0

~

-..J

48

particles, N, each point on the curve is modified by the

factor by which the K corresponding to the new total number

of particles differs from the K given for the curves~ For

example, for N = 10 32 and X = 1.5, the K is dropped by a fac­

tor of 10 to 1.7 X 10 32 from 1.7 x 10 33 since N is reduced by

the factor of 10. The relationships between the exponent X,

the intercept K, the total number of particles N and the

total energy ~E, are tabulated in Table 4.1 below. In the

table, No is the total number of particles greater than some

reference energy Eo, which will be taken as 1 MeV (all ener-

gies are given in units of MeV) .

TABLE 4.1RELATIONSHIPS IN THE INVERSE POWER LAW DISTRIBUTIONS

X K (at 1 MeV) N {> Ed ~E (E2 > Ed

1.5 No lEo No 2K [1/1E1 - 1/1E2] 2K [1E2 - lEI]2 -+ 2

2.0 No Eo -+ No K{1/E 1 - 1/E 2) K In E2/EI

3.0 2NoE o -+ 2N o K/2 [l/E~ - l/E~] K{1/E 1 - 1/E 2)

Temperature Perturbation of Finite Photospheric Layer

Since we are assuming the region of the photosphere

where the particles which give up their energy are essen-

tially bounded in the transverse direction by the magnetic

fields, there is little transverse mixing of particles. Fur-

thermore, diffusion across the magnetic lines is a very slow

process and we may assume the layer to be essentially

49

co11isiona11y heated and radiative1y cooled. During the

relaxing process, energy is continually being pumped from the

thermal field into the radiation field through the process of

collisional excitation and ionization followed by radiative

de-excitation or recombination, including the H- process.

For a short time a number of electrons lose (or gain)

energy and introduce a change in the velocity distribution.

However, the electron-electron collisions redistribute the

energy among all the electrons so as to re-establish a new

distribution at some lower net internal energy storage level.

This will introduce a discrepancy in energy stored in the

motion of the electrons and protons. Thus, the self-collision

time, the equipartition time and the cooling (or heating)

time must be compared. If the self-collision time and the

equipartition time for the gas are much smaller than the

cooling (or heating) time, then one may assume the cooling

(or heating) process to occur under the condition that the

particles do have a Maxwellian distribution and hence a

temperature.

The range and stopping time may be estimated by approxi-

mating equation D-2.6 to be of the form

-dy/dx = A/y.

Then x = y2/2A is the range.

Since y = mv 2/moc 2 , v = dx/dt =(C/I2)yl/2,

and t = (12/3) (y3/2/CA ) is the stopping time.

(4.1)

(4.2)

(4.3)

Schatzman (1963) calculated these quantities for the electron

and the proton for several non-relativistic energies for the

50

Van de Hulst model (1953). For more energetic particles the

range will be greater and lifetime will also be greater. The

lifetime at the top of the photosphere is 2.1 x 10- 2 and

3 x 10- 5 seconds for 100 and 1 MeV protons, respectively.

The se1f-co1iision time is the time required for the

distribution of kinetic energies to approach Maxwellian for

a gas colliding with members of its own species. This time

is given by (Spitzer 1962)

t c = (11.4Al/27f3/2)/(nz41nA),

where T = temperature in oK,

A = atomic weight in amu,

Z = atomic number, and

1\ = h/po = 6.9(T/ne )1/2 (mv2/ZZ1e 2).

(4.4)

The self-collision time for electrons near the top of the

photosphere is about 10- 8 seconds; and for the proton the

value is 43 times larger.

The equipartition time is the characteristic time re-

quired to bring equipartition of a system made up of two

types of charged particles with Maxwellian velocity distribu-

tion at different temperatures (Spitzer 1962),

(4.5)

where n 1 = common value of charged particle density (assuming

electrical neutrality). The equipartition time for electrons

and protons near the top of the photosphere is about 10- 6

seconds.

Since the stopping time is much larger than the

51

relaxation time for the photosphere, we can assume that the

photospheric gas maintains at least quasi-Maxwellian velocity

distribution during the heating process. On the assumption

that a local temperature can be defined corresponding to the

Maxwellian distribution of the particles to which the gas

relaxes, the radiation relaxation time will also turn out to

be many orders larger than the collisional relaxation times

(Schatzman and Souffrin 1967). Hence it is very reasonable

to assume that indeed a local temperature may be defined for

the gas in the perturbed layer of the photosphere near the

surface.

The energy lost by the beam of particles to the gas at a

given depth in the photosphere may be assumed to go into per­

turbing the local temperature at that depth by the relation­

ship

(4.6)

where NT = number density of all particles (electrons, ions,

and neutrals) ,

= ionization potential of the ith element, and

= number of ionizations due to the change in

temperature.

An estimate at about optical depth T = 1.0 in the photosphere

gives the contributions to the specific heat due to all ioni­

zations, and all kinetic energies at about 23 and 26 ergs/oK­

cm 3 , respectively. If equation 4.6 is mUltiplied by the

thickness of the perturbed slab, we obtain the energy per

52

unit area which must fallon the surface in order to perturb

the temperature of the slab by ~T degrees centigrade.

However, we will see from the next section that the

atmosphere relaxes by radiation and that there is a radiative

relaxation time associated with different heights in the at­

mosphere. This relaxation time changes with the specific

heat, temperature and absorption coefficient. Hence, at

certain depths the atmosphere would store energy and radiate

immediately only part of the energy deposited in the region,

while at greater depths (within the region where the atmos­

phere may be considered to be optically thin), all the energy

is re-radiated immediately. This problem is considered in

the following section.

Enhanced Radiation from Thermally P'e'rtu'rbed Finite

Atmosphere

If we let u(x,t), p(x,t), and Q(x,t) be the internal

energy per unit mass, mass density, and the rate of heat ex­

changed per unit volume, respectively, then we can express

the conservation of energy as

p au/at = Q. (4. 7)

However, the rate of heat exchanged is equal to the rate of

energy input minus the rate of energy output,

Q(x,t) = 4TIK(J - S), (4.8)

where K(X,t) = mean absorption coefficient/em,

S(x,t) = integrated source function, and

J(x,t) = integrated mean intensity.

53

Since du = CvdT, if we substitute this relation into equation

4.7 and equate it to equation 4.8, we have

pCv ClT/Clt = 4TIK(J - S). (4.9)

The mean intensity at a point x is affected by radiation

from all parts of the atmosphere. Assuming Kirchhoff's law,

jv/Kv = Bv(T), and the exponential extinction law, it can be

represented by

J(x,t) = !K(~,t) B(~,t) e-L/4TIr 2) d~, (4.10)

where L(X,~,t) = !K(~', t) dr'

r = ~ - x.

Since we are assuming that Kirchhoff's law holds, the source

function SV(L) is also equal to JV(L). Equation 4.9 becomes

pCv ClT/Clt = 4TIK[!K(~,t) B(~,t) (e- L/4TIr 2) d~-

B(~,t)]. (4.11)

If the various terms in this equation are linearized accord-

ing to T(x,t) = T (x) + 8(x,t)

f(x,t) = fa (x) + [Clf o (x)/ClT o] 8 (x,t)

and if B(L) = O/TIT 4 , the integrated Planck function, the

equation for the rate of change in the temperature becomes

Cl8/Clt = q[!(8e-Kr/4TIr2)K8d~ - 8 + aT/4 !(8-K!8dr') x

(e- K r /4TI r 2 ) d~ ] (4 . 12 )

where q - 16oKoT03/pCv ,

and a - (ClKo/ClT) ITo.

Spiegel (1957) has found a separable solution

(4.13 )

where

54

(4.14)

and K = Kp.

On the other hand, Unno and Spiegel (1966) have assumed the

Eddington approximation and obtained

qnE = qk 2 / [k 2 + 3K ("K + e:)p 2] , (4.15)

where e: is the scattering coefficient. P. Mein (1966) has

calculated the radiative relaxation time t R = l/q for the

lower chromosphere and upper photosphere. This is given in

Figure 4.4.

In both solutions n(k) and qnE' as "Kp/k + 0, n + 1.

This is the optically thin case; that is, temperature smooth­

ing takes place independently of self-absorption and proceeds

almost entirely by emission losses. When ("Kp)/k + 00, n +

k 2 /3"K2 if we neglect scattering (e: = 0). k 2 /3"K2 is a diffu­

sivity. The protons travel several mean free paths before

escaping from the perturbed region. This solution corresponds

to the Eddington approximation to the classical theory of

radiative transfer. Therefore, although the functional forms

of these two solutions are different, they could reasonably

represent the solution over the entire range from the opti­

cally thin to the optically thick cases.

Since we are interested in the top of the photosphere

where the atmosphere may be assumed to be optically thin, it

is convenient as a first approximation to assume an iso­

thermal, optically thin, semi-infinite atmosphere. The par­

ticle energy dumped into a layer near the front of this

lJllJl

10,000100

pC v16GKOT03

t R =

1. a () 10t R secFIGURE 4.4. PADIATIVE RELAXATION TIMES FOR THE

LOWER CHROHOSPIIERE AND THE UPPER PHOTOSPHERE(from P. Hein, 1966)

900 rII

800 II

700 I600 r500 ~

400 l.......

300 ~£::~ I

.c 200 ~I

100 rI

a ~I,

-100 LIII

-200 ~ .Ii

-300 :...

-4000.1

56

semi-infinite atmosphere is assumed to generate a boxcar type

temperature perturbation of uniform magnitude 80 over a dis-

tance 28 thick about height Xo ' (see Figure 4.5). Due to

this temperature perturbation, there is an enhancement in the

emergent intensity from the atmosphere. This would then

correspond to the enhanced radiation we have previously des-

ignated as the white light flare. If this indeed is a re1i-

able model of the phenomenon, then the amplitude and decay

rate for the observed flare should be consistent with the

estimated region over which the perturbation occurs.

The assumption that the atmosphere is thin implies that

k»KP where k is the wave number and KP is the absorption

coefficient per unit length. The change in temperature is

given in terms of the initial temperature distribution by

where

8 (x, t) = J00 G(x , x' , t) 8 (x', 0 ) dx 'o

G(x,x',t} = (1/2n) ~oo eik(x-x'}-qnEt dk,_00

(4.16 )

(4.17)

and nE = k 2 /(k 2 + 3K2 p 2} + 1, when scattering is neglected

and energy transfer is by radiation only. Then

Therefore

G(x,x' ,t} = (1/2n) e-qt ~oo eik(x-x'} dk-00

= (1/2n) (e-qt ) 8(x-x'}.

6 (x,t) = 1;00 e-qt 8 (x-x') 6 (x' ,O) dx'

8(x,t} = .6(x,O} e-qt

(4.18)

(4.18a)

(4.19)

(4.19a)

This says that for the optically thin case, there is no

FIGURE 4.5.

8(x~O)

x'x'-o Xl x'+oo 0 0

SQUARE WAVE TEMPERATURE PERTURBATION

57

spread in the temperature profile as a function of time and

the amplitude of the temperature just decays exponentially

-qtas e .

According to the extinction law, the emergent intensity

at the surface of the atmosphere is given by

S(T)e-TdT = foo KpB(T)e-fKPdx dx.o

(4.20)

If we make similar linear approximations in the temperature

and the other functions in the expression, the expression for

the enhancement in the emergent intensity due to the pertur-

bation becomes

l\I(O,t) = 100[OBO 8 exBo 8 _ Bofex8KOPdx] x

o oT o + KoP KoP

-iKo pdx -e Ko P dx. (4.21)

(4.22)

(4.23)

Letting B = (a/TI)T q, where a is the Stefan-Bo1tzman constant,

l\I(O,t) ~ (4a/TI) )(00 KoPT~ 8(x,t) e-KpX dxo

III(O,t) = (4cr/1T)K,PT; e-qt 1,= S(x,O) e-KPX dx

The integral J:= S(x,O) e-KPx dx for a square pulse function

for 8(x,O) can be readily evaluated and we have

fx~ +a 80

e - K0 Px dx =

xl-ao

80

KoP

58

Therefore

(4.24)

[408 0 Tg -Kop(x'-a)

b.I(O,t) ~ TI e 0 (4.25)

A more rigorous analysis must consider the actual normal

photospheric temperature distribution, which would require

tedious numerical calculations on the computer. This does

not seem to be the appropriate time to carry this out since

the objective here is to determine the plausibility of a pro-

posed idea. The cases of some arbitrary temperature distri-

bution and the accounting for opacity would be more realistic

and should be considered if the theory should prove to be

acceptable.

Example: The White Light Flare of May 23, 1967

A specific application of the theory will be made to the

white light flare event of May 23, 1967. In this case fromo

the enhanced intensity measured at 5800 A, it is possible to

predict the magnitude of the temperature perturbation, the

energy required to raise the temperature of the photospheric

slab, the number of protons and electrons required to heat

the photosphere and the radiation observed, and the specific

particle spectral distribution if one assumes the form of the

spectrum.

The bluish white spots were observed in cove-like re-

gions on the outer edge of the penumbra (DeMastus and

59

Stover 1968; McIntosh 1967). The larger of these two spots

had a projected disk area of about 2 x 10 17 cm2 and the

intensity increased from the preflare minimum value of 0.91 to

flare maximum of 1.06 of the adjacent normal photospheric

intensity. We assume that this enhanced radiation originates

from a region with optical depth of about 1.0 and the atmos-

phere may be described as optically thin.

Since the observations were .carried out over a band ofo 0

about 100 A centered at 5800 A, it would be convenient to

modify the results of the previous section to

(4.26)

where oBA/oT may be considered not changing with distance.

Then

6I1IO,t) • I~Bl/~T} e-qtjOO Slx,O) e-K,x K,dx.o

(4.27)

For a rectangular temperature perturbation of width 20 and

amplitude 80, the expression for the enhanced emergent inten-

sity becomes

~IA(O,t) = [(oBA/oT)8 0 e Ko (x~-o) x

(1 - e- 2KOO )] -qt (4.28)e

oBA/oT ~ (2hc 2/A 5) (a/T 2) e 1-a/T , (4.29)

where a/T = hC/AkT > 2. The normalized enhanced radiation is

~IA(O,O)

IA(O,O)

~IA(O,O)

IA(O,O)

=(oBA/oT)80' e-Ko (x~-o)(l - e-2KOO)

BA (T)

(4.30)

(4.31)

60

The observed change in intensity of 15 percent over the pre­

flare intensity at the spot is

tlIs 800 (0) = 0.15 I s800 (at T = 5790 0 K) ,

where

I s800 (at T = 5790 0 K) =:: 3.20 X 10 6

(4.32 )

°ergs/A-cm2-ster-sec (Allen 1955).

°Therefore, tlIs 800 = 4.8 X 105 ergs/A-cm2-ster-sec. (4.33)

Assuming the normal photospheric brightness temperature

of 5930 0 K and using the Planck function, we can find the

temperature change corresponding to the observed intensity

°change from 0.91 to 1.06 of the disk intensity at 5800 A.

The region of the spot before the flare corresponds to the

minimum temperature of 5790 0 K and during the maximum phase of

the flare corresponds to the temperature of 6010 0 K, which

gives an increase in temperature of 220 0 K. The corresponding

increase in integrated intensity is 1.022 x 10 10 ergs/cm2-

sec. If we assume that the radiation continues for 300

seconds at this maximum rate over the region with the surface

area of 2 x 10 17 cm2, the total energy radiated is about

6 x 10 29 ergs.

Now if we substitute T = 5790 0 K, 8 0 = 220°, and tlI/I =

0.15 into equation 4.31, we have

0.875 = e-Ko (x~-o) (1 _ e- 2KOO ). (4.34)

This relation is satisfied if we take KoX~ = KoO and

KoO = L = 1, that is, the center of the perturbing layer is

at L = 1 as anticipated. In the normal photosphere, L = 1.0

61

occurs at -300 km. However, because of the radiative relaxa­

tion time predicted in Figure 4.4, more energy is stored in

the atmosphere above -300 km. Consequently, the temperature

rises and the opacity also increases. As the temperature,

opacity and specific heat coefficient change, the radiative

relaxation time also changes. The exact solution to this

problem has not been attempted.

The energy density required to raise the temperature of

a volume of the slab by 80 degrees is given by ~E = pCv 8 0 •

The specific heat pCv of the normal photospheric gas at about

-300 km has been calculated to be about 45 ergs/cm 3 -oK. The

energy required to raise the temperature of the gas to 220 0 K

is 10 4 ergs/cm 3• Although the temperature at less depth must

be raised by a greater amount than at T = 1.0, the specific

heat is less at less depth and the estimate of 10 4 ergs/cm 3

is probably a reasonable average value over the entire slab.

If the volume of the slab is 2 x 1025 cm 3, then 2 x 10 28 ergs

are required to heat the photospheric slab from -200 to -300

km to 60l0 o K.

According to the results in Figure 4.4, the radiative

relaxation times in the slab between -200 and -300 km vary

from 0.28 to 4.5 seconds. Hence, most of the energy deposited

in the region is re-radiated almost instantaneously. Thus,

a reasonable requirement for this energy deposited in the

region would be ~w = (1.022 x 10 10 ergs/cm2-sec)/107 cm or

about 1000 ergs/cm 3 -sec. Figure 4.6 gives the combined

800

700I

600 I500 r400

300

200.........E8 100.c

0

-100

-200

-300

-400

-5000.1 1.0 10 100 1000

~W ergs/cm 3 -secF.IGURE 4.6. COMBINED ENERGY LOSS BY EQUAL NUMBERS OF ELECTRONS AND PROTONS

The particles have the same inverse power law distribution and equal nurober of parti- mcles, 1.5 xl0 14 /cm 2

, with kinetic energies greater than 10 MeV. N

63

energy loss curve for electrons and protons such that 1000

ergs/cm 3 -sec are deposited in the slab for an inverse power

law with exponent X = 1.5. Table 4.2 gives the relationships

for the total energy and number of particles involved in the

beam and the relative efficiency for the process. For com-

parison, the cases for X = 2 and 3 are also presented in

Table 4.2 for the condition that 1000 ergs/cm 3 -sec is depos-

ited in the slab. From all considerations (total energy,

total number of particles, and efficiency), an energetic

particle flux incident on the photosphere represented by an

inverse power law distribution with X = 1.5 and K = 6.67 X

10 31 would best produce this white light flare event.

TABLE 4.2. REQUIREMENTS FOR THEMAY 23, 1967 WHITE LIGHT FLARE EVENT

N (>10 MeV (~W*/~E')

X K ~E ~E' = 2~E <1000 MeV) x (100)(erg/sec) (erg/sec) (sec) (percent)

1.5 6.67 x 10 31 3.04 X 10 27 6.08 x 10 27 2.1 x 10 31 33

2.0 6.67 x 10 32 4.9 x 10 27 9.8 X 10 27 6.67 x 10 31 20.4

3.0 5.0 x 10 34 1.6 X 10 28 3.2 x 10 28 2.5 X 10 33 6.3

* ~W = (1000 erg/cm 3-sec) (2 x 10 24 cm 3 ) = 2 x 10 27 erg/sec.

CHAPTER V

OTHER MANIFESTATIONS

The white light radiation, the microwave burst and the

x-ray events are very strongly time correlated in the very

early stages of the Ha flare (see Appendix B and Table 5.1).

Because of this very close time correlation, it would be of

interest to check whether the energetic particle flux re­

quired by our model is consistent with the observed microwave

and x-ray bursts. It should be noted that the May 23, 1967

event is the first where x-ray data are available for a

known white light flare. The many other types of radio burst

and geophysical manifestations will not be considered here

because they are not directly relevant to the description of

the flash phase of the flare.

In this chapter, brief reviews will be made of the x-ray

and microwave bursts. Then calculations predicting the

possible radiations will be presented in accordance with the

various models. It will be shown that the numbers of ener­

getic electrons required to generate the emission for the

x-ray and microwave bursts are in accordance with the number

of particles required to generate the white light flare by

the proposed model. Finally, an energy spectrum for the par­

ticles consistent with the radiations during the flash phase

will be given.

X-radiati'on

In general, the solar x-radiation may be classified as

TABLE 5.1COMPARISON OF THE 1835 AND THE 1936 UT EVENTS

Soft x-rg.ys Hard x-rays MicrowaveFlare Hex. 2 - 12 A > 80 keV 8800 MHz White LightEvent 1835 1936 1835 1936 1835 1936 1835 1936 1835 1936

event event event event event event event event event event

Start time* 1835 1936 1835 1936 1835 1937 1835 1936 1838 ---Max. time* 1844 1947 1846 1953 1842 1943 1840 1845 1840 ---End time* 1930 2130 >1936 >2100 1900 2030 1850 2036 1845 ---Duration SSm 2 hr. ~2 hr. ~2 hr. 25m 53m 25m

~1 hr. 7m ---Decay rates ~20m ~40m ~30m ~45m ~4m ~5m m ~7m ~2m~1. 5 ---Peak flux 3B 2B 0.65 0.28 1.6 x 10- 5 4500 23000 2 erg/ ---

er'j/ erg/em 10- 4 er'j/ FU FU em 2 secem sec em 2 sec em 3x10 4 FUsec I sec

* Time given in UTe

0'111l

66

of thermal, quasi-thermal or nonthermal origin. The quiet

sun and the slowly varying x-ray emissions can best be ex-

plained in terms of thermal radiation by free-free, free-

bound, and bound-bound electronic transitions (Elwert 1939;

Kawabata 1960). On this basis it has been estimated that the

temperature of the source would have to be several million

degrees. Thus, it has been hypothesized that the source of

these radiations must be in the solar corona (Elwert 1961;

Mandelshtam 1965) .

The normal thermal solar x-radiation by electronic tran-

sitions in the K, L, and M shells is predicted to have a

°lower wavelength limit of 1.4 A, corresponding to the complete

ionization of abundant heavy elements such as iron (Elwert

°1961). Recent spectroscopic measurements in the 20 to 8 A

range during quiet and moderate solar activities indicate

that most of the x-radiation in this range is in the lines

(Rugge and Walker 1967). It is believed that the radiation

°with wavelengths shorter than 1 A lie in the continuum and

are produced by nonthermal processes, since a thermal model

for such energetic photons would require prohibitive temper­

atures of about 10 8 °K. Such temperatures would pose diffi-

cult problems with respect to the relaxation time of the

corona (Friedman 1960).

The solar x-radiation bursts during flares are either of

quasi-thermal or of nonthermal origin. The quasi-thermal

x-ray emissions extend over a period of tim~ comparable to

the lifetime of the associated Ha flare. The relatively

67

long period over which the radiation takes place suggests

that the emitting particles are confined in a magnetic bottle.

The usual lifetime of the nonthermal x-ray emissions is of a

few seconds to a few minutes and the energy lies in the 10 4

to 10 6 eV (1 to 0.01 A) range. These emissions are assumed

to be due to high velocity streams of electrons radiating as

a result of deceleration either in a magnetic field or by

interaction with the surrounding dense medium (deJager 1967).

There are several different types of these hard non­

thermal x-ray emissions. About 80 percent of them coincide

with impulsive centimeter and decimeter radio bursts occurring

during the flash phase of large optical flares. The other 20

percent are either coincident with Type III radio bursts or

with distinct radio events at the very beginning of minor

optical flares (deJager 1967).

Several mechanisms have been proposed for the production

of these energetic photons in the solar atmosphere. Shklovsky's

(1964) proposal that hard x-rays may be generated by the in­

verse Compton effects has been criticized by Acton (1964),

who shows that relativistic electrons producing measurable

amounts of inverse Compton photons will always produce greater

x-ray photons by bremsstrahlung. The inverse Compton effect

would be an important consideration for x-radiation from an

extremely rare medium. An extension of the results of Chapter

III to shorter wavelengths immediately points out that the

inverse Compton effect and the synchrotron mechanism would

68

require extremely energetic electrons. Peterson and Winckler

(1959) have proposed bremsstrahlung due to braking of high

velocity jets of particles in the dense flare region or in

the photosphere.

The results presented in Chapter III for the inverse

Compton effect, bremsstrahlung, and the synchrotron radiation

have been extended to the x-ray region. On this basis the

number of electrons required to generate the observed

x-radiation has been estimated. The applicability of these

analyses for the soft x-ray region is questionable, since

much of the power in this range may be due to line radiation

resulting from K-shell ionization. The nonthermal bremsstrah­

lung appears to be the only mechanism which would generate

the observed hard x-radiation by a reasonable number of ener­

getic particles. These calculations are presented in Table

5.3.

Microwave Burst

Most microwave burst phenomena are combinations of the

three basic morphological types proposed by Kundu (1959,1965):

(a) simple burst, (b) post burst, and (c) gradual rise and

fall. The simple and post bursts can be readily identified

for the 1835 UT event, while this distinction is not clear in

the 1936 UT event on May 23, 1967 (see Appendix B, Figure 11).

These types of microwave bursts are characterized by the

properties summarized in Table 5.2.

On the other hand, Wild et ale (1963), do not consider

the post burst and the gradual rise and fall as distinctly

69

TABLE 5.2PROPERTIES OF MICROWAVE BURSTS

Properties Simple' Burst Post' BurstGradual Rise

and Fall

Nature Rapid rise to Slow decay fol-peak value and lowing a groupsubsequent decay of simple bursts

Slow rise andslow decay

Duration ~ 1 to 5 min.

Equivalent 10 6 - 1090 KTemperature

Several minutes 10 min.to several hours

Polarization Partially circu­larly polarized

Source Size 1 - 1.6' diam.at 3 cm

Mechanism Nonthermalbremsstrahlungor synchrotronradiation

Partial (similarpolarization tothe simple burstpreceding it)

,...,2.5' diam.

Thermalbremsstrahlungor synchrotronradiation

None or par­tial (circular)

,...1' diam.

Bremsstrahlungor synchrotronradiation(preheatingand compressionof flare area)

7~

separate types. They claim that the post burst has a larger

source diameter because of the greater energy involved. Thus,

post burst and gradual rise and fall are grouped together and

designated gradual burst. This type of burst is generally

more prominent at the lower frequencies and is regarded as

part of the generalized Type IV phenomenon which is believed

to be due to synchrotron radiation.

The impulsive burst makes a sharp rise to a single maxi­

mum in about one-fourth of the total duration, followed by a

slower monotonic decay to the pre-burst or the post-burst

level. It occurs near the start of the optical flare. It

has a broad band continuum but does not extend into the metric

domain (see Figure 3.1) .. Usually the radiation is partially

circularly polarized with the degree of polarization never

reaching 100 percent above 2000 MHz (Wild et al. 1963). The

circularly polarized component of the microwave impulsive

burst is assumed to be due to the weakly relativistic elec­

trons, as calculated by Takakura. However, linearly polarized

components have been measured (Akabane 1958). The linearly

polarized components may be due to synchrotron radiation from

either weak or ultra-relativistic electrons, and the non­

polarized component may be due to all three possible mechanisms.

There is only one white light flare associated microwave

burst (known to the author) for which detailed polarization

measurements have been pUblished. This is the November 15,

1960 white light flare. During the impulsive phase of the

3 cm burst, the polarization was weakly left circularly

71

polarized at not more than about 15 percent (Nagasawa et a1.

1961) .

The impulsive characteristic and the high degree of time

correlation between the hard x-ray and the 3 cm burst is

illustrated by the September 28, 1961 flare (optical continuum

radiation was not recorded for this flare). Although the

start of the x-ray and the 3 cm burst coincided, the duration

for the radio greatly exceeded the former (Wild et a1. 1963).

There are other events, such as the March 20, 1958 hard x-ray

flare observed by Peterson and Winckler (1959), which had the

same duration for both the x-ray and the 3 cm burst.

The May 23, 1967 event contradicts the usual observa­

tions as it displayed a much shorter duration for the micro­

wave pulse than for the hard x-radiation (see Appendix B,

Figures B-11 and B-13). Furthermore, the hard x-radiation

built up to its peak value at a much slower rate than the

3 cm burst.

The nontherma1 bremsstrahlung radiation is not strongly

dependent on the energy of the incident electron if the elec­

tron energy is much greater than the photon energy. Thus,

the radiation would depend primarily on the total number of

particles with energies greater than the approximate electron

rest energy. On the other hand, the synchrotron radiation

would be highly dependent on the energy and the magnetic

field. If the 8800 MHz radiation were due to gyro-synchrotron

radiation (as in the Takakura model), then the x-ray and the

3 cm radiation should be of similar duration. This suggests

72

that the impulsive 3 cm burst may be a measure of the number

of ultra-relativistic particles in the source region.

According to the calculations in Appendix C-4, the peak

of the synchrotron radiation shifts to lower frequencies as

the angle ~ = ~ (v,H) decreases. Accordingly, if a stream of

energetic particles gyrates along a magnetic tube with very

small pitch angles, then there would be a peak impulsive

radiation in the microwave frequency range with negligible

continuum visible radiation. These energetic particles will

rapidly lose their energies by radiation and collision with

the ambient gas and generate large numbers of quasi­

relativistic particles. These particles may then continue to

radiate x-rays by nonthermal and quasi-thermal bremsstrahlung

and post burst cm radiation by the quasi-relativistic gyro­

synchrotron radiation as hypothesized by Takakura.

Calculations for the required number of ultra­

relativistic electrons with inverse power law to radiate the

8800 MHz peak burst is carried out in accordance with the

derivation presented in Appendix C-4. The results are tabu­

lated in Table 5.3. As pointed out by equations C-4.43 and

C-4.44 in Appendix C-4, the contribution to the synchrotron

radiation at a given frequency would be negligible if the

electron energy does not lie within a specified energy inter­

val. The number of particles calculated would be greater

than if one were to optimize the energy distribution of the

particles.

TABLE 5.3REQUIRED ENERGETIC PARTICLE FLUX

Radiations Soft X-;;rays Hard X-rays Microwave W'nite Light2-l2A > 80 keV 8800 MHz

Measured 0.65 ergs/cm2 -sec 1.6 x 10-1+ 5 X 10- 16 ergs/cm2 - 2 ergs/cm 2 -secPeak Flux ergs/cm2 -sec sec-Hz

Radiation Quasi-thermal Non-thermal Synchrotron Enhanced blackMechanism & nonthermal bremsstrah- radiation body radiation

bremsstrahlung lung from photosphere

Number of n*Net = 2.50xlO sO n*N T-4xlOI+ S Ne § = 4.5 x 10 30 N =N #=6xlO 31e - e p

ParticlesRequired

* n = number of protonst Ne = number of electrons with Ek>lO keVT Ne = number of electrons with Ek>.5 MeV§ Ne = number of electrons with 10 MeV<Ek<l BeV, power law with X = 5/3# Ne = number of electrons with 10 MeV<Ek<l BeV, power law with X = 5/3, k = 4xl0 32

-.JW

74

Energy Distribut'i'onfor' En'er'ge'ti'c Part'icl'es

In the previous chapter it was shown that the inverse

power law distribution, KE-X, with exponent X = 5/3, best ex-

plained the temperature perturbation of the photospheric

layer near unity optical depth. It would not be reasonable

to extend the inverse power law to very low energies and it

would be necessary to level off the spectrum at the low energy

end.

Since the nonthermal bremsstrahlung of x-radiation is

not strongly energy dependent, it may be possible to estimate

the total number of energetic electrons associated with the

flash phase. If it were assumed that the earliest phase of

the hard x-rays is generated in the region of the release

point of the energetic particles, it would be reasonable to

truncate the inverse power law distribution in such a manner

that the total number of particles above .50 MeV, for example,

would be that required for the hard x-radiation measured.

From Table 5.3, the number of electrons required for the

hard x-rays is about 10 33 , if we assume 4 x 1012 nuclei per

cm 3 in the ambient medium. This number is in close agreement

with 8 x 10 32 particles with kinetic energies between .50 MeV

and 1 BeV for the distribution shown in Figure 5.1. Thus, the

energy distribution of energetic particles for the May 23,

1967 event has the analytical form

dN4 10,32, .5 MeV < E < 1 MeVdE = x

= 4 X 103 2E- 5/ 3 , 1 MeV < E < 1 BeV,

75

where E is given in units of MeV.

E(MeV)10001010 2 B '------f-------i--.....

0.1

FIGURE 5.1A TRUNCATED INVERSE POWER LAW DISTRIBUTION

FOR ENERGETIC ELECTRONS IN THE MAY 23, 1967 EVENT

CHAPTER VI

SUMMARY AND CONCLUSIONS

This study has shown that the white light flare can be

explained in terms of the enhanced black body radiation from

the photosphere. For the analysis it was assumed that a

sudden release or acceleration of energetic electrons and

protons takes place in either the chromosphere or lower

corona. These particles give up most of their energy in a

selected region of the photosphere, which may be described in

terms of a temperature perturbation in the region. Calcula­

tions based on the model (see Chapter IV) indicate that the

enhanced radiation satisfies most of the descriptive charac­

teristics of a white light flare (described in Chapter II).

In particular, the model calculations indicate similarities

in the observed color, intensity, impulsiveness, and the rela­

tive position with the sunspots.

Furthermore, the model allows for the generation of asso­

ciated radiations in the radio and x-ray regions. Assuming

an inverse power law in energy for the energetic particles, a

distribution was derived for the May 23, 1967 flare which

could also account for the radio and x-radiation by the syn­

chrotron and bremsstrahlung processes, respectively (see

Chapter V). In its analytic form

dN 32 1dE = 4 x 10. , 2 MeV < E < 1 MeV

4 x l0.32Ek-s/3, 1 MeV < E < 1 BeV,

77

where Ek is in units of MeV. This spectrum accounts for the

6 x 10 31 electrons (and equal number of protons) with kinetic

energies between 10 and 1000 MeV required for the peak opti­

cal continuum radiation, and 8 x 10 32 electrons with kinetic

energies between .50 and 1000 MeV required to generate the

peak hard x-radiation.

The model proposed here is competitive with the synchro­

tron mechanism proposed by Stein and Ney, in terms of the

number of energetic electrons (between 10 31 and 10 32) required

to generate the white light flare. Moreover, this model could

explain the heating of the photospheric layer in terms of

protons only. Although presently available observation data

strongly favor the photospheric radiation model, final veri­

fication of the model awaits availability of decisive obser­

vations including (a) observation of an over the limb white

light flare event, and (b) observations showing the polariza­

tion of the flare's optical continuum radiation. Observations

of the solar photosphere in the continuum of the visible

spectrum are being carried on at Ha1eaka1a Observatory in order

to establish the polarization. However, to date, Ha1eaka1a

observers have not witnessed a white light flare event.

APPENDIX A

TABLE A-I. FLARES IHTII CONTINtMI OPTICAL RADIATION AND ASSOCIATED EVF)ITS

Remarks

PCA, GLE.(A24, A26); IV '(A3S);Surge (A26)

PC\, GLE (A26);· SID, II, IV(A3S)

Date

I I I I I I I I II i I 8 I I I I I I 4> II e 6 I la I 6 I Color I Intensity I Observed I Observer I g II e:l '" I ~ I .... I I I' I I 4> II :3 ';;; 6 e (Per~ent of ' in .eI- ~ .3 I ~ I a I I Contmuum) I I I & II I I I I I I : t

". Sept.l, 1859 I 20~1Z\'/ I3+ I 7 I Intense white l I Total light I Carrington I (A 5) II" I I Hodgson I (A18) I,

Nov. 13, 18721 10:-.1001'/ I I I I I Ferrari I (A36) I". June 17, 18911 2W80i~ I 3 I 6 I Ye1101iish I Total light I Trouvelot I (A39) I

July 15, 18921 10X9D:,/ I I I Blinding \~hite I I Rudaux I (A35) I.,. Feb. 21, 19211 7S42\'/ I 3+ I 8 I Nagenta I .Total light I }lartheray I (A12) I". Sept.22, 1928 I OO:-lOo\\, I 3 I 2 I \'rnite I Total light I ~lartheray I (A12) I". July 26, 19371 32N31E I 3 I I Equally visible I Total light I Waldmeier I (A4l) I

I j I I in red, green I I I II I' I & b'lue filters, 0 I . I I

". lolar. 31, 1938120S86E I 3 I 6 I 27\ at A 3220A I U.v. l!ght I Dobbie I (AlO) I SID (AI0)

I I I ' I A 3220A 0 I I Ii I I I LiA=20-50A I I I .

". l-lar. 5, 1946 I 28N10S I 3+ I 3 I\'illite _ 0 I Total light I Hartheray I (All) I No neutron increase (A24)". July 25, 19461 l6E I 3+ Short 10% at ~ 6200A I Spectrum ! Ellison I (Al4) I 15% proton. lZ00% neutron

I I I I I I I I increase (A24)Dec. 11. 19'48\ 9S4SE I l-\'illite I Total light I r.Killer I (A29) I Surge 250 kr.1/sec (A~9)

". Nov. 19, 19491 2S7~1 I 3+ I Short I ~% I Spectrum I Ellison & I CAlS) I PCA, GLE. SID, Surge (Al5). I I I I I I Com~ay I . I

". r.1ay 18, 1951 I 18~35\1 I 3 I 1 I I Total light I Porret I (A33) I .* Feb. 23, 1956123;\174\\' I 3+ I 5 I \'illite Disk brightnessI Total light I Unno I (A3l) I PCA, GLE. (A25, A28); SID, IV(A38)

Aug. 27, 19561 22S461'/ 11+ I I I I Severny I (A22) I". Aug. 31. 1956\ 15N15E 3+ I 1 I Total light I Greem~ich I (A17) I

I I I I I I I Observatory I I". Aug. 30. 19571 ZOE I 10 \'lhite I Total light I :-lcNarry I (A27)

I. I 20 I I I I U. Becker I (A3) I". Sept. 3, 19571 24~30\'/ 3 I 6 I Bluish white I ITotal light I U. Becker I (A3) I SID. radio burst (A22)

Sept. 7, 1957 lSN90l'/ \ I Intense white I Total light I Kiepenheuer I (AZO) I SID, radio burst (AZ2)I I I I I I & Kuenzer I I

". l-1ar. 23, 19581 l4S74E 3+ I 8 I Bluish white I 50% I Total light I 1'/a1dmeier I (1\42) I PCA (A26) ; SID, IV (A21, A38); LPSI I r I I I I I (M)

TABLE A-l. FlARES \~ITII CONrINUlL\! OPTICAL RADIATION N.JD ASSOCIATED EVFNI'S(Continued)

34N63E I 2NE I 1

JWle 9, 1959

2 I \\lhite I I Total light I Denecke I (A3) I SID, IV (A38);' LPS (M), Surge (A3)I I I Several A I Cragg (A8) I SID, radio burst (A26)I I from " ~832,1 I II I " 6365 I

01S90E I 2 I 13 . I I I " 3600-3170 I Dunn, I (Al3) II 1 I I I Jefferies I II I I I I I & Orrall I "\

25N85F. I 2+ I I I Gennan (Al6) SID, IV, (A38)I I I I I I Capri Sta- I I

I I I tion I11 I I I I Orrall, DunnI (A13) I SID, IV, (A38)

I I I I I I Jefferies I I.July 14, 19591 22S36E 2+ I I I I Hrbik I (A19) I PCA (A34).; SID, II, IV (A38);

I I I I I I I I I LPS (1\4)Oct. 23, 19591 j 1 I 4 I I U I IJ. Becker I (A2) I* Sept. 3, 19601 18~88E 2+ 15 \'lhite I 15\ I .Total light Angle (AI) PCA, GLE (M3); SID, lv (A38);

I I I I I I I LPS (M)l1li Nov. 15, 1960 I 265331'1 I 3 I 3 I Pearl white I I Total light I Koyama I (A30) PCA, GLE (A26); SID (A38); II,

I I I I I I 0 I I I IV (A30. A23); LPS (M)IIIl Nay 23, 1961 I 21NZ4E.! 3B I 1 . I \'lhite I 15\ I " 5800A I De.\!astus & I (A9) IPCA, GLE (A31); SID (A1, MO); II,

I I I I I I Stover I IV (A6)JUly 8, 1968 I 13N56E I 3B I 15 I I I 4A away I Pannenter .1 (A32) I SID (A31a)

I I I I I I from Ha I I IL_ I.. I __ IL 1 L 1__ I

* Mar. 30, 1958\June 19, 19581

I?-Iar. 1, 1959 I

* Apr. 8, 1959 I

l1li Considered 'to be flares with contimlum optical radiatl.on of Type 1. Events up to and including Nov. 15, 1960 are after SVestkci, '1966.

-...J\0

80

Referen'ces to' Accompany Table' A-I

Al Angle, K. 1961, Observation of a White-Light Flare,P.A.S. Pacific, 73, 227-229.

A2 Beckers, J. M. 1962, A Search for White Light Flares,Observatory,' 82, 66.

A3 Becker, U. 1958, Beobachtungen von drei Eruptionen imWeissen Licht, Zs. f. Ap.,' 46, 168.

A4 Bruzek, A. 1964, Loop Prominences and Flares, Ap. J., 140,746-759.

A5 Carrington, R. C. 1859, Description of a Singular Appear­ance Seen in the Sun on Sept. 1, 1859, M.N.R.A.S., 20, 13.

A6 Castelli, J. P., J. Aarons, G. A. Michael 1967, The GreatBurst of May 23, 1967, AFCRL 67-0622.

A7 Cline, T. L., S. S. Holt, E. W. Hones, Jr. 1968, SolarX-rays> 80 keV, May 23, 1967 (private communication).

A8 Cragg, T. A. 1959, Photograph of a White Light Flare,P.A.S. Pacific,' 71, 56.

A9 DeMastus, H. L., R. R. Stover 1968, Visual and Photogra­phic Observations of a White Light Flare on 1967 May 23(personal communication).

A10 Dobbie, J. C., M. Moss, A. D. Thackeray 1938, Two SolarEruptions, M.N. R.A. S. ,~, 606.

All duMartheray, G. 1946, Orion, 11, 192, citing U. Becker1958, Beobachtungen von drei Eruptionen im Weissen Licht,Zs. f. Ap., ~, 168.

A12 duMartheray, G. 1948, Orion, 18, 403, citing U. Becker1958, Beobachtungen von drei Eruptionen im Weissen Licht,Zs. f. Ap.,46, 168.

A13 Dunn, R. B., J. T. Jefferies, F. Q. Orra11 1960, Line andContinuous Emission Observed in Two Flares, Observatory,80, 31.

A14 Ellison, M. A. 1946, Visual and Spectroscopic Observationsof a Great Solar Flare, 1946 July 25, M.N.R.A.S., '106, 500.

A15 Ellison, M. A., M. Conway 1950, The Solar Flare of 1949November 19, Observatory,' 70, 77-80.

81

A16 German Capri Station, Quarterly Bulletin Solar Activity,citing Svestka, Z. 1966, Optical Observations of SolarFlares, Space Sci. Rev.,"~, 385-418.

A17 Greenwich Observatory, Quarterly Bulletin Solar Activity,citing U. Becker 1958, Beobachtungen von drei Eruptionenim Weissen Licht, Zs. f. Ap., "46, 168 and Z. Svestka 1966,Optical Observations of Solar Flares, Space Sci. Rev., 5,388-418. -

A18 Hodgson, R. 1859, On a Curious Appearance Seen in theSun, M. N• R. A. S . ,~, 15.

A19 Hrbik, et ale 1961, B.A.C., 12, 169, citing A. A. Korchak1965, Electromagnetic Radiation with a Continuous Spec­trum During Solar Flares, Geomagnetism and Aeronomy,S,467-484. -

A20 Kiepenheuer, K. 0., G. Kuenzer 1958, Uber die Beobach­tungen Eines Solaren Auswurfs am Sonnenrande im Integral­licht, Zs. f. Ap., 44, 138.

A2l Koeckelenbergh, A. 1958, L'Eruption Solaire Remarquabledu 23 Mars 1958 a 09h 49mT.u., Ciel et Terre, l!, 450.

A22 Korchak, A. A. 1965, Electromagnetic Radiation with aContinuous Spectrum During Solar Flares, Geomagnetismand Aeronomy, ~, 467-484.

A23 Kundu, M. R. 1965, Solar Radi"o Astron"omy (New York:Interscience Press).

A24 McCracken, K. G. 1959, The Production of Cosmic Radiationby a Solar Flare on August 31, 1956, Nuovo Cimento, 13,1074-1080.

A25 McCracken, K. G. 1959, A Correlation Between the Emissionof White Light and Cosmic Radiation by a Solar Flare,Nuovo Cimento, 13, 1081-1085.

A26 McDonald, F. B. 1963, Sol"ar Proton Manual, NASA TRR-169,NASA, Washington, D. C.

A27 McNarry, L. R. 1960, The Observation of a Solar Event inWhite Light from Solar Event Resolute N.W.T. on August30, 1957, J. Roy. Astron. Soc. Canada, 54, 273.

A28 Meyer, P., E. N. Parker, and J. A. Simpson 1956, SolarCosmic Rays of February, 1956, and Their PropagationThrough Interplanetary Space, Phy. Rev.," "104, 768.

A29 Muller, R. 1951, Im Integralen Licht Sichtbare Sonnen­eruptionen, Die Naturwissenschaften, 38, 545.

82

A30 Nagasawa, S., T. Takakura, A. Tsuchiya, H. Tanaka and H.Koyama 1961, Flare on November 15, 1960, P.A.S. Japan,13, 129-134.

A31 Notuki, M., T. Hatanaka, W. Unno 1956, A Very UnusualFlare on February 23, 1956, P.A.S. Japan, ~, 52.

A32 Parmenter, B. C., Sky & Telescope, 36, 190-191.

A33 Porret, M. 1952, Communications Ecrites-So1ei1, L'Astro­nomie, 66, 22.

A34 Ried, G. G., and H. Leinbach 1959, Low-Energy Cosmic RayEvents Associated with Solar Flares, JGR, ~, 1801-1805.

A35 Rudaux, L. 1892, L'Astronomie, 11, 342, citing U. Becker1958, Beobachtungen von drei Eruptionen im Weissen Licht,Zs. f. Ap., ~, 168.

A36 Secchi, P. 1872, Compte Rendus, Acad. Sci. (Paris)(Letters), 75, 1581.

A37 Solar-Geophysical Data, IER-FB-274, June 1967, U. S. Dept.of Commerce, ESSA, Boulder, Colo.

A37 Solar Geophysical Data 1968, IER-FB-288, August 1968.a

A38 Svestka, Z. and J. 01mer 1966, Type IV Bursts, Bull.Astron. Inst. Czechoslovakia, 17, 4.

A39 Trouve1ot, N. 1891, L'Astronomie, 10, 287, citing Z.Svestka 1966, Optical Observations-of Solar Flares,Space Sci. Rev., ~, 388-418.

A40 Van Allen, J. A. 1968, Solar X-Ray Flares on May 23, 1967,Ap. J. (Letters), 152, L85.

A41 Wa1dmeier, M. 1941, Chromospharische Eruptionen II, Zs.f. Ap.,20, 46

A42 Wa1dmeier, M. 1958, Die Weisse Sonnen-eruption vom 23Marz 1958, Zs. f. Ap., 46, 92.

A43 Winckler, J. R., P. D. Bhavsar, et a1. 1961, Delayed Pro­pagation of Solar Cosmic Rays on Sept. 3, 1960, Phys. Rev.(Letters), ~, 488-491.

APPENDIX B

OBSERVATIONS ASSOCIATED WITH THEMAY 23, 1967 WHITE LIGHT FLARE

N N N

April 30, 19671900 UT

April 28, 19671305 UT

~

8785

April 27, 19671310 UT

8782 WE

.',

N N N

Hay 23, 19671215 UT

May 22, 19671315 UT

Hay 21, 1967"1220 UT

.__ UUL~ ~~~~ UUL~ 8811WE

FIGURE B-1. CALCIUM PLAGE REPORTS FORMC~~TH-HULBERT PLAGE NO. 8818

Plage Nos. 8793 and 8785 collided about April 28 and on the following cycle they re­turned on the east limb as 8818. Plage 8818 is moving faster than 8817 and begins tohave log jam effect by May 22. Flares appeared in 8818 throughout its transit of thesolar disk (from Solar-Geophysical Data, IER-FB-274, July 1967). co

~

N N N

E ---- -----

Hay 24, 19671310 UT

N

May 25, 19671330 UT

N

Nay 26, 19671250 UT

E ---.Hay 27, 1967

1300 UTMay 28, 1967

1330 UT

FIGURE B-1. CALCIUM PLAGE REPORTS FORMCMATH-HULBERT PLAGE NO. 8818 (Con~inued)

coU1

86

FIGURE B-2. LONGITUDINAL SUNSPOT ~'ffiGNETIC FIELD FORNCHATH PLAGE NO. 8818 (1967)

The data is from Rome. There seems to have been an apparentchange in polarity of the magnetic field of the encircledspot. The Russian report seems to concur with the May 27 po­larity. The dominant pole is leading in the northerm hemi­sphere and has south magnetic polarity for this solar cycle(from private communication, J. V. Lincoln).

ISN '-35 , :2:1. S'

i7N y21S 22N 1 23S n:;~ .,.P~d :liN17N 16N '.!.. -?)I/WJ'-...;;}'1}.,; 105 .v( 'f;:f.:, -', ~'l\'"E; ~ /

i' 1';/ ISSjf ~.j ~~:~ 17S 1" It .;.,ij 195

11N .f:.;"'" ,:;~ II 11S 21 /)-');../ ~ / )qS .'r:~~ :~: '.. J1'"5- ~./ /l::'r·)~·;/ 1'1$ /' •.,: >..r..;:::::::. 155 . " 211J ", •20ti 2.3N' 1~N ~'\.~ 235 ....,.:.::-

1's :' ,

" •-.-.'7J};" )0 S 13 ~ ~"/llfS. .....--21N 17.1/: -';' ....,,/2:l.f.j ':.~~: ~2.7.S-71) ". 'J.I?S

'10,t:iZ ,~,.':~ "'; ·r". ! ,..: .,' ;.. lSI( lOll

1111 / f7r.:"';; :", IllS I'·\;:~~~:/.I(/ /35 2,S.' r;',' \ .;-••".",. 175

11N '" 't '. /9N 205 I \1\ \ /I,N ~3H '/ I"~, ': ,I lZ3S • r\":~'. 1 ~ '-35 11>5 ' • {;;~ 2.IN I ~.., .

215 HI/ 210N \.~,~"< SN ION ' •• 'j" ISN ';',(l~':t.1 ., ',:,,,,.. ; I&N-V '.

': • :2 S.

'I~G).~ \I V20vcr

R1o

{I

R4 " ""

\ ., \ m

" j It.''',' \ \A---. "" · (. (SJ-

• v _ ~ ""\~" Vf>

VI4-R4-

Vl3

FIGURE B-3. THE SUNSPOT GROUPS IN PLAGE 8818OBTAINED BY MOUNT WILSON OBSERVATORY, lfillY 22, 1967

The complex group is made up of three separate groups 16372, 16373 and 16368. Datafor May 23were not taken (courtesy of Mount Wilson and Palomar Observatory) •

co.....:I

V:1.l

@./. °

1.,. \G

~VVI-

V2lf-

\,--

. Rl. .

t ,q

~

\D

,

Rr2

VIO

\."..-

e~\.... LV

/~1",--. C

) C

"Vb

RI!J.

RIc.

VIVVIr.

Rl3

"~

FIGURE B-3a. THE SUNSPOT GROUPS IN PLAGE 8818OBTAINED BY MOUNT WILSON OBSERVATORY, ~L~Y 24, 1967

The complex group is made up of three separate groups 16372, 16373 and 16368. Datafor May 23 were not taken (courtesy of Mount Wilson and Palomar Observatory).

CDco

2SS

.-._---..--- -...., --....-....... ---.-.... \

') \r J

/ I/ II I\ I

J S)( I, '"-- ./- ......_- ~ ------'"

N

~_ ..\

;;', I/ /I /,,-_..... /

2(,5

2.!)"N

.2(,$

2.SN _:~b: I_'" 2.SN~~ '\ ,~----~"

'" 1 ....... - ..., '/ ........"," ...~, / I ..... ,

, ::::... _./ I ,...... ....'" ,.. '" \ "X;' ~e1i\ ;,/,,--, \ 22N "

t'"~.. 1/", /;," "I ~, ... / / . '~'. II I \ \. S \

I ( )/' ',r .... ,I ~. '/ '-- ;/.(1 \

~t III -""~._ / '--- \

I 'VI (:'~\.,",,~ .... ,.II-S .2.2$ \

I (J .;fPS ,i I: ~ ) j 1.l1-S ~ \ J\ /Ij))~\ p" J \ ~J\ .r"\ \ \fi\~ "tJ.., D ! / /'~-'l ; ",/'....~ ....~... C) ,. '---~)~ ~-- l \Y N

\ ii'\ \ / ~ - /'''' (../ /

N I I" / ...._~, ;, -----........I '" ----- ./ " ,/ , ./........ r-J -,,,,'" "........_ I." ....---- / 1..../ "';, ,

... ----... I / 0 ;' -' \

I;' / I \

1I / / ,\ / I I

N ........ N ;' ( I

S.......... "', S I- / I I

(IbN) ---" ~ J'",_, ',I..... -;l..3S /

.......-.... /............. -.__ .... .",

19/.!·

FIGURE B-4. SUNSPOT CONFIGURATION AND PillGNETIC DENSITY FOR MAY 23, 1967The dotted outline is,the author's attempt to separate the regions of one polarity ofsolar bipolar magnetic fields (from data by Ussuzijsk, Kislovodsk and Rome). Accord­ing to N. R. Sheeley, 1966, all solar magnetic fields are bipolar. There are tinymagnetic patches of about 111 to 2" diameter with densities up to 700 gauss. Thesepatches diffuse away from the active regions after a flare and are found everywhereon the' sun.

co\.0

90

1815:30 U.T. 1830:10 U.T.

1836:10 U.T. 1838:10 U.T.

1839:10 U.T. 1840:10 U.T.

FIGURE B-5. SEQUENCE OF Ha FLARE PATROL PHOTOGRAPHSFOR·MAY 23, 1967. They show the pre-existing structures

in the region where the white light flare occurred(courtesy of Institute of Astronomy, Ha1eaka1a Observatory).

1829:50 U. T.6A =± 21

1840:50U.T.~).= ± 2A

1844:00 U.T.LINE-CENTER

FIGURE a-6. Ha FLARE PATROL PHOTOGRAPHS FOR THE MAY 23, 1967 WHITE LIGHT FLARE(courtesy of Sacramento Peak Observatory, Air Force Cambridge Research Laboratories)

\D.....

1844:05 U.T.-0.5 ~

1844:15 U. T.+0.5 ~

1844:50 UT6'A=:!2A

FIGURE B-7. Ha FLARE PATROL PHOTOGRAPHS FOR THE MAY 23, 1967 WHITE LIGHT FLARE(courtesy of Sacramento Peak Observatory, Air Force Cambridge Research Laboratories)

\0N

1844:00 U:r:lED)

---" ------

1840:00 U.l

•-'\.

oFIGURE 8-8. SUNSPOT PATROL CENTERED AT 5800 A FOR THE MAY 23, 1967 WHITE LIGHT FLARE

(courtesy of Sacramento Peak Observatory, Air Force Cambridge Research Laboratories) \0W

~

Ha. Integrated Intensity

.4

1.01 ."'7?,I. ... • ,

t.··.~ "\, 3B+, r'\ ...... \.8,.- (\.1 ",\

[fl.;"" .

· 6~ .~.~\\ f',I .... , .t " !"••1

I I ~ \.':i·\ . .'~:·~:·:;r~·~~·:F~l~\' I4 ....... 4. ~.' '.. ' I : .'\ ". ~ /'" I· r \,i;'~ !(.... ~ry.~~ ",:'''''. .

J' \..... j · '\,.':;. ,I

.. II. 2 ,.. ),.",} U\. - :'\p .. ,12' "'f';".;..:~~ BIll ;.v:,\......,. I~' ll,;.~j· rii'1

. ·r.7.~~'\;{":U~.'!' I} I V • ";;t:li';~:~I';:~/;~~' . ~ '1

~.ft' d orr 1 ~1::r \li':~~l"t~"·\-::~-.,--.J..::==----.l.----H-a.-p-e-a-k-I-n-t-e-n-s-i-t-y-----

r /.\. ,~ .. \I J. t....J \ ~ .~ ". t.( I.6r ! '''~" ..~ /.. 1,(."' .. .1 I .2B

jJ t .......t\~. I~!I I" 0';..... _.f, •• ~'",

~(I i·.. lt ,)I, 'r''',\J ::.i\·...~·•• 'i·~.;·~.~

4 (...1,. • ,./ :0' i ~ll" ".'11 1.

• i-.'., ,.r!.. J\ 'l {"" .. l.:.\.~ ,':.: ' .:: t r r'(1'1'\. '.".. I

L I J '. ~ '~:'>:... { :~'rt~ ..!~:", .. ",,!2 r~'\':.. 'J~r~1 I i ,-1.1 l~"":' l)i' II: .,:· I t!'''1 IfC .....'t"':'\.~ ~ t"\'!' ,· A, r.o ,f 'i I

)

;" I ·,·.;.;./l.,.. J r::.. •• , .:,i".';'-.•;(' ,I I ~.~ ".1 r' 01l.. ".-.......,, 'J

1. 0 I=~ 11 ...:':-' ( L...-.J I~ Iv' f.""

.8 SFn SFD, t/' "'\'~\:...!., SFD Ba Area I6 [ 1. 6 Hz 16 . 4 HZr\t "'\I.e.'.:-'\ 5 . 2 Hz j ../r:. ;\:{..l":!J''\'r\ ....;;. I

• I r-" ~ ~~'\

/

)•..,.} ;\~\~~~ ~ JIY ~ 't\..~ I~ \.~ I If ,\-rr::~ ~.~ ...;\~ r: ~ 1

2 I . • "\ /',. I. " -:,~",\.. .'''. I.'

• r- ~ 'f"~'j...~ r: I' . '''1 c ~

" ~~ .t·· 'f ! I tj r I ! , : J 1 f !

oi~06 16 26 36 46 56 1906 16 26 36 46 56 2006 16 2022

TIME U.T.

FIGURE B-9. VIDEOMETER TP~CINGS FOR THE MAY 23, 1967 FLARE(from private communication, P. E. Tallant, Sacramento Peak Observatory, Air ForceCambridge Research Laboratories).

\.0.t::-

95

50

(].J fS40r-Irt:lt>til

:>iH 10rt:lH.!J.r-!.Q 5H~

21002030

_11-_-'-__~__'----"

200019001830

1 '--_l..--_-'-_'--_-=-_-'-_--'__--'-_--'

1800 1930U.T.

FIGURE B-10. SKETCH OF INTENSITY CURVE FOR \\1IHTE LIGHT FLAREOF ~ffiY 23, 1967. The curve was sketched from relative intensi­ties at the·start, maximum and extinction times (from private·communi cation, H. L. DeNas tus anQ._~.R .~tove~r~)_'.'-- .,

2100203020001900--,-'--'

1830100

180010

1930U.T.

FIGURE B-11. SOLAR X-RAYS> 80 keV FOR }~Y 23, 1967According to Cline, the 1940 U.T. burst was harder than the1840 U.T. burst (from private cOMuunication, T.L. Cline, 8.S.Holt, E.W~ Hones, Jr.).

1000 10000

t>Q)tilI

C'l

S .!Jt> ~

""- 0:>Q) 'd~ (Tj

Q)0 HOJ ""-100 til 10001\ +J

~til ~~ 00 U.!J0.c:P-l

1835

I1000, 1846

1

0 rQ)UJI

~ loJHQ)I

.r-l

f 1817~

~ 1.r-lE:........

rx..

210020001900U.T.

10' , I I ! I

1800

oFIGURE B-12. SOFT X-RAYS 2-12A FOR ~AY 23, 1967

(from private communication, J. A. Van Allen 1968).\.0O"'t

1000

N::c:I

C'l

S.........Ul+l+lro~

C'l

C'l

I0~

~100

i23,000

97

1930U.T.

10 1.--__-1--1- 1 -1._

1800 1820 1900 2000 2030 2100

FIGURE B-13. 8800 HHz .. BURST FOR !-lAY 23, 1967Data from Sagamore Hill Radio Observatory, Hamilton, Mass.(from-So1ar-Geophysica1 Data, IER-FB-274, July 1967).

100,000'

10,000'

1,000'

100·

100,000'

10,000·

1,000'

100·

I I I I

606 HHz

I I

1415 HHz

98

1,000'

100 .2695 HHz

....1 '-.:.....1 --'---'----1...1-L-.!--,---,-,'l---ll--..-L-L-I

10,000'

1,000'

100 .

10,000

1,000

100 .

4995 HHz

8800 MHz

1830 50 1910 30 50 2010 30 50 (UT)

FIGURE B-14. THE RADIO BURST ASSOCIATED WITH THESOLAR PROTON FLAP~ OF ~ffiY 23, 1967

Data from Sagamore Hill Radio Observatory, Hamilton, Bass.Flux density is given in units of 10- 22 watts/m 2 -Hz (from'Solar-Geophysical Data, IER-FB-274, July 1967).

Off Scale

Detail ofStart

8on,64 ~

::~ II16r )j

0[-rrnTi18:30 18:35 D.T.23 DT2221201918

120e I IIB~~9 0 ~ [ljI pi .

60 I- I \''Y . ~\~~ ~------;---;---;-- ,. ,

Flux180·

150

Flux

8000r2300f1000

I

16027

01 1.tI" I t J J I , , r 1 I I ,

18:30 19:00 19:30 20:00 20:30 D.T.

FIGURE B-15. DETAILS OF THE 2800 MHz SOLAR RADIO BURSTSOF ~..AY 23, 1967

Data from ARO-Ottawa and Drag-Penticton, Canada (from Solar-Geophysical Data, IER-FB­274, July 1967) •. \0

\0

10

25...N:c 50~->t 100ucQ)

180='0-Q)w

330~

580 , i i i I

1832 U.T. 34 36 38 40

---....

42 44 46

180 -' •

330-_

>.u 100­c(l)

='0'(l)w~

N:c r. .•. T .....~ '.. . J l=~j::z ..- _.- --50 - ._~....;. - 5 ill

======.==...._-~_.: ...::.:.... :.=.=~~~;..;-- .....580 -

48 50 52 54 56 58 1900 02FIGURE B-16. DYNAMIC SPECTRAL RECORD OF THE MAY 23, 1967 WHITE LIGHT FLARE

(courtesy of A. Maxwell, Harvard Radio Astronomy Station, Fort Davis, Texas) ~C)C)

APPENDIX C

RADIATION DUE TO SELECTED MECHANISMS

10210gauss

1001000

// /

/ // /

/ /

/10 4 / // / // / // / // / /

/ / // / 1 fYciuss

/ / /n==~ 3/ cn 3 / /

/ // / /

/ / /// // n h==/.lO 12

TZOOOK :/./

/ /1// /:

/ / ./ 3 y//crn //. /

/ / // / /// / /

102!t / // / /// / /

/ . //

. / // // / // / /

/ / // / /

/ ,// ,/

,/

n==)..'611/cm 3 .: n;;10 10/cni 3 / n==10 9 /cm 3/

/I -.J-rLtL~_ /

10 L_.L-L_J I J '..-LUlL- I I I '-1

10 6 10 7 10 8 10 9

Electron energy in eV

10 5------,-------

oQ)U)

..........:>Q)

FIGURE C-l. RADIATION ENERGY LOSS RATES PER ELECTRONIN SOLAR HAGNE'l'OPLAS?·lA

(from V. L. Ginzburg and S. I. Syrovatskii, 1964)Synchrotron radiatibn----~~ = 10-3H2(E/mc 2)2 eV/sec

----- Bremsstrahlung radiationdE 1 •

-dt = 8xlO- ~nE eV/sec

Inverse Compton scatteringdE _ 25 2 /.,:" dt = 1.35 :<10 nphE eV sec, \1here E is in eV.

103 .-39

7:OJ

s::Q) -40:z;

'-oJ

........

.-.

;>......u -41Pol

tTlar-I

-4211 12 13 14 15 16 17 18

log v

FIGURE C-2.1. INVERSE COMPTON RADIATION BYMONO-ENERGETIC ELECTRONS (from A. A. Korehak, 1967)

Pc (v) _N n ~ 1.24 0 eh Fe (v/ve )

e ph = 1.65x10-4oFe(V/Ve)' where

F (v/v e ) ~ 1.65 v/ve for v/vc«l

~ ~ e-v/ve for v/v c »l, andve4kT 8nv e =~ (E/rne2)2, 0 = ~re2 = 6.65x10- Z5 em 2

•

1817

5

16

3.62

14

. v I-Xf(X) -­vel 2

o~9011.:8

I "I U!JI--ll.-J-J

13

f:Xl! 0.:1 I

12LLLUU

15log v

FIGURE C-2.2. INVERSE COMPTON RADIATION BY ELECTRONSWITH INVERSE POWER LAW DISTRIBUTION o

T~e distribution is chosen to have peak radiation at 4500 Awith Ek1 = 172 MeV (from A. A. Korehak, 1967).

Pc (v) X- 1· ..:l-,N n = 2.71x10- 4o X+2 vel

e phPe(v)Nenph

tTl-42~ 11

~-40 r=-----OJ

s::J Ek1 =172 MeV'-oJ

........::: -41 .;>......

UPol.......

APPENDIX C-3

BREMSSTRAHLUNG RADIATION

When a particle encounters the nucleus of the stopping

gas, it deviates from its initial direction of motion and

radiates energy in the direction of its motion. If the

nucleus has charge Z2 and mass M, then this acceleration of

the particle with charge Z2 is proportional to ZlZ2e2/M,·and

the radiation intensity is proportional to the square of the

acceleration. Thus, the radiation losses are 3 x 10 6 times)

as much for electrons as for protons and the radiation losses

for heavy particles (M»m) are negligibly small compared with

the ionization losses. However, for electrons with high

energies moving in heavy media, the radiation loss is a large

portion of its total energy loss and may completely dominate

at ultra-relativistic energies.

The effective cross section for the emission of photons

with energy between hv and hv + dhv by an electron with

energy Ek»mc 2 is given by

- dhvorad. (E,hv) dhv = 40 hv f (E,n) (C-3.l)

where - 2 [21Te21fe2]2=° - z hc 1~C2

f(E,n) is a slowly varying function of the initial energy of

the electron E and of n = (h~)/E and depends on the parameter

y = air, where a is the radius of the atom and r is the

effective distance from the nucleus at which the radiation

105

event takes place. According to the Thomas-Fermi model, the

radius of the atom is a = ao/z 1/

3, where ao = l37~/mc. From

the uncertainty principle, if p is the minimum momentum

transferred to the nucleus in collisions, then r -~/p. From

the conservation laws, the minimum momentum transferred to

the nucleus is

p = (mc 2 /2E) (hv/(E-hv»mc. Therefore,

y= (137mc 2/2E) (hv/(E-hv» .<l/z) 1/3. (C-3.2)

Usually y is written with the factor 100 instead of 137/2.

If Y » 1 (i.e. r « a), then the effective radiation dis­

tance is much smaller than the radius of the atom and screen­

ing may be neglected. If y « 1, then the variation of the

momentum of the electron occurs far from the nucleus and

screening is large (Bethe and Heitler 1934).

For the relativistic case, the values of f(E,n) for

various y were determined by Bethe and Heitler (1934) under

the condition of the Born approximation and are given below:

y » 1 (no screening) :

f(E,n) = [1 + (1-n)2 - 2(1-n)/3] x

[In (2E/mc 2 (l-n)/n) - 1/2]

Y = a (total screening):

f(E,n) = [1 + (l-n) 2 - 2(1-n)/3] x

[In (183z- 1 / 3 ) + (1-n)/9]

(C-3.3)

(C-3.4)

y < 2 (intermediate case ) :

f(E,n) = [1 + (l-n)2] [«h (y»/4 - (1/3)ln z]

- (2 (1-n ) /3 ) [( <1>2 (y) ) /4 - (In z) /3 ](C-3.5)

f(E,n) = [1 + (1-n)2 - (2/3) (l-n)] x

[In( (2E/mc 2) (l-n)/n) - 1/2 - C(r)-1/2]

106

(C-3.6)

orad.

where ~l' ~2 and C(r) do not have analytical expressions and

are represented either graphically or in tabular form (Bethe

and Heit1er 1934; Sturodubtsev and Romanov 1962) •

The bremsstrahlung cross section for emission of photons

cannot be represented analytically in the weakly relativistic

case Ek ~ mc 2 •

For the non-relativistic case the Born approximation

does not apply and a more exact evaluation is necessary. This

derivation was carried out and given by E1wert (1939) as

= 8z2amc

2[ln l:E:k + 'Ek-hvlx

3Ekhv IEk IEk-hv

[1Ek 1 - exp(-12 TI a z/mc 2 j Ek) ]1Ek-hV 1 - exp (-12 TI a z/mc 2

/ (Ek-hv» . (C-3. 7)

The power spectrum for bremsstrahlung radiation from an

electron with kinetic energy Ek impinging on a stopping

(hydrogen, proton) gas of number density of nuclei n is given

by

(C-3.8)

where the radiation cross sections orad. for the relativistic

and non-relativistic electrons are given by equations C-3.3

to C-3.7 above.

For the relativistic electrons, the expression app1i-

cable for the complete screening assumption will be used for

107

convenience. This expression is actually applicable for

ultra-relativistic electrons and tends to predict a larger

radiated power at the higher frequency x-rays than if the no

screening or the intermediate screening expressions were

used. For the non-relativistic electrons, Elwert's expres-

sion for orad. will be used. No particular effort will be

made to find the correct form for the mildly relativistic

regions; the relativistic and non-relativistic expressions

will simply be extended from the high energy and the low

energy regions into the transition regions. In most cases

this will not create additional problems and will give

reasonable estimates for the resultant radiation.

For the relativistic complete screening case, n = hV/Ek

is small and the expression for the radiation cross section

is given approximately as

orad. = (16/3) (o/hv) [In 183 + 1/9], (C-3.9)

where ° = 0.58 X 10- 27 cm2 /nucleus. Then the power spectrum

for a single electron with kinetic energy Ek is

= (16o/3)nch [In 183 + 1/9] ergs/sec-Hz.(C-3.l0)

The total power spectrum for electrons with an energy distri-

bution n(Ek)dEk is given by

(C-3.ll)

p(V) = (16o/3)nch [In 183 + 1/9] Ne erg/sec-Hz(C-3.l2)

108

where Ne =rE2n(Ek)dEk is the total number of electrons in

lEIthe energy interval El to E2• This expression is independent

of frequency or distribution. It is valid for particles with

relativistic energies much greater than the energy of the

electromagnetic radiation.

If we let x = Ek/hv, then the power spectrum for an

electron with non-relativistic kinetic energy Ek becomes

p(v,x) = (812/3) crhcn (rnc 2/hv) 1/2 L(x,v) (C-3.13)

where L(x,v) ==1 [1- exp (-12

I x-1 1 - exp (-12

IX + "/X=T •1nt= ~vx - vx-1

1TCt z v'mc"z /hv 1/IX) ] )(

1Ta zlrncz/fiv 1/1X=l

For an inverse power law distribution of electrons, with ex-

ponent X, the total power spectrum is

p(v) ::: bnv'mc 2 (hV)I/(2-X) Khoo

L(x,v)x- X dx

where b = (812/3)crhc = 4.34 x 10- 43 and

(C-3.14)

(C-3.15)

(C-3.16)

Korchak (1965) has numerically evaluated both L(x,V) for dis-

crete values of hv and x and the integral J(X,v) for discrete

values of hv and X. His results are for x = 1 to 50 and

x = 2, 3, 5 and 7. These calculations were extended to in-

clude higher values of x for L(x,V) and also for X = 1.5 for

J(X,v). These results are combined with Korchak's calculations

109

and given in Table C-3.1.

Plots of the power spectrum for a single electron with

kinetic energies Ek = 10 keV, 100 keV and equal to mc 2 are

graphed in Figure C-3.1. If we divide p(v,Ek)/n by Ne , the

number of mono-energetic electrons with energy Ek' this

figure also presents the power spectrum for a mono-energetic

beam of electrons with energy Ek. The cutoff for the curves

essentially takes place at Ek = hv. The curve Ek + 00 gives

the power radiated by an ultra-relativistic electron.

Figure C-3.2 presents the total power spectrum in the

form p(v)/(nNe ) for the power laws X = 1.5, 2 and 3, on the

assumption that the electrons have inverse power law distri­

bution with these exponents down to energy Ek. The curves

are calculated so that the total number of electrons with

energy greater than E1 is the same for all the different

cases. Consequently, the case with larger X will have more

low energy electrons than the case with smaller X, and thus

the contribution at the lower frequencies is larger for X

large than for X small.

TABLE C-3.1. VALUES FOR L(x,V) AND J(X,v)(from A. A. Korchak 1965)

110

I ' hv (keV)x 1.5 2 5 10 I 20 50 I 100 I 200

L(x,v)

1 0.8936 0.8026 0.5534 0.4094 0.2992 0.1958 0.1398 0.10022 0.3495 1.3364 1.3042 1.2876 1.2757 1.2651 1.2596 1.25583 1.3720 1.3658 1. 3507 1.3429 1.3373 1.3323 1.3297 1.32794 1.3458 1.3421 1.3331 1.3284 1.3251 1.3222 1.3206 1.31965 1.3160 1.3081 1.3020 1. 2989 1.2967 1.2947 1.2937 1.29306 1. 2751 1.2733 1.2689 1.2666 1.2650 1.2636 1.2628 1.26237 1.2415 1.2406 1.2398 1.2351 1.2339 1.2328 1.2322 1.23188 1.2105 1.2094 1.2068 1.2054 1.2045 1.2036 1. 2032 1.20299 1.1820 1.1811 1.1789 1.1779 1.1771 1.1764 1.1760 1.1758

10 1.1557 1.1550 1.1532 1.1523 1.1517 1.1511 1.1508 1.150612 1.1092 1.1086 1.1074 1.1067 1.1062 1.1058 1.1056 1.105514 1.0691 1.0687 1..0677 1.0672 1.0669 1.0666 1.0664 1.066316 1. 0312 1.0338 1.0331 1. 0327 1. 0324 1. 0322 1. 0320 1.031918 1.0033 1.0030 1.0024 1.0021 1.0019 1.0017 1.0016 1.001520 0.9758 0.9756 0.9751 0.9748 0.9746 0.9744 0.9744 0.974330 0.8718 0.8717 0.8714 0.8713 0.8712 0.8711 0.8711 0.871140 0.8009 0.8009 0.8007 0.8006 0.8006 0.8005 0.8005 0.800550 0.7482 0.7481 0.7480 0.7480 0.7480 0.7479 0.7479 0.7479

100* 0.601 0.601 0.601 0.601 0.601 0.601 0.601 0.601500* 0.339 0.339 0.339 0.339 0.339 0.339 0.339 0.339

1000* 0.260 0.260 0.260 0.260 0.260 0.260 0.260 0.2605000* 0.140 0.140 0.140 0.140 0.140 0.140 0.140 0.140

X= J(X,v)

1.5* 1.735 --- --- 1.5936 --- --- 1.470 ---2 1.1238 1.106 1.065 1.044 1.028 1.016 1.009 1.0053 0.531 0.518 0.488 0.472 0.458 0.451 0.446 0.4435 0.233 0.224' 0.204 0.194 0.185 0.180 0.177 0.1747 0.143 0.137 0.121 0.117 0.110 0.104 0.101 0.099

* Author's extensions of Korchak's table.

111

LL-..L..-l.....LJ..J.lll..L-.l-U..LLU '

21 2220

10

100

510 keV

19log v

C-3.1. NON-THER}l~L BREMSSTRAHLUNG DUE TO SINGLE ELEC­Electron of kinetic energy Ek incident on a gas withdensity n (number/cm 3

). p(v,Ek) is the equation C-3.13.Ek = 10 keV, 100 keV, and 510 keV

-42

- 4 3l.-1JLWll-.LLLLUl.I1.--L.L...J..16 17 18

- 4ac=------

FIGURETRON.nuc'lei

-40 r=-----__._-I;------.-------------,

-41

E -)-00

k- - - - -- - - - - - - - - - -_.

-e. -42,.....~r::l..;>-

..... -43tylo

M

-44

X=3

-45 llLL ! I I 1/ II

16 17 18 19 20 21 22log v

FIGURE C-3.2. NON-THER~ffiL BREMSSTRAHLUNG DUE TO ELECTRONSWITH INVERSE POWER LAW DISTRIBUTION. Ne is the nunilier ofelectrons with energy greater .than Ek = 10 ~eV..p(v)/nNe isthe equation C-~.15.

APPENDIX C-4

SYNCHROTRON RADIATION

Syn'chr'o'tr'on' Ra:di'a't'i'o'n' 'f'r'om 'a' S'i'n'g'le' Ext'r'eme'1'y' Relat'ivistic

Ele'c't'ron

Synchrotron radiation from relativistic electrons with

helical orbits in a magnetic field will be considered in this

Appendix. Inasmuch as there have been several serious mis­

interpretations of the expressions for the power derived by

Westfo1d, it may be appropriate to present some of the impor­

tant ideas and formula associated with this mechanism perti­

nent to this thesis. The mathematically detailed analysis

will not be repeated here. In this presentation, the results

are given in terms of the power received by a distant observer

and not in terms of the power emitted by the electrons.

Feynman (Feynman et a1. 1963) explains relativistic

radiation and, in particular, applies it to synchrotron radi­

ation from a single charged particle as follows. At a large

distance from a moving positive charge q, the electric field

is given by

(C-4.1)

where r is the unit vector pointing in the apparent direction

of the moving charge from the point of observation. Letting

the coordinate system be chosen such that the xz-p1ane passes

through the observation point and the z-axis in the direction

toward the central region of the movi~g charge (i.e. the xy­

plane perpendicular to the direction of radiation), then

113

there are two electric field components

Ex = (-q/c2. Ro ) (d2. x/dt2.) (C-4.2)

Ey = (-q/c 2. Ro ) (d 2. y /dt 2.) , (C-4. 2a)

where Ro is the fixed distance between the observer and the

origin of the coordinate system for the moving charge. If t

and T are the time of observation and the local time of the

moving charge particle, respectively, then these times are

related by

t = T + Ro/c + z' (T)/C. (C-4.3)

Neglecting the fixed delay time Ro/c, the solution x(t) is

sought in terms of the particle motion x' (T), which is the

time varying x-component of the particle position. This

solution x(t) is obtained by translating each point of x' (T)

by Z' (T) according to

ct = CT + z' (T) • (C-4.4)

Thus, when z' (T) is positive, the time axis is stretched and

when z' (T) is negative, the time axis is compressed. The

compressed portions of the time axis correspond to large

second derivatives in the coordinates of either x or y and

consequently large electric fields in these components.

For example, if a positively charged particle is going

through a counter-clockwise motion in the z-p1ane, then x(t)

as seen by the observer along the z-axis describes a hypocy­

cloid with a sharp cusp centered at the time when z' (T) takes

on the greatest rate of change in the negative direction

(i.e. at the top of the circular orbit). The second deriva-

114

tive of x(t} is very large near the cusp and there will be a

burst in the electric field as seen by the observer. Since

z' (T) becomes negative once per cycle, there will be one cusp

and one associated burst of radiation per cycle. The polari-

zation of this radiation is linear and in the x-direction.

For more, general particle motions, there will be both

Ex and By components present and the radiation will be a

general elliptical polarization, depending on the relative

phases of Ex and Ey . If the components are in phase, the

polarization will be linear. If Ex/Ey = e jTI/ 2 , the radiation

will be circularly polarized in the counter-clockwise direc­

tion and if Ex/Ey = e-jTI/2, the circular polarization is in

the clockwise direction. Nevertheless, irrespective of the

polarization, there will always be a pulse of radiation asso-

ciated with the maximum rate of decrease of z' (T).

Now consider a relativistic electron with velocity v

following a helical orbit with angle ¢ with respect to the

magnetic field H, and the plane wave radiated being observed

at a large distance in the direction k making an angle ¢ with

respect to the magnetic field, as shown in Figure C-4.la.

For an electron moving with a velocity component along

the direction of the observer, the gyro-frequency Wo = eH/mc

is Doppler shifted. This frequency shift is derived from the

observation that the phase of a wave is reckoned by a simple

counting of the number of wave crests passing a point in a

certain time interval, and therefore, it must be invariant

under a Lorentz transformation. An observer at point P in L

115

J

FIGURE C-4.laANGULAR RELATIONSHIP FOR ELECTRON

FOLLOWING A HELICAL ORBIT IN A UNIFORM MAGNETIC FIELD H

records the number of wave crests which reaches him in a cer-

tain time interval. At time t he will have counted (1/2TI)

(k·x + wt) wave crests. In another reference frame L', which

moves relative to L, with velocity v with both origin coinci-

dent at t = 0, another observer starts counting when the wave

crests passing the origin reach him and continue until t' ,

when the point P' coincides with P. In both cases the ob-

servers will have counted the same number of wave crests. For

the condition in Figure C-4.lb, the Lorentz invariance of the

phase is given by

(1/2TI) (k·x + wt) = (1/2TI) (k·x' + w't'). (C-4.5)

If this equation is expanded in terms of its perpendicular

and parallel component terms, substituting the general Lorentz

transformations

x' = x.l (C-4.6).1

k' = k.L (C-4.6a).1

,= 1/II-v2/.c 2 (xII + vt) (C-4.6b)XII

t' = 1/ll-y2/c2 (t + voxn /c ) , (C-4.6c)

116

x' x

V'~

l' . P

• p' J

:/' LL

FIGURE C-4.lbTHE REFERENCE FRAMES FOR THE RELATIVISTIC ELECTRONS

AND THE STATIONARY OBSERVER(L' and L are the coordinate systems for the moving electrons

and the stationary observer, respectively)

and equating the coefficients for xII and t, one obtains the

system of equations

k n 11 - V 2 /C 2 = kd + w'V/c

wll - V 2 /C 2 = k n ·v + w'.

(C-4.7)

(C-4.7a)

Solving these equations simultaneously we have

w,/l-v 2/c 2w =

1 - vJl /c(C-4.8)

where k n = w/c was used.

For the angular relationships between the velocity vJl

and direction of propagation k with the magnetic field H in-

dicated in Figure C-4.l, ~I = v cos~ cos8. Therefore, the

Doppler shift in the gyro-frequency for a relativistic elec­

tron going through a helical motion is given by

wH = (eH/mcy)/(l - Scos~cos8). (C-4.9)

The radiation is confined to a conical beam of half angle

l/y = moc 2 /E, centered about the direction of the velocity

vector v. For ultra-relativistic electrons, moc 2 /E becomes

117

very small and there is little radiation detectable except

when e ~ ¢. Thus, if we set ¢ = e and 8 ~ 1, the angular

dependence factor becomes 1/sin2 ¢ and

(C-4.10)

The period between the pulses of radiation from an e1ec-

tron with total energy E is T = 2~/wH. The pulses of electric

field radiation from the relativistic particles may be ex-

panded in a Fourier series

00

E (r ,t) = L En (r) exp (inwHt) '.n=-oo.

where the coefficient is

(C-4.11)

(C-4.12)

This last integral has been evaluated by Westfo1d (1959) and

is given for an electron by

2ewH . r) n 2 2 A

= - exp(J.nwH- -- (E;; +ljJ )K 2 / 3 {gn).t 1 +l3~cr c sine

(C-4.13)A A

where .t l and .t 2 are mutually perpendicular unit vectors in a

projection plane perpendicular to the direction of radiation

indicated by the propagation vector k. If we let k = k/lkl,A A A

then (.t l , k, .t 2 ) form a right handed system of unit vectorsA

with .t 2 along the projection of the magnetic vector H on the

projection plane. K1 / 3 (gn) and K2 / 3 (gn) are modified Bessel

functions of the second kind, which decay with increase in

the argument

n (l; 2 + .ljJ 2) 3/ 2

3 sin<j>

118

(C-4.14)

For each of the harmonics, the electric vector describes

an ellipse as a function of time. The minor axis of theA A

ellipse lies along ~2 and the major axis along ~l. The ratio

of the minor to the major axis of the ellipse is given by

(C-4.15)

The angle ljJ is positive if the k and H vectors are on the

same side of the velocity cone and negative otherwise. When

ljJ is positive, the direction of rotation of the E vector is

right-handed (clockwise as seen by the observer) and when ljJ

is negative, the rotation is left-handed. For the nth har-

monic, the energy flux density of the radiation averaged over

a period is

Pn = (c /8 7f ) IEn I 2 • (C-4.16)

At the very high harmonics, the spectrum is very close

and the usual practice is to treat it as a continuum radia-

tion. In order to carry the expressions into the continuous

representation we let

A critical frequency Vc = (3/47f) (eH/mc) 1 2 sin<j>

(C-4.17)

(C-4.18)

(C-4.19)

is defined. Then the spectral densities of the radiationA A

flux in the directions ~l and ~2 are given, respectively, by

(C-4.20a)

119

p(~) = (3/4'lT 2R 2 ) (e 3H/mc 2i;sin 2<j» (vlv c ) 2 ( 1 + ljJ2/i;2) 2 X

K~/3 (gv), (C-4.20)

p~d = (3/4'lT 2R2 ) (e 3 H/mc 2i;sin 2<j» (vlv c ) 2 (ljJ2/i;2)

(1 + ljJ 2Ii; 2) K~ I 3 (gv) •

For small angles of ~ =!(v,k), the main contribution comes

from p~l); i.e., the radiation with the electric vector

normal to the projection of H on the projection plane. Thus,

for ultra-relativistic electrons, the radiation is essentially

linearly polarized, independent of angle <j>.

The spectral distribution of the total radiation over

all directions o~ a single relativistic electron is obtained

by integrating p~l) and p~2) over all solid angles. In

these integrations p~l) and p~2) vanish rapidly outside an

angle ~ljJ - mc 2 /E and the contribution comes primarily from a

narrow ring ~n = 2'lTsin<j>~ljJ about the velocity cone. Thus,

the integral to be evaluated reduces to

R2 Jp~ 1,2) dn = 2'lTR 2sin<j> 1: pjl,2) dljJ

J;~d dljJ l3e 3 H 1Fl (vlvc )=

2'lTmc 2R2 sin2<j>-00

J;~ 2) dljJ l3e 3 H 1 F2 (vlvc )=2JImc 2R2 sin 2<j>

-00

where

(C-4.2l)

(C-4.22)

(C-4.22a)

(C-4.23)

120

F 2 (v/v c ) :: (v/2V c ) [rOO KS / 3 (x)dx - K2/ 3 (V/Vc )] (C-4.23a)v/vc

and F(V/V c ) - F 1 (v/v c ) + F2 (V/V C ) = v/vc foo Ks / 3 (x)dxv/vc

(C-4.23b)

Fp(V/Vc ) :: Fl (v/v c ) - F2(V/VC) = (v/v c ) K2/ 3 (v/v c ).(C-4.23c)

These integrals have been evaluated partly by Dort and

Walraven (1956) and extended by Westfold (1959). The table

and graph of the functions F(V/Vc ) and Fp(V/Vc ) are repro­

duced in Table C-4.l and Figure C-4.2.

The spectral distribution for the total radiated power

from a single electron received by the distant observer is

p(V) = 2TIR 2sin¢ f(p~l) + p~l» d~

l3e 3 Hp(v) = F(V/Vc ) ergs/sec-Hz.mc 2sin¢

(C-4.24)

(C-4.24a)

The polarization of the total radiation is given by

f [p~ 1) _ p~ 2) ] dQ K2/ 3 (v/v c ) Fp(V/Vc )IT = J [p~ 1} p~2)] = = .+ dQ

JOO Ks / 3 (x)dxF(V/Vc )

V/Vc(C-4.25)

The curve for TI is also plotted in Figure C-4.2. According

to the expression (C-4.l3) , the component of the electric"-

field En in the direction ~2 (along the projection of the

magnetic vector on to the projection plane) is proportional

to the angle ~ = ~(v,k). This angle ~ ~ mc 2/E goes to zero

for highly relativistic particles. Hence the radiation"-

would be strongly linearly polarized in the direction of ~l'

TABLE C-4.1THE FUNCTIONS F(x) AND Fp(X)

(from K. C. Westfo1d, 1959)

F(x) = x f: KS / 3 (n)dn

Fp (x) = XK 2 / 3 (x)

x F Fp x F Fp

0 0 0 1.0 0.655 0.4940.001 0.213 0.107 1.2 .566 .439

.005 .358 .184 1.4 .486 .386

.010 .445 .231 1.6 .414 .336

.025 .583 .312 1.8 .354 .290

.050 .702 .388 2.0 .301 .250

.075 .772 .438 2.5 .200 .168

.10 .818 .475 3.0 .130 .111

.15 .874 .527 3.5 .0845 .0726

.20 .904 .560 4.0 .0541 .0470

.25 .917 .582 4.5 .0339 .0298

.30 .919 .596 5.0 .0214 .0192

.40 .901 .607 6.0 .0085 .0077

.50 .872 .603 7.0 .0033 .0031

.60 .832 .590 8.0 .0013 .0012

.70 .788 .570 9.0 .00050 .00047

.80 .742 .547 10.0 0.00019 0.000180.90 0.694 0.521

121

101.00.1x

0.01

.2

o I 'I I ! I I I II I!! J ! ! I I J J "!!! I I! ! I \I 1>1.. I ! II

0.001

1.0

1 F(x)

.9f

.8i

.7rI

.6

.5

.4

FIGURE C-4.2. THE FUNCTIONS F(x), Fp(X) AND THE DEGREE OF POLARIZATION, ~

(from K. C. Westfold, 1959)

I-'NN

123

independent of the pitch angle ¢.

Radiation from an Aggregate of Particles

Radiation from an aggregate of particles received by a

distant observer will beJconsidered in this section. Con-

sider a collection of'~~c.trons all gyrating along the~.~

magnetic field lines at a pitch angle ¢ and the distant ob-

server is looking at the collection of electrons at the

angle ¢. The streaming electrons will be within the range of

observation for a short time L sin 2¢/c cos¢. The number of

electrons entering and leaving the region per second is pro-

portional to (c cos¢)/L. Hence the effective number of

particles as far as the observer is concerned would be pro-

portional to

[(c cos¢)/L] (L sin 2¢/c cos¢) = sin 2¢.

The number of particles in the volume element dV =r 2drdQ, and with velocities within the solid angle dQ

sin

the neighborhood of the direction ~ is given by n(E,r,~) x

dEdVdQs. To obtain the number of particles contributing to

the radiation received by a distant observer, this must be

mUltiplied by sin 2¢. Assuming the radiation from the indi-

vidual electrons is incoherent, the intensity of radiation

received by the observer along the direction k from the sys-

tern of particles is then

J(v,k) = sin2¢fj(v,E,r,¢,w)n(E,r,~)dEdQsr2dr (C-4.26)

where j(v,E,r,¢,w) = pel) + p(2) is the total radiation in-v v

tensity for a single particle. In general the integration

124

over r is carried out along the line of sight in the direc­

tion -k. However, if the source has a small angular size,

the quantity measured experimentally is obtained by inte-

grating over the entire volume of the source,

Since the contribution to the radiation comes from par­

ticles moving in the small angle ~~ ~ mc 2/E, the integration

over dns

reduces to an integration over d~, or

J(v,k) = sin2~fn(E,r,~) [fj(v,E,r,~,~)2TIr2sin~d~] dEdr.(C-4.28)

J(v,k) = sin2~fn(E,r,~) p(v) dEdr (C-4.28a)

J (v ,k)

dEdr. (C-4.28b)

Occasionally J(v,k) is written in terms of the volume

emissivity

(Note:

ergs/cm 3 -sec-Hz.

_ _ no (E, r,~)n(E,r,s) = 4TI )

(C-4.29)

For example, for a mono-energetic distribution of u1tra-

relativistic electrons, the intensity of radiation is

J(v,k)

fn(E,r,~)dEdr

J(v,k) = sin2~p(v)n(k) ,

(C-4.30)

(C-4.30a)

125

where n(k) is the total number of e1ectrons/cm2 in a unit

solid angle along the line of sight with the velocities

directed toward the observer, k.

Another example is that of an energy distribution of

electrons along the line of sight represented by the inverse

power law within a limited energy interval El < E < E2'

given by

(C-4.31)

where K(E/Eo)-X is the number of electrons along the line of

sight moving in the direction of the observer per cm2 per

unit solid angle per unit energy interval. The number of

electrons n(E,k) along the line of sight k may be evaluated

from the number density of electrons n(E) with arbitrary

direction of motion and in energy interval E to E + dE given

by

n(E/Eo) dE/Eo = K(E/Eo)-XdE/Eo,

- fL - -_ fLand n(E,k) = 0 n(E,r,k)dr 0 n(E)/4TI dr =

(C-4.32)

Ln(E)/4TI.(C-4.33)

Therefore, K(E/Eo)-X = Ln(E/Eo)4TI

LK (E/Eo)=

4TI(C-4.34)

or -K = LK/4TI. (C-4.34a)

Since J(V,k) = fnv(<j»dr, it is sufficient to determine the

volume emissivity nv(<j»,

(C-4.35)

Making the transformation between frequency and energy

according to x :: 2v (C-4.36)

neE/Eo) = K[ 2v J-X/23VHosin</>xj

126

(C-4.36a)

(C-4.36b)

Ke2

VH o (3)X/2( V )(I-X)/2. (X-)/= - __ (sJ.n</» I 2 x8nn 2 VH o

(C-4.37)

[G(V/Vc ) - G(v/vc )]2 I

where

G(v/vC> = foo X(X-3)/2 F (X)dxV/Vc

(Note: (FI + F2) = F (x)

G (V/V c ) = Joo X(X-I)/2 J: KS / 3 (I;;) dl;;dxV/Vc

(C-4.37a)

(C-4.38)

(C-4. 38a)

(X + 7/3) 2 (v/vc ) (X-I)/2

= + 1 Gp (v/vc ) - X + 1 xX .

(C-4.38b)

and

(C-4.38c)

This last integral cannot in general be evaluated in terms

of known functions and must be evaluated numerically. For

the particular case X = 5/3, the evaluation of this integral

127

can be made in closed form and is given by

(C-4.38d)

Thus Gp(x) and G(x) have been evaluated and given by

Westfold (1959). The functions are tabulated in Table C-4.2

and plotted in Figure C-4.3.

The volume emissivity nv (¢) has been evaluated for the

case X = 5/3 for various magnetic fields, energy intervals

E 1 and E 2 , and helical pitch angle ¢. These results are

graphed in Figures C-4.4 to 4.7.

If one is interested in the mid-frequency range such

that the radiation from electrons with energies E < E1 and

E > E2 contribute only negligibly to the volume emissivity,

then the limits of integration may be extended over the

entire energy interval. Ginzburg and Syrovatskii (1966)

have taken this approach. The volume emissivity is now

= ~r(3X-lJr(3X+19) ~ ( 3e ~X-l)/2~ xX+l 1 1271 12 mc 2 2TIm2c 5

) 4TI

H(X+l)/2 (sin¢) (X-l)/2 V-(X-l)/2 ergs/cm 3-sec­ster-Hz

(C-4.39)

and the degree of polarization is

TI = X + 1X + 7/3

(C-4.40)

It is assumed that X > 1/3. As in the single electron case

the polarization would be primarily linear.

When the radiation source covers a large region of

128

TABLE C-4.2THE FUNCTIONS G(x) AND Gp(X) FOR X = 5/3

(from K. C. Westfo1d 1959)

G(x) = (3/2 ) Gp (x) - (3/4 ) X1 / 3 [F (x) - FP (x) ]

Gp(x) =.X 1/

3 K 1 / 3 (x)

x G Gp x G Gp

0 :2.531 1.688 2.0 0.172 0.1470.2 1.585 1.158 2.5 .097 .0860.4 1.170 0.888 3.0 .056 .0510.6 0.891 0.696 4.0 .019 .0180.8 0.690 0.551 5.0 .0068 .00641.0 0.537 0.438 6.0 .0024 .00231.2 0.425 0.351 7.0 .00087 .000821.4 0.338 0.281 8.0 .00031 .000291.6 0.271 0.. 226 9.0 .000112 .0001061.8 0.215 0.182 10.0 0.000040 0.000038

3.0r---------------------------.

2.5

1.0

0.5

oL---'-_-L-_.L-.=:L::::::::c::~E:::?::::::::r:====x..___l

o 0.5 1~0 1.5 2.0 2.5 3.0 3.5 4.0 4.5x

FIGURE C-4.3. THE FUNCTIONS G(x) AND Gp(x) F9R X =5/3

16'

129

14

sin ep = 0.1

11I LLW.lL-..L-l...LU..UlI I' I I II 111..-L-lJ..l....II.......IJu-'--,-_LLLLWL.--l--L'JI_J-.U..L.U

1510 12 13log v (cis)

FIGURE C~4.4. VOLUME EMISSIVITY nv (¢)'CALCULATED FOR 10 2 GAUSS

El = 10 MeV, E2 = 1 BeV, and X = 5/3

- 25 l--J'---1-U I I III

9

-21 1::".:-------------------------

-22

tJ1o -24

r-I

-20 ,..-------------------------

= 1.0

sin ¢ = 0.1

16151410 11 12 13log v (cis)

FIGURE C-4.5. VOLUME EMISSIVITY nv (¢),CALCULATED FOR 10 3 GAUSS

.El = 10 MeV, E 2 = 1 BeV, and X = 5/3

_ 2 3L.---L--I--!.-I--L..Ull-L_LLUlllw.r--Jl-l-!LLWII I / J ! ( J.1IL-L..l.LlJ.UlL-l.-LLUll.u..1I _.L-J'-LI.UJl

9

-19 F_.~ ._1_3---,0

~ -20..........

,-..-e-

;:>~ -21......tJ'I0

r-I

-229

Ll..UIII' I II II II L.LLUIII I I I U.l1l! I I IIIIJI! I I "I III

10 11 12 13 14 15log V (cis)

1.0

= 0.1

16

FIGURE C-4.6. VOLUME EMISSIVITY n~(</»,CALCULATED FOR 10 4 GAUSS

El = 10 MeV, Ei = 1 BeV, and X = 5/3

1614 15

..LLL1U.L.--L-L.LLlll1.-L....LLLllUL---l-LU..lll.w..1I---,--,-I......1 uu " I I I

10 11 12 13log v (cis)

FIGURE C-4.7. VOLUME EMISSIVITY nv(</»,CALCULATED FOR 10 3 GAUSS

E 1 = 100 MeV, E 2 = 1 BeV, and X = 5/3

-269

-25

-21sin </> = 1.0

sin </> = 0.1-22 .

</> = 0.01

-23

tJ'Io

r-I

l-24......

131

space, where the magnetic field orientation differs over

different regions, it would be appropriate to assume the

magnetic field to be randomly oriented. The result of the

homogeneous magnetic field case above may be averaged out

over all magnetic field directions,

= fi r[ (X+5)/4]2" r[ (X+7) /4] .

(C-4.41)

With this averaging, the volume emissivity becomes

I31T r[(3X-1)/2] r[(3X+19)/12] r[(X+5)/4]nv(<j» = 2(X+1) r[(X+7)/4]

e3

[ 3e ] (X-1)/2 [~1 H(X+1)/2 V-(X-1)/2mc 2 2~m2c5 4n

x

ergs/cm 3 -sec-ster-Hz. (C-4. 42)

In extendi~g the limits of integration in equation

C-4.38 over the entire energy range, errors are introduced.

This error has been estimated by Ginzburg and Syrovatskii

(1966) not to exceed 10 percent at frequency v, if the fre-

quency is related to the energy interval over which the

inverse power law is applicable by

E1~ mc 2 (4nmcv/3eHY1 (X»1/2 ~ 2.5 x 10 2 [V/Y1 (X)H]1/2 eV

(C-4.43)

E2~ mc 2 (4nmcv/3eHY2(X»1/2 ~ 2.5 x 10 2 [V/Y2 (X)H]1/2ev .

(C-4.44)

The numerical values for Y1 (X) and Y2(X) have been evaluated

for various power laws by the same authors. These are given

below:

132

X 1 1.5 2 2.5 3 4 5

Y 1 (X) 0.80 1.3 1.8 2.2 2.7 3.4 4.0

Y2 (X) 0.00045 0.011 0.032 0.1 0.18 0.38 0.65

For X = 1.5, the range E1

to E2

over which the power law must

hold extends over a factor of a hundred. For X > 1.5, more

than 80 percent of the radiation at a given frequency comes

from electrons with energy differences of about factor of

ten and for X < 1.5, the energy interval increases rapidly

and becomes infinite as X approaches 1/3.

i ...... ".:::aw,

APPENDIX D-1DIVERGENCE OF ENERGY FOR ENERGETIC PARTICLES

TABLE D-1.1. MODEL ATMOSPHEREDensity Distribution of the Van de Hulst (1953)

Model for the Photosphere and Chromosphere .

h (km) ~x(cm) Ne = Np No Nn = No + Np

6000 0 3.98 X 10 9 2.5 X 10 8 4.23 X 10 9

5000 10 8 6.4 X 10 9 1. 58 X 10 9 8.0 X 10 9

4000 10 8 1 x 10 10 1 X 10 10 2 X 10 10

3000 10 8 1.58 X 10 10 3.98 X 10 10 5.56 x 10 10

2000 10 8 2.5 X 10 10 2.5 x lOll 2.75 X lOll

1500 5 x 10 7 3.16 X 10 10 2 X 10 12 2.03 X 10 12

1000 5 X 10 7 5 X 10 10 2 X 10 13 2 X 10 13

750 2.5 x 10 7 6.4 X 10 10 7.94 x 10 13 . 7.94 X 10 13

500 2.5 X 10 7 1 x lOll 3.16 X 10 14 3.16 x 10 14

250 2.5 x 10 7 2.5 x lOll 1. 58 X 10 15 1. 58 X 10 15

0 2.5 X 10 7 1 X 10 12 5 X 10 15 5 x 10 15

-50 5 x 10 6 2.08 X 10 12 1.04 X 10 16 1.04 x 10 16

-100 5 x 10 6 3.16 x 10 12 1.58 x 10 16 1. 58 X 10 16

-150 5 x 10 6 5.56 X 1012 3.29 X 10 16 3.29 X 10 16

-200 5 x 10 6 7.95 X 10 12 5 X 10 16 5 X 10 16

-250 5 x 10 6 1.98 x 10 13 7.5 X 10 16 7.5 X 10 16

-300 5 x 10 6 3.16 x 10 13 1 x 10 17 1 X 10 17

-350 5 x 10 6 3.31 X 10 14 1.29 X 10 17 1.29 x 10 17

-400 5 x 10 6 6.3 X 10 14 1.58 X 10 17 1.68 x 10 17

134

APPENDIX D-2

DIVERGENCE OF ENERGY FOR ENERGETIC PROTONIN SOLAR ATMOSPHERE

The divergence of proton energies due to elastic colli­

sions which the free electrons is given by (Sturodubtsev and

Romanov 1962)

where

-dE/dx = neJQdo = ne Be 1n Qmax./Qa'

2 2 1+ 2rre l+rrz 1 z 2 e m1Be = = --2 ,T m2 m2 v 1

(D-2.1)

Qa =m1 z 1 Z2

m2

Ry - e 2 /2a o = 13.6 eV = 1 Rydberg,

T(Ry) = kinetic energy of incident particle in labo­ratory system expressed in Rydberg.

Subscripts 1 and 2 stand for incident and target particles,

respectively.

Equation D-2.1 may then be written in terms of electron

volts as

(D-2.2)

For the inelastic collisional loss, the only modifica-

tion to equation D-2.1 required is in the logarithmic term.

Here

135

Thus, the inelastic collisional loss per unit length becomes

(D-2.3)

Letting E = T

(D-2.6)

where no is the number of neutral hydrogen atoms.

Therefore, since it is necessary to consider only these

two processes in the energy range of interest, the expression

for the average energy loss per unit length is given by

27Te4rrp [= Trne {ne 1n[2(rne/rrp)2 (TeV/13.6ev) x

(1 + E/npc2)]} + 2n, In [ (4me/np> (Til) 1]. (D-2. 4)

+ rrpc 2 = total energy, then dE = dT since rrpc 2

is constant. Then if we divide through by the rest energy

of the proton we have

Now we let y = T/rrpC 2 and we have

_~ = 27Te4

1 { 1n f2 (ffie C2) ev)2 ( y2) [2+Y1] +

dx (rne c2 ) (rrpc 2) y no l 13.6eV

4rne c2y }2n o 1n I .

It has been normally accepted that the mean ionization poten-

tia1 I ~ 11.5 eV for a hydrogen gas. If we use this value

and evaluate the constants in the equation we have

- ~ = 2.56 ; 10-28

{ne In[2.8 x 109 y 2(2+y)] +

2n O In(1.77 x 10lty )}.

136

(D-2.7)

1000

MeV

10 100!:iW (keV/km)

800

:::~500 L400

300

200 L 75 MeV-e:~

...... 100..c::

0

-100

-200

-300

-400

-5000.1 1.0

FIGURE D-2.1. ENERGY LOSS FOR ENERGETIC PROTONSEk = 25, 50, 75 MeV

I--'W-J

900

800

700

I600t

500

I400

- 300S~...... 200..c:

10: ~

-100r-200 ~

-300

r-400

0.1 1 10 100 1000/::"W keV/krn

FIGURE D-2.2. ENERGY LOSS FOR ENERGETIC PROTONSEk = 93.8, 300, 900 MeV

.

I-'Wco

APPENDIX D-3

DIVERGENCE OF ENERGY FOR ENERGETIC ELECTRONSIN SOLAR ATMOSPHERE

For the relativistic electrons, the energy divergence is

made up of losses due to ionization collision, elastic co11i-

sions with field electrons and protons and atoms, and

bremsstrahlung losses. There should now be an obvious re1a-

tivistic correction because of the high velocity of the ener-

getic electrons. The mean ionization energy loss per unit

path length is given by (Sturodubtsev and Romanov 1962)

(D-3.1)

where y is the ratio of total energy to the rest energy of

the electron. The mean bremsstrahlung radiation loss for

relativistic electrons for total screening is given by

(Sturodubtsev and Romanov 1962)

(D-3.2)

where orad. is given in Appendix C-3. Hence

L dE) = ~(l+b)\ dx rad. Xo (D-3.2a)

where Xo = avalanche length = distance along which a fast

electron loses l/e of its energy,

1 4nn o 1n(183z- 1/

3)

Xo=

-Z2 (27Te 2 /hc) (e 2 /mc 2

) 2 = O.S80z 2 10- 3barns/nuc1eus° = x

nn = number of nucleus (atoms and protons) per cm 3

140

The expressions for the elastic collisions between the

incident electrons and the field electrons and protons are

similar to the expressions given in D-2, with the proper

interpretation for the subscripts.

The total mean energy loss per unit length by re1ativ-

istic electrons due to ionization, bremsstrahlung and elastic

collisions with free electrons and protons are given by

-dE/dx = -[(dE/dx)ion + (dE/dx)rad. + (dE/dx)e1as.e. +

dE

dx

(dE/dx) e1as. p. ]

2TIe 4 [T2 (l+Y) 2y-1= 2 no 1n 21 2 + (l-S 2)- ----y21n2 +mev 1

(D-3.3)

Here again the energy loss due to elastic collisions with the

free electrons is much greater than that due to the free pro-

tons by about the mass ratio. The remaining three terms will

be competitive according to (1) energy and (2) height in the

atmosphere because the contribution depends on the density of

the different particles. For example, at very high altitudes

where most of the hydrogen is ionized and no is negligible,

bremsstrahlung and elastic collisions may be the important

contributing factors in the divergence expression.

Now if we normalize in terms of the electron rest mass

141

and let E = T + mec2 we have

dYe---dx

2y-1+ (1-B 2

) + --2-1n2 +y

(D-3.4)

Evaluating the constants we can render the equation for

computation into the form

dYe 4.98 x 10- 25 { 9 2- -dx- = B2 2. 3ne log (1. 4 x 10 Y ) +

+2 2 1+y

Y mec 2I 2

2y-1 + y1 (Y_y1)2]}n o [1-B 2 -O.693 --2-y

2.3n o log

. ' .. '.'

(D-3.5)

1.15 X 10- 24 {= B2 ne1og(1.4 x 10 9 y 2) +

n o1og[y2(1+y)9.85 x 10 8] +

llY-1)2 }+ -- ]Y Y

1090100/:'W (keVjkm)

10

75 MeV

50 MeV

1.0

800

700

600

500

125 MeV

400iI,

300

l? 200~-.c

100

10

-100

-200

-300

-400

-5000.1

FIGURE D-3.1. ENERGY LOSS FOR ENERGETIC ELECTRONS.Ek = 25, 50, 75 MeV

~

~

tv

800

700

600

500

400

300......E:8. 200..r::

100

0

-100

-200

-300

-400

-5000.1 1.0 10 100 1000

/),W keV/km

FIGURE D-3.2. ENERGY LOSS FOR ENERGETIC ELECTRONSEk = 100, 300, 900 MeV

f-'~

w

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