D I A G N O S T I C S OF S O L A R F L A R E H A R D X - R A Y S O U R C E S

PETER HOYNG

Astronomical Institute, Space Research Laboratory, Utrecht

JOSHUA W. KNIGHT

Institute for Plasma Research, Stanford University, Stanford, Calif. 94305, U.S.A.

and

D A N I E L S. SPICER

Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md. 20742, U.S.A.

(Received 12 August, 1977; in final form 29 March, 1978)

Abstract. The dynamics of hard X-ray producing electron beams in solar flares can be strongly affected by the occurrence of a reverse current. The parameter diagram for a beam can be divided into three regimes, one of which is the usual thick target case, the two others being due to two different possible consequences of the reverse current. The use of this parameter diagram as a possible diagnostic tool for solar flare hard X-ray sources is discussed, together with the necessary observations and their inter- pretation.

The forthcoming Solar Maximum Mission, complemented with concurrent ground-based efforts provide the next possibility to obtain these observations, given a good coordination of observing programs. We stress the importance of microwave (GHz) ratio observations with good temporal (-<few sec) and spatial resolution (~<1") in one dimension, and of reliable spectroscopic methods to determine the density in solar flare hard X-ray sources.

1. Introduction

It is a ccep t ed n o w a d a y s tha t ha rd X- rays ( 1 0 - 1 0 0 keV) f rom solar flares a re

g e n e r a t e d as b r e m s s t r a h l u n g f rom fast e l ec t rons wi th c o m p a r a b l e energies . The

unde r ly ing acce le ra t ion process is f r equen t ly o b s e r v e d to o p e r a t e on very shor t

t imesca les ( - 1 s), to last for ~ 1 0 2 s and to consist s o m e t i m e s of a ser ies of s imi lar

burs ts (Van B e e k et al., 1976; H o y n g et al., 1976; M~itzler et al., 1977). A f t e r the

work of B r o w n (1971), the so -ca l l ed t a rge t mode l s b e c a m e fash ionab le , w h e r e one

visual izes e lec t rons ( local ly acce l e ra t ed by an u n k n o w n me c ha n i sm) to s t r eam ou t in

beams . The fast e l ec t rons lose the i r ene rgy par t i a l ly o r to ta l ly in o t h e r a t m o s p h e r i c

layers by C o u l o m b in te rac t ions , bu t a smal l f rac t ion ( ~ 1 0 -5) of the ene rgy lost is

r a d i a t e d in the fo rm of X- rays by b remss t rah lung . T h e p r o b l e m assoc ia ted with these

m o d e l s is that one needs very m a n y e lec t rons , and the i r to ta l k ine t ic ene rgy can be of

the o r d e r of the flare ene rgy spent , whence efficiencies of 1 0 - 5 0 % are n e e d e d

(Brown, 1976). I t has been k n o w n for some t ime (e.g. Brown, 1974) tha t a way ou t of this

p r o b l e m is to a b a n d o n the idea tha t real acce le ra t ion occurs (i.e. the gene ra t i on of

s u p r a t h e r m a l tails on the local e l ec t ron veloci ty d i s t r ibu t ion) , and to suppose tha t

Solar Physics 58 (1978) 139-148. All Rights Reserved Copyright (~ 1978 by D. Reidel Publishing Company, Dordrecht, Holland

1 4 0 P E T E R H O Y N G E T A L .

the flare instability causes intense heating (up to - 1 0 - 1 0 0 keV) in a few relatively small volumes. Because Coulomb losses are in principle absent amongst the 'equally hot ' electrons in these hot nuclei, it is possible to generate the same hard X-ray emission with many fewer electrons and thus thg required efficiency for electron acceleration goes down. It is necessary of course that the individual electrons are confined for some time in the hot nucleus. It is presently unclear how and to what extent this can be achieved�9 Two possibilities exist: (a) If one is dealing with a beam of fast electrons, then instabilities of the reverse current limit the rate at which beam electrons stream out and hence, provide some containment (Brown and Melrose, 1977). (b) It is not always possible to separate clearly the beam from the background plasma. In particular this situation arises if one starts with a locally very hot plasma. In this case, too, the reverse current can become unstable and in the literature this situation is referred to as an electrostatically unstable heat flow (Mannheimer, 1977; Spicer, 1977b; Smith, 1977c). Note that the difference between both situations is mostly one of nomenclature; the physics is the same in both cases.

It will be clear that - if only because of, e.g., expansion - an effective and prolonged confinement of the hot plasma (electrons) is not possible, but one might be able to explain in this way the recurrent 'spikes' seen in the hard X-ray timeprofile (Van Beck et al., 1974, 1976; M~itzler et al., 1977).

With the coming Solar Maximum Mission in view, the purpose of this paper is to suggest a possible method for diagnosing the physical state of solar flare hard X-ray sources�9 Section 2 discusses a few details of the reverse current and Section 3 the required observations and their interpretation�9 Section 4 then deals with the use of a diagnostic diagram from two examples.

2. Origin and Consequences of a Reverse Current

If an electron beam is forced to pass through a plasma by a non-electromagnetic force, a reverse current can be generated in the plasma, such that the total current

�9 3 . . . . 3 density J d v vftot(v) almost vamshes (m comparison wlth J d v Ivl/tot(v)), whereas the energy flux density, ~ d3v i 2 ~mv V/tot(V), does not;/tot(v) is a skewed distribution. This phenomenon is well established experimentally (Levine et al., 1971; Klok et al.,

1974) and theoretically (Cox and Bennett , 1970; Hammar and Rostoker, 1970; Lee and Sudan, 1971; Chu and Rostoker, 1973).

The physical origin of a reverse current is straightforward and for the sake of argument we consider the situation of a beam, see Figure 1.

As the electron beam is building up, it causes a magnetic field to increase, which induces a decelerating electric field Eind. At first, Eind causes the thermal background electrons to flow backwards, and, due to the usually small beam to background plasma density ratio, a small backward flow velocity is sufficient for compensation of the beam current�9 The beam electrons are only slightly affected by the electric field. In a quasi-stationary beam where induction is not important, the

D I A G N O S T I C S O F S O L A R F L A R E H A R D X - R A Y S O U R C E S 141

C

BLACK BOX PRODUCING FAST ELECTRONS

l ; \1 �9

1 t Eind ~'rev

§ § §

§ § §

1 t Estat ~rev

Fig. 1. A black box produces fast electrons, which stream out. Two situations are sketched. Left: the transient state during beam formation. Right: the quasi-stationary state after beam formation is

completed (thick target case).

beam will sustain a (small) charge separation, just sufficient to drive the reverse current. Note that the entire process is self-regulating towards almost zero total current and charge density. One can simply say that the system beam + background plasma is seeking its lowest energy state by eliminating the self magnetic field of the beam. To a good approximation the net current will remain constant, determined by the pre-existing value of V z H.

The concept of a reverse current in astrophysical plasmas is not new; it was already postulated in connection with beams causing type III radio emission by

Melrose (1970). It should be noted here that non-relativistic beams of astrophysical interest cannot be expected to have a sharp front due to velocity dispersion, and therefore excitation of coherent plasma oscillations at the beam front will be unimportant (cf. Smith, 1972; Melrose, 1974).

In the case of solar flares when the emergent hard X-ray flux is caused by bremsstrahlung from a beam of fast electons (so-called thick if thin target models (Brown, 1975)), the inferred beam strength F can be as high as ~ 1 0 36 electrons s -1

Th.e energy in the self magnetic field of such a beam [~(eF/c)2xbeam length] is orders of magnitude above the total beam kinetic energy and the total flare energy. This argument has been used against beam models (Colgate, 1978); however, other authors have pointed out that it underlines the need for a reverse current (Hoyng et al., 1976; Brown and Melrose, 1977; Hoyng, 1977; and Knight and Sturrock, 1977).

142 PETER HOYNG ET AL.

It follows that the fast electrons cannot be accelerated by one single, large-scale DC electric field (pointing upward in Figure 1), because it would accelerate all electrons in the same direction, and a reverse current is then impossible. Possible processes in the black box of Figure 1 are either bulk energization of all electrons to KT, --~ 30-60 keV e.g. by joule heating in a composite of randomly oriented electric fields generated by resistive MHD instabilities (Spicer, 1976, 1977a), or accelera- tion of a specific fraction of the electrons (for example, the suprathermal electrons by Langmuir wave turbulence (Pikel'ner and Tsytovich, 1976; Benz, 1977; Hoyng, 1977; Smith, 1977a, b)).

Incidentally, the existence of a reverse current solves the 'fast electron number problem' (Brown, 1976), which says that since ~ F d t is so large, there is the problem how to supply the electrons to be accelerated if the acceleration has a preferred direction: these electrons are essentially all supplied by the reverse current. Of course, there remains the problem of the energy supply.

2.1. C O N S E Q U E N C E S OF A REVERSE C U R R E N T

Electron beam dynamics is governed by the equation

0 0 0 (D .0 -~+A) f for [v]~>v,, ( _ _ + v _ _ _ e E r e v O_yO___~f= 0 ~Ot Oz op~/ O-pp" (1)

1

where v, = ( xT , /m ) 2= electron thermal velocity (a homgeneous magnetic field H~ results in a zero Lorentz force term in (1)). D and A include Coulomb as well as wave-electron interactions. The existence of a reverse current introduces two new elements of importance for the dynamics of the primary beam:

(a) Microscopic stability. The reverse current will be stable against excitation of ion-acoustic or electrostatic ion-cyclotron waves if (Kindel and Kennel, 1971; Figure la; note the factor x/2 involved in the definition of their electron thermal velocity a , )

[ 0.47 T, = T~ /

vr /v ,<~0.19 T, =3Ti (2)

1,0.063 re = 10ri

where v, = reverse current velocity and Te., = electron, proton temperature. eErev < A z , leading to (Hoyng and Melrose, due to Coulomb losses (collisions) if cou~ �9

(b) Dynamical effects of E .... The electric field Erev that drives the reverse current (against ohmic, i.e. Coulomb losses), will decelerate the beam (Lovelace and Sudan, 1971; Knight and Sturrock, 1977). Provided the plasma is micro- scopically stable, the reverse current Jr~v = noev, and Er~ are related through the Spitzer (1962) conductivity,* which enables one to express vr in terms of Er~v. For a given beam electron with velocity v, this deceleration by Er~v is smaller than that

�9 This point is not trivial. An Ohm's law of the type J = cr �9 E holds in the restframe of a plasma if (1) the only secularly disturbing external force is E, (2) the plasma has an internal mechanism restoring thermal equilibrium. In the present situation the plasma as a whole is reacting to the beam injection as an external agent and hence an 'Ohm's law' -/'tot = o ' totE~ for the plasma as a whole does not exist.

D I A G N O S T I C S O F S O L A R F L A R E H A R D X - R A Y S O U R C E S 143

15

v / v t

t 10

5

I I I I I I I [

R E V E R S E CURRENT \ELECTRIC FIELD \ DOMINATES

Te-~lOTi

\

Q\ \ \ \ \ \ \

.qp-\---~

I I I I I I

REVERSE CURRENT

STABLE ~ UNSTABLE

Te --'=" 3 T i T e ~ T i

COLLISIONS DOMINATE

b~

\

\

\ \ \ \ \ \ \ \ \

0.0 5 0.1 0.5 1.0 VrlV t

Fig. 2 Diagnostic diagram for hard X-ray sources. The stable and unstable regimes are indicated; the former can be subdivided in a collision-dominated and a 'reverse current electric field'-dominated regime. Knowledge of the absolute velocity or energy distribution results in a vertical velocity band at the relevant value of v,/vt. Four such bands are drawn and discussed in the main text, Section 4. The width of the band is a rough indication of the relative energy distribution of the beam electrons. The arrows indicate movements of the source made while positioning it on the v,/vt-axis, as is explained

in Section 4.

1977) :

( v / v t ) 2 (v , /v , )< 6.4. (3)

R e q u i r e m e n t s (2) a n d (3) a re p u t t o g e t h e r in F i g u r e 2.

W e n o w discuss first t h e o b s e r v a t i o n s n e e d e d to p o s i t i o n a h a r d X - r a y s o u r c e in

F i g u r e 2, a f t e r w h i c h we br ie f ly d e a l w i th t he i m p l i c a t i o n s fo r t he phys ics of t he

s o u r c e r eg ion .

3. Required Observations

F i g u r e 2 ho ld s fo r o n e l o c a t i o n in t he b e a m , bu t in a p p l y i n g it to so la r h a r d X - r a y

s o u r c e s o n e m u s t be sa t i s f ied wi th , at leas t , l ine of s ight a v e r a g e s . I t wil l n e i t h e r be

144 P E T E R H O Y N G E T AL.

easy to obtain the necessary observations nor will it be easy to interpret them reliably. For this reason, too, it seems appropriate to stimulate a timely discussion. The following is needed:

(a) Electron temperature Te in the hard X-ray source (107 K and higher). The best way appears to be spatially resolved broad-band soft X-ray observations combined with simultaneous observations of line intensities of successive ionization stages, e.g. of iron (Gabriel and Jordan, 1972; Widing, 1975). There is an obvious problem with the mutual identification of the source plasmas emitting continuum hard X-rays, soft X-rays and X-ray lines.

(b) Proton temperature Ti, is not directly measurable (unless Ti >> Te). However, one might identify the proton temperature with the ion temperature, and the latter can be obtained from linewidth measurements.

(c) Electron density no in the hard X-ray source region. This will be a crucial quantity and a systematic determination of no will already by itself reveal much about the physical conditions in such sources (e.g. by answering questions like whether no is a reasonable thick target density, or whether it is too low/high).

Two methods are in principle available: (1) For higher densities: observation of (quasi-stationary) density dependent line

intensity ratios (Gabriel and Jordan, 1972). These techniques have been applied for instance by Feldman et al. (1977).

(2) For lower densities (~1011 cm-3): observation of non-stationary ionization equilibria, i.e. observation of rise times of spectral lines from successive ionization stages in a transient plasma (Mewe and Schrijver, 1975, 1978; Shapiro and Moore, 1976, 1977).

Above temperatures of a few times 107K spectroscopic determination of no appears to be impossible, because of the absence of appropriate line pairs.

(d) Spatially resolving hard X-ray observations with sufficient temporal and energy resolution will determine the beam area A, the fast electron energy distribu-

tion in the beam, which translates into a vertical velocity band in Figure 2 (see however Craig and Brown, 1976; and Gabriel, 1977) and, finally the thick target

electron beam flux F, e.g. computed from Hoyng et al. (1976; relation (2)). The reverse current velocity v, itself cannot be determined, but one can obtain a

measure ~, for it:

vr ~ vr = F /noA . (4)

If the actual fast electron influx into the target were F, then vr = ~Tr and then (4) just expresses the requirement of zero total current. However, the actual fast electron influx, and therefore vr, can be larger as well as smaller than F, depending upon mostly unknown circumstances.

There are three reasons why the actual influx of fast electrons could be larger than F, even if they form a beam (for details the reader is referred to Hoyng et al.

(1976) pp. 248, 249): (1) The actual target could be thin.

D I A G N O S T I C S OF S O L A R F L A R E H A R D X - R A Y S O U R C E S 1 4 5

(2) The problem of the value of the low energy cut-off.

(3) The net single electron energy losses can be larger than the collisional losses on which the expression for F is based. This occurs, in Figure 2, in the regime where the beam is stable, but Er~v is dynamically dominant over collisional losses.

For the following reasons, the actual influx could be smaller than F: (4) The hard X-ray flux is almost insensitive to the angular velocity distribution

of fast electrons in the target (Elwert and Haug, 1971; Brown, 1972). In other words, a beam is not always necessary, vr = t;, in the case of an unidirectional beam with known low energy cut-off, impinging on a cold (v, -~ 0) thick target. If the beam velocities would be isotropized, F would remain practically the same, but the actual influx, and hence v,, would be zero.

(5) F as given by Hoyng et al. (1976, relation (2)) presupposes only collisional losses in a cold target [(v/v,) z >> 1]. In partly thermal hard X-ray sources (see next section), (v/vt)2>> 1 is not true any more. The collisional losses are then smaller, and so is the actual influx of electrons. This point is closely related to point 3 above.

4. Practical Use of Diagnostic Diagram

At present, we have no information at all concerning the position of hard X-ray sources in Figure 2. Bearing in mind the above mentioned limitations, we shall now deal with two fictitious cases that cover the various possibilities expounded above. Of course, many other scenario's are possible. Much of the discussion concentrates on the question of the source position on the vr/vt-axis in Figure 2. The discussion is deliberately kept brief, as our purpose is solely to illustrate how Figure 2 can be used, and many details are omitted.

(1) Suppose one finds that during the early stages of hard X-ray emission a source is located at, say ~r /V t = 0 . 1 , <(1)/Vt) 2) >> 1 and T e = T/, as indicated in Figure 2, at a.,The question of the value of v,/vt depends, in the first place, on the density no. If it is found to be very high, then a virtually isotropic fast electron distribution is very well possible (reason 4), whence v,/vt << ~,/vt may hold. One would then have formally a thick target situation in the sense that collisional losses dominate. Suppose now, for the sake of argument, that the hard X-rays are found to originate from a relatively low density region (too low to be reasonable thick target density), then streaming of fast electrons must occur. Reason 1, 2 and 3 mentioned at the end of the last section, now move the position of the source to the right in Figure 2. In particular, because of reason 3, there is a displacement to the right by a factor about equal to the ratio of the expected thick target density and the measured density no. It is possible that this brings the source into the reverse current instability regime, and we shall discuss this case further under (2) below.

Let us suppose now that the source position does not move into the instability regime. Then there isa stable beam, decelerated by the reverse current electric field instead of Coulomb collisions. The possibility that this situation could occur in solar flares was first pointed out by Knight and Sturrock (1977). Its importance lies in the

146 P E T E R H O Y N G E T AL.

fact that it- provides a mechanism for heating a low density plasma via joule dissipation of the reverse current (Erev �9 Jrev). It will be a challenge to try and prove observationally whether this state actually occurs, but we are somewhat sceptical ourselves as to the prospects.

As time progresses, one must expect the situation to change rapidly (e.g. in a few

sec) because the ambient electrons will be heated to, let us suppose, Te = 10Ti. If the other parameters do not change at the same time, the source position moves to

the left in Figure 2 and its vertical extent is reduced, both by a factor 101/2 --- 3.2, as indicated in Figure 2 at b. As a consequence, the source is now in the collisional- dominated regime. We speculate that this would show up observationally by an increase in vt and no at the location f rom which the hard X-rays originate (there remains the uncertainty how far the source yet moves to the right because of reason

2). The possibility that the familiar thick target picture of solar flare hard X-ray sources could prevail was pointed out in Houng et al. (1976, pp. 250, 251) and Hoyng (1977, p. 29); these authors adopted a high electron tempera ture (vt/c 0.1), and implicitly, a large beam area (A ~ 1018 cm2).

(2) It may turn out that the product of ambient density no and area A is so small, that ~r/vt is much larger than 1. The reverse current must then be unstable. This will result in isotropization of the fast electrons and v,/vt will become smaller than

~/vt such that the source position moves to the left to one of marginal stability. This is indicated in Figure 2, at c, supposing Te -~ Ti.

This situation is very interesting: the fast electrons generate a reverse current which turns unstable and drastically reduces the outflow of fast electrons, and hence, of energy. The possibility of such a situation was implied by the studies of Brown and Melrose (1977), and has been developed further by Spicer (1977b) and Smith (1977c) for the case of an electrostatically unstable heat flow.

The interesting aspect is that the released energy is prevented from escaping

freely to a considerable degree. As a consequence, the source tempera ture must have an extremely high value and the emerging hard X-rays can to a large extent be due to

thermal bremsstrahlung of a plasma with KTe -- 30-60 keV. This in turn reduces the collisional loss rate (reason 5) and therefore F: Electron acceleration did not really take place, but bulk energization occurred and produced these high temperatures in the first place.

The attractive point of such a thermal model is that it reduces the rather extreme requirements on electron acceleration efficiency, hopefully, to acceptable levels (Brown, 1974). One visualizes a very hot plasma (KT~ ~ 30-60 keV) of small size; the hard X-ray emission measure is known to be ~ 5 • 1045 c m -3 (Hoyng et al., 1976), hence no = 2 .2• 1011 cm -3 and V = 1023 cm 3 would do, for instance. To

illustrate the gain in efficiency, the internal energy 3noxTV is of the order of 1027 erg only, compared to -1030 erg for the flare. Detailed studies of the time

evolution of such a plasma are needed for realistic efficiency estimates. The crucial question is at what rate energy can escape from the volume V by

streaming of fast electrons, hampered by an unstable reverse current. In addition to

DIAGNOSTICS OF SOLAR FLARE HARD X-RAY SOURCES 147

this, shock formation could be an important factor too in bottling up the hot electrons. We are currently investigating if it is possible that such a thermal source

model explains the sharp 'spikes' in the hard X-ray time profiles studies by Van Beek et al. (1974, 1976) and M~itzler et al. (1977).

After this digression we have to retrace our steps concerning Figure 2, since the source temperature is now suspected to be higher than can be measured with spectroscopic methods. Starting again from the position gr/v,, one must first move the source position to the left from ~r/vt (by at most a factor - 3 since we are dealing with a factor ~< 10 in temperature ratios) and shrink the extent of the source along the v/v,-axis by the same factor. Additional displacement to the left in Figure 2 is possible, because also the (spectroscopically) measured density no will cor- respond to that in the adjacent plasma. Finally, from that position the source must be moved to the left further (by fast electron isotropization as explained above) to a marginally stable position, Figure 2 at d.

5. Closing Remarks

In order to be able to prove the existence of a thermal hard X-ray source, one must have a spatial resolution of 1" or better, to measure small areas A. Such a resolution is not always available in hard X-rays during the Solar Maximum Mission. There- fore, it is very important to plan concurrent radio observations at a few frequencies in the GHz range having one-dimensional spatial resolution of < 1" combined with good temporal resolution (~<few sec).

The hard X-ray source density no emerges as a crucial quantity from the above discussion, illustrating the importance of reliable spectroscopic techniques for its determination.

Acknowledgements

In carrying out this reseiarch, the authors have been benefitted considerably from their participation in the Skylab Solar Workshop Series on Solar Flares. The workshops are sponsored by NASA and NSF and managed by the High Altitude Observatory. We are indebted to Drs J. C. Brown and D. F. Smith for their constructive criticism.

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