STUDIES on HARD X-RAY EMISSION from SOLAR FLARES and on Cyclotrop ~- RADIATION from a COLD Magnelt)Pllima

STUDIES ON HARD X-RAY EMISSION FROM SOLAR FLARES AND ON CYCLOTROp ~- RADIATION FROM A COLD MAGNElt)PLliMA

PETER HOYNG 30VST: Solar flare of 7 August 1972, taken 0.5 A offband Ha at UT 1520:20. field of viev 3' x 4' (courtesy Big Bear Solar Observatory) STELLINGEN

Het elektromagnetische veld in een willekeurig medium is op na- tuurlijke -wijze splitsbaar in stralingsvelden (index s) en Coulomb- velden (index c) -.

->->-->-->- 1 S •"*• **•_ p = D [p ^ o] ; J=J + J ; J =--— -r— D [div J - o] ; c s sec 4TT dt c s

-> -+.-•-»•-* -+• E - E + E ; B = 3 [B Ho]. se s c

Beide deelvelden met hun bronnen voldoen aan de volledige Maxwell vergelijkingen:

div. D = o; div B ~ o div D = 4TTO s 3 c ' c

-* 13-* rot E = -—•=— B rot E = o s c at s c

* 4T -* 1 J •* rot H = J + — T— D s c s c ót s

Het verdient aanbeveling bij de behandeling van elektromagnetische problemen deze splitsing vóóraf uit te voeren en (zonodig) beide vergelijkingssystemen gescheiden op te lossen.

II Het leidt ir. principe tot onjuiste resultaten wanneer men bij de berekening van het uitgestraalde vermogen per eenheid van ruimtehoek een scalaire uitdrukking als uitgangspunt neemt -- zoals bij- .-r -f 3-f- •+• voorbeeld . E"J d r, de arbeid verricht door het door J opgewekte elektrische veld.

dit proefschrift, hoofdstuk VI. UI Het onlangs door Molodensky gegeven bewijs van de stabiliteit van het randwaardeprobleem van een krachtvrij magnetisch veld is onjuist; veeleer wordt nogmaals aangetoond dat zeer kritische beoordeling onontbeerlijk is om een tijdschrift op peil te houden.

Molodensky, M.M. : 1974, Cjilar Ihua. 39_, 393.

IV Op grond van het thans beschikbare waarnemingsmateriaal moet he.t waarschijnlijk worden geacht dat de afmeting van de bron van zachte en, a fortiori, harde Röntgenstraling in normale 4 zonnevlammein hoogstens van de orde van 10 km is, in alle richtingen.

De meest frappante gevolgtrekking uit waarnemingen van harde Röntgenstraling van de zon is niet zozeer dat de tijdens zonnevlammen versnelde elektronen een machtsspektrum zouden hebben, als wel dat hun aantal zeer groot is.

dit proefschrift, hoofdstuk II.

VI Optimale informatie over een tweedimensionaal beeld, xraarvan spleetaftastingen in een aantal verschillende richtingen bekend zijn, kan worden verkregen door middel van het principe van maximalisatie van informatie-entropie. Deze methode leidt tot een eenvoudige operationele konstruktie voor het ensemble- gemiddelde van alle door de randvoorwaarden toegelaten beelden. VII Regelmatig blijkt opnieuw dat problemen van gegevens-verwerkende aard tijdens de loop van een ruimteonderzoekprojekt ten onrechte zeer grote organisatorische knelpunten vormen.

VIII Het onlangs door O'Neill voorgestelde ontwerp van leefgemeen- schappen voor kolonisatie van de ruimte lijkt meer op een goed uitgewerkte mechanika opgave dan op een serieuze bijdrage tot de oplossing van het vraagstuk van de dreigende evenwichtsverstoring van de aardse biosfeer.

O'Nei.lI, K.G.: 1974, Nature, 250, 636. O'Neill, K.G.: 1974, Phy3ics Today, septenbernusaner. TIME, 26 mei 1975.

IX De manier waarop de Rijksdienst voor het Wegverkeer heeft omge- sprongen met Ie door middel van het kentekenbewijs deel III verkregen gegevens onderstreept nogmaals de grote achterstand die er in Nederland bestaat bij de invoering van wettelijke regelingen aangaande de registratie van persoonsgegevens.

Een analyse van het slingersysteem in de bij vele uurwerken ge- bruikelijke uitvoering leert dat de relatieve frekwentiestabili- -4 teit van de orde 10 is. Dit is in overeenstemming met de waarneming.

P. Hoyng 23 juni IS STUDIES ON HARD X-RAY EMISSION FROM SOLAR FLARES AND ON CYCLOTRON RADIATION FROM A COLD MAGNETOPLASMA

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE RIJKSUNTVERSITEIT TE UTRECHT, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. DR. SJ. GROENMAN, VOLGENSBESI.UIT VAN HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP MAANDAG 23 JUNI 1975 DES NAMIDDAGS TE 4.15 UUR

DOOR

PETER HOYNG

GEBOREN OP 20 JULI1941 TE 'S-GRAVENHAGE

DRUKKERIJ ELINKW1JK BV - UTRECHT PROMOTOR: PROF. DR. C. DE JAGER Aan mijn vader, ter nagedachtenis aan mijn moeder aan Vivienne Veel dank ben ik verschuldigd aan alien die aan de voltooiing van dit proefschrift hebben bijgedragen. In het bijzonder koint mijn dank toe aan mijn promoter, professor \C. de Jager, sn aan Dr. G.A. Stevens, Dr. J.C. Brown en Dr. H.F. van Beek, die ijeder op hun eigen specifieke wijze hun stempel op dit werk hebben gedrukt, Ik ben erkentelijk voor alle hulp die ik ontving van de heer Ch. Lapoutce bij talloze problemen van data-verwerkende aard, en van mej. C.F.M. Jansen bij het opsporen van vele artikelen. Ik prijs mij gelukkig met de vakkvindige metlewerking van mevr. H.E. Velders-Geurtse bij de uitvoering van het typewerk, en van de heer J.M. Braun bij de verzorging van de vele figuren. Het was niet te voorkomen dat mijn vrouw Vivienne en onze beide kinderen het slachtoffer werfien van het vele werk dat moest gebeu- ren. Vermoedelijk zullen zij Bevrijdingsdag voortaan vieren op 23 juni in plaats van op 5 mei. CONTENTS

CHAFTEK I INTRODUCTION AND SUMMARY. IT

CHAPTER II HIGH TIME RESOLUTION ANALYSIS OF SOLAR FLARES OBSERVED ON THE ESRO TD-1A SATELLITE. 14

2.1. Introduction 14 2.2. Instrument and satellite 16 2.3. Event selection 19 2.4. Spectral analysis 21 2.5. Time series analysis 51 2.6. Correlations 64 2.7. Implications for X-ray flare models 68

CHAPTER III INTERPRETATION OF HARD X-RAY AND MICROWAVE EMISSION OF THE FLARE OF MAY 18, 1972, UT 1406. 79

3.1. Discussion of the X-ray data 79 3.2. Radio data and their relation to the X-ray data 84

CHAPTER IV 3ETATR0N ACCELERATION IN THE LARGE SOLAR FLARE OF AUGUST 4, 1972. 86

4.1. Introduction 86 4.2. The August 4, 1972 event 88 4.3. The betatron model 55 4.4. Collisional and other corrections 102 4.5. The magnetic field changes 104 CHAPTER V PHYSICAL CONDITIONS OF ELECTRON ACCELERATION IN SOLAR FLARES. 109 c. .1. Introduction 109 5.2. Implications and assumptions for the hard X-ray source region 110 5.3. Generation of Langmuir waves from currents sheets 112 5.4. Acceleration of electrons 118

CHAPTER VI RADIATION FROM A SOURCE IN A COLD MAGNETOACTIVE PLASMA, REVISITED. APPLICATION TO CYCLOTRON AND MULTIPOLE RADIATION. 123

6.1. Introduction 123 6.2. Outline of the problem 126 6.3. Computation of Green's tsnsor G 131 6.4. Cyclotron radiation 140 6.5. Multipole radiation 146 6.6. Concluding remarks 147

SAMENVATTING 158

CURRICULUM VITAE 160 C:HAPTI:RI INTRODl'CTIOX AM) SIMMARY

The plasrr.a i.n the ;y.:t (-r lay.-r-: c* :.h>- ." ;r. IF atror,"ly ir.r.'-.v.o-j'.-r/.-.v.-.-, and, as i rule-, pt-rn.<-i!it_-.rit iy out L, ? UOJI: ii.n^ri. This is t. rj. home out r,y the appearance oi' active rc.-q ;cr.".;, and oni5 cf the most extrf-iy.f- rr.ar.: f eK'a' : ons

l is the- sclar f'rir.- pi.'.'iyj-f-.-.o:., .s-Vl:_:. i j -.-.-iS'T;'-.:-J! iy i 1 ar 7<_- -:; _dl'.- . 1'," >,-., plasraa-instability, regularly occurring in active regions. Among other 2 things, hard X-rays are emitted during flares for ""10 s - in some cases up to u 10 s - in the energy range 10-250 keV and higher. These X-rays are generated as bromsstrahiung ir. collisions of accelerated electrons with ambient protons. During the years VJ(JH-1'<~2 a spectrometer was developed .t Utrecht for detection of this transient solar X-ray emission (Van Beek, Ij73). The instrument was flown by the ESRO TD-1A satellite and it has operated successfully for two years after launch, in March 1372. From the measured spectra and intensity one can derive the energy distribution of the accelerated electrons and, in addition, an impression of their total number is obtained. This serves as a basis to study electron acceleration mechanism(s) operative during solar flares. This thesis consists of two parts. The first and larger part com- prises the chapters II to V, inclusive, and centers round the interpretation of solar flare hard X-ray emission. The second part, chapter VI, is a theoretical study on cyclotron radiation from a cold magnetoplasma.

In chapter II (Hoyng et at., 1975) the hard X-ray observations from the first two scans are presented and analyzed together with various derived parameters. The emphasis is on presentation in a self-contained format allowing independent usage by other workers; and in the analysis general issues are stressed such as spectral properties, time series analysis, presence of periodicities and correlations. Two conspicuous results are: (1). The number of accelerated electrons is so large that they can hardly constitute a beam; their velocity distribution must rather be roughly isotropic. This is elaborated in chapter V. (2). A remarkable correlation was found to exist between the spectral parameters of the large August 4, 1972 flare.

-11- CHAPTI Kl INTRODI'CTION AND Sl'MMARY

The plasma in the outer layers of the Sun is strongly inhemogeneous_^ _ -Sf . «! and, as a rule, permanently out of equilibrium. This is e.g. borne out by - j\ .r T y* the apoearance of active regions, and one of the most extreme manifestations i% 4 ~- r is the solar flare phenomenon, which is essentially a large-scale (^ 10 km)_ plasma-instability, regularly occurring in active regions. Among other * 2 " % things, hard X-rays- are emitted during flares for "^ ID s - in some cases J* 3 - 1 '•> up to u 10 s - in the energy range 10-250 keV and higher. These X-rays 1 I are generated as breir.sstrahlung in collisions of accelerated electrons with -z. • ambient protons. ~i , During the years l'Jbii-lj~2 a spectrometer was developed .t Utrecht for detection of this transient solar X-ray emission (Van Beek, 197 3). The in- struraent was flown by the ESRO TD-lA satellite and it has operated successfully for two years after launch, in March 1972. From the measured spectra " and intensity one can derive the energy distribution of the accelerated electrons and, in addition, an impression of their total number is obtained. This serves as a basis to study electron acceleration mechanisni(s) operative during solar flares. —, This thesis consists of two parts. The first and larger part com- ' prises the chapters II to V, inclusive, and centers round the interpretation of solar flare hard X-ray emission. The second part, chapter VI, is a theoretical study on cyclotron radiation from a cold magnetoplasma. -"- In chapter II (Hoyng et al., 1975) the hard X-ray observations from ~" the first two scans are presented and analyzed together with various derived parameters. The emphasis is on presentation in a self-contained format "^ allowing independent usage by other workers; and in the analysis general issues are stressed such as spectral properties, time series analysis, presence of periodicities and correlations. Two conspicuous results are: (1). The number of accelerated electrons is so large that they can hardly constitute a beam; their velocity distribution must rather be roughly isotropic. This is elaborated in chapter V. (2). A remarkable correlation was found to exist between the spectral parameters of the large August 4, 1972 flare.

-11- This is discussed further in chapter IV. In the chapters III and IV two models are discussed: In chapter III (Hayng. assi Stevens, 1971) a set of consistent parameters are derived for the flare of May 18, 1972 using the hard X-ray observations in combination with observations of the simultaneously emitted centimeter radio waves. In chapter IV (Brown and Hoyng, 1975) a model is suggested for the large flare of August 4, 1972, on the basis of the above mentioned correlation between its spectral parameters. The idea is that a coronal magnetic trap is filled with fast electrons by a brief, unspecified acceleration process. Trap eigerunode oscillations are excited and in addition, expansion occurs. Both movements are visible in the observed X-ray time profile by the action of betatron acceleration on the fast electron population. Chapter V, finally, goes into the problem of the electron acceleration mechanism in ordinary solar flares. Using a few crucial observational facts as a basis, it is suggested that electrons are accelerated by collective interactions, that is electrostatic interaction between one single electron with a group of electrons partaking in a coherent wave notion. These phenomena are presently studied intensively in laboratory plasmas. It is suggested that in the case of solar flares these waves are generated in thin current sheets; these waves propagate laterally from the sheet, causing electron acceleration in a relatively large adjacent plasma volume.

The second part, chapter VI (Hoyng and Stevens, 1975) is an outgrowth of attempts to reconcile solar flare hard X-ray and radio emission. It is a theoretical study on the generation of cyclotron radiation by a source in a cold magnetoplasma. In this work, the emphasis has been on a correct definition of the radiation flux as a function of direction and frequency, using Poynting's vector. A new result is found for the emitted power per unit frequency and per unit solid angle; among other things, an expression is found for the harmonic frequency that differs non-trivially from the one commonly used. The origin of these differences is rather technical.

-12- REFERENCES. van Beek, H.F.- 1373, . .- v.' z".'".' z*ui :•'>'? r'i:K?# / ; /.'.•:- •. ;.»j X-ray Spectrometer, Utrecht University (Ph.D. ThPsisi. -Jf— Hro^n, J.C. and Hoyng, P,-• 1975, Aatpophys. J., to be published. Jc

Hoyng, P., Brown, J.C. and van Beek, H.F.: 1975, Solar Phys., to be submitted. i,

Hoyng, P. and Stevens, G.A.: 1973, in J. Xanthakis (ed.), Solav Activity and Related Inter- 1 f platetiry tni2 Tarrestrio.l Phenomena (Proceedings First European Astronomical Meeting, i Athens 1972}, vol. 1, p. 97. %

Hoyng, P. and Stevens, G.A.: 1975, PhyB. Fluids, to be published. -

-13- CHAPTER II .UTION ANALYSIS OF SOLAR FLARES THE ESRO TD-lA SATELITE

ABSTRACT. The Utrecht hard solar X-ray spectrometer S-1QO on board the ESRO TD-lA satellite covers the energy range abova 25 keV in 12 logarithmically spaced energy channels. The time resolution is 1.2 s for the four low energy channels (25 - 90 keV) and 4.8 s for the remaining. Observations of a number of solar flares in the period 12/03/72 - 01/1G/73 are presented, among which four flares from the highly active 1-8 August 1972 period. The observations are entirely free from instrumental defects.

Spectral parameters (thick target parameters) for these flares are presented and analyzed. The suggestion is made that the hard X-ray observations admit an classification in two groups; the interpretation of observed breaks in the X-ray photon spectra is discussed. A time series analysis of the hard X-ray time profiles has been made. Specifically this includes (1! Noise properties. (2) The question of a correct definition of time scales involved in the hard X-ray emission and a discussion of their meaning and application. (3) Existence of periodicities. (4) Timing differences between short "spikes" at different energies. Correlations between various (derived) hard X-ray emission parameters have been looked for. A remarkable new feature is the good correlation between the thick target parameter of the large August 4, 1972, event. Finally, a few implications from the observations for existing (X-ray) flare models are discussed. In particular, the possible physical differences between the two groups of hard X-ray bursts is looked into. Also the existence of streams of fast electrons is discussed; as these streams are very strong, they have important jonsoquences for flare theory.

2.1. INTRODUCTION. It is now widely accepted that acceleration of electrons is a basic feature in the flash phase of many solar flares {Kane, 1974; Lin, 1974a,b). These fast electrons emit hard X-rays by the process of brerasstrahlung and since there are no propagation effects at all, it is believed that observations of hard X-rays provide the most direct clue to the spectrum and flux of fast electrons during flares, and their evolution. In this paper we report and analyze observations made by the Utrecht Hard Solar X-ray spectrometer S-I00 on board the ESRO TD-lA satellite in the period March 12, 1972 until October 1, 1973. The observations were made above 25 keV in 12 logarithmically spaced channels with a time resolution

-14- of 1.2 s in the four low-energy channels (25 - 90 keV) and 4.8 s in the others. With one exception, this paper includes only events large enough for a reasonable determination of the photon spectrum. '-- Observations of many smaller solar flares have been made by spectrometers --• flown by OSO-3 (Hudson et at., 1969), OGO-5 (Kane and Anderson, 1970) and OSO-7 (Datlcwe et at., 1974a,b). These observations were made in the energy - range from a few keV up to ^ 100 keV and mainly have been used to study f general statistical characteristics. The scarcer large events have on the other hand been observed at higher energies (,> 20 keV) by 0S0-5 (Frost, 1969; Frost and Dennis, 1971). The main assets of the present observations are the continuous high time resolution of 1.2 s together with the absence of any saturation effects during even the largest events. The purpose of the present paper is twofold. At ona hand observations are presented in a self-contained format allowing independent usage by other workers; on the other hand a general analysis is made, avoiding specific models for specific events, but emphasizing general issues. In view of the direct relevance to acceleration processes, a detailed time series analysis making full use of the 1.2 s time resolution was a major aim, in addition to a spectral analysis. In our opinion this has not yet been satisfactorily achieved. No attempt was made *..o invoke other flare observations (radio-, optical-, EUV- and particle radiation) in the analysis, since we take the view that the comparative fruitlessness of many previous joint data studies in terms of definite physical models may be attributed to insufficient analysis of individual data before intercomparisons are made. The outline of the paper is as follows. In section 2.2. a few essentials on instrument and satellite are reviewed, and in section 2.3. we discuss our selection of events. The spectral analysis is discussed in section 2.4. After a few remarks on our data reduction technique, the so called thick target parameters are introduced. We have opted for the convention to express the obtained spectral fits in these parameters. Their advantages as well as their interpretation under general circumstances is reviewed. Next, the observations and thick target parameters are presented and ana-

-15- lyzed in ths context of existing observations. The possibility is pointed out that the X-ray bursts can be classified in two groups. Breaks in the X-ray, phatQu spectra are observed and their interpretation is discussed. Section; 2.5. is concerned with a, time series analysis of beth the count rates and; the; inferred parametee-rs. In particular, the statistical properties of the count ra,tes are investigated and we connect this to the issue of timescales involved in the electron acceleration process. Finally, we inquire into the possible presence of periodicities in the time profiles using Fourier analysis and into the significance of timing differences of short "spikes" at different energies. Xn the last data analysis section (2.6.) the presence of correlations between various parameters is examined. A remarkable correlation is found to exist between the thick target parameters of the large August 4, 1972 flare. In section 2.7. implications for X-ray flare models are formulated. Speci- fically we discuss (1) possible physical differences between the two types of hard X-ray bursts. (2) The question of thermal v. non-thermal X-ray emission. (3) The existence of streams of fast electrons during flares. These streams would have a Very large number current, typically of the order of ^ 10 electrons s ; this point has important implications for the mag- netohydr©dynamics of the acceleration region.

2.2. INSTRUMENT AND SATELLITE. On March 12, 1972 the ESRO TD-1A satellite was launched into a near polar, circular orbit at a height of about 550 km. The orbital inclination is 97.6°, and this causes the orbital plane to precess 1° a day. Thus the satellite was sunlit along its whole orbit for a period of about seven months after launch. After an hibernation period of about three months, during which the se-'^ite eclipsed periodically and the scientific instruments were switched off, a new observation period started, in early February 1973. Reference is made to a description of the satellite by Tilgner (1971). The satellite is three-axis stabilized. One axis is kept sunpointed with an accuracy of better than 1'.

The Utrecht Hard x-ray Spectrometer S-100 (Van Seek, i973) has its main axis mounted parallel to this axis and hence the instrument is permanently viewing the Sun. Measurements of the solar radiation above 25 keV are made in 12 (roughly) logarithmically spaced energy channels. On May 18,

-16- Be entrance window (j'iource oj g source Sn.Cu shielding telemetry command;

impulse heigtn njjTTe- analyser

scintillation crystal < p m tube

coincidence 12 channels circuit 20 - 800 uv ( logaritmicaiiy spaced

particle identification measurement

Normal mode X-radiat>on sun Test model x-radiation a.gsource Test mode 2 x-radiation sun • electrons {5"source fig. 1. Simplified outline of the hard solar X-ray spectrometer S-100.

1972 the channel limits were 28, 39, 51, 71r 102, 141, 193, 266, 363, 488, 641, 834 and 1037 keV; these channel limits increased very slowly as a function of time. The permanent sunpointing permits a continuous use of the fine time resolution , set tc 1.2 s for the four low energy channels and to 4.8 s for the others. Basically the detector consists of a Csl (Na) scintillation crystal 2 (sensitive area 5 cm ; thickness 1.5 cm) optically coupled to a photomulti- plier tube, followed by a pulse heighu analyzer. Passive shielding has been applied to suppress the background X-radiation generated in satellite and instrument. This shield ends in a collimator. All charged particles that enter within the acceptance cone of the colliiaator (geometry factor ^ 1 cm • sr) and penetrate into the crystal, are also detected by the two solid state detectors (SSDs) and hence are rejected in the normal mode of operation. Once every 38.4 s, however, the number of particles detected by both SSDs is recorded during 4.8 s (particle counter). The particle counter is subcommutated with channel 1, causing regular datagaps of 4.8 s in each stretch of 38.4 s X-ray data of channel 1 (cf. e.g. fig. 2 and fig. 16. Datagaps occurring in all channels, e.g. fig. 16 at UT

-17- Table I. SENSITIVITY AND EFFICIENCY OF THE S-tOO

energy photon flux equivalent phofcopeak. of equatorial background efficiency (cm -"1

25 6-1Q"3 0.5 50 8M0~4 0.7 100 3'10~ 0.9 300 4'SO"5 0.5

0631, can be due to telemetry breakdown). The particle counting rate is mainly due to electrons from the radiation belts with energies exceeding *\» 1 MeV. The particle counter is a useful diagnostic tool to distinguish solar flares from phenomena caused by particles: hard X-rays (from a flare) will not affect the count rate of the particle counter. It is important to note that the two SDDs (thickness: 360 Urn silicon each) provide a sharp cut-off of the photon efficiency below ^ 25 keV. Even during the largest solar flares this allows the instrument to measure spectra free from any saturation effects, such as pulse pile-up and DC shift. The sensitivity under low background conditions and efficiency of the instrument are given in table I. Because of data compression requirements the registers for channel 1 through 7 have a variable counting stepsize, defined as follows: 6U)63; 64(4)316; 320(16)1328; 1344(64)5440. These numfiers match to Poisson Statis- tics i The induced stepsize at a read-out c is always < c and usually < "j c . The information content of the signal is therefore not affected [but sometimes visible effects show up, e.g. fig. 16, channel 1, around UT 0633:30]. By the same token, Poisson statistics does not apply exactly to measured count rates, but only to a very good approximation. For inflight calibration two radioactive sources, 90Sr and 241Am have been mounted in front of and between the two SDDs, respectively. On ground command coincidence measurements between the SDDs and tha scintillation counter can be initiated. In these calibration modes of operation, only the 241 90 photon spectrum of the Am source or the electron spectrum of the Sr

-18- source will be analyzed, thus providing an accurate inflight measurement of the efficiency, the resolution and amplification of the scintillation counter and the energy calibration of the discriminator lev.els. These mea-^ surements are performed a few times a week. The instrument was subjected to an extensive ptefflight calibration program, providing rdetailed "knowledge of all detector characteristics. For a more extensive description of the instrument,' see Van Beek (1973) and Van Beek and De Feiter (1973). The instrument has operated successfully during the two years' lifetime of the satellite and it has observed a considerable number of solar flares,. The actual observation periods of the instrument are: March 18 - October 25, 1972 and February 15, 1973 - May 5, 1974 (during the second hibernation the instrument was kept switched on and sunpointed). Due to failure of both satellite taperecorders, from May 23 1972 on, observations were possible only during real time telemetry contact. Data coverage initially dropped to about 30%, but increased gradually up to 60% and more in August 1972 by extension of the ground station network.

2.3. EVENT SELECTION. The observations presented in this paper are summarized in Table II. Potential hard X-ray flare emission was located in the satellite data using the criterion that the 1-8 A soft X-ray flux measured by SOLRAD should be above a minimum level. This turned out to be a reliable and effective method. A flare was included in this paper when its hard X-ray emission was well covered, sufficiently free from background contamination and detectable up to channel 4 at least, so that a reasonable determination of the photon spectrum was possible every 1.2 s. The period covered for this paper is 1972: 18-3/25-10 and 1973: 15-2/1-10 (beginning of second hibernation). In this way ti;e events # 1-9 and 11 from Table II were found. Many more events have been observed having detectable X-ray flux in, say, channel 1 and 2 only. Event # 10 is an example. Table II also serves as a cross- reference to all figures pertaining to one and the same event. Throughout, we will consistently refer to the event number as given in Table II, first column. Table II. SURVEY OF OBSERVATIONS.

event date tine of max optical / position X-ray taick tar- background photon power timing correlation * count rate sofjj X-ray timeprofile get para- subtracted spectra spectra of F and y in channel 1 class (fig #) meters (fig it) (fig #) spikes (fio #) (fig #) (fig *}

1 21-3-72 0112:51 1B/.M1 N0SE42 2 3 yes 2 18-S-72 1406:12 1B/M4 4 5 yes 18-5-72 1617;35 1B/M4 S16E24 6 7 yes 2-8-72 1839:31 1B/M4 N13E25 8 9 yes 26 7-8-72 0252:25 SB/M2 N14W33 11 to iO yes O 19-5-73 2244:08 1B/X4 N09E20 12 13 yes

7 2-8-72 0330:02 1B/X2 N13E35 14 15 no 8 4-8-72 0626:11 3B/X5 N13E08 16 17 no 23 24,25 27 9 7-8-72 1521:28 3B/X5 N14W39 18 19 no 28

10 18-5-72 1954:46 SN/C2 SO^H09 20 11 23-8-72 1913:03 1B/M2 S14E42 21 26

Solar Geophysical Data (Prompt Reports).

Preliminary Report and Forecast of Solar Geophysical Data (NOAA); For the C/M/X-claesification see Simon and Mclntosh (1972), 2.4. SPECTRAL ANALYSIS. 2^4. l^_Data_Reduction. The response function of the instrument was rep./ved from the data re- taining full time resolution, using a reduction method t'nat requires much t 2 less computing time than the usual two parameter X -fit (e.g. to a power -Y law spectrum a£ ) and, moreover, actually reconstructs the photon spectrum l (Hoyng and Stevens, 1974). No a priori assumption is needed on the shape of the photon spectrum. The method easily establishes, for example, devia- ~ tions from a single power law behaviour, if present (cf. fig. 23). The response function of the instrument was taken accurately into account using : inflight and preflight calibrations [included were effects of efficiency, escape peak, compton scatterings, crystal nonlinearity and crystal dead layer; see Van Beek (1973), p. 43-62]. We routinely also compute the best single power law fit to the obtained -Ij photon spectrum (to avoid confusion- not to the measured pulse height dis- a tribution). Tv's single power law fit was needed for model calculations, ' such as determination of thick target parameters as described in the next section. Fig. 22 and 23 give a few more details on our data reduction method. For the smaller events # 1-6 background subtraction could be achieved straightforwardly. In the long lasting events #7-9 background subtraction was very involved due to lack of proper background data or to changes in the background during the event. No background was subtracted in these three events. However, neglect of background is presumably no serious ^ problem in this case because of the large excess of X-ray over background » counts. "•

2^4^2. JThicfc Target Parameters. In the thick target model one conceives fast electrons being in-jected into a "target" region having a relatively high density (Brown, 1971). These electrons loose their energy mainly by electron-electron collisions and a minor fraction (^ 1G ) of their initial energy goes into hard X-rays, emitted during electron-proton collisions (bremsstrahlung). If the target density is so high that the injection rate of fast electrons is virtually constant over an energy loss time, then the emerging X-ray photon spectrum

-21- is no longer a convolution of the -electron injection spectrum with respect to time,, and also it becomes independent of the target density. In the nonrelateivistic range, the Bethe-Heitler cross section allows analytic integration if the electron energy spectrum is a power law. Representing the photon spectrum at 1 a.u. (see end of section 2.4.1.) by

1 I(e) = a£~^y (cm •s'keV)" (1)

then one finds (Brown, 1971):

FCE ) = 4.15-1033 a(Y-l)2 E(Y-h,h)E ~Y s"1 (2) o o

P(E ) = 1.6«10~9 —^r- E F(E ) erg's"1 (3) o Y~* ° o

[B(x,y) is the beta function, and E is in keV]. F(E ) and P(E ) are the o o o number flux and energy flux of fast electrons into the target region, having energy 5s E . Henceforth E = 25 keV is taken: a small extrapolation

below the low energy limit of channel 1, and we introduce the notation F?t_

and P25. Because the photon spectra we observed are always close to a single power law, the parameters (a,Y) provide a sensible and complete description of an X-ray burst. However, instead of the pair (a,Y) we will use the parameters (F ,Y): they are just a more convenient set than (a,Y) and will be referred to as thick target parameters. It is stressed that F. and P purely formal parameter's, representing required number anrd energy fluxes into a hypothetical target, for the electron energy loss function given by Brown (1971), formula (10).

A few connnents on the interpretation of F?t. and ?,,-• If a given X-ray burst is due to fast electron injection into a target, then F__ represents a lower limit to the required number flux, as is further discussed in section 2.7. Other parameters pertaining to the X-ray emission, expressible in a and Y/ can always be re-expressed in terms of F and Y- E.g. the nonthermal emission measure of the (instanteneous) collection nonthermal electrons emitting hard X-rays is given by

-22- cm"3 (4) n is the (homogeneous) density of the background protons and N is the total number of fast electrons with energy 3* 25 keV. It is reminded that, whereas F depends on the energy loss of a fast electron per unit time, n N does not. It is stressed that (4) only holds if F is evaluated according to (2) . In the thick target model each electron is accelerated and dumped into the target only once. Reacceleration therefore reduces the actual value of F . However, if there is significant and continuous reacceleration, then target and acceleration region become identical and F looses its meaning: In this case the hard X-rays are produced in some region in the solar atmosphere, where, by some mechanism, a nonthermal electron population is maintained and contained, characterized by n N _. This situation will be referred to as the "containment model". For both models mentioned above (for the thin target model there is an additional function of y of order unity), P as evaluated from (2) and (3) is equal to the rate by which energy is dumped by collisions.

In fig. 2-21 we have put together the hard X-ray observations of the flares in Table II and their thick target parameters when available. The sequence follows that of Table II:

- fig. 2-19: flares # 1-9 from Table II. For each event, the page to the left shows the X-ray time profile. For channel 1-4 the total number of counts acquired in one integration period of 1.2 s is plotted. The page to the right shows the computed thick target parameters; to facilitate intercomparison, the timeprofile of channel 1 is reproduced at the top. In each case the time integrals /F dt and /P dt over the entire event are given. - fig. 20 and 21: last two flares from Table II.

(text continues on p. 44)

-23- O 0112 0113 0114 -TIME ( UT) fig. 2. Event of March 21, 1972. Regular datagaps of 4.8 s in channel 1 are due to subcoimnu- tation (section 2.2.). The signal of channel 2 and 3 appears to be periodic, period "-9 s (section 2.5.3.).

-24- MARCH 2\ 1972 100 1 1 1 1

1 u in 75 a

5O o u

25 24-34keV

1 1 1 1 1 1 1 1

JD 10x1034 t = 2.3 x1036 electrons \P^5dt= 1.3 x 10 29 erg

4 -

3 -

i i I I 1 I I I 0113 0114 TIME ! UT) fig. 3. Thick target parameters of the March 21, 1972 event. The fluctuations in F._ and Y are unreal. The qualitative behaviour of F (F ^ t if y t and if count rate channel 1 t) is clearly visible.

-25- evew m z May 18,1972

0 1406 1407 1408 a*- TIME ( UT ) fig. 4. Event of May 18, 1972. Channel 1 shows a single peak of ^ 31 p width, preceded by a precursor, and ending in a prolonged tail, that could be of '.hermal origin (section 2.7.). x-ray soorce parameters for this event are derived in the appendix. The photon spectrum of this event at UT 1406 is shown in fig. 22.

-26- MAY 18,1972 36O -1 1 r —i

CM 300 u If) u 240 ill IS) M i 180

28-39 keV 60

0 J I I U _1 1 [_

37 lf2b d t = 5.0x10 electrons 36 30 2x10 - iP25dt=2.3xi0 erg

j 1405 1406 1407 •TIME (UT) fig. 5. Thick target parameters of the May 18, 1972 event. Fluctuations in Y are minimal at maximal count rate in channel 1. Y increases linearly with time throughout the event; analogous behaviour of Y is seer, in all IXBs.

-27- EVENT # 3 May 18, 1972

1617 1618 1619 TIME ( UT) fig. 6. Event of May 18, 1972. This flare occurred 2 hours after event 8 2, fiq. 4, in the same active region. Both flares have qu'te the same emission features and in particular the tail in channel 1 can be of thermal origin. MAY 18,1972

u w in ™ iao -

in z

o 120 2B-39 keV

1 r 1 i r

*• 37 2x10 36 = 4.8x1O electrons i = 2 3x1O3Oerg

U in in 1 h (M l

4 * t 5 6^

i 4 r-

L. 1 .. 1 I 1617 1618 TIME £ UT) fig. 7. Thick target parameters of the May 18, 1972 event. Again > increases linearly with time and the fluctuations are minimal at maximal count rate in channel 1. The apparent decrease of Y at UT 1618 is unreal. Upward fluctuations in Y are seen to cause largo peaks in F .: these are unreal.

-29- EVEJNLT # 4 800 August 2,1972

—— TIME ( U T ) fig. 8. Event of August 2, 1972. The X-ray emission consists of a series of spikes with a duration of a few seconds each. This event had probably a small electron acceleration 9 region (< 10 cm; section 2.5.2.) and also a small X-ray source region, whose position was fixed in time (Neupert et ;'. , 1974). Peterson et al. (1973) find that the hard X-ray photon spectrum for this flare preserves its power law character down to •^ 5 keV. The vertical arrows identify the spikes used in the analysis of section 2.5.4.

-30- AUGU5T 2.1972 800 1 1 r~~T

O in u 600

Z 400 o u

200

2x1036 _

i/) fi i -

1839 1840 • TIME ( UT ) fig. 9. Thick target parameters of the August 2, 19~2 event. Deviations from the linear in-

crease of V are largely real.

-31- fig. 10. Event of August: 7, 1972.

-32- AUGUST 7,1972

electrons 3O $P25dt=2.Ox1O erg 1 x 1O3e

~ 0.8

2.5-

1 I I L. 0252 0253 mt- TIME (UT) fig. 11. Thick target parameters of ths August 7, 1972 ev-ant. fig. 12. Event of Kay 19, 1973. The X-ray emission shows a precursor and a few pronounced spikes. Note that a large spike in channel 1 was (probably) not observed, due to subcommutation. The arrows identify the spikes used in the analysis of section 2.5.4.

-34- MAY 19, 1973

OJ 400 u

u w 300

z 200 8 37-51 keV f 100

1.4x1038electrons 4x1036 3o ^P?5dt=72x1O erg

4 0

2244 2245 TIME ( UT) fig. 13. Thick target parameters of the May 19, 1973 event. All datagaps in the count rates are linearly interpolated before computation of the thick target parameters; at the vertical arrow an incidental bad result is pointed out.

-35- T " T [ 1 I 80 r EVENT U 7 £ eot- AUGUST 2 .1972

\ -140

\ O 0313 0315 0317 0319 »-TlME ( UT)

I T""'n r*>Tt r IT T I I It • I T 1 EVENT n 7 AUGUST 2 , 1972

0327 0329 0331 0333 0335 —»-TIME ( UT)

fig. 14. Event of 2 August 1972. This is the first and smallest of a series of three extended bursts (EBs) all occurring in the same highly active region of August 1972. The observations are incomplete, see insertion.

-36- AUGUST 2 ,1972 "T"

160 -

CM 2 u tn 120 u ui C/1

80 COU N k 40

8x1035

6 - u LU if) in 4 u.

0327 0330 0333 »»TIME ( UT ) fig. 15. Thick f.arget parameters of the August 2, 1972 event.

-37- 4000

3500 EVENT * 8 AUGUST A ,1972

3000

2500

2000 t-

) 500 r o u 1000

500 - 400

300

- 200

-,oo

0620

fig. 16. Event of August 4, 1972. This flare and that of August 7, 1972, fig. 18, are extremely large. Datagaps occurring in all channels at the same time can be due to telemetry breakdown. The constant amplitude fluctuations in channel 1 at UT 0633:30 are of instrumental origin (section 2.2.).

The X-ray time profiles show quite pronounced oscillations that gradually Jie out towards the end (section 2.5.3). The X-ray source of this flare probably consists of a large, vibrating and expanding coror.al trap (section 2.6.; chapter IV). Photon spectra of this event are shown in fig. 23, at times indicated by vertical arrows. Features in the time profiles are seen usually to occur progressively later at higher energies !cf. section 2.6.).

-38- AUGUST 4,1972 T r r'"•

i 29- 41 keV -

D E

5 =3.5x1039 elec trons - 32 5 ^2Ox1O erg

1 .... : .-1.

0620 0625 0630 0635 *• TIME ( UT) fig. 17. Thick target parameters of the August 4, 1972 event. The accuracy in F and has increased to a few percent in this case. F , and y show a remarkable correlation 27); at the same tine, y and the count rate in channel 1 anticorrelate in the region A-D (section 2.6.). The letters A-E refer to those in fig. 27. Only every fifth datapoint Fjc and Y is shown.

-39- 1 T i I i r T •' ' T" "-' i ' ' •|-T~r-T-r-r--r~T~T'y

EVENT •# 9 AUGUST 7,1972 4000

m 3000 h u W

2000- o

- 800 1000-

400

200

I • . I • . I • . I 1513 1518 1523

TIME ( UT)

fig. 18. Event of August 7, 1372. This LS the other extremely large flare? from the highly active August 1972 period.- Unfortunately, the end part of the flare is missincj.

fig. 19. Thick target parameters of the August 7, 1972 event, y is seen to decrease, on the average, as it does in all EBs, cf. fig. 15 and 17. Again an ant.icorrelation between Y and the count rate is seen to exist, cf. fig. 28 and section 2.6. Only every third datapoint F and Y is shown.

-40- AUGUST 7, 1972 T •- r

4000- w in 3000

3 2000 o o

1000

I 1 1 1 10x10.36'' _

= 2 Bx1039electrons

6x1032erg U Ixl "In

J L 1514 1516 1518 1520 1522 ( UT)

-41- EVENT 4* 10 MAY 18,1972

u If) u

if)

3 o

1954 1955 TIME ( UT) fig. 20. Event of May 18, 1972. It is an example of a smaller event from our data, exhibiting detectable flux in channel 1 and 2 only. This event is classified as a subflare and its hard X-ray emission consists, of a single peak of ^ 10 s duration only. The X-ray emission is very hard in the stnse that the ratio of nard X-ray v. soft X-ray flux is large in this event, cf. section 2.4.3., end.

-42- 60h EVENT % 11 AUGUST 23,1972 -

1912 1913 1914 —*- TIME (UT) fiy. 21. Event of August 23, 1972. The arrow identifies the spike used in the analysis of section 2.5.4.

-43- Before entering into a general discussion four remarks on the figures:

(1). The behaviour of P, as a function of time: F(E ), cf. (2), contains -Y 2 ° the factors a E ' and (y-l) BCy-'s/'s) • *f follows that F will be roughly O i-J proportional to the count rate of channel 1 (^ 30 - 40 keV) and that F^ increases with increasing y. This is clearly visible in the figures. (2). All datagaps in the count rates were linearly interpolated (in particular the regularly recurring gaps in channel 1) before computing the thick target parameters. This is necessary because the values of F^ and Y depend most strongly on the lower energies. (3). The accuracies in F and y are determined by the received flux. Fig. 3 and 7 give an impression of what accuracies are reached in smaller events.

The values of /F? dt and /P dt are usually correct to a few units of the second decimal. In large events, e.g. August 4, fig. 17, the accuracy of F _ and y amounts to a few percent. (4). All times mentioned in this paper are expressed in Universal Time, with an absolute accuracy of better than 0.1 s.

(i) Discussion of time profiles and thick target parameters. Inspection of the X-ray timeprofiles suggest that they can be classified phenomenologically in two types: impulsive X-ray bursts and what will be called extended bursts.

(a). Impulsive x;;ray_Bursts_|IXBs^: The hard X-ray emission lasts from ^ 10 to ^ 100 s and the timeprofile exhibits either a single peak of ^ 10 s duration (fig. 20) to ^ 30 s (fig. 6), or a quasiperiodic series of peaks lasting from a few seconds (fig. 2 and 12) to ^ 10 s (fig. 8 and 10). These IXBs are by far the most common type of hard X-ray flare emission in our data. Many more were observed than included here. Sometimes a precursor is visible before the main event (fig. 4 and 12) The photon spectra of these IXBs are usually simple power laws. Ob- servations of this type have been reported before by Hudson et al. (1969), Kane and Anderson (1970), McKenzie et al. (1973) and by Dat- lowe et al. (1974b). (b) . Extended Bur.sts_^EBF^, comprising the three large events of August 2 (fig. 14), August 4 (fig. 16) and August 7 (fig. 18). The hard X-ray

-44- emission extends over '^10 s and the flux can be an order of magnitude larger than that of IXBs. Because of their long duration only the August 4 event was observed completely. Tlvis type of hard X-ray emission is mostly associated with very large solar flares and occurs therefore only rarely. The three EBs reported here were all produced by the highly active region of August 1972 (Zirin and Tanaka, 1973) . The time profile of the August 4 event shows a periodic signal, quite pronounced in the beginning and disappearing towards the end, where ail three flares exhibit an almost featureless decay. For the August 4 event, there exists a marked correlation between F and y (section 2.6.).

The photon spectra are near power laws throughout the event. Other observations of this type are scarce: a flare on March 1, 1969 (Frost, 1969) and the well-known March 30, 1969 event reported by Frost and Dennis (1971). All these EBs are associated with large radio bursts and, contrary to IXBs, there is usually a type II burst.

We point out that care must be taken with the above classification because of the small number of events on which it is based; certainly no general validity is claimed. Yet, experience with our observational material as

Table III. AVERAGE THICK TARGET PARAMETERS.

36 evcf:t # •.F25>-10-

i 3.1 5-10-2 2 5 0.8 3 5 0.9 4 3 0.3 5 3.6 0.5 6 4.2 2

7 4 0.25a 8 3.3 3.5 h. 9 3.6 5"

probably too low because main part event is missing.

too high because decay phase is missing.

-45- well as other observations published so far do give support. The difference between the two groups is of phenoffienQlogieal nature and phrfsieal difference remains to. he proven. A few suggestions in this direction will be made in section 2.7. In Table III the average thick target parameters for each of the

< > events # 1-9 are summarized. The ratio of F25 for event & 6 to that of # 1 is about 40. Note that this - apart from differences in the average count rates in channel 1 and in y - is mainly due to the shift of the channel limits (from 24-34 keV to 37-51 keV for channel 1). In Table IV we have summarized characteristics and "typical values" of IXBS and EBs. The basic difference between IXBs and EBs are given in the first two columns in this table; all other differences can be seen as con- sequences : (1) IXBs have a duration of at least an order of magnitude smaller than EBs. As , like , is not too different for both groups, the large differences in the last two columns are main3y due to duration effects. (2) In all IXBs >' increases about linearly or remains constant as a function of time through the event, whereas in all EBs Y decreases on the average. This systematic softening of the spectrum of an IXB from beginning to end as we observe it contradicts former results (Kane and Anderson, 1970; McKenzie et al. , 1973) . This may be due to the fact that these experiments suffered from instrumental problems and also because their energy range extends to lower energies than ours. All authors, however, agree that towards the end of IXBs the spectrum softens. The usually very detailed (spiky) structure in the timeprofiles of IXBs and in parts of those of EBs provides evidence for the existence of continuous acceleration in these flares. There seems to be general agreement on this point.

Table IV. TENTATIVE GROUP CHARACTERISTICS.

< type duration (s) (cm 3) (erg)

36 1XB > 0 1•io 5-iO45 5-1037 2.5-1O30 X 3 < 36 46 39 32 EB ° io 0 3•io 3-IQ 3-l0 1.5-10

-46- In section 2.7. we will c.^me back to the main points made above concerning IXBs am! EBs, when possible physical differences between them are discussed^ i

(ii) Spectra. The photon spectra we derive for IXBs are generally compatible with a single power law spectrum. In no case did we find clear evidence for a softening or steepening of the spectrum towards higher energies; note, however, that: this conclusion is based en the use of 4, (in a few cases 5) ~

100 i -r-ry

EVEUT s 2 MAY 18 .19^

UCSD U7&ECHT OSO-7 TD-lA 10 I 1 1 -i integration time > 1405 52 0 1405 52 5 06 02 2 06 03 4

best (it •4 6

10"

10r" 3 10 100 400 ENERGY i keV) fig. 22. Thi hard X-ray photon spectrum of event * 2, fig. 4, as measured by TD-lA and OSC-7. The OSO-7 fit is based on five channels from 21-155 keV, the TD-lA fit on the four channels shown. The difference between both results is inconsequential. For the sake of the figure, the OSO-7 data were fitted in the same way used throughout in this paper [ the log of the photon flux was least mean square fitted to the log of the ensrgy; this method differs from that used by Datlowe et al. (1974)].

-47- AUGUST 4,1972 EVENT «•• 8

-r-Ar

too

'« 3 : Z.9 >

oc i UT 0624 : 52 UT 0625 40 1.0 o I Q.

01 -

10 40 100 400 10 40 100 400 10 40 100 400

ENERGY (Ke«) AUGUST 4 ,1972 EVENT tt 8

100

CM E i o 0.

10 100 400 10 100 400 - ENERGY ( K.eV ) fig. 23. Photon spectra of the large August 4, 1972 event at different times (arrows in fig. 16). The first three spectra exhibit -i break around 60 keV. The apparent hardening of the spectrum at high energies is unreal and due to the fact that no background was subtracted. Attention is called to the fact that in spectra, tig. 22 and above, the lower limit of our channel 1 is lower, and the upper limit of our highest channel is higher than stated in fig. 4 and 16. This is connected to the fact that channel limits 7n the photon energy Scale can in principle be chosen freely. This freedom was used to incorporate the effects of low and high energy photons (see further Hoyng and Stevens, 1974). energy channels, extending over less than one energy decade. In fig. 22 a photon spectrum for event # 2 {May 18) is shown as derived from simultaneous observations by the OSO-7 experiment of the DCSD groupt and by our TD-lA experiment. This event was well within the dynamical range

twe gratefully acknowledge extensive contacts with Drs. H.S. Hudson, D.W. Datlowe and M.J. Elcan on the subject of instrument intercomparison.

-49- of both Instruments and the observations show surprisingly good agreement, which, fact - as it was felt by both groups - adds to the credibility of hard X-ray observations. In EBs the spectrum often softens towards higher energies, see fig. 23, for the August 4 event. During the first part of this event the spectra are compatible with two power laws with a break around ^ GO keV, and Ay j> 1. A sharp break is however not positively indicated. Such a steepening of the spectrum towards higher energy has been reported before by Frost (1969), Frost and Dennis (1971) and by Kane and Anderson (1970); Elcan (1975) reports that these spectral breaks occur regularly in the OSO-7 observations, and that breaks as large as Ay > 2 around 50 keV have been observed. Recently, Petrosian (1973) has explained this spectral steepening by the effect of relativistic beaming of the emitted breinsstrahlung from a beam of fast electrons shot downwards to the photosphere. Although the effect is undeniable, it is doubtful whether this explanation holds, because:

(1) The model predicts Ay x 1 around 100 keV, whereas observed breaks amount to Ay > 2 between 10 - 100 keV. (2J In terms of the electron trap model for the August 4 event (Brown and Hoyng, 1975) there are no electron beams, although admittedly as yet no effects of geometry were taken into account. It is interesting to note that the trap model does provide the possibility of a break in the photon spectrum, in the right sense. (3) Petrosian takes an exact power law electron spectrum at the onset of the beam. This is an arbitrary assumption and therefore one must be extremely careful in drawing conclusions from deviations from power laws. This matter is further discussed in section 2.7. (4) Brown (1972) has treated the same problem, but included eollisionai scattering of beam electrons; no break was found.

The break could therefore equally well be present in the spectrum of accelerated electrons.

(Hi) Ratio of soft and hard X-ray flux from IXBs. The soft X-rays emitted in IXBs (say e < 10 keV) are generally interpreted as brernsstrahlung from a thermal plasma [ kT '^ 1-4 keV; emission

-50- 47 49 -3 measure Y ^ 10 -10 cm ; Kahler et al. , 1970; Horan, 1371; McKenzie et at,, 1973? Culliane ana Phillips, 1970]. Attention is called to the fact that the ratio of hard X-ray flux in channel 1 (28-39 keV) to rhe soft X-ray flux ii». the 1-8 8 band can differ greatly between different IXBs. In our data, the two extremes happened to be separated by only 64 hours: the IXB on May 18, 1972 (fig. 20), classified as C2 (1-8 X flux = 2*10 erg/cm -s) and on the other hand a SN/M5 -2 2 flare (5-10 erg/cm *s) on May 16, having no detectable hard X-ray flux. Estimating an upper limit of ^ 5 counts/5 cm -1.2 s in channel 1 for this flare, one finds the soft v. hard X-ray flux ratio for both events to differ by a factor 125. It is pointed out here that these large differences must be explain- able in terms of different hard X-ray source parameters. From a preliminary analysis we found that variations in hard X-ray source density would have by far the greatest effect. This holds for the thick target model as well as for a containment model, if the soft X-ray emission is taken to be due to heating by nontherma] electrons.

2.5. TIME SERIES ANALYSIS. The following issues will be discussed here: Noise properties of observed count rates; timescales involved in the hard X-ray emission; the possible presence of periodicities in the observed count rates and, finally, the question of timing differences in short spikes. The analysis in this section is based entirely on the observed count rates in the channels # 1-4 as a function of time, and statistical considerations will be important. A few notions and definitions are introduced: The number of counts acquired in a channel register in one integration period of 1.2 s (loosely speaking the count rate) is denoted by c., where i labels consecutive read outs, c, is a realization of a random variable i c. having a Poisson distribution (cf. section 2.2.) with expectation value X. (that is E c, = A,}; whereas the c. are measured, X. is unknown. We introduce c. = A. + d.: The series {X.} represents the actual timeprofile -II-I i that one would like to measure but due to fluctuations d. only the series {c.} is achieved. Consecutive c. - as well as c. from different channels - are statisti-

-51- cally independent (Hoyng and Stevens, 1974). The following notations are used- \c = c -c (c , c . from the same channel ), idem for c., d. and i i i-1 i i-1 -i -i A,,- the integration time (1.2 s) is denoted by T . For expectation value and variance cf a random variable, e.g. c.f the symbols E c. and D*"c. are — 1 — 1 —1 used.

2.5.1. Noise Properties of the Series {c }.

Here we check to what extent the observed time series {c.} consists of Poisson noise, superimposed on a smooth series {X.}. A quantity ^ is introduced, the sum of the squares of the (normalized) differences of c. and the running mean p. = tc +c.+c . ) /3 :

2 n -> X = I (c.-u.r/p. (5)

i=l y X X

The statistical properties of this x differ from the usual: 2 2 2 EX = E I (c.-.J /^ . f n+ i l{±^ - >,i) /Ai (6) i=l

2 2 2 4 1 DX = (ECx^-EX ) } •- (2n)' (7)

2n/3 instead of the usual value n appears in (6) because c. and U. have a -l -l 2 tendency to fluctuate in the same direction, lowering the value of (c,-U.) . 2 ii As the variance of X was difficult to evaluate we just estimated in (7) 2 its order of magnitude by the ordinary X variance. Table V gives some results. The time series of events * 1, 2, 3, 5 and 7-10 are consistent with the hypothesis of Poisson noise superimposed on a slowly varying systematic changet. From Table V it is inferred that the second term in (6) is small

tWe correct here an earlier statement that the August 4 event would exhibit "intrinsic white noise in its emission at all periods from 15 s down to the Nyquist period, 2.4 s" (Hoyng et al. , 1975). There is no such intrinsic white noise.

-52- UT of remarks rh 1

; f: -i 6 • II identical results for ever.tr. " 1 , .'.' , and '."-,. i, similar results fc event " t, be it less extreme.

for cver.tf

compared to 2n/3, or, on the

(8)

In other words, an observed c. does - on the average - not differ significantly from the previous c.; their difference are attributable to noise. This implies that these events do i.ot have significant tiraescales on a 1.2 s basis, see next section. It does not imply that the events are "structureless", because e.g. the sign distribution of the series {c.-U. .- does r.oL enter in (5) at all. ror the events # 4 and 6, one cannot decide between enhanced noise or real structure on the basis of (6) only. Fig. 9 and 13, show that the large 2 values of x" are, very probably, due to systematic changes, or to the second terra m (6) . For the extreme case of event # 4, channel 1, one then finds: \ 1 2.4 X ' (9) - x. l

The conclusion is that only events ** 4 and 6 have much real structure ">: a 1.2 3 basis, all others do not (this statement is instrument dependent)

-53- .1.5.2. Timescales. Ti.aeaca.les involved in the hard X-ray emission are important to establish as they directly reflect timescales involved in the electron acceleration region and hence contain physical information on this region. Various timescales can be defined. E.g. the total duration of the hard X-ray emission or the duration of a "spike". This timescale is simply a measure of how long accelerating structures can exist. It is however very difficult to relate this timescale directly to physical quantities in the acceleration region. Presumably that is more easy upon introducing a timescale T that measures the time necessary for a characteristic change to occur in the hard X-ray emission. If c(t) is the count rate, then

T = ctdc/dt)"1 (10)

T measures how fast a "typical change" in the acceleration region us a whole takes place. On the grounds of dimensional analysis [cf. (18) and following discussion] it is expected that x, to order of magnitude, is equal to the length scale divided by a characteristic velocity in the acceleration region. In another application, a minimum density estimate for the hard X-ray source region is found from T. Examples are given at the end of this section. The derivative dc/dt is computed from two consecutive c. , giving

Ti = CiTo/fici C11)

In this way the best time resolution is achieved. Statistical aspects are important, however, and the question is under what conditions T. from (11) equals the true timescale f. = X.T /AX.. It is obvious that when , i I o I |Ac.| = |c.,-c.| £ c. one is unable to distinguish a possible real change in the count rate from a random fluctuation [hence, from (11), x. can only be meaningful when x < x c. ] . It turns out that two conditions ^ l o I must be met, c. » 1 and c. */|Ac.| << 1. In practice we adopted (a) c. > 75. (b) c^/JAcJ <0.43, and also <0.31 (probability l.Q'lO""1 and 2.15-10"2 respectively).

-54- Table VI. TIMESCALES T FOR CHANNELS 1, ^ AND 3.

T. (B; Cs) event 9 tijne event * time (UT) ch 1 ch 2 ch 3 (UT) ch 1 ch 2 ch 3

18.39:03.7 1.5 1.? 2.1 0623:07 .5 e.e ^, 4 7 13.3 -3.6 -3.6 -3.7 8 08.7 • .:- e.c 4 29.0 L.i 4.'. l.Z (August 4; 2i .9 lc ;c 7.r • August 2) 30.2 6.5 j.Si ". 1 cont . j5 . 2 20 8.9 5.6 d . p - p. : 1 1 . 4 -.',..* - /, 0 ~4. C 24: 16 • i -XJ -10 -S.2

:.3'.< 32. fa -,?.$ -.-.•* -<.: 54 12 C. J J.8 8.9 t,1 4,3 15-5:22 .7 7. 1 4.8

1 " : Ott 3 7.7 6. 8 6.3 .iSt.e -;. .• -:... -: .' 31.6 17 6.9 8.3

•;• i. 2 23 19

•*'..'• . ' - . H ." . O

1 . - - , . . <- . . - A !;:.« is pnnnei cursiveiy if c'/ .'.c"S 4B.2 a., i.£ 4./" 0.31 for all three entries. 49.4 11 12 4.6 50.6 12 7.3 13 Missing due to subcommatation, but "*?."!; 54.2 -14 -12 -4.1 pr-obably satisfying c /'icj<0.31, cf. fig. 12.

-55- Because i8) implies that, on the average, c. /!Ac. J "»"• 1, it follows that the majority of the T. computed through an event - with the jjossxbiUj .exutsy- tion of events * 4 and 6 - are meaningless Sue to fluctuations•<. To further minimize the influence of fluctuations and increase the physical significance, we require (a) and (b) to hold for channel 1, 2 and 3 at the sans tine [probability - when (a) already holds - 10 for c.°V|Ac.j ^0.43 and 10 for the stronger requirement <0.3l]. The results are given in Table VI. It includes a1I T, of all events * 1-9 satisfying (a) and (b) , using probability 10 ". Imposing the stronger requirement of probability 10 , the cursively printed subgroup in Table VI is obtained. It is seen that only the events # 4 and 6 (as expected) and 8 and 9 from Table II have a few significant timescales T.. For each event also the a postei*iori probability is given (under a.p.p.) i.e. the ratio of the number of lines in Table VI and the number of times c, ^=75 holds for all three channels at the same time. Values for x. range from ^ +_ 2 s in IXBs to ^ + 20 s in Eflsf. Striking is the strict sign correlation, the August 2 event having 7 positive and 6 negative sets, spread more or less evenly over the whole duration of the event, whereas August 4 and 7 have almost only positive timescales, clustering at their beginning. It is noted that the timescales for the two EBs of August 4 and 7 in general decrease with increasing energy. Two applications are discussed:

(i) Length scale for acceleration region. One has T *v« Z/v > £/v where %, v and v are equal to the length scale ^ A A of, a typical velocity in, and the Alfven velocity in the acceleration region [cf. eq. (19) and following discussion], or:

£ < 2«10U H n T (12)

tin the computations and in Table VI, actually I. = *3T (c.-rc. ,)/Ac. is i o I l-l i used instead of (11). This detail is mentioned here because it follows that |TJ > 4t = 0.6 S. This absolute lower limit is seen not to be approached in Table VI.

-56- where K and n are magnetic field and density of the -acceleration region I*,, H, n and x in c.g.s. units) . Adopting e.g. H ^ 500 G, n 'v* 10 (see o o second application below) and T ^ 2-4 s (Table VI) for the IXB of August 2, t ^ 5-10 x 10 cm is obtained, a rather small quantity. In the containment model this would be also a typical dimension of the hard X-ray source region as acceleration and source region are identical in this case. In the EB of August 4, (12) together with other available estimates 10 for this event give the very large value £ > 10* cm (Brown and Hoyng, 1975).

' ii / Lower Unit s> the density : f thi: h:rd X-ray 3J'

is the observed hard X-ray power law index, one finds

T ^ f Of/it)"1 = f (rr- (Ef)]"1 = E/YE numerically:

T ^ 2'108 E^/Yn (13) ~ keV o

(x and n in c.g.s. units). It is reminded that this derivation supposes the absence of any acceleration. For the IXB of August 2, Y ^ 3 (Table III) , T ^ -2s at E ^ 60 keV, and n > 10 cm is found (in the containment model, n is also the den- o ^ o sity of the acceleration region). The timescales in Table VI for both IXBs in Table VI are about constant as a function of channel number, that 3/2 is they do not show any trace of a E dependence. This is an additional strong argument for the density n in IXBs to be actually higher than the derived lower limit. Concerning the size of the hard X-ray source, it is interesting that g existing observations suggest that it be small (< 10 cm) in small flares. In particular, Neupert et al. (1974) find that the soft X-ray source of

-57- the IXB of August 2, as observed near 1.9 A, is always smaller than one 9 S 2 resolution element of the spectroheliograph, 1.5*10 x 1,5*10 em , and that its position remains fixed in time [the size of the acceleration region was found above to be small, too: ^ 5*10 cm].

We summarize our conclusions for the IXBs of August 2 and May 19: g (1). they have presumably one or a few small (< 10 cm) acceleration regions . (2). their hard X-ray source region should have at least a density of n ^ 10 cm and, in the case of August 2, a size ;< 10" cm.

2.5.3. Periodicities. Some of the hard X-ray timeprofiles appear to carry a periodic signal, notably the flare of August 4 (fig. 16) and also that of March 21 (fig. 2). A periodicity in the timeprofile has potentially important implications because if it is inferred that they are due to real physical oscillations of some kind in the flare region, then the choice of possible flare confi- gurations is restricted. In addition, the period is expected to have a simple relationship to other parameters of the acceleration region like density, dimension, etc. In this section we therefore inquire into the reality of such periodicities, using power spectra. The fourier transform of a time series (e.g. the count rates c ....c . of one channel) is constructed, resulting in o N-l the series a ..a .. The power spectral density (power spectrum) is defined as |a | for p = o,...,N-l. [In all figures the quantity iog|a | is plotted.] Periodicities in the time series will show up as paaks in the power spectrum. Fig. 24 gives the result for the August 4 event, for tho count rate in channel 2, F and y. In all three power spectra large peaks show up at p = 10, 20 and 35, corresponding to periods of 124 s, 62 s and 35 s and suggesting that these periods are really present in the data. In case Poisson statistics applies to the series c ...c., ,, the relative „ o N-l uncertainty in the power ja | due to fluctuations is given by (Hoyng, 1975):

(2x-x2)\- x= N"2(Ec.)/|a |2 (14) K p -58- POWER -iPECTRA EVENT » 8 AUGUSI 4 197? UT 0620

100 150 200 250 FREQUENCY P

fig. 24. Power spectra of the count rate in channel 2, F_>5 and y for the August 4, 1972 event. The usual technicalities were applied before the actual Fourier transformation (Brault anr". White, 1971). Due to this the power spectra are distorted for p £ 5. Frequency p and period T are related by p'T = 1024 x 1.2 s. In the top figure, the straight line represents the average high frequency power (see text).

-59- DYNAMIC SPECTRUM RAW DATA CHANNEL 2(41-53 keV ) AUGUST 4 . 1972 EVENT # 8

240

06251-:

20 30 Frequency P fig. 25. Dynamic spectrum of the count rate of channel 2 of the August 4, 1972 event. The indicated resolution is set by the lineprinter output, and is much better than the actual resolution (see text).

* _2 N Ec is the expected average high frequency power level, drawn in as a k horizontal line in fig. 24. The ratio x can be read easily from the figure as a difference. For the peaks at p = 10, 20 and 35 in fig. 24, top, the relative error ranges from 5% to 15% and these peaks are therefore quite real (no such analysis for F,._ and y is immediate as their statistical properties are unknown). Power spectra of other events have been obtained. In the March 21 event (fig. 2) there is an indication of a periodicity in the timeprofile

-60- of channel 2 at p ^ 17, or 9 s [width Ap/p ^ Q.I; relative peak accuracy 40%, from (14)]. The other events of Table 1J do jripfc show. any|:h;ipog> For the August 4 event, a dynamic sp.egt.rum was constructed ;fpr channel 2, fig. 25t. In brief outline, the method applies a gausslan window of cor.a* ant relative width Ap/o to the fourier series a ,..a, ,, after o N—1 which it is transformed back to the time domain. In this way a wave packet is constructed with central frequency whose power at any time is known (Dziewonski and Hales, 1972). Time and frequency resolution At and Ap satisfy the relation (Ap/p)•(At/T) i T [Ap, At = half width at 1/e of maximum; T = period; p-T = 1024x1.2 s], and in fig. 25 Ap/p ^.0.2 and At/T 2i 2, the best possible combination for the method used. In view of the time resolution obtained, the local maxima at p^ 10, ^ 20 and ^» 50 cannot be said to occur at different times; the p ^ 50 maximum is rather weak. Our conclusions are: (1). In the August 4 event significant and well separated periodicities of ^ 120 s, ^ 60 s and ^ 30 s are indicated at the onset of the hard X-ray emission. The two higher frequencies die out and the lowest one seems to shift towards lower frequencies as time progresses. The vibrating coronal trap model for this flare (Brown and Hoyng, 1975) admits a possible interpretation in terms of trap-eigenmodes, but this needs further theoretical substantiation. (2). With the possible exception of event # 1, on March 21, no other flare shows la trace of a periodic X-ray emission.

2.5.4. Timing Differences in Spikes. The hard X-ray tijneprofiles of IXBs sometimes exhibit very narrow spikes in all four channels at almost the same time. We analyze here to what extent apparent timing differences are real and indicate possible implications.

tThe help of Dr. G. Nolet, Vening Meinesz Laboratory, Utrecht .is gratefully acknowledged.

-61- Seven such narrow spikes were identified, in the events # 4 (August 2), # 6 (May 19, 1973) and # 11 (August 23), indicated by arrows in fig. 8, 12 and 21. For purposes of intercomparison, a central time t is defined for eac?h sgiteej

n, ,n t = £ .t.c./E c. (15) c i=l * ^1=1 x where c., t. are count rate and the time halfway its integration period. In all four channels the summation runs over the same time interval, comprising from n = 3 up to n = 9 integration periods of 1.2 s. The uncertainty in t due to Poisson noise follows by computation of the (expectation value and) standard deviation of the random variable t = Zt.c./Zc.:

Et ^ Et.X./EX. a- Et.c./Ec. = t -C — XX X ~~ XI X C

Dt 2l [EX.(t.-T )2}h/T.X. 2t [Ec.(t.-t )21h/1c, (16) C X X C X X 1 C X

(T = It.X./ZX.). Dt^ turns out to be typically of the order of 0.05 s. Because a spike could be due to a change in only a part of the hard X-ray flare region, t is also calculated subtracting the background flare emission by using (15) with c. = c,-c.-(c -c ) (i-l)/(n-l) instead of c. i i *. n l i Errors in these t_ have not been evaluated; they will be substantially greater than (16). Results are shown in fig. 26. Channel number is converted to mean channel photon energy using the relevant power law weighting function, t is seen to either increase or decrease systematically with energy and subtraction of the background flare emission does not alter the energy dependence. The three spikes from the May 19 event are not shown; in each case, all t are equal within the error limits. A detailed analysis of this timing difference phenomenon is rather involved and outside the scope of this paper. It depends critically on unknown factors such as the behaviour of the acceleration as a function of time and the magnetic field geometry. The purpose of this section is mainly co establish the phenomenon, as it is not easily deri^/able from the figures.

-62- EVENT tt 11 ALMOST 23

o 191314 77 ••••

EVENT ** 4 2

1839 45.17

1839 3B 31 «•—|

1B39 3063

3O ?>0 70 100 •- ENERGY ( keV ) fig. 26, Central times t as 3 function of photon energy for the events of August 2 and 23, 1972, with and without subtraction of background flare emission (crosses and dots, respectively) . The total length of the error bar is 2 Dt . The number of integration

periods, n, used in (15) is indicated, as is the UT of tc of channel 1. Because n = 3 in the top figure, the crosses all have the time of the one integration period left over with c. =* o (see text).

An obvious and simple explanation could be the following, Suppose a bunch of electrons is accelerated "instantly", with mainly downward velocities. If the acceleration region is high up in the atmosphere, then due to velocity dispersion, the fastest electrons are stopped first. In this case t is expected to increase with decreasing energy, as it does in two spikes (fig. 26 middle). Xf the travel distance H is about the same for all energies (equivalently, the hard X-ray source is formed by a relatively sudden increase in density), then t tt S.E . We estimate > from the timing

-63- difference At between channel 1 and 4 (3 = v/c):

1.1 (17)

9 Taking At ^ 0.2 s from fig. 26 I "° 7'10 cm is obtained. The reverse situation, increasing t with increasing energy, could perhaps arise if the acceleration takes place in a high density region, such that low energy electrons virtually do not propagate, but the high energies need some time to slow down.

2.6. CORRELATIONS. We have searched for the existence of correlations between the thick target parameters, timescales and count rates.

CORRELATION of FJS and 8

3 8x10 6 AUGUST 4 , 1972 UT 0621 -0639 EVENT # B

6 -

If)

2 -

0 -

25 30 35

fig. 27. Correlation diagram for the August 4, 1972 event, obtained by eliminating time be-

tween F25(t) and Y(t) in fig. 17. Only every fifth data point is shown for clarity. The solid line (drawn on the basis of all data points) indicates the smoothed path of the event development (see text). The letters A-E are time markings and they refer to those in fig. 17. Note tliat point C is passed several times and so does not refer to a definite time in fig. 17, but to an extretnum flux.

-64- CORRELATION of F?t, ono 6 AUGUST 7, .972

36 Ul J&13-1623 10*1Q EVENT # 9

o in tn 6

O - 30 35 40 •s ftg. 28. "Correlation diagram" for the August 7, 1972 event, obtained by eliminating tine between F (t) and y(t) in fig. 19. Only every third datapoint is shown for clarity. The solid line (drawn on the basis of all datapoints) indicates the smoothed oath of the event development (see text).

(i). Correlations between the thick target parameter's y and F c. A "marked correlation was found to exist in the case of the EB of August 4, see fig. 27. During the brief rising phase of the event - cf. fig. 17, top - "the dog-leg" part AB is swept out, and all later (F2t-« Y) are seen to exhibit a correlation.

F Following the actual behaviour as a function of time, the point ( 7t-» Y) moves on the average along the drawn-in line, that is neglecting scattering due to statistical fluctuations and to small digressions. Each of the three counter clockwise "ellipses" corresponds to one of the three large oscillations visible in the timeprofile, fig. 16. After the oscillations have died out there is a monotonic decay down the line DE. The velocity along the curve is inversely proportional to the density of points.

-65- The existence of this correlation has stimulated the coronal electron ti.vo .r.teipretauo:-, lor this flare lBrown and Hoynq, 1975). The reasoning is tr.c toLlowir.i. in the first place, the occurrence of "ellipses" is im- puted to secondary, deviating effects and the existence of a strict correlation line is assumed along which the point (F , y) moves up and down and finally down along DE. Then it would follow that the hard X-ray source region must be describable in terms of one parameter that determines the position on the correlation line. Specifically, the system must be able to return to the same physical state at later times, implying that there is no continuous injection (but only a brief initial acceleration) and negligible collisional losses. This points already towards a large, low density trap; the (relative) magnetic field changes, associated with eigen- mode oscillations of the trap, serves as the free parameter. Betatron acceleration is continuously acting on the fast electron population, causing the X-ray flux modulations (fig. 16) the trap eventually expands and oscillations die out (part DE).

The details are worked out in chapter IV and the correlation line is reason- ably well predicted. Of course it is always possible to explain any event, also this one, with the thick target model. The attractiveness of the present model is however, that for the first time it has been possible to give a simple analytic treatment of a (re)acceleration mechanism. The corresponding diagram for the other large flare of 7 August is given in fig. 28. A correlation between F and y is seen not to exist in this case (in the sense that knowledge of either parameter enables one to do a probability statement on the other). The existence of a well-defined

path in the (Fo_, Y)-plane need not be a surprise: Any well-behaved photon flux implies continuous functions F _(t) and y(t) - in other words, a path in the (*"„_, y)-plane. The flux must just be strong enough to eliminate the influence of statistical fluctuations. In this respect it is mentioned that the "correlation diagrams" for the other events # 1-7 are virtually structureless (scatter diagrams). No path exists because fluctuations dominate. Therefore, fig. 28, if suggestive, does not necessarily contain some- thing that has to be explained. There are, however, a few similarities between fig. 27 and 28. Both events begin in the same way, although the

-66- A-.i-juat 7 f.-vent rr.or'.- .slowly, an..": tr.trre is also a tendency towards cour.ter- c-iockwise movements in fig. 28. It is a pity that the end phase of this ilare was not observed. A coronal trap interpretation for this event seeraa therefore impossible, perhaps until an explanation is given for the counterclockwise movements in the August 4 event.

(ii). AKtieorrelatioK between y and count rate in the August 4 and ? events, t In fig. 16 and 17 a correlation is seen to exist between a timeprofile of any channel and -Y, provided the latter is shifted backwards in time some 15 s at most. This point is closely related to Ci): The titneprofiles of the August 4 event, fig. 16, show that a "temporal feature" in general occurs progressively later at higher energy, totalling up to a shift of ^ 15 s; and this holds until point D in fig. 17. As a consequence, channel 1 and Y will oscillate, with a phase shift. As variations in Y are small, F is just proportional to the count rate in channel 1. Counter clockwise "ellipses" in fig. 27 appear as a result and their axes are approximately aligned with the co-ordinate axes because of the relative size of "F?I- and AY, and because the phase shift is about a quarter period. The relation between Y and a timeprofile "I'l^iifesis itself as an anticorrelation because that requires the smallest time shift. For the August 7 event also this anticorrelation between channel 1 and Y is seen to exist, fig. 19. There is no indication for a time shift (possibly because the features are much less pronounced).

(Hi). Correlation between y and the ti-mesoale T. In no event of Table II we have been able to establish any correlation between Y and the timescale T (cf. Vorpahl and Takakura, 1974, who compared risetimes and y for 36 different events).

tWe are grateful to Dr. A.O. Benz for drawing our attention to these points.

-67- - •~" • IMPLICATIONS FOR X-RAY FLARE MODELS. Despite all efforts invested, flare models are in a poorly developed state and a ."•." ,-J >: .'i % evaluation of electron acceleration .and hence of their hard X-ray emission is out of the question for some time to come.

This is true even for the best elaborated flare modelr Petschek's wave assisted diffusion mode (Petschek, 1964) and follow-ups (Syrovat-skii, 1966; Green and Sweet, 1967; Petschek and Thorne, 1967; Sonnerup, 1970; Yen and Axford, 1970; Coppi and Friedland, 1971: Anzer, 1973; Priest, 1973). The present observations therefore do not have discriminating power against existing flare models (this situation could change considerably when hard X-ray observations with good spatial resolution become available). However they do so for the existing X-ray source models such as thick and thin target model (Brown, 1971; Datlowe and Lin, 1973). Four issues will be concentrated on in this section: (1) The question of possible physical differences between IXBs and EBs. (2) The possibility of a thermal contribution to Y in IXBs. (3) Evidence for in situ power law electron spectra in flares. (4) Inferences from the present observations for thick and thin target models.

2.7.1. Physical nifference between IXBs and EBs? It was suggested that the observations can be divided, phenomenologically, in two groups, irnpulsive X-ray bursts (IXBs) and extended bursts (EBs): - EBs have a much longer duration than IXBs and are usually associated with a type II radio burst. Only a fev; observations of this type exist. - The X-ray timernrofile of EBs exhibit a much more "gradual" development than those of IXBs, and especially during later stages of the X-ray emission a prolonged, featureless decay is observed. IXBs on the other hand consist of one or a few brief peaks [this is e.g. confirmed by the discussion en timescales, section 2.5.2.]. - On the average, y decreases with time in EBs, but increases in IXBs.

It is possible that these different characteristics reflect physical differences in the X-ray source region of these flares. The characteristics of EBs indicate a large,, low density coronal X-ray source region. This can be substantiated for the EB of March 30, 1969

-68- (Frost and Dennis, 1971); this event occurred far behind the limb and therefore the coronal interpretation is rather cogent, as was pointed out by Hudson (1973). In addition to this, it has recently been possible to account for the major characteristics of the EB of August 4, 1972 (Brown and Hoyng, 32 3 1975; see also section 2.6.), and this also involves a large Cv 10 cm ) , 7 -3 low density ('« 10 cm ) coronal volume. On the other hand, IXBs would then originate in a relatively small, high density region in the chromosphere or low corona. The present observations confirm this foi the IXBs of August 2 and May 19 (cf. section 2.5.2., end). As a further support it is pointed out that existing observations relevant to size and density of smaller X-ray flares suggest that it be small and high, respectively (Walker and Rugge, 1970; Takakura et al., 1971; Purcell and Widing, 1972; Catalano and Van Allen, 1973; Neupert et al. , 1974; Phillips et al. , 1974; Widing and Cheng, 1974; Alissandrakis and Kundu, 1975). See also beginning of next section 2.7.2.

2.7^2. Possibility of a Thermal Contribution to Y in_IXBs. The fact that > 0 in IXBs can be due to an increasing contribution of thermal bremsstrahlung in the lower energy channel(s) [note that this will only work if the X-ray source density is sufficiently high]. In fact, it is shown in the appendix that the "tails" in channel 1 occurring in events # 2 and 3 on May 18 (fig. 4 and 6) can be of thermal origin. It is emphasized that firm settlement of this question requires further investigations, as well as very precise, concurrent measurements of both soft and hard X-ray flare emission. These measurements must be really free of any instrumental defect.

2/7^3. Evidence for in situ Power_Law_Electron Spectra_in_Flares. Models for hard X-ray flare emission often assume power law electron spectra in the flare region. It is pointed out that there is no direct and very little indirect evidence for this assumption. The spectrum of fast electrons is relatively best (but still very poorly) determined by hard X-ray observations: An integral equation must be solved in which the bremsstrahlung cross section serves as the kernel.

-69- The condition of this equation is such that small variations in the photon spectrum induce very large distortions in the electron spectrum. Given the present status of hard X-ray flare observations, a great variety of electron spectra - among which sometimes a single power law - is compatible with observations. In addition, the deduced electron spectra are spaae- aVeragad, Direct observations of power law electron spectra in e.g. the cosmic ray spectrum and in solar cosmic rays (Lin, 1974a) may give some comfort, but at best single power law electron spectra in solar flares provide reasonable working models only. Model calculations assuming a single power law must be considered with great caution insofar conclusions are draip-i fpjm deviations from vouser laws. In situ fast electron spectra may well be no power law at all, or only so on very limited energy intervals. In particular, a thermal explanation of hard X-ray bursts up to higher energies that usual (say e.g. ^ 100 keV) by a suitable temperature distribution cannot be excluded on the basis of the present observations (Chubb st al., 1966; Chubb, 1970, Milkey, 1971; Brown, 1974). Such a thermal explanation has not yet been worked out vigorously, and it is very attractive in that the actual electron flux is much smaller than the values given for F__ in Table III, thereby relieving some of the difficulties formulated in the next section.

Further investigations are badly needed to settle this matter, as we^ as the above mentioned high quality concurrent observations of isoft and hard X-ray flare emission.

2.7^41_Electron_Beams, Thick and Thin Target. Thick and thin target model invoke the existence of streams of fast electrons propagating upward or downward in the solar atmosphere. The

< > following discussion is largely based on the values of F9c (Table III). As it Will be argued that these are very large, it is relevant first to review the interpretation of F (cf. also section 2.4.2.),

If an X-ray burst is due to an influx of fast electrons into a targets then F usually represents a lower limit to the electron flux, for the following reasons: (a) if the target is thin, the actual flux should be higher to reproduce the observed X-ray burst.

-70- (b) 25 keV was taken as low energy cut-off, just for instrumental reasons, but the true figure could easily be lower, say '•' 5 - 20 keV [iCahler and Kreplln, 1971; McKenzie et al. , 191'3; for event # 4, on August 2, see Datiowe and Peterson, 1973 and Peterson et al. , 1973]. On the other hand, there is the above mentioned possibility of a power law resulting from spatial superposition of maxwellians, in which case the actual flux is much smaller. {c\ Expression (2) for F. is based on the expression for zcllisional energy loss of a single electron (Brown, 1971). To obtain the actual flux one needs an expression for the total energy loss of an average beam electron, which is always larger than or equal to single electron collisional losses, hence causing the flux to be larger than F-,r-

As the values of F?I. (Table III) are very high (see below) it is noted here that they would go down if somehow the total electron energy losses are smaller than the collisional losses. This seems to be almost impossible; resistive instabilities (Bekefi, 1966, Furth et al. , 1963) provide the only theoretical possibility for this, but it is unclear if any efficiency can be reached. (d) Directivity effects. The distribution of fast electrons is supposed to be isotropic in deriving (2), but is of course anisotropic when a beam is considered. This can increase as well as decrease the actual flux, but only little; less than a factor 10 (Elwert and Haug, 1971; Brown, 1972) .

We now inquire into the physical nature of the electron streams. First, the electrons will be supposed to have about parallel velocities. It is noted that F 'WO s implies an enormous current I = e F "WO A, and essentially because of this large value, the following can be concluded:

(a). It seems unlikely that the fast electrons form (part of) a current system. This follows by considering the time evolution of a current J, given by

(v x H) (18)

The first term describes the evolution of a current when little or no mass 2 2 2 2 motion occurs. The associated timescale T ^ ad /c > W d /c is very long P

-71- and this term can be neglected [a JP conductivity; u = plasma frequency; the current is taken to have a perpendicular area A = £*d with £>d]. The second term has a much shorter timescale x~d/v ;> d/v (v = A.lfv€n velocity] . Estimating the self field of the current by

H r^ 41/cK- (with I > e F ) (19) one obtains 6 •t ^ 7-10" A18 n10V25 (20)

(T in sec; A „ in 10 cm ; n.. = ambient density in 10 cm ' ; F_g in

10 s ). Therefore T << 1 s, and since observations require T r> few sec. at least (section. 2.4.3.) a quasistationary solution of (18) is indicated. However, this is impossible because if one requires e.g. H ^ 10 G, then 12 (19) implies £ ,> 7*10 cm! Only by allowing extreme field strengths of ^ 10 G one can bring £ down to ^ 10 cm. Although one is dealing here with chromospheric or coronal magnetic fields during flares, about which nothing is known, such large fields are impossible on various grounds. This admits the interesting conclusion, tnat observations exclude the possibility that the fast electrons are (part of) a runaway current e.g. set up by the appearance of a large scale transient electric field in the -4 -3 local restframe of the fluid. Only a small fraction [10 - 10 at most] of the fast electrons can do so. The rest must either have an isotropic velocity distribution, or their current must be neutralized by the background plasma:

(b). The next possibility is that the fast electrons constitute a beam. A reverse current is set up in the background plasma, neutralizing the beam current (Benford and Book, 1971). An electric field is supposed to be absent in this approach. The reverse current is excited upon passage of the beam front. The beam density is given by:

nb * 27TT F25/A V25

(A is the beam area; v 2l °-3 c is the velocity of a 25 keV electron), from which, for A ^ 1018 - 10*9 cm2 , beam densities n > 10 - 10 cm are obtained. The beam could therefore be dilute.

-72- The ratio of the reverse current velocity v (> F__/n ?.) to the sound ve- r j zb o locity v must satisfy the inequality

3 5F /n A T V <1 (22) - 25 10 18 7^V S

(Fpj. in 10 s ; the background plasma density n in 10 cm , etc) . Beam instabilities will keep v /v ^ 1 [ in any case ^ 43, but in solar flare conditions it seems likely that T >> T. holds]. In this context it is reminded that the above cited observations indicate a small flare area,

The occurrence of beams during flares is not ruled out by (22) , though it is rather strongly restricted; as soon as a beam enters a region violating (22), it can be largely destroyed. In particular, it is possible that beams of fast electrons exist in flares, propagating in any direction, but travelling over a limited distance determined by (either/or): - violation of (22). As (21) and (22) imul" njn < 3-lCf3 T_ (or < 10~1x . b o / T in case T ^ T.), this happens before the beam density surpasses the ambient density. - indefinitely increasing collisional losses (thick target case). If an upward travelling beam is stopped, then, especially in large bursts, a shock wave could develop separating the quiet coronal plasma from the flare plasma. Finally, two remarks: - The required beams are much stronger than those in type III bursts, ^10 s , compared to ^ 10 s at most in type III bursts. The sta- bilization of such strong beams is not considered here, and this could easily impose further constraints. - It is reminded that, apart from the question whether beams can exist, section (a) essentially concludes that beams of the required strength cannot be accelerated. (c). In view of the above points it seems attractive to suppose that the fast electrons have an isotropic velocity distribution. It is necessary, then, to assume that the fast electrons are contained in some volume and constantly reaccelerated in there for the duration of the hard X-ray flare (^ 10 s). Such a containment as well as reacceleration can in principle

-73- be achieved in a Weakly turbulent plasma. If a current is invoked to sustain the turbulence (which is very like- lv•), then the foIlQwiag. eonsequerxce is pointed out. Because o£ the

n r high, magnetic field gradient (Vft > 6-10 iQ 1 G/cm) only a sheath-like structure is possible, with a thickness of at most 1 km, say. Either the density n in the sheath, or its lateral "active" area A, or both, should be large in order to contain a sufficient amount of fast electrons: From 45 Table IV, "^ 5*10 ; estimating the fast electron density to be a ° 2 43 fraction 10~2 of n , one obtains n A > 10 Even for a very large area 20 2 ° ° of n" 10 cm (21 x 2s; cf. above cited observations on X-ray source size) at least a density n ^ 10 cm is required. o In summarizing (a), (b) and (c) it is concluded that the observed values of , when translated in terms of a current, a beam or an isotropic distribution of fast electrons, are very large and put quite stringent requirements on flare parameters. In principle three possible ways out suggest themselves: - a more efficient mechanism for generation of X-rays than bremsstrahlung. - the effective fast electron energy losses are smaller than the! collisional losses. - currents become unstable long before their velocity reaches the velocity of sound. However, the present interpretation is as yet likely to be correct (if not, point (d), below, would become a coincidence}. Fwther work on flare theory has to explain how - loosely speaking - so many electrons can be accelerated^ as expounded above.

(d). The values of /P? dt as given in figs. 2-19 and summarized in Table IV represent the total energy dumped in the flare region by collisions of fast electrons in any model, cf. section 2.4.2. These values are also large, from ^ 2*10 erg in IXBs up to ^ 10 erg in EBs, and in addition they are lower limits because of the cut-off at 25 keV. It follows that the energy put into fast electrons is about equal to the total energy released (Brown, 1974? Lin, 1974a, Petrosian, 1973). It is therefore possible that all flare energy is channeled through fast electrons into other forms such as mass motion, heating, optical and EUV emission.

-•74- APPENDIX. TENTATIVE HARP X-fAY SOURCE PARAMETERS FOR EVENTS » 2 AND 3.

A set of |>Q^s.i;feig:jgggr,c.e,;;pa]|aineter5 for _the event #" 2'is qiven^in Table VII. For this event we obtained detailed SOLRAD 10 soft X-ray observations (Dere, 1974), The reasoning below probably holds for both event # 2 and 3, as they occurred in the same active region separated by two hours and had quite the same emission features (homologous flares).

Table VII. POSSIBLE X-RA¥ SOURCE PARAMETERS FOB EVENT * 2, OK MAy jq, 1972.

kT count rate VsAl A*5 UceV) (cm"3) channel 1 (cm i (cm J (cm! (cm) 2 (cour.ts/S cai *1.2 s)

b 2.9° •2.3-10"* 1C 1.6.1010 !028 10* io10 (0.1-100!

a0.5-3/1-8 S observations from SOLR.V3 10 at OT 1406:50 {Dere, 1974).

Observed, at UT 1406:50 above background: ">» 20 (fig. 4). The reported error of 30% in kT introduces the indicated error limits.

The first two columns of Table VII give observed temperature and emission measure. The count rate in channel 1 follows from Bekefi (1966) eqs. (3,65) and (3.82). Because of the large error limits, the agreement with what is observed is rather accidental. The density (and hence the volume V) follows by supposing that the plasma.is collisionally heated by the nonthermal

electrons, or 3n kT 2L ^^At- — 2*10 erg 'fi9- 5 and 7)••

The argument" is pushed further by taking a cylindrical voliamer axis parallel to the magnetic field, of length L and area A. Cooling by con- duction along the field lines has a characteristic time t i 10 * n L*"x T l (Culhane e-.t at. , 1970). For the characteristic time for expansion perpendicular to the field lines we take A divided by the sound velocity, t i. A (m /kT) . By requiring t = t it follows that t = t = 20 s, A = 109 em and L = 10 cm (Table VII). Now 20 s is a reasonable decay time of the hard X-ray emission of channel 1, cf. fig. 4 and 6. Of course t need not be equal to t , but if they are not about e c 2 h equal, then either of the two drops below 20 s, since t t is a constant

-75- for given n^, T and V. To be consistent, A and L should therefore have about the values from Table VII.

" ACKNOWLEDGEMENTS.

The results described in this paper have be'en. ootained thanks to the contributions"of many individuals, wnose efforts, the authors gratefully recognize. .

For the development and manufacturing of the instrument our thanks are due to Messrs. P.J. de Groene, J.j. van der Laan, W.A. Mels, W. Zandee and many other members of the Physics, Electronics and Mechanics Department of the Space Research Laboratory at Utrecht.

The software development for. datahandl. ug was mainly a responsability of Mr, Ch. Lapoutre.

Messrs. W.J.J. van Iersel ard G.W. Geytenbeek have assisted with the data analysis.

We are indebted to Dr. K.P. Dere

We thank Professor C. de Jager and Dr. G.A. Stevens for their critical remarks.

RSPEREHCES. Alissandrakis, C,E. and Kundu, M.R.: 1975, Solai' Thys. 41, 119.

SneSr, U.: 1'973, Solar Vhys. M, 459.

v*»n seek, li.F.: J973y Development and Perfoi-manes of a Solar Hard X-ray Spectrometer, Utrecht Uni%'ersity (Ph.D. Thesis).

var. Beek, H.F. and de Feiter, I.D.: 1973, la J. Xantbakis (sd.), Solar Activity and Related int&Fplcffietary mid Terrestrial Phenomena (Proceedings First European Astronomical Mee- ting, Athens :272;-, vol. 1, p. 103.

ReSi^ri, G.: S966, Radiation Processes in Plasmas (John Wiley and Sons, New York, 1966),

Besi*»r<3, G. laid Beak, D,L.: 1971, in A. "Simon and W.B. Thompson (eds.), AdvatiCes in Plasma Ph&gisa ilntssrsiionce Inc. . New York) , vol. 4, p. 125.

Br-auS.'.-., J,W. and Wfiite, o.R. : 1971, Astron. Aatrophya. j_3_, 169.

Aj J,C.: 1971. Solar Phys. 18, 489.

S», J.C.J 1972, :k?tar Phyo. 2i>, -141.

-76- Kane, S.R.: 13"M, i-. o. Newkirk, Jr. (ed.i, .'rrcial Disturbances (Proceedings IftU Symp. 57, Surfers Paradise li'~3: , f. 1C1-.

Lm, R.P.: 1974a, Sf^.'e $? -ev. 16^, 189.

Lin. R.P.: 1974b, in :.. Newkirk, Jr. ;ed.), Coronal Disturbances (Proceedings IAU Symp. 57, Surfers Paradise 1973:, p. 201.

McKenaie, D.L., Datlowe, D.W. and Peterson, L.E.: 1973, Solar Phys. 2E^, 175.

Milkey, R.W.: 1971, Selzs Phys. 16_, 465.

Neupert, W.M., Thomas, R.J. and Chapman, R.D. : 1974, Solar Phys. 34_, 349,

Peterson, L.E., Datlowe, D.V1. and McKenzie, D.L.: 1973, in R. Raraaty and R.G. Stone (eds.), High Sr.ergy Ph-SKormns SK The Sim (Proceedings Goddard Symp. Greenbelt, Md. 1972) NAJA SP-342, p. 132.

Petrosian, V.: 1973, Asircphys. J. U36, 291.

Petschek, H.E.: 1964, in W.N. Hess (ed.), The Physics of Solar Flares (Proceedings AAS-NASA Symp., Greenbelt, M<3., 1963) NASA SP-50, p. 425.

Petschek, H.E. and Thome, R.M. : 1967, Asirophys. J. 142, u57-

Phillips, K.J.H., Neupert, W.M. and Thomas, R.J.: 1974, Solar Fkys. 36_, 3B3.

Priest, E.R. : 1973, A'8trophyS. J. 181, 227.

Purcell, J.D. and Widing, K.G. : 1972, Astrophya. J. V!6_, 239.

Simon, P. and Mclntosh, P.S.: 1972, in P.S. Hclntosh and H. Dryer (eds.). Solar Activity Ob- servations and Predictions (Progress in Astronautics and Aeronautics, vol. 30; MIT Press, Cambridge 1972) p. 343.

Sonneru^, B.U.O. : 1970, J. Plasma Phys. 4_, 161.

Syrovat-skii, S.I.: 1966, Astro*. Ah. 43_, 340 [Soviet Astron. - A* \0_, 270].

Takakura, T., Ohki, K., Shibuya, N., Fujii, M., Matsuoka, M., Miyamoto, S. , Nishimura, J., Oda, M., Ogawara, Y. an<5 Cta, S.: 1S71, Solar Phys. _16_, 454.

Tilgr.er, B.: 197i, Eldo-Cecles/Esro-Cers Saient. and Tech. Rev. 3_, 567.

Vorpahl, J.A. and Takak.ura, T.: 1974. Astrophys. J. 191, 563.

Walker, A.B.C. and Rugge, H.R.: 1970, Astron. Astrophys. 5, 4.

Widing, K.G. and Cheng, C.C.: 1974, Astrophys. J. 194, Llll.

Yeh, T. and Axford. W.I.: 1970, J. Plasmc. Phys. 4^, 207.

Zirin, H. and Tanaka. K.: 1973, Solar Phys. 32, 173.

-78- CHAPTER III IN1W1PRETATION OF HARD X-RAY AND MICROWAVE EMISSION OF THE FLARE OF MAY 18,1972, UT 1406

ABSTRACT. We present an analysis of the flare of May 18, 1972, UT 1406, observed with the Utrecht hard solar X-ray spectrometer on board the ESRO satellite TD-lA. In this analysis we include the pertinent GHz radio data of this event. The proton density in the source region is in- 9 -3 ferred to be at least 5.10 cm . The radio spectra are found to agree with calculations of Holt and Ramaty. The radio and X-ray data together result in a consistent set of parameters.

3.1. DISCUSSION OF THE X-RAY DATA. On May 18, 1972 a IB flare of X-ray class M4 occurred in McMath region 11883. The event started at UT 1406, reached maximum intensity in Hot at 18 2 UT 1409, and its area was 3.7»10 cm [Solar Geophysical Data (prompt reports) , June 1972] . No type III burst has been observed in connection with this flare. In fact, this event appears to be a complex event with a faint precursor flare, followed by a major event 5 degrees SE of the precursor (Mclntosh, 1972). Fig. 1 displays time profiles of the X-ray burst associated with this event as observed with the hard solar X-ray detector on board the ESRO TD-lA satellite (see chapter II and Van Beek and De Feiter, 1973). In channel 5 (100 - 140 keV) and the higher channels the solar radiation did not exceed the instrumental background. The X-ray flux during the first 35 s is probably due to the precursor event. The spectrum fits very well to a (steep) power law at all times, as illustrated in fig, 2, The interpretation is well known by now: electrons are accelerated in the sol atmosphere and injected into a region with a magnetic field and a certain ambient density, giving rise to synchrotron radiation (GHz radio burst) and bremsstrahlung by electron-proton collisions (X-rays). The fast electrons loose their energy by electron-electron collisions. The average energy loss is given by (Brown, 1971) :

-79- UT 140609 MAY 18.197?

1 300 f- t

200)- j Z7- 3B keV i 100- V,w - —| 1 1 +—+—-1- ( 1—*. 1 1 =— 50 100

>" 80

z O u

-+•--+—i—i—i—'-^~-i-——~- ^#-^H

50 100 TIME ( SEC) —

fig. 1. Count rates of channels 1, 2 and 3 as a function of time. Missing data are dae to bad telemetry and for channel 1 also to subcommutation.

dE 55.7 IT e -h. -— = - n v <* n E (1) dt , E p p where e, E and v are the electronic charge, kinetic energy and velocity; n is the ambient proton density.

-80- 10- 14 06 09 UT 140559 ~\ 110619 1406 48 MAY 18,1972

lltl „ , PHOTONS/CMZSEC KeV \ &«44 6-55 \ J.59 I B -66

001 — POWER LAW FIT TO PHOTON SPECTRUM

• E(keV) 0001 n]—^ i i 1111—\ ' 20 5010020 50100Z0 5OtX>20 50 HO fig. 2. Power law fit to the X-ray spectrum at four different times. Background is subtracted.

—v For av initial electron spectrum of the form N(E,0)dE = cE dE Ve define a half-life time t, foorr decay due to collisions \?ast'as give given nb by] formula (1)] by N^t+t^) = . The following expression is derived:

t - _ (2)

The value for y needed in (2) can be found from that for the photon spectrum (fig- 2) by subtraction of h. The time variations in fig. 1 are too big to be of statistical origin (e.g. up to "" Aa in channel 3). A lower litait to the ambient density if found from formula (2), noticing that the decay time at "V 60 keV is less than 2 s:

9 -3 n > 5*10 cm P - The actual density in the source region is probably much higher: as the time resolution of hard solar X-ray detectors improved during the last ten years, they continued to detect time variability in the X-ray flux up to the limit set by their time resolution. In all our observations up to now

-81- we found these fast-decay fine structures. We conclude therefore that the so called impulsive injection models (Takakura and Kai, 1966) are untenable, if more easily tractable. We can compute the electron injection function f(E), if we take the photon _Y 2 spectrum I(e) to be a power law at any instant tl(e) = a E^v phot, (cm -s* keV)~ ] and if we assume the ambient density to be sufficiently above its

MAY 18.1972 SAGAMORE HILL RADIO DATA

340 15 4 GHz

225

112

128 2.7 GHz

I 2.9 14 GHz

J L 1403 4 5 6 7 8 9 1410

fig. 3. Time profiles of the microwave flux density at five different frequencies associated with the flare of May 18, 1972, UT 1406.

-82- TIME EVOLUTION OF RADI MAY 1H. W? . JT -a 0615 «oo-

200-

I 70K IS sec

9 (•

Z O

4 *

, 1 ! 17 10 20 4710 20 *- FREQUENCY (GUI) fig. 4. Microwave spectra at five different tines. The curves at t = t + 1b, t +60 and t + 180 s are spectra calculated fcy Holt and Raraaty ri9f.9) and the other two are hand drawn best fits.

lower limit (thick target assumption; Brown, 1971):

-1 f(E) =7. (keVs) (3)

F(x) is the gamma function; E is in keV. Thus vie find the total injected energy due to electrons with E >_ 27 keV between t = 0 and t = 100 s to be 1.5.10 erg; the total number of these injected electrons: 3.10 . The maximum injection rates occur at UT 1406:09 viz. 6.3*10 erg/s and 1.2*10 electrons/s, respective"y. In section 3.2. we need the total number of electrons in the source region, N, with energies above 100 keV, at t = t + 15 s (t is defined in fig. 4). By extrapolation of the injection function to higher electron energies, 34 this number is estimated to be N ^ 10 . In this calculation, collisional decay of electrons already injected in the source region was accounted for, 9 -3 taking n =5*10 cm

-83- 3.2. RADIO DATA AND THEIR RELATION-TO X-RAY DATA. In fig> 3 the micr.aw.aNEe emission, at several frequencies associated with this flare is shown. The shape of this burst exhibits the same features as seen in fig. 1. The only conspicuous difference is the long-lasting tail in the radio flux density. At 8.8 GHz the decay of this tail obeys:

F8.8 GHz - 100 keV. According to formula (2), 9-3 taking n =5*10 cm ,t=0 and t, = 15 s, one needs energies of the P T order of ^ 150 keV. Hence, it is possible that the tail in the microwave time profile is due to electrons with energies > 100 keV slowly decaying in the source region. There remains the problem that these electrons must be shown not to give rise ~o a dp*" JU le X-ray Sux. Fig. 4 sh;«;s the t:\ie evolu'. :_" o . the radio spectrum. xne spec:.-a at c - t - 15, tQ + 60 and t.. + 180 s show good agreement with the Hoi. om< Raanty ca2'_ Nations. Surprisingly, this fit is better than for fie flare stut.: sd i.y Molt and Ramaty (1969) themselves and it contradicts th= remark;-- mad-j oy Zi-in et at. (1971) on the steepness of the Holt and Ratnaty spectre . Unfcrtu: atelv, v-.e H 1 R spectra assume -i low energy cut-off at 100 keV in thefc.'lr.-c'.ror. s^ec-trjr. therefore, a comparison of tha spectra for t <_ t is impcisibl-2 0,ar t~ thi copious presence of electrons with energies below

A luant .-,a' Ive„ '.hough not exclusive, fit of the soectrum at t = t + 15 s 9-3 "0 with n^ = 5*10 cm yields the following values for the magnetic field B, the ar.a A of the emitting region and the total number N of electrons with energies above 100 keV:

B = 95 G; A = 1.3'IQ17 cm2; N = 1034

-84- The H & R spectra do not depend strongly on the parameter Ct = 3& /2ui_ ; we used Ct = 0.7. The value of N is the same as derived above. Since 34 10 electrons with E >_ 100 keV, given the steepness of their spectrum and the background levels, indeed cannot give any detectable X-ray flux, we can say that the X-ray and microwave data confirm each other. A weak point is that the value of Y used by H & R does not apply. It is therefore desirable that their calculations be extended to other values of Y and lower energy cut-offs. The area A is a factor 20 smaller than the maximum Ha area: however, they apply to times separated by ^ 3 min., and, presumably, to entirely different flare regions. If one accepts this value for A, then the hard X-ray/microwave flare would be very small: 5" x 5".

ACKNOWLEDGEMENTS. The radio data included in this analysis have been kindly placed at our disposal by Drs. J.P. Castelli (Air Force Cambridge Research Labora- tories), D.L. Croom (Radio and Space Research Station, Slougb), P. Kaufmann (Centro de Radio-Astronomia e Astrofisica, Sao Paulo), and E. Schanda (Institut fur Angewandte Physik, Bern) .

REFERENCES. van Beek., H.F. and de Feiter, L.D.: 1973, in J. Xanthakis (ed.j, ScZar Activity and Reli- ted iKterplamlary and Terrestrial ther'tomana (Proceedings First European Astronomical Meeting, Athens 1972), vol. 1, p. 103.

Brown, J.C.: 1971, Solar Phys. 1£, 489.

Holt, S.S. and Ramaty, R.: 1969, Solar Phys. 8_, 119.

Mclntosh, P.S.: 1972, private communication.

Takakura, T. and Kai, K. : 1966, Publ. Astron. Soo.

Zirin, H., Pruss, G. and Vcrpahl, J.: 1971, Solar Phys. lj>_, 463. "-''"'

-85- CHAPTER IV BETAMOM ACCELERATION IN TME LARGE SOLAR FLARE Of AUGUST 4,1972

ABSTRACT. The problem of diagnosing flare particle acceleration mechanisms from hard X-ray burats is discussed and it is argued that the electron trap -nodel of bursts is more amenable to observational investigation at presunt than models of thick-target type. It is then shown that data for the large X-ray burst of August 4, 1972 are consistent with the source electrons being trapped in a very large vibrating coronal magnetic bottle. Futhermore the observations show that the burst time profile is not dominated by colliaional losses. It is proposed instead that the entire profile is essentially determined by betatron action of the varying trap field on the electrons. This betatron model is then analysed in detail and shown to predict very well the observed correlation of electron flux and spectral index in this event when it is supposed that ths electrons are initially produced by a brief, unspecified acceleration. Comparison of the model with observations permits inference of the approximate form of magnetic field evolution in the trap.

4.1. INTRODUCTION. Solar hard X-ray bursts are widely recognised as a potentially valuable key to understanding particle acceleration processes in flares and as being particularly important since the emitting electrons ("" 20 - 200 keV) comprise the bulk of the energy in flare particles and sometimes in the entire flare (see e.g. reviews by Kane* 1974, Lin, 1974, Brown, 1974). Neverthe- less, hard X-ray burst studies have in the main not led as yet to any major steps forward in quantitative understanding of electron acceleration, for several reasons. Firstly the great majority of bursts studied are small and of short duration so that the count statistics, and the instrumental time resolution used, limit interpretation mostly to establishing general trends in the spectra and intensities of bursts (e.g. Kane, 1974, Datlowe et al., 1974). Secondly, when large events are observed with good count statistics and high time resolution, it is necessary to use a rapid reduction technique in order to obtain the dynamic X-ray spectra, with full instrumental time resolution, which are necessary for detailed quantitative inference of electron acceleration parameters (Brown, 1*971). This procedure

-86- has not generally been followed but rather replaced by consi derations of only general burst trends over sample time intervals Ce.g. Frost, 1969) Van Beek et at., 1974). And, above all, on the theoretical front there is a lack of quantitative models predicting the intensity and spectral evolution of bursts, to which observations can be compared. In this chapter we develop such a model of electron acceleration to describe the main phase of a very large burst observed by the hard X-ray spectrometer aboard the EJ3RO TD-lA satellite (Van Beek. 1973; Van Beek et al. , 1974). The event in question was sufficiently large and prolonged to provide observations with excellent statistics, permitting accurate spectrum determination with the full instrumental time resolution of 1.2 s. Actual reduction of the data to over 10 instantaneous spectra was made feasible by utilising the rapid pulse height inversion technique established by Hoyng and Stevens (1974), Two distinct types of model of electron acceleration in X-ray bursts prevail in the literature. Firstly there are those involving continuous injection of electrons into a target from an unspecified 'black-box1 acceleration region. In these the burst time profile is determined purely by the modulations of the black-box electron source since the X-ray emission lifetime after injection into the target is very short due either to energy loss in a dense plasma {thick target - e.g. Brown, 1971) or to escape from the emitting region {thin target - e.g. Datlowe and Lin, 1973). Thus though the thick target model, for example, is of interest in connection with optical and UV flare heating and in providing a useful reference standard (cf. section 4.2.) it does not lead to a model for the electron acceleration because this process is decoupled from the x-ray source (this is equally true for a thin target).

The second class of burst model is that in which energetic electrons are initially produced in a rather brief acceleration phase (Takakura and Kai, 1966), again unspecified, and subsequently emit X-rays while trapped in a coronal magnetic bottle. According to this electvon-tvap model, the burst time profile is governed, after initial acceleration, by the development of the fast electron flux and spectrum within the X-ray source itself, and so provides a direct diagnostic of conditions in the source. In particular, the simple impulsive (single spike) time profiles of some small

-37- events (Kane, 1974) may be attributed to the collisional decay of the fast electrons in a static trap, the decay tj.me yielding the ambient plasma density. Secondly the softening of some such events in their decay phase (Kane and ftndei?s©n,197/Q ) may be Attributed to non-uniforraity of tha am- M.en,t plasma density in comfeinafcion with an energy dependent pitch angle distribution of the injected electrons (Brown, 1972) . Larger and more complex events, however, cannot be interpreted in this way since their time profiles display multiple peaks, contrary to the monotonic decay characteristics of collisions. Brown (1973) has pointed out that, in these cases, the X-ray burst profile may be determined by oscillations of the trapping field which modulate the density of the trapped plasma and, more importantly,- drive the trapped fast electrons. That is, the trap model must in fact invoke continuous acceleration of electrons to explain the profiles of large bursts but, distinct from continuous injection situations, the acceleration occurs as part of the behaviour of the X-ray source itself and not as a decoupled mechanism. In this chapter we elaborate the electron trap model further. First we show that the general features of the large August 4, 1972 event are compatible with a trapped electron X-ray source modulated by Alfven oscillations, subsequent to a short initial accaleration. Then we derive quantitatively the effect of field changes on the trapped electron flux and spectrum in the trap model and show that these predictions are in good agreement with the observed interrelationship of the instanteneous flux and spectrum in the August 4 event (Hoyng et al. , 1974; chapter II) during the trapping phase. (The mechanism of acceleration in the initial phase is left unspec ~ied.) This enables us to estimate the magnetic field and density evolution in the trap directly from the burst observations. Finally we briefly discuss possible origins of these field variations and their relationship to other observations.

4.2. THE AUGUST 4, 1972 EVENT. At 0620 UT on August 4, 1972, one of the largest events of the present solar cycle occurred. The entire event was observed by the hard X-ray spectrometer, aboard TD-lA, which has twelve channels, with approximately logarithmically spaced boundaries viz. 29, 41, 53, 71 etc. keV, this burst being detectable up to channel 8 (i.e. about 350 keV). Time profiles for channels

-88- 4000

3500

3000

2500

2000

1500 •

uo 1000

500 •

0620

fig. 1. Observed count rates (channels 1,3,5 and G) for the large event of Augasc i, l'.'~2. Time resolution is 1.2 s for channels 1-4 and 4.6 s for higher channels.

1, 3, 5 and 6 are shown in fig. 1. Hoyng et at. (1974) have carried out a detailed tenporal analysis of this and other events using the full 1.2 s time resolution of the instrument. Firstly, using the technique established by Hoyng and Stevens (1974), the counts were converted to photon spectra for each instrumental integration period. Next, though slight deviations from a single power-law were present in these spectra (cf. Van Beek et at., 1974; see also section 4.4.), the instanteneous best fit power-laws

= a(t)E-YU) (1)

were determined. As part of the routine data reduction, the thick target electron flux was then derived according to (Brown, 1971)

-89- AUGUST -4,i972

3OOOI-

\ keV V 2OCK> f! \ lOOO

8xlOJsr T dt = 3 5x 10"'9 elec trons "P dt = 2Ox1O3? erg

Oi-

4 2-

, 38- U-

3 Of

2 6-

12r

1 I 08}- 14 04 r

0 2j 0620 0635 •— TIME (UT) fig. 2. Time dovfsicprnent of the X-ray spectral index Y and thick target electron flux F cotn- cated from the data in fig. i. using linear interpolation to fill small data gaps. In the F profile, the integrals show respectively the total number and energy of electrons injected (E "• 2r, keV) on a thick target interpretation, o Channel 1 data are again shown at the top of the figure for comparison purposes and lecters A-F refer to corresponding points in fig. 3. The bottom profile refers to ambient field and density variations interred from the betatron model (see sections 4.3. and 4.4.).

-90- F(t) = 4.1-1033 (25) ^ a(Y-l)2 B(y-h,\; (2) with a(t), y(t) obtained from (1). F is the instanteneous rate of injection of electrons with energy exceeding 25 keV, into a thick target, which would be needed to produce the burst. (The figure of 25 keV is adopted merely as a convenient reference energy not involving significant extrapolation beyond the directly observed energy range.) Use of the thick target model in the initial reduction was motivated by two considerations. In the first place the thick carget model yields a figure for the continuously injected electron flux needed for the burst, independent of the target density or its distribution (Brown, 1971). Secondly, since the electrons injected into a thick target loose all their energy collisionally, the bremsstrahlung efficiency j.s a maximum and the model sets lower limits to the total number and energy of electrons needed for burst production (Brown, 1974; Hoyng et 2'. , 1974). Thus we adopt the procedure of analysing bursts first in terras of the thick target electron parameters and later convert to parameters appropriate to the model we wish to consider using the relations derived by Brown (1971). Fig. 2 shows the time development of F and of y through the August 4 event, together with the count rate in channel i. Mote that, for clarity, only every fifth data point (having 6 s spacing) has been included in these profiles. Also shown in fig. 2 are the total number and energy of electrons above 25 keV which have to be injected into a thick target to produce the entire event. The relatively minor influence of statistical fluctuations in the observations is evident from the smoothness of variation of the fitted spectral index y. In particular, this permits comparison of instanteneous values of the electron flux F and the X-ray spectral index y with the result shown in fig. 3 as a plot of F(t) against y(t) (after Hoyng et al. , 1974; chapter II). What is most striking in this figure is its division into the 'dog-leg1 portion AB, swept out by the event during the brief initial rising phase AB in fig. 2, and the portion CE indicating strong correlation of if and y throughout the remainder of the event. The time sequence through this latter (main) portion, is also quite distinctive. After the initial rise AB, the event follows a series of counterclockwise loops

-91- through fig. 3, centred about the general line of correlation, each loop corresponding to one of the large oscillations visible in the original time profile (fig. 2). As the event profile decays (subsequent to about UT 0632) and the oscillations die out, the loops in fig. 3, transform into a monotonic path down the curve DE. Physical interpretation of the part CE of fig. 3 in terms of an electron trap is the principal aim of the present paper. As already indicated in section 4.1., interpretation in terms of a thick target will require theoretical treatment of, for example, stochastic acceleration in a plasma turbulent region of field reconnection. No solution of this problem seems imminent. In an electron trap, on the other hand, the electron acceleration is governed only by time variations in the trapping field intensity and this is amenable to a much simpler description. Before proceeding to obtain this description in section 4.3., we present some general evidence that an

i—i—|—i—i—i—i—;—i—i—i—i

CORRELATION of T and Y

AUGUST A . 1972 UT0621-0o39

_ 4 b

fi I I I U 30 35 40

fig. 3. Correlation diagram obtained by eliminating time between Fit) and y(t! in fig. 2, again showing only every fifth data point for clarity. The solid line (based on all data points) indicates the smoothed path of the event development, discusce^ fully in the text.

-92- electron trap may Indeed have been involved in tne August 4 event, and derive some constraints on the parameters involved. Firstly tne sharp division in form between the main event CE in fia. 3 (lasting about 15 minutes) and the rapid initial rise AB (lasting about 1 minute) is indicative of distinct physical processes governing the two phases and, in particular, is consistent with the trap model postulate of an impulsive acceleration phase followed by prolonged burst development under quite different conditions. Secondly Hoyng et at. (1974) have found real periodicities in the time profile of this event with periods up to about 120 s, the period apparently increasing with time. (Periodicities have previously been reported in large bursts - Frost, 1969, Parks and Winckler, 1969). According to our trap interpretation this 120 s period should correspond in order of magnitude to the travel time of a magneto-acoustic wave across the typical dimension L(cm) of the trap. We take this for the present to have the Alfven speed (cf. section 4.5.) which, if the field is H(gauss) and the plasma density n(cm ), implies the relation

Ln /H 2l 2.6-1013 (3)

General oscillatory behaviour is in fact observable in the original time profiles (fig. 1) and is seen to persist, though with decreasing amplitude, right to the end of the event (note especially the 53-76 keV channel). As discussed in section 4.1., such behaviour is incompatible with a collisionally governed time profile. Therefore we have the further condition that the mean density n should be sufficiently small not to totally obliterate the burst oscillations observed right down to 30 keV. This requires that the energy loss time t of a 30 keV electron should not be 3 less than 10 s so that, using the formula for t given by Brown (1972) ill

n ^ 2'107 cm"3 (4)

We will see later that the model requires that n should decrease substantially toward the end of the event. We estimate that the initial density n of the plasma in which the electrons are initially accelerated (phase o

-93- AB) could be as high as

n '- 4*10 cm " ir-) o — without violating the secondary role to be played by collisions throughout the event. This figure determines the total number N of electrons above 25 keV which must be present, in the trap to produce the observed burst intensity. The non-thermal emission measure nN deduced for this event (Hoyng et al., 46 1974) was about 5*10 so that

N \. 1C39 (6) which corresponds to a total energy of about 5*10 ergs of electrons. If the magnetic field is to be capable of containing these particles, we obtain a final condition that its energy density H ,'B^ should be least comparable to theirs, viz.

The conditions (3) - {7} imply L > 1.1'IC ^. However, if we take L '_± 10

then the number density of fast electrons n1 = N/L" IS larger than the ambient plasma density n, which seeras inplausible. We are therefore driver to adopt a large value for L, giving, as a typical set of parameters

L ^ 5-2O10 cm H ± 12 gauss 7 -3 ( ' n = 4*10 cm ° 6 ^ n1 = 8-10 cm

These parameters, though involving a strikingly large volume, are quite compatible with direct observations of coronal active features obtained by various means - cf. Newkirk (1967), Vaiana st al. (1973), McQueen et a.1. (1974). Such a trap geometry is also consistent with our assumption, in (7), that the trap volume is L - i.e. that the trap has a thickness

-94- comparable to its length. Furthermore, it must be reiterated that this flare was one of the largest on record. Interpretation of certain large events in torms of coronal traps has already been suggested by Hudson flS'V3) on the grounds of the visibility of X-rays even when the associated chroinospheric flare is far behind the limb, a prime example being the event of March 30, 1969 (Frost and Dennis, 1971) which shows similarities to the event we are studying here. Both the limb occultation and data on the closely related radio phenomena (Hudson, 1973) indicate an X-ray source extending very high in the corona. We show, further, in the Appendix that for the above parameters, the model predicts the GHz radio observations quite well. More definite figures for the trap parameters will, however, have to be based on a proper assessment of the motion of waves along the trap. In particular, the -iKei'oy density n E of fast electrons is seen from (8) to be at least comparable, and probably in excess of, the thermal energy density n kT of the ambient plasma. Parker (1965) has shown (in connection with o galactic cosmic rays) that under thsse conditions wave modes exist propagating at near the sound velocity in the suprathermal gas. With the parameters given in (8) and with E ^ 20 keV, this sound velocity exceeds the Alfven speed by an order of magnitude. To obtain the same oscillation time [cf. (7)] , the trap would then also have to be that much larger. However, this would implv n E -'•< n kT and a situation again controlled by Alfvenic 1 o motions, so that the true dimensions of the trap must be somewhere between these limits.

4.3. THE BETATRON MODEL. For the purpose of describing the effect of the changing magnetic field on the trapped plasma and energetic electrons, we will take the field H to be spatially uniform, within the trapping region, but time varying. In fact, for standing waves to be set up along the tube, its ends must be tied to the solar surface, as shown in fig. 4. Therefore our assumption corresponds to supposing the electrons to be confined in the upper part of the tube in fig. 4, in which case one- imensional geometry should be an ade- quate first approximation. The actual presence of a time varying transverse field component has the additional effect, however, of producing some Farmi acceleration of the trapped particles. We shortly show this to be negligible compared to the effect of the longitudinal field changes.

-95- H

H U)

fig. 4. Impression of the evolution of the coronal trap after electron injection (top). The middle figure shows the small amplitude field oscillation (dashed line) set up, superposed on the overall expansion of the bottle responsible for the event decay (bottom).

-96- Since we are neglecting collisional energy losses and since the electron Larmor radii are very small compared to the scale distance of field variation, the electron energy changes due to longitudinal field variations are given by the adiabatic invariance of their magnetic moments (e.g. Spitzer, 1962) - just as in a betatron accelerator. That is

E./E, = H/H (9) X xo o where E• , E, are the energies in transverse motion (= p,/2rn) when the field has values H, H respectively. Thus betatron acceleration occurs on o the time scale T of magnetic field changes in the trap, that is 3

TB < L/Vft CIO) where v is the Alfven speed. Fermi acceleration by reflection between the travelling transverse field changes, on the other hand, will occur on a time scale x given by (Sweet, 1969)

from which it follows that T >> T and Fermi acceleration can be neglected. F B To relate eq. (9) to the changes produced in the flux and spectrum of all the fast electrons, we require to know firstly the flux and spectrum at one instant, and secondly the relationship between the transverse energy E. and total energy E of the electrons (i.e. their pitch angle distribution) at some instant. We take this reference instant to be the time at which the initial phase of acceleration of electrons in the trap ceased - i.e. the end of phase AB in fig. 3, namely at 0622:54 UT. Quantities pertaining to this instant are given the subscript o, i.e. the field is H , tho plasma density n and so forth. Directly from the observations we in- ° 36 -1 fer Y =3.4, F =4.6*10 s while the distribution of energetic elec- o o trons (electrons per unit total energy E ) is N -6 f (E ) = (6 -1) -2- (E /E) ° (12) o o o E. o 1

-97- where N is the total initial number of electrons present with energies o greater than the reference energy E. (which we later set = 25 keV to correspond to FJ while the eleots-pjh spectral index <5 is related to the X-ray spectral index y by (Brown, 1971)

<5 = y - h . (13) o o

The effect of the field changes on the total anergy E is given by

E +E = 1+hE = f- 1O l)o V ic/V o

where

h = H/H - i (15) o 2 and E, /E = sin a where a is the initial electron pitch angle, lo o o o As regards the initial distribution of pitch angles, we cannot predict this in detail without a quantitative model of phase AB of the burst. Here we suppose the initial acceleration to have occurred by the action of a large scale electric field. Under these conditions the acceleration occurs by increase of the energy E. along the magnetic field only, Ei being unchanged. It does not seem unreasonable, therefore, to treat E, o-O as a constant for all fast electrons, to be interpreted as the mean transverse energy cf the electrons that run away under the action of the electric field on the plasma. Since it is outside the scope of the present papsr to treat this process quantitatively, we henceforth treat E. as a J-o free parameter and write (16)

Combining equations (12), (14) and (16) we thus obtain, by continuity, the spectrum f(E,h) of trapped electrons at any time subsequent to the injection phase (h parametrising time)

dE N . -6 f(E-h) =fo(V sr- (6o-1} if {5i7-} °

-98- From Eq. (17) we now obtain the total number of electrons and their effective spectral index, at any time. The total number of electrons above energy E, now becomes

« .-6 +1 N(h) = / f(E,h)dE = N (1 - he/E,) ° (18) El Since the spectrum (17) is not exactly a power-law, we must define a power law index 5 to be associated with it. For this we take the logarithmic point slope at an energy E^ near the lowest channel boundary of the X-ray spectrometer used to obtain the data - this corresponds as closely as possible to the technique actually used in fitting the X-ray data to a power-law. That is, we take [using: (17)]

J |E=Ei

which is related to the X-ray index Y at the same instant by [cf. Eq. (13)]

6 = Y " h (20)

For purposes of comparison with fig, 3, we first need to relate N to the thick target flux parameter F used in reducing tho observations. Using the equations of Brown (1971) we have:

p/p - Y"3/2 n N F/Poo " 7^572 J o As far as the ambient density n is concerned, with our assumption of one-dimensional trap geometry, this is simply given by

n/n = H/H = 1 + h (22) o o

Now N/N is given by (18) as a function of h; substitution of this and o (22) in (21) would give F/F as a function of h. However, we actually require to have F/F as a function of Y- We can express h in Y by using (19) o and (20), viz.

-99- E _Y - Yo -j h = (23) Y - and, consequently we obtain -Y +3/2 'o (24) Y - h

Using the initial values of Y , F (at 0622:54) already quoted and

setting E = 25 keV (which defined F in fig. 3), Ei = 30 keV (to correspond to\the lowest channel boundary), we show in fig. 5 the results of prediction (24), ~e.s compared to the data, for values of the parameter £ = 7.5,

-| 1 r—l 1 1- "«—I—"" = 15 keV CORRELATION of T and Y ./E =20keV 1 8x10J AUGUST 4 , 197? UT 0621 - 0639

START OP Z 4 BETATRON PHASE (0622.54)

I I—I I I l__J t I i i i i I i 2.5 3.0 3.5 4.0

fig. 5. Theoretical (7,y) correlation lines, based on the betatron model for three values of the parameter E, superposed on the observed data. According to the basic model, the point (Y,F) is constrained to move on one of these lines, the actual motion depending on hit).

-100- 2.0

h-.Q2 iH»12 Hoi

tl.00 ( H.H,)

1.0

h.-08 (H-02

25 50 75 1OO 150 200 --ELECTRON ENERGY E--(keV) fig. 6. Electron spectra [Eq. (37)]..due to betatron action on the trapped electrons for initial and extreme values of the field ratio.

15 and 20 keV. Evidently .the model predicts the observed (F,y) correlation well and is not highly sensitive tc the choice of e. The best agreement with observations is obtained for e ^ 15 kev" which, interpreted as a runaway energy, implies an initial plasma with a thermal energy kT a few times 7 smaller than e, i.e. T ^ 3 - 5 x 10 K,a result in satisfactory agreement with, soft X-ray measurements of the high temperature plasma of large flares (Neupert, 1969). In addition, the range of h covered by the correlation in fig. 5 permits us to calculate the extremes of deviation of the electron spectrum (17) from a pure power-law. In fig. 6 we show that the electron spectrum at- tained at the extreme peaks of the burst (h 2L °-2> is very slightly concave upward while near the end of burst decay (h = - 0.8), and to a lesser extent at burst minima, the electron spectrum shows an inerea.se in y of

-101- about 0.7 in the index over the energy range 25-150 keV. Bearing in mind that these have to be integrated over the bremsstrdhiung cross-section to obtain X-ray spectra and that we have neglected relativistic effects (among others - cf. section 4.4.), we conclude that the predicted spectrum does not differ significantly from observations,

4.4. COLLISIONAL AND OTHER CORRECTIONS. We conclude from fig. 5 that betatron acceleration of electrons injected into a coronal trap by a direct electric field could be a realistic model of the August 4, 1972 event. The basic reason for the qualitative agreement of the model with the observed (F,y) correlation is that, since electric field acceleration results in smaller pitch angles for higher energy electrons, these are least affected by betatron action on the transverse energy. Thus as the magnetic field increases, lower energy electrons are accelerated most and the spectrum softens (y increases) while the flux increases - and conversely for decreasing field - just as observed. However, the agreement with our prediction of F(y) (Eq. 24) can be seen to be imperfect. In particular, at higher values of the flux F (i.e. of the field H), the data points mostly fall at smaller y than our predictions - i.e. the burst does not simply sweep back and forth along the predicted correlation line. Though we do not propose to pursue refinements and variations of the model in detail here, we may check whether the realization of various idealizations we have made could explain the form and magnitude of thii deviations from the modal. (i) Collisions. A complete treatment of the effect of collisions would require solution of the full continuity equation for the energetic electrons, allowing for collisional energy loss and scattering and for the time dependence of the ambient density. Since, however, collisional effects are, by hypothesis, small, we can obtain an estimate of their effect from a perturbational correction to eq. (14). This perturbation solution yields

(Eo ~ C) E ^ EQ + h£ - K n —2__— t m(t) (25) E o where n is the mean plasma density through the event and m(t) is a factor

-102- of order unity, depending on the detailed form of 'n(t), t being the tir;e 4 h after initial injpcti<"">n an^ K - ^TTR A(2/m ) is the constant ir: the Cculcib ' e energy loss cross-section, with c,m the electron charge ar.d mass respectively and A the Coulomb logarithm. If m is approximated to unity, inclusion of the collisional term in (17) leads to the rusults

(Y " 3/2)T ! F /F = 1 ° -1/? (26) (1-hE/E ) and

._ = _ I !l,3/2 V y Y ( ; Y 2 (E ; ,. ,5/2 2 E i (1-hE/E ) l with

x = Knt/E^/2 (28)

Here F1, y' are the results including collisions, F, Y the results without collisions, and T is a dimensionless time. Elimination of T between (26) and (27) shows the effect of collisions on fig. 5 to be displacement of eacli point on the correlation curve (24) along a curve [in fact a straight line in a (log F,y) diagram] to smaller F and y - i.e. to the left and downwards, the displacement increasing with time. Taking n 2i 2*10 cm and t = 200 s in the above equations yields Y' - Y 2L - 0.5 and F'/F 2l 0.7 as the order of the displacement due to collisions during the three highest peaks in the burst (fig, 2). Thus collisions move the upper part of the predicted correlation curve (fig. 5) in the right sense and by the right order of magnitude to explain the spread of data points to the left. (ii) Non-Uniform Field. In section 4.3. we took the trap field to be spatially uniform. The curvature actually present in the trap field will modify some of our equations based on one dimensional geometry [e.g. both (9) and (22)] . We do not attempt to evaluate this effect here but merely note that spatial variation of our equations through the X-ray source will add to the spread in fig. 5. As noted in section 4.3., however, the existence of a transverse

-103- component in the oscillating field also results in some Fermi acceleration of the trapped electrons. Thouqh this was shown by eq. (11) to be slow compared to the betatron process, the high Alfven velocity involved (2i 5000 km s~ ) prevents its effect being entirely negligible. Since the Fermi process increases the longitudinal component of electron energy its effect will be to enhance the existing pitch angle/energy relationship -- equivalent to a decrease in c - and so again move points to the left in the upper part of fig. 5. The actual pach taken by the burst through the correlation diagram (fig. 3) is not explained by these general considerations of effects ft) and (ii), but we note that the incorporation of these refinements in the model appears capable of explaining the small discrepancy of the results of section 4.3. from the observations.

4.5. THE MAGNETIC FIELD CHANGES. The actual behaviour of h, F and y as functions of time has so far not been brought into our physical description of the betatron model, time entering the equations only via h. If the actual burst data followed the predicted correlation line in fig. 5 exactly, however, we could derive completely the evolution of h(t) = H(t)/H(o) by evaluating h from eq. (23) and the observed y(t). In practice this procedure will lead to some spurious element in the deduced field variation since individual data Y(t), F(t) do not conform in detail to their theoretical interrelationship derived in section 4.3., due to the idealisations discussed in section 4.4. Since, however, the necessary corrections are not too large, the field development h(t) derived in this way should be indicative of at least the general trend in field behaviour required to explain the burst time profile. We have chosen to derive h(t) by expressing h as a function of F by elimination of Y between (23) and (24) and so computing h(t) from the data F(t). In fig. 2 we have added the derived variations of H/H (and o hence n/n ) throughout the event, subsequent to the initial acceleration phase AB. The general behaviour is seen to follow that of the electron flux as expected. However, one can also derive h(t) from Y(t) directly using (23). One then gets a behaviour resembling that of Y(t). The reason for the differing results is of course that the correlation (F,Y) is not per- fect and the true behaviour of h(t) can only be derived form the observa-

-104- tions once we have obtained a model for the complete detailed behaviour of F(t) . In fig. 2 it is very notable how small a field variation (1.0 H tc o 1.2 H ) is needed to produce the very pronounced main peaks in the burst profile. This is because only a quite small increase in individual electron energies is needed to produce a large change in the total flux above any fixed energy, due to the power-law spectral distribution and, secondarily, because the field variations change the X-ray flux by changing the ambient density n as well as the fast particle flux. Thus only compression waves of rather small amplitude have to be set up in the trap to explain the burst oscillation (contrary to Brown, 1973). The form of field variation is thus seen to correspond to a small (standing wave) oscillatory component superposed on a more or less monotonic field decay - i.e. expansion of the trap as the event proceeds. This expansion may be compared with the expansion of coronal magnetic bubbles on a larger time and distance scale observed directly by the ATM corono- graph (McQueen et al. , 1974). Field oscillations have, on the other hand, been invoked in relation to radio pulsating structures (Rosenberg, 1970) . As to the origin of these field changes, they might be attributed to an external controlling factor - e.g. motions of the photospheric feet of the bottle (fig. 4) associated with the changing field which initiated the flare. On the other hand, in this particular event, the energy density of trapped electrons must be comparable to that of the trapping field, unless the trap dimensions are even larger than the values we adopted ieqs. (3)~ (8)]. Therefore it seems likely that the field variations in fig. 2 ar^, at least in part, determined by the dynamic response of the trapping flux tube to the injection of a large amount of energy in the form of fast electrons. This is an intriguing aspect of the problem, which needs further investigation. Specifically, though the response may be expected to indeed comprise an oscillatory component (due to wave reflection at the feet of the tube) superposed on an overall field expansion (due to the fast particle pressure), it is unclear why the expansion time scale should be long compared to the oscillation time, (fig. 2), since naively those might both be expected to be ^ L/v . Proper analysis of this problem, will, however, have to take account firstly of the interaction of the flux tube and the surrounding active

-105- recjion atr.ic^phere a::d secondly of Hie presence of the sunratnormal wavn ..,,,;.,.: ;p,-ukor, '. ''-'• in the trap. Finally it is of interest that, in the inodt;l we have proposed, the energy initially released in the form of energetic electrons, presumably by dissipation, of the active region field, is ultimately returned to the field (minus a small collisional loss) when it expands under the pressure of the trapped particles. This state of affairs reduces the amount of energy which has to be dissipated ultimately from the active region field as a whole in order to produce a hard X-ray burst. The problem of the huge number of electrons involved in the burst remains, however, (cf. Brown, 1974, Hoyng -.: -i'-., 1974) and suggests that th° electrons may be supplied from the low dense regions of the arch in the initial phase (cf. fig. 4).

APPENDIX. Evaluation of_Expected GHz Radio_Flux. Takakura and Scalise (1970) have computed the volume emisslvity of cyclotron radiation of an isotropic electron distribution in a homogeneous field. Since these computations were performed for a spectrum N(E) rv E , we may readily adapt the results to estimate the expected flux density from our electron trap model. Ti.v ing an aspect angle of 60 and neglecting re- absorption and plasma effects (which limits the application to high frequencies) one finds:

F 0, 2-1O"33 N V (V /v) (29) V n ri —22 —2 —1 where F is in solar flux units (10 Wm s ) and v , the cyclotron fre- 39 quency, in MHz; from (6) and (8): N ^ 10 and \> = 34 MHz. In (29) an extreme relativistic approximation is made, limiting its validity to v/V > 2 H % 10 or V > 4 GHz. Comparing the observations (Croom and Harris, 1973) with the predictions one gets

F 4 1Q ,,„ 2il 1.2-10 (observed: ^ 2-10 )

4 4 16-10 (observed: ^ 1-10 )

-106- The agreement is within an order of magnitude, which is very good considering the number of appruxiiiwi.iuns involved. In this frequency range the similarity in the observed hard X-ray and GHz time profiles themselves is also very good. Less good agreement between both the time profiles and the fluxes obtains at both higher and lower frequencies (the latter partially attributable to absorption effects). By and large, however, it can be said that the GHz observations are consistent with the electron trap model. The radiation at these high frequencies ori- ginates from harmonic number '•-- 10 and is emitted exclusively by the (extreme) relatxvistic electrons, £ > 500 keV (the X-ray emission of these electrons was below the detection limit- ).

ACKNOWLEDGEMENTS. We would like to thank Drs. M. Kuperus, J. Kuijpers, A. Benz, H. Rosen- berg, A. McClymont and J.-R. Roy for helpful discussions. One of us (J.C.B.) wishes to acknowledge the support of an ESRO Fellowship and of the Dutch National Committee for Geophysics and Space Research.

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Spitzer, L.w. : 1962, Ph^3;ce of Fz

Sweet, P.A.: 1969, Ant. R*V. As'.i^or.. AstvzpkjS. 7, 149.

Takakura, T. and Kai, X.; 1966, i~*b'. Astvon. .'•:•. ,*3p2>; 1_3^, 57.

Takakura, T. and Scalise, E.: 1970, .'clar rhys. 11, 434.

Vaiana, G.S., Davis, J.M., Giacconi, R., Krieger, A.S., Silk, J.K., Timothy, A.F. and Zcaibeck, M. : 1973, Astrophys. -J. 185, L47.

-108- CHAPTER V PHYSICAL CONDITIONS OI; ELECTRON ACCFXHRATION IN SOLAR FLARES

ABSTRACT. On the basis of a few crucial observational facts a discussion is given or. what physical conditions might prevail in the electron acceleration reg'.on of ordinary solar flares. It is argued that electrons are probably accelerated by resonant interactions with electron plasma- waves (Langmuir waves), and the possibilities of generating these from current sheets are discussed. Because the sheet volume itself is orders of magnitude coo small to contain the observed number of fast electrons, their acceleration must take place ::>T.j.'it the sheet. It is argued that nonlinear interactions are negligible and as a consequence, efficient electron acceleration can occur.

5.1. INTRODUCTION. The problem of fast particle acceleration in astrophysical plasmas is virtually unsolved. After the original contribution of Fermi (1949, 1954), who for the first time traated a statistical acceleration mechanism resulting in a power-law particle soectrum, no real progress has been made. In the last years, however, collective wave-particle interactions (usually, if vaguely, indicated by the term "plasmaturbulence") have been forwarded as providing a very promisinq acceleration mechanism. This view has been expressed with particular vigour by the Russian school (Kaplan and Tsytovich, 1969, 1973; Tsytovich, 1970, 1972; Kaplan et :zZ. , 1974). It is however fair to say that notwithstanding the large efforts invested, the problem of particle acceleration in plasmas of astrophysical interest has eluded any sound, quantitative treatment. In this chapter an attempt is made to define the physical conditions occurring during acceleration of electrons responsible for \Jhe observed impulsive hard X-ray emission in smaller flares (e.g. events # 1-6, 10, 11 from chapter II); the very large long-lasting flares (chapter II, events # 7-9) are not considered here. The discussion below is based on the following observational facts, which have been fully discussed in chapter II:

(a). The velocity distribution of the fast electrons is roughly isotropic in the ion-frame. The nonthermal emission measure nN is of the order

-109- 5*10 cm J; the background plasma density n is at least r'» 10 cm 30 lb). The total collisional loss of these electrons is aoouc iu erg over "^ 10" s. This energy is of the order of the total energy liberated during these flares. The energy release is very non-uniform in time and consists of a series of peaks ("spikes") of < 10 s duration. (c). The soft X-ray observations indicate a hot plasma, T ^ 1-5-10 K, and 47 49 -3 emission measure Y ^ 10 -10 cm 9 (d). The dimension of the X-ray flare is probably small, say <^ 1-2*10 cm. The suggestion will be forwarded that during flares generation of electron plasma waves (Langmuir waves) occurs in thin current sheet(s). These waves propagate laterally from the sheet, causing electron acceleration in a relatively large adjacent plasma volume. The scope of this chapter is best described as a feasibility study. A few remarks on notations. n,T denote electron density and temperature; when necessary T and T. are used to distinguish electron and ion temperature. To facilitate numerical evaluation use is made of the notation n,_,T_, etc, indicating density and temperature measured in 10 , 10 times the c.g.s. unit, respectively. m,e = electron mass and charge; M = proton mass; c = velocity of light. v ,v = sound velocity, electron thermal velocity. w ,ui = electron plasma-, electron cyclotron frequency. P H Ion-sound and Langmuir waves are indicated by s,Jl {usually as a superscript), their number density pet unit volume of wavenumber space by N s (k)•+ , N 3,(k) -*• . £ c; I s The quantities W '" ik) and W ' are defined by

/ dk w£'s(k) = w£/S

%,s -1 W is tne wave energy density; the Debye length is given by k - v /us . D t p

5.2. IMPLICATIONS AMD ASSUMPTIONS FOR THE HARD X-RAY SOURCE REGION. (1). Because of their roughly isotropic velocity distribution, the fast electrons must be contained in some volume in the solar atmosphere and con-

-110- stantly being reaccelerated. A stochastic acceleration mechanism based on T collective wave-particle interactions is a very likely candidate. 1 In fact it is the only one: Direct acceleration of particles by an electric 1 field [e.g. along a neutral sheet (Friedman and Hamberger, 1969)] cannot f-r be operative as an acceleration mechanism on the required scale. What happens £ is that a current builds up, which becomes unstable. Longitudinal wave tur- \\ bulence usually results and this dominates in the dynamics of individual If \ 1 particles (cf. discussion on the occurrence of runaway in section 5.3.). j« (2). Since the energy content of the soft X-ray emitting plasma, ntcTV = \ 27 1 1.4*10 Y.^T /n,n erg, is of the order of the nonthermal electron energy f 30 I {r^ 10 erg) , it is reasonable to assume that both components coincide spa- fj tially. 1 This assumption is also attractive, because then the ratio of nonthermal to " thermal emission measure equals the fraction a of the background electrons f| that is accelerated (above 25 keV, see chapter II); one finds a %, 10 f —3 f 10 , a very reasonable number. * 2 2 3 (3). Nature of acceieratinq waves. It will be supposed that UJ << UJ 1 H P $ [magnetic energy density << electronic rest-mass energy density; numerically: " 2 5 : H /n._ << 10 ]. This enables one to neglect the influence of the magnetic •

field on plasma waves and wave-particle interactions. t -+• -*• = For a resonant interaction between a wave (u,k) and a particle (velocity v) , ' -+• -+• the component of v along k must equal the phase velocity of the wave, or a) = -*k*v• -•-. Hence, for electron acceleration waves with phase velocity oj/k "\> c 2 2 --c are required. When w << to only electron plasma waves (Langmuir waves) are f H p i elegible; their dispersion relation, phase and group velocity are given by I ,2 3 2,1! W = (ui + 3k v) P t I w /k 2i co /k = v (k^/k) p t D I dw /dk ± 3vfc(k/kD)

(4). Generation of Langmuir waves. Generally speaking, some kind of anisotropy is needed. Because flares are exclusively associated with regions of large magnetic field gradients, it is natural to invoke a plasma current unstable against wave excitation (Petschek, 1964; Syrovat-skii, 1966;

-111- Friedman and Hamberger, 196S, 1969i Tomozov, 1971). When T >> T , ion-sound waves are yenerated if the drift velocity v of e i a tee electrons surpasses the velocity of sound v (Stringer, 1964; Sagdeev and Galeev, 1969). Ion-sound waves can in turn be converted into Langmuir waves. When the drift velocity surpasses the thermal velocity v , Langmuir waves are excited, in the electron-frame (Buneman, 1959; Hair-berger and Jancarik, 1972). The required magnetic field gradients are very high. Using VH ^ 4imev,/c, one finds in both cases: d

2 T VH > 6«10" n 7 (ion-sound instability; Tg » Tj (2a)

VH > 2.5 n T (Buneman instability) (2b)

Very thin, sheet-like structures or shock waves are therefore indicated. If such sheets are required to contain all fast electrons, then

rn.-A.-T ~ > 6-10 (ion-sound) (3a) _ I 10 Id 7 an AH /VH ^ nN._ •*< m -h 6 1 *-n,nA1oT_ > 2.5*10 (Buneman (3b) an = density of accelerated electrons; A = lateral active sheet area; the sheet thickness is given by d^H /VH, where H is some maximum fields m m strength. The inequalities to the right follow by requiring Ct < 0.1; H < 45 m 500 G and nN25 = 5'10 . From (3a,b) it is concluded that the sheet cannot contain enough nonthermal electrons? either its density or its active area become unacceptably high. Hence, the possibility is inquired into that Langmuir waves are generated in a sheet and propagate outwards, enabling acceleration to take place in a much larger volume.

5.3. GENERATION OF LANGMUIR WAVES FROM CURREMT SHEETS. The terra current sheet is used here to indicate a thin plane in which a current is maintained by a gradient in the magnetic field; it can be a

-112- shock wave as well as a neutral sheet proper. SL + •> A detailed analysis of the number density N (k,r) of Langmuir waves in a current sheet is very difficult and has not yet been made. It is supposed here that :owards the sheet boundaries the angular distribution of waves becomes asymmetric, such that waves can leave the sheet (fig. 1)• Depending on their wavenumber distribution and their "history" after leaving, these waves may accelerate electrons. The required wave energy density in the sheet can be estimated when it is supposed that all wave energy is. eventually spent in electron acceleration (arguments for this are given in section 5.4.). The emitted power per unit area is given by:

fig. 1. Sketch of a- current sheet (shock wave or neutral sheet) with lateral "active" area A and thickness d "^ H /VH. Electron plasma waves are generated in the sheet and propagate outwards. An .impression of their angular distribution for fixed |k| is drawn in. j. -* At c the wave spectrum has relaxed to itii thermal equilibrium value N (k) ^ KT/tiu) .

-113- g ^ ^ o/V(k)dk i %v/ (4)

The < sign obtains by supposing a half-sphere angular emission pattern (fig. X). It is assumed here that this yields at least a correct order of magni- £ t tude estimate. The final results (4) is computed by taking W (k) = const. I -1 Equating dP /do to the energy loss of fast electrons per unit area, A • dE/dt, one obtains: £ -1 dE , . . -29 dE/dt (5) 3/2— n T i0 7 \8 Takina dE/dt ^ 1028 erg s"1, A = 1 {A"1dE/dt ^ 1Q10 erg s"1 cm"2), n = 2 £ -A 10 , T = 5, one finds W /n

generated per unit area are estimated by (Kaplan et at.f 19"/4) :

equating (4) and (6), one finds y ^ v /d; and therefore, using (2a):

ws „ vt \ ' }

This spectrum is often found in model calculations (Kaplan and Tsytovich, 1973). It will be used here in order of magnitude' estimates to exemplify wave spectra for which £ k . If on the other hand condensation of Langmuir waves occurs, < 10 k * and (4) will be a factor 10 smaller. It is assumed here that condensation does not occur in the sheet.

-114- S Taking again n1Q ^ 10 ; T? = 5? Hm = 500 G, one finds W /nKT < 0.02. Values of this order of magnitude are also found in experiments (Daughney et al., 1970; Hainberger and Jancarik, 1972). Emission of Langmuir waves from the sheet presumably occurs in the large wavenumber regime k ^ k , because (a), a detailed analysis shows that y is maximal at k ^ k . (b). In a field of strong ion-sound waves, Langmuir waves are rapidly isotropized and they diffuse radially in wavenumber space towards k ^ k (Davidson, 1972). The diffusion time T, can be of the order I of the travel time of a Langmuir wave across the sheet, T > d/(du> /dk) = ufX(k d)(k_/3k).

(2). Generation of Langmuir waves by a beam of runaway electrons in the sheet. It is argued here that this phenomenon is unlikely to occur (contrary to the findings of Tomozov, 1971, 1973a and Kaplan and Tsytovich, 1973). Strictly speaking, in fig. 1 runaway cannot occur because E-H = U in the restframe of the fluid. Of course there can be a component of E parallel

-*• -+• to H in a different geometry, allowing runaway along H after a sufficient lapse of time (Kruskal and Bernstein, 1964; Rudakov and Korablev, 1966; Sagdeev and Galeev, 1969). The physical picture in the ion-frame is as follows. The ion-sound waves have a broad angular spectrum (Tsytovich, 1972; Sleeper et al. , 1973). Resonant electron-wave interactions cause an electron to move randomly in velocity space over the sphere |v| = const, [the magnetic field causes in addition velocity vector precession around H; therefore, the runaway cone is "washed out"]. The effect of the electric field is twofold (Kruskal and Bernstein, 1964): (a), the sphere |v] = const, is displaced in the -y direction -E by a small a:uount v,(v, > v << v ) , hence there is a current. d d ^ s t (b)* electrons slowly diffuse to spheres with progressively larger radii v, hence electron heating occurs. Because the coupling constant of the electron- _3 wave interaction is proportional to v , an electron eventually moves freely -v -»• under the action of E and H, and runaway results. This can be visualized by a sphere with contracting radius v; electrons outside the sphere run away:

Runaway perpendicular to H is only possible when Ei > H, or e.g. E, .> 300 V/cm for H = 1G.

-115- (this result is arrived at by requiring uT ^ 1 in the work of Kruskal and 2 -5 Bernstein, 1964). E is expressed in the Dreicer field E = 47rek - 3.85*10 n T V/cm (Dreicer, 1959). A consistent estimate of E is obtained by 10 - 2-1 requiring that the total power generated, OE d, is larger than A dE/dt (but not much larger, see point (b) of introduction). The approximate relations given by Kaplan and Tsytovich (1973,- §4', and n = 10 , T^ = 5 result in E./E ^ 10 (or E "« 1 V/cm) . Requiring v/v ^ 3 for noticeable runaway, t "^ 5*10 s is found from (8). This time is much longer than the transport time of matter through the sheet} in addition, H would have to lie in the sheet plane. It is concluded that runaway does not occur.

In the extreme case of a Buneman instability, Langmuir waves are excited, in the electron-frame. These waves are probably unable to propagate out of the sheet: In the ion-frame their frequency equals ^ ^(m/M) UJ , which is P below the plasma frequency outside the sheet unless the density drops by a factor 4(M/m) = 600. Therefore, if this instability occurs at all, it will heat the electrons to T >> T., after which the more "relaxed" ion- e l sound instability takes over. Yet, it is noted that the power generated per unit sheet area can be of the required order of magnitude, 10 erg -1 -2 s cm :

l 9 (£> 'Wd «, 10 <^> (nioV\ (9)

(Buneman, 1959; Hamberger and Jancarik, 1972).

In closing, four remarks:

- The occurrence of unstable current sheets requires very extreme conditions. For the ion-sound instability the sheet thickness d ^ H /VH is of the order m h of 50 cm [a few times the ion gyroradius r = 8.5 (to /w )(T /n ) cm]. The p H i/ 10 hard X-ray observations, however, only require that these conditions are "squeezed into existence" for a few seconds, possibly repeatedly over a period of ^ 10 s.

-116- - Magnetic field annihilation is in principle able to provide sufficient energy, as follows from:

riio A (v. and v are inflow and Alfven velocity, respectively).

- A difficult point is the fact that the energy ir. accelerated electrons is of the order of the total flare energy released. The implication is that at least, say, ^ 10% of the total energy generated in the sheet is in outwardly propagating Langmuir waves. Although this is passible if the electric field in the sheet has the right value (^ 10 E for the parameters adopted here, cf. discussion on runaway), it must be confirmed by a complete analysis of the current sheet. n - An independent handle on the value of WYnKT in the sheet is obtained considering the radio emission at the second harmonic of the plasma frequency. Two Langmuir piasmons "merge into" one transverse plasmon; neglect of selfabsorption is presumably not too serious in this case. £ Taking W (k) = const, and estimating a bandwidth of ^ tu , the flux density is given Dy (Kaplan and Tsytovich, 1973; §8): /2 2 F ^-E-l Z- = loV A H (£—) (11) 2v 4, 2 2 XU ll A18 m nKTJ ' ' p TTnmc k R V,R = volume, distance of source. The result to the right is expressed in -02 -2 -1 solar flux units (10 Wm Hz ), and V is set equal to the sheet volume, I 2 from which most of the emission will come due to the (W ) dependence. i, -4 4 Taking W /n

at 2v ^ 18 GHz). Since F? depends very sensitively on the wave distribution over angle and wavenumber, (11) can go down by as much as a factor 10 (cf. Kaplan and Tsytovich, 1973; §9). £ th. It is concluded ,-tha4 t W /nicT cannot be much larger than the value derived above, ^ 5*10

-117- 5.4. ACCELERATION OF ELECTRONS. The foregoing suggests to consider the following problem: A fully ionized plasma moves parallel to the z-axis with velocity v^, while the plane z = o forms a stationary source of Langmuir waves with known source strength (the attention is focused on shock waves here). Find the electron distribution f(r,v) and the wave distribution N (r,k). In the ion-frame, an electron sees Langmuir waves of increasing intensity and, after passing through z = o, again of decreasing intensity. As a result, f(r,v) will depart from a maxwellian around z = o, for v ^ few times v ; Coulomb collisions provide the mechanism restoring equilibrium. The time evolution of f and N can be described with the following equations, in the ion-frame:

-iL) N= 3r

(•£- + L)f = 3. (D. ,3 +A. )f 3. = 3/3v ; v > v (13) at i 13 3 i 11 o

and s Q is the source term; it contains the factor 6(z-v.t). Yv.rY,, - v describe linear, nonlinear Landau damping and spontaneous emission; these quantities are functionals of f (Tsytovich, 1970).

-+->-*--•• -> -*- The operator L = v*3/3r + (K /m)'3/Bv describes free streaming in (r,v)- L space. D.. and A. include both the contribution of Coulomb collisions (Montgomery and Tidman, 1964) as well as the contribution from resonant wave-particle interactions (Tsytovich, 1970). Eq. (13) holds only for |v| > v , where v equals a few times v . It is made plausible by Montgomery and Tidman (1964;§§3.2 and 7.4) that the relatively few electrons with v >> v can be considered as a low density group of ttst-particles with negligible mutual interaction; the interaction with the main body of thermal electrons can be described by a Fokker-Planck equation (13). Eqs. (12) and (13) will have a stationary solution in the frame of the shock wave. Because in practical cases the source term Q is unknown, the discussion below is restricted to establishing that (12) and (13) indeed describe electron acceleration. Two conditions must be met for this:

-118- (1). Nonlinear Landau damping must be negligible, y ' « y . Otherwise, there will be a transport of Langmuir waves in wavenumber space from k ^ k to k « k (Tsytovich, 1972; Kaplan and Tsytovich, 1973) and electron acceleration is then impossible. The case T >> T. is considered in some detail. The transport to small e i „ k is determined by the decay process £ ->• £'+s; y and y are given by (Tsytovich, 1972):

(27T)3e20J2 „ k £

mk c *

[normalization: /f(v)dv - n]. Both expressions are strictly valid only in an homogeneous, isotropic plasma; as in the present case no large deviations from these conditions are expected, they will be sufficient for order of magnitude estimates. For a maxwellian, Y. < Y. implies k > k /7 for N /nKT ^ 5"10 and !L 3 2 -10 k ,> k /9, only, for W /r.KT ^ k /6u n ^ 10 , the thermal equilibrium value. Therefore, whatever the value of W /nKT, y can be dropped for k ,> k /8 (as an average value) and Langmuir waves with k ,> k /8 will inter- act resonantly with electrons in the velocity range v < v < 8v , When rather T ^ T., y is determined by nonlinear scattering of Langmuir waves by thermal electrons or ions (Tsytovich, 1970; 1972). It turns out that again for a maxwellian y is small in comparison to y for k > k /8.

(2). Langmuir waves must be mainly generated at large wavenumbers, k > k /8 (there are indications that this is actually true for the case of an ion- sound unstable current sheet). Then, as a consequence of (1), electron acceleration occurs in the velocity range v < v < 8v , i.e. to energies of the order of 32 KT. Usually, this will manifest itself by the occurrence of a nonthermal ta.'.l or plateau in the distribution function. In that case, the combined effect of presence of a tail and of collisional interaction will be that f(v) is "lifted" at velocities above ^ 8v , and acceleration occurs to energies beyond ^ 32 KT because y > Y. holds for wavenumbers smaller than

kD/8, cf. (14).

-119- It is instructive to consider this in more detail by deriving a relation from (12) and (13) expressing conservation of energy. Take a large volume V, containing at least the region where the Langmuir wave number density 8. ->•->• N (k.,r) differs from its equilibrium value. For simplicity it is assumed now that Y? is negligible for all k for which Qi= o. In a stationary state, the following relation is a direct consequence of (12) and (13) together with explicit expressions for y , D , etc.:

2 fdr I —• fiu) 0 = a fdr 9 dfi 'jmv [ -z— + l] f ' J 3 ^k •'Jv vdv V (2TT) V v=v o

+ ma /dr / dv dH [ v 2 |- + v] f (15) ' ' -> ' V t OV V v ^ v o where a = (e'j /m) log A (log A = Coulomb logarithm). P The left hand side of (15) is the total wav° energy generated in the sheet per unit time, P in the notation of section 5.3. The first term at the right hand side expresses the outflow of energy due to electrons crossing the boundary v = v in velocity space, "t can be o made plausible that this term is negative, that is, electrons in fact flow in from the reservoir of thermal electrons. This fact underlines that: a proper definition of the boundary condition at v = v is important when o (12) and (13) are actually solved. This point is left aside here. The second term is the total energy loss of electrons with v ^ v due to collisions with thermal electrons. It is noted that the differencial operator yields zero when applied to a maxwellian distribution, which has of course no collisional losses. The collisional loss term can be rewritten as ma C/O dv v~ IV -3—+ vl F(v) (16) t ov V o

t 2 By taking formally a cold plasma, v +0, or by arguing that v 3/clv is o an of the order v /v which is small compared to v, (16) reduces to / dv (ma/v)F(v). This expression is normally used in computing collisionai energy losses.

-120- with

F(v) = v j dfi /dr [ f-fj (17) f is a Maxwell distribution normalized to v and density n. F(v)dv = total number of nonthermal electrons in V with velocities between v and v + dv. - since (16) is positive, it follows that F(v) must develop a nonthermal tail, such that v F1 + vF > 0. - (16) is in fact larger than or equal to (15), left hand side, which term, as shown in section 5.3., can attain values actually observed for Coili- ng _ i sional loss (^ 10" erg s ). From this it can e.g. be shown that the nonthermal emission measure, nj F(v)dv, can be of the order of the observed value. It follows that a sufficient number of electrons can be accelerated to explain the X-ray observations. The above conclusions follow from simple considerations of energy conservation and it is interesting that they are independent from factors such as containment of fast electrons by a suitable geometry. On the other hand, nothing can be said at this stage about the shape of F(v) [a power law, F(v) ^ v , is observed] , nor about the spatial extent of the acceleration region. These questions require further considerations, probably even solution of eqs. (12) and (13), and, in e.ny case, knowledge of Q, and of the magnetic field geometry. JC

In summarizing, the following conclusions are drawn. It seems possible that Langmuir waves are generated in ion-sound unstable current sheets, in sufficient quantity and with the right wavenumbers to explain the hard X-ray observations in smaller flares. Electron acceleration then occurs to energies of at least ^ 30

ACKNOWLEDGEMENTS. This paper has benefitted from discussions with Drs. G.A. Stevens, J. Kuijpers, R. van den Nieuwenhof and J.-R. Roy.

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-122- CHAPTER VI RADIATION FROM A SOURCE IN A COLD MAGNETOACT1VE PLASMA, REVISITED. APPLICATION TO CYCLOTRON AND MULTIPOLE RADIATION

ABSTRACT. An analysis is presented of the radiation emitted by an arbitrary source in a cold, magnetoactive plasma. In spite cf the ajrople coverage this subject has received in the literature, ieconsideration of this problem in our opinion was necessary. The plasma is supposed to be Lnfinite and homogeneous; its dielectric properties are described by a dielectric tensor c. Expressions for the radiation fields are derived using the technique of Fourier-decomposition. An expression for the vector potential is constructed and elaborated as far as possible for an arbitrary current source. Our approach differs from that in previous work on a technical point, namely the sequence in which the various integrations are carried out. The radiation fi'jx is defined on the basis of Poynting's vector S; in the definition we distinguish between current sources behaving as a given function of time and randomly fluctuating sources. In the latter case an ensemble average is used rather than a time average. The general result for the radiation flux is then specified for cyclotron radiation from a stationary ensemble of electrons and for multipole radiation. For both types of radiation expressions are obtained that differ from those previously given. In the case of cyclotron radiation we find e.g. a difierent harmonic frequency. The cause of these differences is indicated. Throughout the paper a compact notation could be used on the basis of the work of Bremmer.

6.1. INTRODUCTION, The problem of radiation from a source in a magnetoactive plasma has a long history. Tha earliest attempt to solve problems of this kind dates back to 1940 when a paper by Ginzburg1 appeared on "Electrodynamics of Anisotropic Medium". From about 1950 on, there has been a steadily expanding literature due to the general interest in this subject in thermo- nuclear research, radiophysics and astrophysics. In particular much attention has been given to the problem of cyclotroi. radiation in a magnetoactive plasma. A useful general reference is the bibliography compiled by Marr, Munro and Sharp.2 Multipole radiation on the other hand has received much less attention.3 The majority of the authors takes the macroscopic point of view in which the background plasma is treated as a continuous medium whose proper-

-123- -.ies are fully described by a dielectric tensor c. The problem of radiation by a source then becomes essentially an excercise in classical electrodynamics in a complicated medium. There are two techniques in use to obtain the radiation fields: The oldest one used is the so-called Hamiltonian method, known from quan- tum electrodynamics,1* in which the source is enclosed in a large box and the radiation field is represented as a superposition of box eigen modes. Although in this way the radiation of an arbitrary source could in principle be found, the method has been applied only to derive the radiation from one point charge in rectilinear or helical motion.1'5 9 The other technique is to consider an infinite plasma and to make a space-time Fourier-decomposition of the radiation field. In this way both the radiation from a point charge as well as multipole radiation has been treated.3cI°-27 We mention here that radiation problems in a plasma have been treated using a kinetic description.28 Mainly problems concerning breiasstrahlung and longitudinal wave emission have been considered but some attention was paid to cyclotron radiation.31' 34 In this work we consider an infinite, homogeneous cold plasma and make a space-time Fourier-decomposition of the radiation fields (section 6.2.). Next, in section 6.3., an expression is derived for the vector potential due to an arbitrary current distribution. All previous authors dealing with cyclotron radiation have computed the radiation of one point charge in a helical motion. In that case one of the several integrals that must be computed can be (and is) carried out first in a trivial way due to the occurrence of the delta function. In the remaining integrations an error is made which in our opinion can only be avoided by postponing as long as possible use of any specific properties of the current. This point is rather technical and it is basically a question in what sequence the various integrations have to be carried out in order to handle each one correctly. We obtain new expressions for the cyclotron radiation emitted in a magnetoactive plasma; among other things we find a different expression for the harmonic frequency (section 6.4.). Our results differ, too, from those obtained with the Hamiltonian method and we indicate one reason for that. Because we derive an expression for the radiation of an arbitrary current

-124- distribution we find as a by-product an expression for the radiation from a multipole source in a plasma (section 6.5.). Radiation problems in a magnetoactive plasma generally give rise to very lengthy and complicated formulae. In the present work a compact and transparent notation was possible by using Bremmer's inversion of the tensor Q (see below).35 In section 6.2. we alsc pay attention to a correct definition of the frequency distribution of the emitted radiation- A distinction is made between a current source that behaves as a given function of time (such as a multipole source) and a randomly fluctuating current source (such as a collection of electrons emitting cyclotron radiation). This approach has the advantage that in the computation of cyclotron radiation an SKSe^bI & average over a stationary particle distribution can be used instead of a time average over a one-particle orbit followed by a summation over all radiating particles. It appears that this ensemble method actually is much more simple and powerful than the time averaging method used up to now. We base our definition of the radiation flux or. Poynting's vector S, rather than on the work done by the electric field per unit time on its generating current, / E*J d r, as some authors11"19'21'27 do. The latter quantity cannot be expected to give the correct expression for the emitted radiation per unit solid angle although it does give a correct result for the total energy loss due to radiation. A few comments on notations and conventions. Throughout we use the following Fourier-transform convention;

g(r,t) = / d jc / dw exp[-i(k*r - wt)] g(k,u>) (1) -4 r 3 r r l

g(k,w) = (2TT) J d r J dt expl i .KT - art)] g(r,t)

For any physical quantity g(r,t), g and g always stand for g(r,t) and

VL . ** g(k,co), respectively. In mixed transforms, the symbol is used and at least one of the arguments is written explicitly, e.g. g(u)) ~ g(r,oj) = (2TT) / dt exp(-iu)t) g(r,t). However, this g, g, g notation is only used when at 1 east two out of the three actually occur in the text. All tensors are denoted by symbols like Q, K, etc. The hermitian adjoint = t T and the transpose of the tensor A are denoted by A and A ; complex con- -125- jugation by ft = A = A . Unit vectors are indicated as k = k/k, p = p/p, etc

6.2. OUTLINE OF THE PROBLEM. b.2.1. Basic Formulae.

We consider radiation from a localized charge and current distribution in an infinite, homogeneous plasma that is permeated by a stationary, homogeneous magnetic field H . The Fourier transforms of the macroscopic Maxwell equations are given by:

k*D 4TTiT (2)

k-H = o (3)

ck (4)

ck H = - ioD 4niJ (5)

The plasma dispersive properties that we consider are summarized by a. The plasma is fully ionized, with stationary ions. b. Spatial dispersion is neglected ("cold" plasma). The first assumption is only convenient; the second one is essential here, D and E are connected via the dielectric tensor £(w):36'??

D = £'E ; e = I - (6)

with

a (l-i5) a = b = d = a (l-i6)2~B2 (l-i6)2-B2 1-16

(7)

a = v/ai

2 ~H I is the unit tensor? a> = (4TT n e /m ) is the plasma frequency,-

W e H m c is tne 37 H = o/ o gyrofrequency and V is a collision frequency. In writing down (6), the positive z-direction of the coordinate frame

-126- is taken parallel to H . The plasma is considered nearly collisionless (6 "^ 1); in fact collisions are retained only for computational convenience. t T z satisfies the relation E(O)) = e(-oj) . This property reflects that e is a real opei .tor in the t-domain.. The external sources T and J are as yet arbitrary. Only aL a later stage a specific choice will be made. E and H can be solved from Eqs. (2) through (5) in terms of a tensor Q:

E = 4iTi(jj Q~ «J ; H = 4TTic k * Q-1*J • (8) W = *A- R/V. /w =: "A- with Q = c2k2(K-I) + LU2E (9)

The tensor K is defined by K'a = k(k'a) for any vector a,- and k=k/k. These E and H satisfy Eqs. (2) through (5) identically. It is convenient to introduce a vector potential A:

A = - 4TTc Q~ *J (10) in terms of which

E = -(iU)/c)A; H = -ik X j\ . (11)

One must be careful and make sure that all the Fourier transforms used above do indeed exist. A sufficient condition for this is integrabi- lity of the absolute value. This refinement is usually disregarded, but it is indispensable for a correct definition of the radiation flux. The field sources T and J can be made to satisfy this condition by cutting them off to zero outside some given time interval. For simplicity, the zero of the time axis is chosen half way this interval. These cut-off sources are given an index T:

fJ(r,t] , T(r,t) for Itl < T ,t), T (r,t) (12) v-o, o for jt| > T

All quantities that depend on J , T via the Maxwell equations are likewise given an index T, e.g. A^,, H , and they now also satisfy the above integra-

-127- bility condition [it is pointed out that A , H , etc., are not given by relations analogous to (12)1• These quantities have mixed or complete Fourier transforms that are again given by (1), e.g.

00 E (r,w) = (2TT)~ f dt exp(-iwt) E (r,t) (13) CO E (r,t) » / dw exp (iwt) E

Henceforth, all field sources occurring in Eqs. (8) through (11) are truncated.

6.2.2. Definition of Radiation__Fiux.

A measurement can be quite naturally defined in terms of truncated fields, because physically it involves always an averaging over a finite time.39 To be specific, a measurement of the energy flux ~~at t = t corresponds to a measurement of the time average of Poynting's vector~~

S = (C/4TT) E x H^: t +T t +T -1 ° = (2T) f dt _ST(t) = (c/8TTT) J dt ET x HT (14) *"• t -T **• t -T ~ ~ o o t is equal to the time needed for the signal to reach the observer. At o one hand, the point of observation is far away (many wavelengths) from the source, on the other hand it is so near that the dispersion in arrival times is « T. It is emphasized that Eq. (14) holds only if S (t) is due to a stationary current distribution. In a non-stationary case - e.g. a source that moves as a whole - the signal at the observer will not extend over the time interval. {-T, T] due to the Doppler effect, and in (14) S (t) must than be averaged over [-1", T1] , where T1 depends on T and the ^A i, radial source velocity. This applies in particular, for example, if the source consists of one electron gyrating in a magnetic field.1*0"1*1* If the electron is relativistic, the Doppler corrections to (14) can be large: _2 In vacuo an extra factor (sin9) appears and in a background plasma very elaborate corrections due to plasma effects are necessary.**s In our opinion these efforts are unnecessary because one always observes the ra-

-128- diation of a collection of electrons. In this case it is much more straight- forward to use Eq. (16 ' ), where the single electron corrections do not appear. Ir non-stationary cases like moving sources one can simply first transform to another frame of reference. ^ If the current system J behaves as a given function of time (as is - e.g. the case for a multipole source oscillating sinusoidally with a given frequency) a suitable expression for the frequency distribution of can be derived by substitution of the time Fourier transforms (13) of E and H in Eq. (14); if one chooses T such that T >> (ALO) , where Aw is a bandwidth corresponding to a significant change in E (to) and H (to) , one finds:39

~~= j P(r.iolduJ ; P(r,to) = he Re — - (15) nn. **. -v» nn V. T O~~

However, if the source consists of a large collection of electrons, the current represents rather a randomly fluctuating source of which only average properties can have physical meaning. In that case the Wiener- Khinchin theorem relates £ to the cross-correlation of E^ and H :

~~P(r,w) = i Re / dx exp(-iojx) ~— < E (t) * H (t-T)> (16a)~~

~~= ~ Re / da)1 exp[ i(u)'-w)t] < E (u)') x H. (w)X> (16b)~~

Eq. (16 } again follows from (16 ) by using the Fourier time transforms. The average in (16 ' ) is now an average over a stationary ensemble. In (16 ), <•> will contain S(m-U)1) so *hat the t-dependence drops out. T must now be large with respect to the correlation time of the current system. P contains all information on angular and frequency distribution of the emitted radiation, and it can be measured directly: The power, per unit a), passing through a surface element dO with normal e equals e*£ dO. P will be proven to have the same direction as the direction of observation r = r/r. The definition of the radiation flux is based here on the vector of Poynting, SL,. Some authors11"19'21'27 start from an expression for the

~~-129- aveirage work done by the electric field on its generating current-~~

~~CO 00~~

~~= (2T)""1 f dt / d3r E'J = (2ir)4 f d k f dw RefE -J*/T)~~

One then postulates that in the last expression the integrand of /dSl represents the emitted power per unit solid angle. It is pointed out here that this approach muse be wrong. The standard interpretation in electrodynamics is that the emitted power per unit solid angle is given by 2 lira r and that the direction of this vector gives the direction of r "" ?i i 4 2 energy flow. In fact the quantities lim r""|[ and (2TT) / k dk / du X _x r «• Re(E •J_/T) cannot be identified a priori because the former depends on r = r/r and the latter on k = k/k. If r and k coincide - which means that fl/V AA. the energy flow in the direction of k is carried by waves with the same wavevector - this identification is justified. In other words, this procedure is only correct in the case of isotropia media. The authors applying the Hamiltonian method1'5 9 base their computation on the average increase of the field energy in the box per unit time, (2T) / dt / d r (E *D + H *H )/4TT. For this approach the same objection holds, as pointed out by Ko."8 for further evaluation of (15) and (16 ) we need an expression for A (UJ) . It can be found from (10) by application of the convolution theorem: T 1 I I G(r-r',u) • / dt exp(-iu3t )J(r'/t ) (17)

~~where Green's tensor G(p,u)) is given by = on.~~

~~G(p,u) = / d3k exp[-ik-p] Q'^k,'^) (18) — i\f\ tin. AA iw s M.~~

In the next section the integration over k in (18) is performed and an explicit expression for G is obtained, correct to lowest order in p, viz. p . We stress that this integration over k is done without making use of any specific property of the current density J. For example, if J is due to a point charge J = -e v 6(r-r ), then one can trivially carry out the in -wO AA nAO integration over ^r' in (17), next perform the t'-integration and only then integrate over k. All authors who treated the problem of generation of

-130- cyclotron radiation with Green's function technique have done so. It is argued here that this procedure - although in principle correct - leads to erroneous evaluation of the integral over k (see end of section 6.4.). As a consequence, we obtain qualitatively new results.

~~6.3. COMPUTATION OF GREEN'S TENSOR G.~~

~~We shall use an explicit inversion of Q, given by Bremmer:35~~

~~w^lele"1- (ukc)2[(k'|'k)I + (Trg)K - |«K - K«e] + (kc)4K " = — —-—- ^ ^ r ^ , (19) a) I m e + (tjakc) k'T*k + (kc) k*e*k]~~

~~where ,£~~

~~|g| = det E; Tr E = Z £.,; T = E - (Tr E)e and k = k/k. J~~

~~The denominator of (19) equals det Q. Expression (19) is valid for an arbitrary tensor e. In the present case, we consider a cold plasma and det Q can then be factorized straightforwardly:~~

~~2 2 2 2 2 2 2 2 2 det Q = w k'£"k(k c - u) n. ) (k c - w n ) (20) = = 1 I~~

~~35 37 119 The refractive indices are given by the Appletcn-Hartree formula: ' ' M~~

~~-k*T-k _ [ (k*T-k) - 4|e| (k*£*k)] 1~~

~~if2 2 k-E'k •~~

~~(21)~~

~~2 2 1 _ 2a (l-a -i~~

~~(n2-n2) k-e-k - -^-5-[S4 sin46 + 4S2(l-a2)2 cos26]h (22) 1 l 2~~

-131- fr In carrying out the integration over k in (18), only the zeros of the deno- - rainator of (19), det Q, do contribute. These 2eros are given by 7> 2 2 "> ~ A. A « /s (1). k c = a) n." , provided that k«e*k and k*T»k are r.ot both equal to zero. (2). k*e»k = o and k'T'k = o. It follows that det Q = o in this case, because k'C *k = o, or |e*kj = o and so |e| = o (see appendix A for a — — ~ ^\ J\ further discussion). It is emphasized that k*|'k = o as such is not a zero of det Q. n and n correspond to the ordinary and extraordinary mode, respectively (there are, however, other conventions; see ref. 49). For the integration over £ in (18) many authors3'l 3'15''a'21'2k'2h start with the following factorization of det Q (see also end of section 6.4.): 2 2 22 22 22 det Q = A(k(jc - w Uj )(k(/c - w M2 )

2 2 2 2 2 with U = U (k, ,a ,B ); k.. = k cos9; k• = k sin6 (see fig. 1). The integration over k is then performed in cylindrical coordinates in the sequence k.., $, k• . Integration over k.. is done by contour integration and the integration over $ is done with the method of stationary phase38 (one is only interested in 1/p-fields). The result is then integrated over k. by first transforming to an integral over 9 with the substitution k. = (ca/c)n (8) sin6 (in general 8 will be complex) and then by performing the 8-integral again with the stationary phase method. This substi-

tution ki = (to/cln, ? (8) sin0 - apart from being very involved - is questionable because the mapping 9 -*• k., may not be one to one. For the integration over k, in ref.34 the k -axis is chosen along p instead of H (fig. 1) . Using spherical coordinates (k, arccos y, <$) an asymptotic series is derived by first integrating by parts over M. However, this procedure fails for anisotropic media, because then the integrand of the residual term contains poles of the second order due to the differen- tiation to p. This gives a contribution proportional to the derivative of the analytic part of the integrand at the pole, and this contains parts proportional to p. The final result is that the residual term contains parts of order p which are omitted by the authors.s ° In view of these remarks, we decided to use spherical coordinates as

~~-132- fig. 1. Position of the vectors D and k and their unit vectors 0 and k. The positive z-direction is taken parallel to H ; the x-axis is chosen so that 0 is in the xz-plane. :. = cos8cosUi~~

~~indicated in fig. lf and to perform the integrations in the sequence k, $, 8,~~

~~From (9) one observes Q(kfaj) = Q(-k,cu) and therefore one has, with U = k'p (p=O/p):~~

~~G(p,w) = / d k exp[-ik-p] Q (kfu))~~

~~-1 /kdk exp[-ikpy] Q (23)~~

6.3.1. Integration over k. -1 The integral over k in (23) does not exist, because Q " Aj constant as |k( t °°. In appendix A it is shown that the terra of highest order in k from -1 2 Q corresponds to "electrostatic" 1/P -fields. This term can be subtracted from Q without affecting the radiation fields, and as a result Q ^ k , effectively. The integration over k is now conveniently done with contour- integration. The formal retention of collisions gives an easy clue to the positions of the poles in the complex k-plane. Two cases arise (the state- ments below hold for n as well as for n ):

~~-133- U). n" < o for ^ = o. Then two poles appear on the imaginary axis at~~

(2). n* > o for 5 = o. From (21) one derives that, to first order in 6, Im n~ =» -c6, where c is a positive definite function of a , 6 and 9. Ob- serving the definition of the square root,51 it follows that Imusn is pro- •ortional to 6 and always negative, independent of the sign of J, whereas Reuin may have either sign.

~~Fig. 2 illustrates two possible pole geometries.~~

~~complex kc plane |u)|n2 -wn, -urn, -um2 •am~~

~~fig. i- Possible contours for the integration owr k m ;2J;. ^. is taken positive in t;.e toft~~

~~figure, negative m the figure to the ri^ht.~~

~~The integration over k in (23) can now be performed by closing the contour with a semi-circle in the upper half-plane (if u < o) or lower half-plane (if p > o); the result is:~~

~~G(p,w) = ^ ~ I f dfiK n expt -i(w/c)n p|u|] ~ "^ 2c P P~~

~~e|e~~ -n [ (k-e-k) 1+ (Trc) K - e»K - K«t.]+n K (24) (n - n ) k*e*k q p where~~

~~2 2 (k-E'k)I + (Tr C)K - e«K - K"E - (n +n )K ~:—:—. ^ "A~—=-^ = 6(u) c p ' k*£'k~~

-134- The term R in (24) is due to the contribution of the semicircles [ it is reminded that these arise because the integrand in (23) is not absolutely 2 -1 integrable as k Q " ^ 0(1)]. R does however not describe any wave pheno- ,__- mena and must be anticipated to have no physical meaning; it will ultima- S tely cancel out. jt The summation in (24) is over p,q =1,2 and p,q = 2,1. We emphasize that J U| appears in the exponent. In (24) it is assumed that '_ n > o. If n < o then one finds that exp[-i (to/c)n pjvl] must be replaced P i P P by —LO I (jj j * exp[ in p|wia|/c] , which, in the limit p t ro, gives an exponen- tially vanishing contribution to G. The interpretation is of course that ~ 2 there is no radiation in a given mode for those (!J,6) for whioh n (w,9) < o [ Razin effectJ3'57] . Henceforth, the integration in (24) is taken only over 2 ngies t for which n > o. These angles can be found from fig. 4 for any 2 2 ^ vai.e of a and 0 .

~~6.3.2. Integration over d>.~~

The integration over 4> is somewhat involved because the absolute value of u occurs. Only the outline is given here; more details are left for appendix B. The only ^-dependence in the integrand of (24) occurs in juj and in K. The integrand of (24) is invariant under the reflection

~~(6,0) •+ (TT-G, 4+TT) . Therefore in (24) one can put (cf. fig. 3):~~

~~/ dU = 2 / d6 sin6 / d$ -i- 2 / d8 sin6 / d(j> + 2 / d6 sin6 / dtp 4TT O -^TT hii-ty -H hv-ty — Jjir~~

(25) lfi is the angle between p and H ; cosn = -cotg9 cotgip and o < r| < IT. The value of n follows from the condition V = o. In the first two terms of (25) y is positive, in the last one U is negative. In (25) , o ^ ip < h^ is assumed. Since we are interested in radiation fields, it is only necessary to retain the first term in the asymptotic expansion of (24) for large p. We therefore proceed by evaluating the (^-integrals in the first part of (24) for large p with the method of stationary phase,38 after which the integration over 6 is done in the same way. In appendix B it is shown that this integration over (j> yields the following expression for G(p,w) [for the denominator of (24) we have used (22)] :

~~-135- fij. i. The unit sphere is pioje. leJ or. t)-.p xz-plane -ji f iq. 1. Th« -.-ase . "j" ;> drawn.~~

~~The surface is Jividt'd ir. six regions. The transformation ! '<, {! - If) Maps ""i'Jr. t< or.tc 1, 2 ur.tc '• ar.i -1 or.tc 3. T'.e -.ritogral wvi \ r-.e ji.it~~

~~equals - ti.-nts t:.jc ovei •!..; !w;.r, : • .< • i. Ir. t!,is way ,jt.. arr.~~

~~de a p~~

~~- n I (k-£*k)I - e-K - ,P _ P = = = =o (26) sin4e + 462(l-a2)2~~

~~K^ stands for K with ] . It is noted that the term R from (24) has been cancelled.~~

~~6^3.3^ Integration over 0.~~

The integrand of (26) is oi type g(8)«exp[ pf(6)], where f and g are analytic functions of 9 and so (26) satisfies the requirements for evaluation with the stationary phase method for large p. The critical point(s)

~~-136- Region 1', i" 4", 6" Region 3", 5*. 7 Region 7" ( n, >n_ in region 1)~~

Region 6",8* Region 5" fig. 4. 2 Aliis, Buchsbaum and Bers divide the is" s - plar.e tl^ft; ir, eight reoior.s and in each region the qualitative solution of Eq. •.;"": for a 'vpr. mode ca:. i^e represented by cne figure, the ;*v_'fc:x sit.'Y'a.'fc'. The troker. lines should be ignored for th-:- region ciivisi^ri. An index surface (right1) is a polar plot of ck, _ :- r. k ir r. k: at constant -. All figures should "* * ~ 2 ^ be rotired ar-^^nd H . rht r.^mbers i through 8 refer to the regions ir. the (.i*", i" 1- plane: - and - refer to ordinary and extraordinary modp. The lengti. of the radius vector to a point c-f the sjitace is inversely proper t ion a 1 to the phase velocity in that direction, and the r.orxal tr, the surface gives the direction of the group velocity. In the index surface for region 6^ and S the angle X is formally negative, because 8 and y point into different half-spaces. The figures are not to scale mutually; the unit circle is drawn in ( ) for reference. The dotted lines (...) are polar plots of a k as is further explained at the end of section 6.4. For a given mode, P 2 2 2 2 the index surface preserves its "character" for all i , 8"~ within a givan region of the (a , 3 )- plane. The + mode does not exist in*" •- o) in region 2, 3 and 8. The resonances (asymptotes of the index sur-ace) are drawn with a broken line in the (a , 8 )-plane for some fixed resonance angle. The angle 6 needed in the evaluation of (28) or (29) can be aualitatively constructed from these ° 2 2" index surfaces: From the mode and the values of a and 8 the appropriate figure is selected; the normal to the surface in the direction of observation (polar angle >;' is constructed and the radius vector then gives 6 . It is observed that: o a. The angle between group and phase velocity, |\-8 |, is always < Vr. This property does not depend on the specific form of n " {LJ.O) , but follows quite generally from the Maxwell equations, bee appendix C. b. To a given group velocity, there corresponds at most one wave vector or phase velocity, except for the extraordinary mode in region 7, where up to three possible wave vectors may exist. -137- are given by O/j;')n cos(O-'j) = o, or, equivalently: P , a© i) (27)

This equation has the following interpretation-.39 For a plane wave with given wave vector angle 0, eq. (27) gives the angle \l> of the direction of its group velocity, defined by grad w(k). It follows that for large D only those waves contribute in (26) whose group velocity is in the direction of P, which coincides with the direction of observation, because D ^ r for large p. Following Allis, Buchsbaum and Bers,"*9 the nature of the solution of eq. (27) can be comprised elegantly in a few simple diagrams for all values of :i" and £ [see fig. 4. We distinguish between the polar angles i|/ of p and X of r. For large p, however, X and i> can be identified] . The final result for G(P,oi) is:

~~> M \ \ '-..,% N I • ,. exp[-i(w'c)a (6 . p ': I -1 J ^ • n «' ' / e~~

~~< (28) I a < o p where we adopted the following notation:~~

~~, 2 jck"1- n (0 )[ (k -ck )I + (Trc)K - e«K - K -E] + n4 (6 )K 0 = (_i)p 1- 0P . ='= po o=o= = =o = =o =o = po=o =P a2 [S4sin46 + 462(l-a2)22 h o . 2 = n cos((?-~~

The vector k is the unit vector in the xz-plane (fig. 1) making an angle 6 with H . K is defined in terms of k by K *a = k (k *a). 'VUD =O O =O vw O O *v\ The summation over 9 in (28) is over all solutions G of the equation ° ° 2 2 da /iG = o. From fig. 4 it is seen that everywhere in the (a , 8 )-plane there is only one such 6 at most, except in region 7 in the extraordinary mode, where up to three solutions may exist. One infers from (28) that t T G(p,w) = G(p,-u>) so that G is a real operator in the (p,t) domain.

~~-138- 6.3.4. Evaluation of the Radiation Flux.~~

In writing down G(r-r',,.), needed to evaluate A !.*,) frorci '17), we make a few customary approximations. The point of observation r is given polar coordinates (r, X» °) » °f- fig- 5. Because r » r' one has 4* r'u X anc* by the same token D is replaced by r everywhere, except in the exponent where the next order approximation p = |r-r'| 'v r-r-i' must be used (r = r/r). The first order term r'r' takes care of interference (the second order term is always small if wa 0/c >> 1). We then find: P iJ1

P 'o (29) T , rCt > O x 0 • / Jt' / d r1 exp[-iu,(f-a rT'/c)] J(r',t') J P ~P __ "VA. D V. ivi V, j " la < o p where r = r/r and a = n cos(6-v). For future use we note that the ex- T P 3 P pression / dt'.' d r1 exp[ ••*],!(•••) occurring in (29) is equal to [cf. (1)1 : (2iT) J CuXX r/c,o;) . ~T p The appearance of Ot = r. cost1: -'< ) instead of the refractive index P P ° ' n in the exponents of (29) has a simple geometrical explanation: usually one measures the wavelength along the wave vector k. In the above expression, the wavelength is measured along r and so along the group velocity; consequently the wavelength is larger by a factor l/cos(5 ~x)• It is emphasized that a is not equal to the ray refractive index as defined 3 9 e.g. by Bekefi. This extra cosine factor in the first, exponent has also been found by other authors, the one in the second exponent, which is important for the determination of tht frequency spectrum, has not been found before. b We end this section by expressing P(oo), defined in (15) and (16 ), in A (GJ) as given by (29). One has J (u)) = t-iu)/c) A (UJ) and _H (to) = rot

A tu>) . Now An]((j) depends on r in a very complicated way; however one need only differentiate the exponent exp[-i (uj/c)a r] occurring in A (OJ) . Diffe- P 2 rentiation to any other r-deper.dence always yields l./r -terms. Because grad (a r) = n k if a (6 ) = o, one finds H (to) => (-ioi/c) I n x J r v^p p o p o '.. *T P P k * A (LJ) . [The unit vector k has been given an index p: It is de- o, p T, p o

~~-139- fined as lying in the xz-plane and making an angle 6 with H , see fig. 5, and therefore it is a function of X« a*~ • &" anci the mode p] , Using these results.,, it follows from (15) that~~

~~2 .-~~

~~P(w) = —• y n • - Re Am (to) x [k * A (UJ)*] (30) Z. 2c L q T ,vff,p o,q ~T,q p»q and from (16 ) •.~~

~~P(OJ) = J— y Re / ui co'dw' expli(aj'-a))t] n < A (u)1) * {k x A («)*]>~~

~~(31)~~

Eqs. (30) and (31), together with Eq. (29) are general expressions for the radiation flux emitted in a raagnetoactive plasma. Further evaluation is possible along two lines. Firstly, one can make a multipole expansion of (29). This subject has been covered in the literature,3 but we briefly touch it again in section 6.5. because it appears that errors have been made. Secondly, in cases where a multipole expansion is not appropriate, one must specify the current distribution; as a specific example cyclotron radiation is treated in the next section.

~~6.4. CYCLOTRON RADIATION. 6^4.1^ CurrentCorrelation Function.~~

~~We now apply the above formulae to the case of cyclotron radiation. Comparison of (31) with (29) shows that the correlation function < J_.(k,w) J (k',(jd') > must be computed. Using (1) one finds:~~

~~8 - - t 2 T T (27,) < J (k,u>) J_(k',w!) > = e T f dt1 / dt"exp[ i(ui't" - ut' )] m,n -T -T~~

~~x < exp[i(k«r' - k'vr")] v1 v" > (32) Aft **Ain ** ^*n <\AIQ *v\n~~

Here r^tt) and ^(t) are the position and velocity of the n-th radiating electron, r' = r (f); r" = r (t"), idem for v1, v". In order to have a <»n -v*n «~n «n -»-n «*in simple picture at hand, it is supposed that the orbit of each radiating

-140- electron consist? of a series of consecutive helices around H . The tran- sition from one helix to the other can be thought of as a collision, accom- panied by a jump in pitch angle, phase, field line, etc. If further no correlation is supposed to exist between different electrons and between consecutive helices of one electron, it then follows:

~~(2TT) <•> = e" / d v f(v) /dt1 /at" exp[i(iu't" - wt1)] v -T -T x < exp[i(k'r' - k"r")] Vv"11" > (33)~~

f (v) is the distribution function of the radiating electrons and / d v f(v) is their total number, v1, v", r', r" now pertain to one single helix, and IW V\ AO. Ml the average in (33) is an average over the volume containing the radiating electrons, and over the distribution of initial phase and duration of the free helix movement. The average over <|> is trivial due to the isotropy of f(v) around H . Averaging over the volume gives

~~< exp[i(k-r' - k'-r")] v!v"T > = 5, , < exp[ik«(r' - r")3 V v"T > nnivv »ft Mi MI Ml K,K~~

~~where r , and v now pertain to a helix centered, around the origin (cf. fig. 5): . '~~

~~V V r (t) = (—i sinco t, l cosio t. v t) AI»O CO O CO Off ° °. ' (34) v (t) = (v, cosco t, v, sinco t, v,,) nfO . X o X Off~~

Vi and v,, are the velocities perpendicular and parallel to H , and co = to /Y; we will use the notations B « = v../c; 8, = Vi/c (but g = co /co) . On ? 2 -V Y is the Lorentz factor (1 - 0^ - (3. ) . Averaging over the length of a helix is done by assigning a weight to a given helix length, e.g. exp[-It'-t"I/T ]; X is the average time between successive collisions, o o We now have the result:

~~-141- H,~~

~~fig. 5. Vector diagram showing the relative position of the vectors r^( r and k , as well as the helical orbit of an electron. At t = o the electron is taken to be on the negative y-axis.~~

~~(2-rr) <•> = e 6, / a v f(v) jdt' jdfexpl i(ti)'t" - ut1) - |f-t"|/t ] KfK AA. aft. -T_ -T_ O~~

~~x exp[ik*(r' - r")] v1 v" (35) -w. "AO *tO WO~~

Further evaluation of (35) is tedious, but standard. New variables are introduced, (t1, t") -*• (tf, t'-t"),and T « T is used to extend integration limits to +^ °°. Both integrals then decouple and substitution of (34) gives, for large T : o

~~(27T)8 < J (k,w) J (k',0)1) > = (2ffec)2 6, / d3v f(v)~~

~~a (36) •wn~~

~~-142- where~~

~~, raw \ I krl Jm \~~

~~J = J (k, v, /a) ) (37} a = ra m m 1 i o~~

J denotes the Bessel function of the first kind and m-th order.55 J1 = m m d J (x)/dx. k., and ki are the components of V narallel and DerDendicular to m // J- w • H . If one retains a finite T in (36), a Lorentz line profile aopears in- IAO O * stead of 6(ui - mu - k,,v ) . o II !l If we now use (36) to evaluate (31), we have k = uxi r/c and k1 = ^ AA p /w u'a r/c and because of the delta functions in (36) only the suramands q p=q=l and p=q=2 contribute. The obvious interpretation is that the two modes propagate independently, without interference. The final expression for P(w) is written as a sum over modes and harmonics:

~~P(u» = (38a) m=-°°~~

~~with~~

~~n" sin9 xx —-~ ~ Re [ C -a (k x 0 'a ) i | . =P -wj] o; p =p /win, p I a I sin>;~~

~~x 5 (mu> - u) + uxx 3 cosX) (38b)~~

a is given by (37) on substitution of k| = (toOt /c) sinX- It is pointed out again that (38 ) only holds for a stationary ensemble of particles and 'not for a single particle, cf. section 6.2.2. In appendix C it is shown that the direction of P (w) always coincides with the direction of observation r. The harmonic frequency is given, or rather, must be solved from:

~~mu) /(I - a (3.,COSY) (39) o p //~~

-143- This exDre = -'ion differs from the usual one, w = mu /(1-n B, cosO ), in that it o p (/ o c^'itain* '<• cos \ = n cos (0 -.} cosv instead of n cosG . For a physical *p • P o • • p o interpretation of the various harmonics see e.g. refs. 8 and 21.

~~6.4.2. Discussion.~~

The emitted power in mode p in the iu-th harmonic, per unit frequency and per unit solid angle, which we denote with P , can be found from (38) : in t p ? 2 , n" sin6 (c • a ) : d v f (v) ,' \r ..•! : * -£ ^ ~£ 0 (w-iii (40) 6 p !!_B cosx 3 (uia )| o // 9OJ p Here we have introduced the notation 3 = Re [ •••] for the vector quantity wm,p in (38^). In (40) the frequency -^ is a pjsziije quantity, cf. (15). The harmonic frequency, i.e. the solution of Eq. (39) is written as Q. The emitted power per unit frequency or per unit solid angle is found by integration of (40) over 2TT sin x dX or over da), respectively. The emitted power per unit frequency, H , is best derived directly from (38): m,p 2 ^ , n sin6 (41) c -' ~ ^ '^'-fl fc |a" sin(6 -: e o o P x=xo X is a solution of (39) for given w. There may be more than o.^e solution y and (41) must then be sumrped over Y (complex y do not contribute) . 9 o o o o in turn is evaluated from da /S6 = o. Use has been made of the relation p o [ G is now a function of yj :

~~d S dQ d — (a cosy:) = -r|~ (a cosv) —2. + |_ ) = in(6 - 2x) d,< p c)B p dx_ dX (a pcosX n pS o~~

~~dQ (a cosy.) = cosx(9a /39 ) = o] . o p p o~~

We now briefly discuss the effect of a cosX in (39). Generally speaking, considerable changes in the harmonic frequency - and consequently in the va- 2 ^ rious emitted power formulae - are expected in those regions of the (a , 3")- plane (cf. fig. 4) where the index surfaces differ appreciably from spheres, because there 6 and n differ appreciably from X and ct . This does happen

~~-144- in all regions of the (rx , t )-piano and ir; par1 icular r.ear~~

To establish the behaviour of ~, :\'-~,r •:>• : :\:i:. •••••••-., v- j..;r. 3;. .':• i- ment >' to be a function of 0 :''; '->r.t /~:c: = o: p da (ia Sa , }•:< d6 ife d>. d6 :j>, d; ''p "±1" • ' ' From this expression it is easily urio-ekr-d *;;.at :i. /d~ < o in region £ ar.d + "• ? -' _ 8 of the {'x", S") pla:.' jr.d •;;«*- L:^/J;: • o ir. rt.-71.cr. 5 , see fig. 4 [it is reminded that in region 6 a:.: -; •..-.•? a:.;jie . is formally negative). It

follows that JI in a: L '_-as<-.> •• •.••-.' •'•'• - r.or.cton^-slv if v aor.roacr.es the p - resonance angle whcre.is r. , the i-.-:.:rr. or the radius v&ctcr tt the irdex P Suiface, approach*"-:; i::i;r.iry, Ir. fn. 4 tr.e E'in."."iour cr "<. ;- drivr: .-..I:; dotted lines. Consequently, large differences are indicated. It must be concluded that, the numerical evaluations of the power formulae existing in i3 i 7 the literature '-'-'' ' - •• are of limited value and that new calculations must be made. It remains to explain h'hy we find a different result. The reason was already suggested at the end of section 6.2. We proceed from (17) by the substitution J = - e v r(r - r 5, oerform the integration over r' and make use of (13) [and of (34) for the second line] to find:

~~^- J dJk expl-ik'r] Q i- /at' exp[ ik'r'-i^t' ] v^ , ,., a ,~~

~~sin[ (maj - co + kv COS6)T] exp[-ik*r] Q i •} v —: —. %- (42 ) (27T) - m -K™ 'm -'s o — uj + IIKV COSCJ~~

Actually, all authors write TT6 (• • •) instead of sin[ (*•• )T] /(•••} in (42 ). v contains - among other things - Bessel functions of integer order, J (kv, sin8/w ). One now performs the integration over k (in cylindrical or spherical coordinates) using contour integration and stationary phase method. In the contour integration a basic error is made: the contour is closed in the usual way with a semi-circle [ in the upper or lower half of •T the k or k -plane] without taking into account the k-dependence in ' _dt' 'a "b expl***] v1 in (42 ) or in £ v sin[ ••]/(•*) in (4?. ). The closing of the wo m "*>m contour is based on the behaviour of exp[-ik'r] or.li/• However, it is

-145- ,. t-ilv checked that in fact the k-depender.ce in e.q. sin[''] in (42 J) r.akes any simple half-circle closure of the contour impossible, because the contribution along the semi-circle does not vanish. Nevertheless proceeding in this way and using the 5-function instead of ' 2 2 '> the sir.us then one finds [the poles are at k"c = J)~n ~ j : 1 ,2- 5 (im^ ~ J + -en ,,S cos'?) which gives just the "old" harmonic frequency o 1, - || eauatior..

6.5. MULTIPQLE RADIATION. Bunk in has treated the problem of muitipole radiation in a magnetoactive plasma.3 Contrary to the case of cyclotron radiation, the .integration over r1 in (17) now cannot be executed firstlv and one really must evaluate G(D,x). This line has been followed by Bunkin; however, an error is made and we therefore present the formulae again. The following A>:satz is made:

~~J = J'r) sin^t; T - T(r) cosj t (43)~~

~~Substitution in (23) and expansion of the exponent gives:~~

* s x 0 • fdf exp(-iwt') sin (oi f )• ^ p2— / d^r1 (r-r'/c)S J(r') "~P _. OS« ^A OA.

The error in Bunkin's work makes him find (k T'/C)" instead of (r'r'/c)S. O,P «. "A We give here the explicit formulae for s = o, corresponding to electric dipole (El) radiation, and for s = 1, corresponding to magnetic dipole (Ml) and electric quadrupole (E2) radiation. The usual electric and magnetic dipole moments p and m and the electric quadrupole moment tensor N are introduced:

~~-146- 3 3 p = j• r T(r)d r = — f J(r)d r~~

~~m = TT / r x i(r)d r (45)~~

~~N.. = ^— / (r.J +r, J.)d3r = - ~ J r. r T(r)d3r lk 2c ' 1 k k i ^ 2c J I k ^v -v>~~

And a computation analogous to that in section 6.4., based on Eq. (30), gives for the radiated power per unit frequency and per unit solid angle: 4 , 2 p A I I Re[O v ((kk x 0"«v' )] iU-^j. ) (46) 3 =p c p =p ^ 8TTC p eo\ !a"!sinv / ^ ' ^

~~where v = p for El radiation, v - rx m >' r fur Ml radiation and v = OL N-~~

6.6. CONCLUDING REMARKS. {'tJ. Validity cf the sold plasma approximation. This approximation is used throughout in this paper, and it is valid if the radiation field contains "almost no" waves (k,w) near (cyclotron) resonance. A sufficient condition is:60

~~k2v^ << 'J (47)~~

~~k2v^ « (OJ-SW )2; s = 0,1,... (48) t H~~

~~v is the thermal velocity of the background plasma, supposed to be non- relativistic. For s = o, (47) and ('3) can be combined into 2 2 2 2 (kc/VO « (c/wv ) min((jj ,u ) , or t ri~~

~~(49)~~

K is Boltzmann's constant, T is the plasma electron temperature and 2 B = ix\,/oi, cf. (7). The resonance regions n t ro in fig. 4 are therefore ii 1 , JL excluded, however by a wide margin. The case of a very weak magnetic field, 8 << 1, is not properly described by (49). It is a separate limiting case that will not be considered here.

-147- For rf - I a:ui higher, (48) requires the frequency u to be sufficiently far fi -T. -\\-i.->tro:i ro:'o;ui:>c«, s-. Ono need not impose (48) tor :•!,.':• ;:'U s: The relative strength of successive resonances for s =** 2 ia at moat of the order (kv /u> )*" << 1 and therefore, up to what s one should go depends on how well (47) is satisfied. If 210 restrictions on the radiation source are imposed (in the case of cyclotron radiation, source and background plasma electrons could then be identical), then in addition to (49) frequency bands of relative width AIO/OJ ^ k v /.u ^ (v /c)n 0 centered at the first few harmonics ;o = S^J must be excluded. On the other hand, it is possible H to satisfy (48) for .;.';' frequencies allowed by (49), by requiring:

v2 >^ v2 (50) v is a typical source electron velocity [ one is led to (50) by combining (48) with (39) , in the form ui = mu; + ;ja 8 COSY] • In other words, the source o p // electrons must be physically distinct from the background plasma in that they have a much higher average velocity. (ii). The prcrlaation of the radiut:on through an inhomcgeneous medium (e.g. the solar corona) is of interest, especially in astrophysical applications. The properties of such a medium change slowly with respect to the wavelength and the geometrical optic« approximation holds.36 Because the density and the magnetic field decrease outwards in the corona, propagation of the radiation from its origin out of the corona corresponds to 2 2 2 2 a curve in the (a , B )-plane (fig. 4) connecting (a , 8 ) at the point of 2 2 generation with (a , B ) = (0,0) . Because the -wave does not exist in region 3, only radiation generated in region 1 can escape the corona. Like- 2 wise, for the +wave to be able to escape, it must be generated in a <1, that is in region 1, 3, 5 or 7. The path described in the (a , 8^)-plane or equivalently the form ot the rays in the medium is governed by the ray equation and the variation of the intensity along the ray is governed by the equation of transfer.3" (Hi). The polarization of the radiation generated has not been discussed here at all. In the case of a multipole source, the emitted radiation

-148- at. thf3 po.i.rit of observation consists of two free wave modes each v;i*-h a well defined phase. The polarization of free wave modes ir a cold rr>.gr.etc- d,;t ive plasma is treated extensively ir. the literature. 'U * In the case of cyclotron radiation by an ensemble of electrons the situation is more "!_; complicated and the reader is referred to refs. 42, 53, 56-59. -fV3 APPENDIX A. 1 For |k| T °° one has Q ^ K/(uT k*e*k) . The field corresponding to J this term is: j

~~K-J ,„, k(k-J) = 4TTiU!~~

~~' = y = k* and H = o because k * K = o. Consider now the electrostatic equations, with their transforms:~~

~~div D = 4ITT ,• k-e-E = 4triT (A2)~~

~~rot E = o ; k>~~

From (A3) one finds E = Ok and by substitution in (A2) one gets exactly (Al). Hence the field due to K/(ui k'C-k) is equal to the instantaneous (T may depend on t) electrostatic field of a localized charge distribution, which does behave like 1/p . For the computation of the radiation fields — 1 —t 2 A ^ —2 one may thus use Q = [Q - K/(u k'£*k)l and this behaves like k as 2 ^ -1 |k| f m. If we now give K/(u> k*e*k) the same denominator as that of Q ~ -1~ 22 22 22 =2 2 one gets in the numerator of Q an extra term -{k c - CJ n )(k c - u n_)K. On performing the integration over k this term vanishes again because the 2 2 2 2 poles are at k c = ui n1 „. In this way one arrives at the conclusion that -1 ' -2 Q , as given by (19) , behaves effectively as k We elaborate this point a bit further. The aomplete electric field is "iven by: _. _ kT E = 4TTiW Q «J = 4TUW Q «J + ~ -P—^r (A4) = - =R ^ k2 k.£.k The corresponding field in the (p,t) domain is built up of all E(k,to) for

-149- whi.-h the iispersion relation det Q - o holds. The zeros of det Q can he •itv.Jov} ir.ro two qv-vips, coi tospond::.q to the two terms in (A4): 225>> ** *. *» * (1), k c = u)~n. ,, provided k*e*k and k«T>k are not both equal to zero. 1,2 — = These zeros of det Q contribute only in the first term of (A4). This term contains the r-adiaticn fields which are the subject of this paper. t,2) - X*t.*k = o: k*T*k = o (from this it follows that det Q = o as shown in che main text). An equivalent definition of these zeros is given by: k*£'k = o; E parallel to k. These zeros of det Q contribute only to the second term of (A4), and Q remains finite in this case. The corresponding longitudinal field is equal to the charge distribution's own instantaneous Coulomb field. In a warm plasma, this field consists of genuinely propagating longitudinal u.ues. The eK^ryy losses associated with tUese waves are usually much larger than the pure radiation energy losses due to the first term in (A4).

APPENDIX B. In this appendix we suppose o < i < ^TT. There are a few trivial changes in (25) and in the derivation below, if this is not the case. Wri- ting down '-- and K explicitly one finds that the following six integrals must be dealt with:

j d(Ji exp[-iX cos$J '{sini; cosf sin$; cos $; sin &} n = 0,1,2 (Bl) -a with X = ('.o/c)n 0 sin8 sinif/ and o < a < IT. Only their asymptotic values for large, p(large X) are required. Because o < a < ir there is always ex-i-'ily one stationary point, given by (d/dtf)) cose)) = o, or = o. Thus one finds, including contributions ^ X : 3a

~~a r ,, r . , ., r . , . , n. 2. -aj a~~

~~If one substitutes (B2) into f^i>), the integral over 8 involves three types~~

~~-150- of integrands [ /.' - '../c)n c] :~~

(1) . A fast-varying one of order ,• ' !''•>'' ' expt -iX' cos'•;-..;] ) (2). A fast-varying one of order A (vA exp[-dX' cos8 costt1] ) :€" , " t (3). A slowly-varying one of order A , without an exponential and origina- =| •^ ting from the case a = 1, which gives |i = o in the second terra of the "t I right hand side of (25) . 1 The integration over 9 of the second type of integrand will result asympto- i -1/-V 2 tically in a contribution 0(p " ), and hence it can be omitted. Having tically in a contribution 0(p ) obtained these results, one finds:

~~/ d Q exp[-iAJu|] {l; K} ^ 4TT~~

~~2 / d6 sin8 (J-^—^Q sfr'r^ e*Pl-~i}-' cos(6-ii»)] {|; K]~~

~~2 / d8 sin6 (.-, ~i I I—-)) exp[+iA' cos (9-Ui)] {i; K> X Sln9 Sinlp ' = =~~

~~+ y~ / d9 (sin* sinn) (B3)~~

In the second integral in (B3) always 19—^ | >1jTr holds. Therefore, this integral has no stationary point (see fig. 4) and it will be omitted because it is 0(p ). The third integral can be rewritten:

~~A ' 1 T * B4 A / d6 (sin* sinn)~ {i; K> = (O)/c), —n -p / d H 6(P) {i; K^ ^ > P~~

Substitution of (B4) irto the first part of (24) shows that this contribution cancels against R. Hence only the first integral in (B3) will remain and this leads directly to (26). 2 The above considerations still apply if n < o for some 6-interval: In that case, as fig. 4 shows, an interval symmetric around 8 = ^TT is to be omitted from the 8-integration interval. The division of the unit sphere (fig. 3) leading to (25) becomes a bit more involved but the essentials remain the same.

-151- APPENDIX C. The j!r,\.'t:o'.!S of the cjroup velocity V v(k) and the phase velocity (w/kjk are related by Eq. (27) and this is visualised with the help of the index surfaces in fig. 4. Prom the discussion following Eq, (2?) It follows that at the Doint of observation, r, G(p,u>) only contains free wave modes with group velocity along r. It Is proved here that the direction of P coincides with this group velocity direction and hence that P is parallel to r. It then also follows that the polarization of the detected radiation from a given direction and frequency is the same as that of the corresponding free plasma mode (more generally, a superposition of them with different phases). In expression (19), Q is written as

~~Q"1 = Q/A with A = det Q (CD~~

When u = o, 0 has only eigenvalue =£ a:22 If we neglect collisions then Q is hermitian and so it has an orthonormai set of .eigenvectors e. with eigen- -1 -1 r1^^,x values u., in terms of which Q -a can be written as Q 'a = ). e, (e.'a)/u. , 1 = *A- = IW "I 1 1

~~He [0«a * (k * 0*«a*)l = -L~^~ le^'al2 Re [ e x {k * e*>] (C2)~~

The direction of P is independent of a. To establish the direction of Re[ e '< (k * e )] we consider all small variations 6k that satisfy A = o at constant w(so 6oi = o). These 6k span the tangential plane to the index surface, fig. 4. From Q'e = o it follows 6Q-e + Q'fie = o and also:

~~-152- T * ,''• „,. T * * If there are no collisions then Q~ = Q and so e *Q"5e = (Q *e )"5e = o, rri V y " ~ ~~ because Q *e = (Q'e) = o. We are now left with i~~

~~e ••Sg-e = o (C4)~~

From (9) we find, using *•<„ = o and using that •:. does not depend '..n k because we consider a cold plasma- <',Q*e = C"'JL k (k# e) - k^e] , where 6 operates only on -k^. This can be worked out and substituted in (C4) with the result

~~Ce -Ck) (e-k) + (e-Sk)(e>-k; - 2!k-:ki = o (C5)~~

~~Or, since k is real:~~

~~Re[ e x Ck « £X}]'6k = o (C6)~~

So Re[ e x (k x e )i is perpendicular to the index surface, and hence coincides with the qroup velocitv. Since e = E/E and k >' E = .JH/C W;= raay as well say that Re[[ E * HH ] is perpendicular to the index surface. Inward or outward normal is fixed by the following property:

~~k'Re[i x HX] = Re[ k-(E x HX)] = Re[HX« (k x i)] = - |HJ > O (C7) pj^ fW V. Art. »VN. /UV AA- ^^- *W- C~~

[ tj > o, cf. '15)] . It follows that group velocity and phase velocity sub- tend an angle < 'jit. It also follows that P has the direction of the group velocity and because the latter coincides with r, so does F.

~~ACKNOWLEDGEMENTS. The authors wish to thank Professor K.P.H. Weenink and Dr. J. Kuijpers for their useful remarks. We are indebted to Ms. Constance Jansen for her valuable library research.~~

~~-153- REFERENCES AND FOOTNOTES.~~

~~1. V.L. Jir.Jburq, Zh. Eksp, Teor. Fiz. K>.i 601 (1940) [English tranrl.. SLA Translations Center; •;o. TT-0r.-ll >"s>l .~~

~~2. G.V. Mats, I.H. Mwira eUi^ J.C.C. Sharp, _'^>i..lJav:.".--: ::.;/\;';-•'..• ; :• .'i.!:'.,;/•.•> *!:,• (Report DNPL/K.i!4,~~

~~Daresbury Nuclear Physics Laboratory, Daresbury, Nr. Warrington, Lancashire, 1972).~~

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~~s 4. W. Heitler, Tie H unrun ~he^i~~

~~5. A.A. KoiomonsKii. Sh. Eksp. Teor. FIE, 2£, 167 \1953) [English transl.: R.T.S. Inc., 855 Btoom-~~

~~field Ave,, Glen Ridge, New Versey 07028; no. RJ-5D8J .~~

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~~7. V.t. Gtnaburg and V.Ia. Eidnian, 2h. Eksp. Teor. Piz. 3£, 1823 (1959) [ Sov. Phys. - JE1P it_,~~

~~13Q0 v 13591] .~~

~~8. K.B. Lieraohn, Radio Sci. 69D, 741 il955).~~

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~~10. A.G. Sitenko and A.A. Koloraenskii, 2h. Eksp. Teor. Fiz. 30_, ill (19S6) ( Sov. Phys. - JETP 2-~~

~~41C (1956)].~~

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15. V.I. Pakhomov, V.F. Aleksin and K.N. Stepanov, Zh. Tekh. Ftz. 3J_, ll^O (196!) [ Sov. Phys. - Tech. Phys. 6_, 856 (1962)]. ID. V.I. Pakhomov and K.N. Stepanov, Zh. Tekh. Fiz. 3_3_, 43 (1963} [Sov. Phys. - Tech. Phys. 8, 28 (1963)]. 17. v.l. Pakhomov and K.N. Stepanov, Zh. Tekh. Fiz. 32, 437 (1963) [Sov. Phys. - Teen. Phys. 8, 325 (1963)].

~~18. J.F. McKenzie, Proc. Phys. Soc. Lond. 8£, 269 (1964).~~

~~19. V.H. Mansfield, Astrophys. J. r47_, 672 (1967).~~

~~20. D.B. Helrose, Astrophys. J. 1S4_, 803 (1368).~~

~~21. J.r. McKenzie, Phys. Fluids U).. 2680 (19671.~~

~~22. D.B. Melrose, Astrophys. Space Sci, 2, 171 (19&8).~~

~~23. P.C.W. Fung, Can. J. Phys. 47_, 757 (1969).~~

~~24. K. Sakurai and T. Ogawa, Planet. Space Sci. ]_7, 1449 (1969).~~

~~25. T. Ociawa and K. Sakurai, J. Geomagn. Geoelectr. 2j_, 705 (1969).~~

~~26. K. Sakurai, Astrophys. J. 174_, 135 (1972).~~

~~27. G. Kalman and S. Yukon, AFCRL Report 72-0290, Boston College, Dept. of Physics, Chestnut Hill, Massachusetts 02167 (1972).~~

~~-154- 1'B. D.C. Montgomery and D.A. Tidnjar,, Pljtmu Ki>:e f'. •• "he'.'-py (McGraw-Hill, .New '/ork, 1964).~~

~~2i(. T.H. Dupr^e, Phys. Fluids !_• -'2i ',1'jC-i',.~~

~~J'J. T. Birmtnrjl , , J. Dawuo;i j:.d :.'. <;bermar., Phys. Fluids S, 297 W'-itV; .~~

~~31. T.J. Birisi nyhar., Plasma Physics Labcra:ory Rep-art JSfc, Princeton '.Ir.ive. iity '. ifeC / .~~

~~32. G. Bekefi, Radiation Pt'Oaeoees in Plasmas (John Wiley s Sons, New York, 196t>S.~~

~~3i. K. P&padopoulos, Phys. Fluids j_2_, 2185 (1S69).~~

~~34. D.E. Baldwin, I.B- Bernstein and M.P.H, Wec-nink, Adv. Plasma Phys. 3_, 1 (1969). [See also p. 109 o£ this paper]~~

~~35. H. Bremmer, Rijnhuizen Report 67-42, FOM Instituut voor Plasmafysica, Rijnhuizen, Jutphaas, the Netherlands (1966).~~

~~36. L.D. Landau and E.H. Lifshltz, £'..e^ircJy^-M^ :..' s of Ccnti*::icus '-'.eaia (Pergamar. Piess, Oxford, 1360).~~

~~I'l. V.L. Gmzburg, (TiJijJt:':1! . .'" A ~ejzfc",c£'~,e"'-e Waves in hlus>",i (Gordon and Breach, Inc., Mew York, 1961!.~~

~~Vd. B.B. Dingle, Asu"!V' :'.''• Kzpiis'' .'.s: The'r def'''ai''.'K -.""..: Cr.tsvzi'i : ni?'. (Academic Press, London, 1973).~~

~~39. Bef. 32, chapter 1.~~

~~40. R.I. Epstein and P.ft. Feldman, Astrophys. J. Lett. 150, L109 '1967).~~

~~41. P.A.G. Scheuer, Asttophys. J. Lett. 15J_, L133 (1968).~~

~~42. V.L. Ginzburg and S.I. Syrovatskii, Aniu. Hev. Astron. Astroptiys. 1_, 375 ,19691.~~

~~43. T. Takakura and V. Uchida, Astrophys. Lett. 1, 147 (1968).~~

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~~45. H.C. Ko and C.W. Chuang, Astrophy;. Lett. ^5_, 125 (1&73).~~

46. The implication is that if one has managed to obtai:. a I complicated) expression for the emission of one single electron, then this expression would subsequently simplify again upon integrating over an enseioble, which muut be doae anyway 'cf. the final expression (38 ) foi. cyclotron radiation) .

~~47. J.D. Jackson, Classical Elecirodynamise (John Wiley s. Sons, New Yoi"k, 1963) .~~

~~4B. H.C. Ko, Proceedings NASA Symposium on "High Energy Phenomena on the Sun", Ed. R.. Ramaty and R.G. Stone (1973), NASA SP-342, o. 198.~~

~~49. W.P. Allis, s.J. Buchsbaum and A. Bers, i&wss in AtiisJtropic Plasmas (M.I.T. Press, Cambridge, 1963).~~

~~50. As a warning example may serve the following integral [ U=cos8] .-~~

~~;r ,k2 dk Jr d „fl expUkpy•- ,)— = lim Jr do Q ij k1 _,dk—^—, explikpu-,)— K k +u k +» K -k k +p* o o~~

~~= 2T"*(e"D +U/D \- 2^':/p~~

~~Following Baldwin et aZ.31*, the long range part is found to be 4TT" e /p.~~

-155- T:;rcu;;::cu- this V-ILH-'I the square root \ a is defined with a cut 'llonc/ the neqative real axis iv. the comp-'.ax :-pU:ie: -7" •'• arg 2 'S + IT, or -"ITV " avg \/~? ^ * ">"- ^eja.;se :;, ..^ .:t:fir.ed as a square root, n is either o.". tht-? positive real or on the

~~:c»:: ivo i~a-.i;r.jry axis; \' j~ = !:ijj, etc.~~

5>. It ir.ay «i!c:. j..-. t ;.ou-)h the pclr geometry is rathe: accijpnr.al boca-jsc it depends on the specific way damping is introduced (collisions). This is however-not true... One. mmy. just require that any plane wave in the plasrca should experience, a (vanishing!'/ small) dan.ping in tinjd, or 1m u = £ > o, with real k. Putting UJ - m + it and xisina the dispersion 2 2 2 2 ° ' " relation k c* = •'.' n,~, one finds iu n = + kc - itdito n!/dss , For tr=insoarent media one !ti O — Q O has Q|U> n!/du 5 o is is discussed by ref. Ii6, chapter IX. S'nce kc is real, it follows o o that the same damping is achieved by taking u> real, but Itn uin ,*. o. If a Laplace transformation is used instead of the Fourier transformation, this kind of damping is implicit in the formalism.

~~53. V.L. Ginsburg and S.I. Syrovatskii, Anna. Rev. Astron. Astraphys. 3^, 297 (1965).~~

~~54. Ret. 32, chapter 6.~~

~~35. M. Abramoviits and I.A. Stegun, Handb:.-yk ~f '•iath&'niiiiaai Fi*n;!tio>l8 (Dover Publication, Inc., New Vork, 1968!.~~

~~56. K.C. Westfold, Astrophys. J. 130. 241 Uy59) .~~

~~57. R. Ramaty, Astrophys. J 159, 753 (1969).~~

~~58. E.G. Mychelkin, Astron. Sh. £5, 408 (1968) [ Sou. Astron. - AJ J_2, 32G (1968)].~~

~~59. V.H. Sazar.ov, Astron. Zh. 4£, 502 {1969! | Sov. Astron. - AJ _13_, 396 (1969)1.~~

~~80, A.G. sitenko, i'lesiromagr.etie Fl.u-.'tuatianr /'»! PlaEnj (Academic Prass, New Vork, )967).~~

~~-.156- SAMENVATTING~~

'Je buitenste lagen van de sor. vormen een sterk inhomogcen plasma dat permanent niet in evenwicht vorkeext. Eer. van de extreme uitinqen hiervan is de sonnevlara, een plasma-instabiliteit op grote schaal (^ 10.000 km) waarbij ondermeer gedurende ongeveer 100 sec. - in enkele gevallen tot 1000 ssc. toe harde Röntgenstraling wordt uitgezonden (10-250 keV). Deze straling wordt opgewekt tijdens botsingen van versnelde elektronen met proconen. In Utrecht is een spektrometer ontwikkeld, bestemd voor de TD-1A satelliet van ESRO, om intensiteit en spektrale verdeling van deze harde Röntgenstraling te meten. Hieruit kan men de energieverdeling van de versnelde elektronen afleiden en krijgt men een indruk over hun aantal. Dit dient als basis voor een bestudering van het versnellingsmechanisme. Het instrument heeft na lancering, in maart 1972, twee jaar succesvol gewerkt.

De inhoud van dit proefschrift valt uiteen in twee delen. Het eerste en grootste deel omvat de hoofdstukken II t/m V en is gecentreerd rond de interpretatie van tijdens zonnen lammen uitgezonden harde Röntgenstraling. Het tweede deel, hoofdstuk VI, is een theoretische studie over de opwekking van zgn. cyclotrcnstraling.

In hoofdstuk II worden de waarnemingen uit de eerste twee scans ge- presenteerd en tezamen met diverse andere afgeleide parameters geanalyseerd. Bij de presentatie ligt de nadruk op onafhankelijke bruikbaarheid en bij de analyse worden algemene aspekten benadrukt, zoals spektrale eigenschappen, tijdreeksanalyse, aanwezigheid van periodiciteiten en korrelaties. Twee in het oog springende resultaten zijn (1). het aantal versnelde elektronen is zo grcot dat zij nauwelijks een gerichte bundel kunnen vormen; veeleer moet hun snelheidsverdeling ongeveer isotroop zijn. Dit wordt nader uitgewerkt in hoofdstuk V. (2). een opmerkelijke korrelatie werd gevonden tussen de spektrale parameters van de grote zonnevlam op 4 augustus 1972, nader uitgewerkt in hoofdstuk IV. In hoofdstuk III worden uit de waargenomen Röntgenstraling en de gelijktijdig uitgezonden centiraeter-radiostraling een aantal consistente bronparameters voor de zonnevlam van 18 mei 1972 afgeleid. In hoofdstuk IV wordt op grond van de gevonden korrelatie tussen de spektrale parameters een model voorgesteld voor de nrote zonnevlam van

-158- 4 augustus 1972: In een "magnetische fles" zijn door een kortdurend versr.el- lingsproces snelle elektronen opgesloten. De fles zet hierdoor uit er. bevindt zich bovendien in een eigentriliing. Betatronversnelling maakt deze beide bewegingen zichtbaar in de (waargenomen) Rontgenfluxvariaties. In hoofdstuk V, tenslotte, wordt ingegaan op het versnellingsmechanisme voor elektronen in een gewone zonnevlam. Gesuggereerd wordt dat elektronen worden versneld door zgn. kollektieve interakties, wisselwerking tussen één elektron enerzijds met een groep elektronen die een geordende golfbeweging uitvoeren, anderzijds. Deze verschijnselen worden thans uitgebreid bestudeerd in laboratoriumplas- ma's. Voorgesteld wordt dat in het geval van een zonnevlam deze golven worden opgewekt in een dunne strooralaag en dat zij vervolgens zijdelings de stroomlaag verlaten, waarna versnelling van elektronen plaats vindt in het aangrenzende plasma.

Het tweede deel, hoofdstuk VI, is voortgekomen uit pogingen een gemeen- schappelijke verklaring te geven voor tijdens zonnevlammen gelijktijdig uitgezonden harde Röntgenstraling en radiostraling. Het is een theoretische studie over de opwekking van cyclotronstraling door een bron in een koud plasma met een homogeen magneetveld. Een nieuw resultaat wordt afgeleid voor het uitgestraalde vermogen per eerlieid van frekventie en ruimtehoek; ondermeer wordt een uitdrukking gevonden voor de zgn. harmonische frekwentie, die niet, triviaal verschilt van de tot nu toe gebruikte.

~~-159- a'RRicrii'M viTAi-:~~

De auteur van dit proerschrift wera geboitón in IJ*I t_'_ j-Jiavt'.'.i.j.'jr', en hij behaalde in 1959 het einddiploma gymnasium-B aan het Camsiuscallege te Nijmegen. In september 1959 begon hij zijn studie in de Wis-, Natuur- en Sterrekunde aan de Rijksuniversiteit te utrecht, waar hij in maart 19b3 het kandidaats- examen vletter D) aflegde. Na een jaar medicijnen te hebben gestudeerd en vervolgens eer. jaar experimentele natuurkunde, werd in september 1965 begonnen met de hoofdrichting theoretische natuurkunde. In juli 1909 werd het doctoraalexamen theoretische natuurkunde met groot bijvak wiskunde behaald. 2owel het experimentele onderzoek als de doctoraalscriptie hadden bet -ekkimj op een zelfde onder- werp uit de microgolfspektroskopie. In augustus 1969 trad hij als wetenschappelijk medewerker in dienst van het Sterrekundig Instituut te Utrecht, afdeling Laboratorium voor Ruimte- onderzoek.

~~-160- The research described in this thesis was carried out at~~

~~The Astronomical Institute, Space Research Laboratory, Benelnxlaan 21, UTRECHT, The. Netherlands.~~