The effects of quasi-linear interactions on thick-target electron beams in the solar corona are investigated. Coulomb collisions produce regions of positive gradient in electron distributions which are initially monotonic decreasing functions of energy. In the resulting two-stream instability, energy and momentum are transferred from electrons to Langmuir waves and the region of positive slope in the electron distribution is replaced by a plateau. In the corona, the timescale for this quasi-linear relaxation is very short compared to the collision time. It is therefore possible to model the effects of quasi-linear relaxation by replacing any region of positive slop in the distribution by a plateau at each time step, in such a way as to conserve particle number. The X-ray bremsstrahlung and collisional heating rate produced by a relaxed beam are evaluated. Although the analysis is strictly steady state, it is relevant to the theoretical interpretation of hard X-ray bursts with durations of the order of a few seconds (i.e., the majority of such bursts)

Brown, J. C.

Emslie, A. G.

Mcclements, K. G.

English

NASA Technical Reports Server

N87"21825
THE QUASI-LINEAR RELAXATION OF THICK-TARGET ELECTRON
BEAMS IN SOLAR FLARES
K. G. McClements and J. C. Brown
Department of Astronomy
The University
Glasgow GI2 8QQ, Scotland, UK
A. G. Emslie
Department of Physics
The University of Alabama in Huntsville
Huntsville, AL 35899, USA
ABSTRACT
The effects of quasi-linear interactions on thick-target electron beams
in the solar corona are investigated. Coulomb collisions produce regions of
positive gradient in electron distributions which are initially monotonic
decreasing functions of energy. In the resulting two-stream instability,
energy and momentum are transferred from electrons to Langmuir waves and the
region of positive slope in the electron distribution is replaced by a
plateau. In the corona, the timescale for this quasi-linear relaxation is
very short compared to the collision time. It is therefore possible to model
the effects of quasi-linear relaxation by replacing any region of positive
slope in the distribution by a plateau at each time step, in such a way as to
conserve particle number. The X-ray bremsstrahlung and collisional heating
rate produced by a relaxed beam are evaluated.
Although the analysis is strictly steady state, it is relevant to the
theoretical interpretation of hard X-ray bursts with durations of the order
of a few seconds (i.e. the majority of such bursts).
I. Introduction
It is widely accepted that hard X-ray bursts observed during solar flares
are produced by the bremsstrahlung of non-thermal electrons, but relatively
few authors have considered the possible consequences of collective plasma
effects on the dynamics of thick-target electron beams. Considerable
attention has been paid recently to the importance of reverse current Ohmic
losses due to collisional resistivity (e.g. Emslie 1980), but the effects of
plasma wave generation resulting from beam instability (as described by quasi-
linear theory) have been neglected by most authors. Emslie and Smith (1984)
pointed out that the effect of Coulomb collisions is to produce regions of
positive gradient in electron distributions which are initially monotonic
decreasing functions of energy: this gives rise to the well-known "bump on
373 arm,_.'_w,r.__,__
https://ntrs.nasa.gov/search.jsp?R=19870012392 2020-03-20T11:43:47+00:00Z
tail" instability, and Langmuir wave generation is set up.
In this paper we examine the effect of quasi-linear relaxation on the
bremsstrahlung emission and collisional heating rate associated with an
electron beam in the corona. In Section 2, the quasi-linear equations and
their asymptotic solutions are discussed. In Section 3, the collisional
energy loss rate of an electron in a warm target is used to infer the evolu-
tion of the electron beam in a collisionally dominated thick target.
Numerical computations of the distribution function, with and without quasi-
linear relaxation, are presented. Computations of the corresponding hard
X-ray spectra and heating rates are presented in Section 4. In Section 5 we
compare our results with those of previous authors and consider their
implications.
2. The Quasi-linear Equations
In the following we will assume that the source region is a homogeneous
fully ionized hydrogen plasma. The beam electrons will be assumed to be non-
relativistic and to be streaming in one direction only: for simplicity, pitch
angle scattering will be neglected. In order to simplify the quasi-linear
equations, only Langmuir waves propagating in the streaming direction will be
considered.
Let f(v) and W(v) denote respectively the electron velocity distribution
(differential in velocity space) and the energy density in Langmuir waves
(differential in phase velocity space). Then the quasi-linear equations may
be written as (Melrose 1980)
+v]df - _ _v v W(v) (I)dt mn
_f
dW _ --_mD v2W(v) ___v (2)dt n
where m is the electronic mass, n is the ambient density and mp is the electron
plasma frequency. In general, d/dt denotes the total (i.e. advective) time
derivative. We will now argue that, if there exists a region of positive slope
in the electron distribution (i.e. positive 3f corresponding to wave growth)
then quasi-linear interactions will dominate over Coulomb interactions in the
corona. The wave growth rate associated with equation (2) is
_w = _ _p v2 _f (3)
n 3v
Now consider the situation shown schematically in Figure i in which a beam
distribution is superimposed on a background Maxwellian, a region of sub-
stantial positive slope lying between v I and v 2. Putting
3f _ f (Av = v 2- Vl) and defining3v -- Av
n I - f dv _ f Av
v I
374
f(v)
fp
I Ii,I !
vA v I v 2 v B v
Figure i. The form of the combined electron distribution giving rise to
the "bump on tail" instability. A region of positive slope lies between
velocities v I and v 2. The plateau of the relaxed distribution is defined
by the three parameters VA, v B and fp.
the growth rate may be written as
Yw =
while the eolllsional damping rate is given by (Ginzburg 1961)
Yc _ 7 %n i0_ _ 70
(4)
where T is the electron temperature, and the logarithmic factor has been set
equal to a constant with T _ IoTK and n _ 1010 cm -3 (i.e. typical coronal
values). From equations (3a) and (4) we obtain, assuming Av _ v (a
reasonable assumption in practice),
Yw /T _3A
r : 2500nlt ) . (5)
The value of n I depends principally on the total injected electron flux. For
fluxes of the order of lol9cm-2s -I (fluxes as large as this are required by
the thick target interpretation of some hard X-ray bursts) it turns out that
nI _ 106 cm -3. Putting T _ IOTK and n _ iO I0 cm -3 as before indicates
375
that r _ 105 , so that the timescale for quasi-linear relaxation is extremely
short compared to the collision time. This justifies the omission of
collisional terms in equation (i) and enables us to model the effects of
relaxation by applying the asymptotic solution of equations (i) and (2) (in
the chromosphere, F << i, so that the effects of quasi-linear relaxation may
be neglected in that region).
The quasi-linear equations can only be solved numerically. Grognard
(1975) obtained a solution of the one-dimensional quasi-linear equations
including spontaneous emission terms, with the initial conditions of zero
wavelevel and a Gaussian electron distribution. As expected, the asymptotic
solution for f(v) is a plateau in velocity space. However, Grognard points
out that the time for the plateau to be formed is considerably longer than
asymptotic solution is only valid for times T _ iOO/Yw: this does not,
however, alter our conclusion that quasi-linear interactions dominate over
Coulomb interactions in the Corona.
Although a numerical treatment of equations (I) and (2) is essential
for studying the details of the relaxation process, the asymptotic value of
the wavelevel may be readily determined for any given initial distribution
f(v, O). Melrose (1980) obtained such an asymptotic solution in the case of
a delta function velocity distribution. An explicit calculation of the wave-
level to be expected is important because of the (possibly observable) plasma
radiation it excites. In fact Emslie and Smith (1984), on the basis of their
calculation, estimated that the wavelevel would give rise to a microwaw_ flux
far in excess of that observed in a typical event, unless the microwaves are
strongly gyroresonance absorbed. A convincing explanation of this anomaly,
consistent with the thick-target model, does not yet exist.
3. The Evolution Of The Electron Distribution with Depth
We will assume that instability (i.e. wave generation) will always
occur whenever a region of positive slope appears in the electron distribu-
tion. The combined distribution function is given by
f(v) = fb(v) + fo(V) (6)
where fo is the distribution function for the background plasma and fb is
the distribution function of a vertically injected beam of electrons.
Following Knight and Sturrock (1977) we will consider the beam distribution
corresponding to the injected differential energy spectrum
Fo(Eo) (6 I) F E°°6-1= - (7)
oo (Eoo+ Eo)6
where Foo is the total injected flux (cm-2), and Eoo and _ are constants.
Neglecting pitch angle scattering, the instantaneous steady state electron
energy spectrum F(E) is given by the continuity equation
F(E) dE = Fo(Eo) dE o (8)
and the beam distribution function is given by
376
i.eo
v fb(V) dv = F(E) dE
fb(v) -- mF(E). (9)
To evaluate Eo from any given E we require the beam electron energy loss rate.
It turns out that the relaxation process affects f(v) at energies of typically
a few kT (depending on the model parameters, such as Foo), and therefore it is
not self-conslstent here to use the cold target formula for the energy loss
rate assumed by, for example, Brown (1971). The energy loss rate of an
electron in a warm fully ionized target, taking into account only electron-
electron collisions (cf Emslie 1978), is given by
dE= 2_ e 4 _nA
nv (_(x) - 2x_' (x)) (i0)dt E
(Spitzer 1962) where £nA_ 25 is the Coulomb logarithm, e is the electronic
charge, x = (E/kT) I/2 and _ is the error function. Writing
_(x) = _(x) - 2x_'(x), K = 2_e _ £nA and defining the usual column depth
variable
N = n dz' = nz
O
equation (i0) becomes
dE--N = - $(x). (IOa)
The numerical solution of equation (10a) yields Eo for prescribed E,N .
dEo/dE is then given by
where x° = (Eo/kT)_.
dE_.__o = E _(Xo) (II)
dE Eo _(x)
Using equations (9), (IOa), and (ii) we can evaluate f(v) (neglecting
the effects of quasi-linear relaxation) for any prescribed set of parameters
(Foo, Eoo , 6, T and n). Quasi-linear relaxation can be incorporated in the
scheme in the following way: if a region of positive slope is found in the
combined distribution, it is immediately replaced by a plateau which
conserves particle number. The three parameters which define the plateau
are, as indicated in Figure I, VA, vB and fp. These are (uniquely) defined
by the condition that
VB(f(v) - fp) dv = O. (12)
VA
Although there are three unknown parameters, only one of these is independent:
they may all be readily determined numerically. The smoothed-out distribution
function minus the background Maxwellian can then be taken to be the new
Fo(Eo) , and the distribution function F(E) corresponding to the subsequent
N-step can be evaluated as before. Eo- E is thus the energy lost by an
377
electron in a single N-step. If Eo lies in the plateau region then
m Fo(E o) = fp- fo(Eo). (13)
Otherwise, Fo(Eo) is given by the function F(E) as evaluated in the previous
step.
In Figure 2 we present numerical computations of the combined distribu-
tion function f(v), for typical thick-target parameters, at a depth of
1021 cm-2. The plateau formed by relaxation extends from 13keV to 130keV.
' " ' ' ' I i , , , ! i , , l
IO-'
f(v)
4 5 6 7 IJ | iO I Z 3 4 | I T • | IO !
E
Figure 2. The combined distribution function for the model parameters
7 ii 3 19 2 1 TheT = i0 K, n = I0 cm- , Foo = i0 cm- s- ,Eoo = 20keV, 6 = 4.
h 21 2column dept is i0 cm- . The dotted line shows the plateau formed by
i 3 I 1relaxat'on. (f is measured in electrons cm- (cms-)- and E is
measured in keY.)
4. The Bremsstrahlun_ Emission and Heating Rate
Figure 3 shows the local bremsstrahlung spectrum corresponding to the
distribution function shown in Figure 2 (the non-relativistic Bethe-Heitler
cross-section, averaged over solid angle, was used). The dotted line shows
the spectrum obtained by including quasi-linear relaxation. It may be seen
378
' ' ' ' ' I ' ' ' ' ' ' ' ' i
dj I°5_
I0 z I_. _
1.0j - _ _ _ _
""
i0 o - _ _'__ -
-\
i0-I --
, , , , , I , i , , , I i I l
4 8 li ? • • i01 ,_ S 4 S • 1' • • 102
C
Figure 3. X-ray bremsstrahlung emissivity spectrum corresponding to the
distribution function shown in Figure 2. (dj/dc is measured in photons
cm -3 s-I keY and c is measured in keV.)
that relaxation has relatively little effect on the X-ray spectrum.
Qualitatively, the emissivity is reduced in an energy range corresponding
roughly to the plateau region in the electron distribution: the reduction
is never more than about 50%. If the X-ray emissivity is integrated over
the source volume, the overall effect of relaxation on the spectrum is much
smaller: the reduction is < 10%.
The above results are in qualitative agreement with those of Hoyng,
Melrose and Adams (1979). They may be attributed to the "filtering" property
of the Bethe-Heitler cross-section. What this means is that the source
function f(v) is very sensitive to small perturbations on the photon spectrum.
Conversely, different electron distributions can give rise to bremsstrahlung
spectra which are almost identical (cf Brown 1975, Craig 1979).
In Figure 4 the collisional heating rate is shown as a function of column
depth with the same beam and source parameters as before. The dotted line
again indicates the case in which quasi-linear interactions are included.
There is a considerable reduction in the heating rate for N _ 1020 cm -2
379
I0 |
In(N)
i01
I0 0 --
' , i i i i ! i ! j
D
10TM
I I I I I I I I I
2 3 4 S S 7 $ 91020
\
\
= 3 4 5 S 7 8 /91021
N
Figure 4. Collisional heating rate as a function of column depth with the
same beam and plasma parameters as before. (IB is measured in ergs cm -3 s-I
and N is measured in cm-2.)
(_ 50% at N = 1021 cm-2): this is to be expected since energy is being lost
from the beam in the form of Langmuir waves. These waves are then damped and
thereby heat the plasma: the total energy deposition rate is therefore greater
than that indicated by the dotted line in Figure 4. The bremsstrahlung
efficiency is consequently reduced and greater fluxes of electrons are
required to explain hard X-ray bursts on the basis of a thick-target inter-
pretation.
5. Discussion
As indicated previously, our results are consistent with those of Hoyng,
Melrose and Adams (1979). These authors used a rather different technique,
involving a Legendre series expansion of the three-dimensional quasi-linear
equations. The form of the initial particle distribution was similar to that
considered in this paper. It was found that bremsstrahlung spectra were not
greatly affected by quasi-linear relaxation. It is quite likely, however,
380
that the total energy requirement of the thick-target model (with the
inclusion of quasi-linear effects) may depend critically on the form of the
injected electron spectrum. Our choice of a modified power law was governed
by the aim of reproducing power law photon spectra, while at the same time
having an acceptably small beam density to plasma density ratio. As mentioned
previously, Melrose (1980) evaluated the asymptotic wave energy density in the
case of a delta function injected particle distribution and sho_ed that the
particles eventually lose two thirds of their initial energy to waves. We
would therefore expect the effects of quasi-linear relaxation on the energy
requirement and the beam lifetime to be quite substantial in this case. For
the beam and plasma parameters assumed in this paper, however, it appears that
wave-particle interactions have an observationally negligible effect on the
integrated bremsstrahlung emission.
There remains the problem of determining the wavelevel generated by a
thick-target electron beam - this requires the numerical solution of the quasi-
linear equations with collisional damping terms. The wavelevel so obtained may
exceed the threshold for strong turbulence, with important consequences for the
stability of both the beam and the reverse current (Vlahos and Rowland 1984,
Rowland and Vlahos 1985).
Acknowledgement
This work was supported by NASA, SERC and Glasgow University.
References
Brown, J.C. 1971, Solar Phys., 18, 489
Brown, J.C. 1975, in S.R. Kane (ed.) "Solar gamma-, X- and EUV radiation",
IAU Symp. No.68, 245
Craig, I.J.D. 1979, Astron. Astrophys., 79, 121
Emslie, A.G. 1978, Ap.J., 224, 241
Emslie, A.G. 1980, Ap.J., 235, 1055
Emslie, A.G. and Smith, D.F. 1984, Ap.J., 279, 882
Ginzburg, V.L. 1961, "Propagation of Electromagnetic Waves in Plasmas"
(Gordon and Breach)
Grognard, R.J.-M. 1975, Aust. J. Phys., 28, 731
Hoyng, P., Melrose, D.B., and Adams, J.C. 1979, Ap.J., 230, 950
Knight, J.W. and Sturrock, P.A. 1977, Ap.J., 218, 121
Melrose, D.B. 1980, "Plasma Astrophysics", Vol.2 (Gordon and Breach)
Rowland, H.L. and Vlahos, L. 1985, Astron. Astrophys., 142, 219
Spitzer, L.W. 1962, "Physics of Fully Ionized Gases" (Interscience)
Vlahos, L. and Rowland, H.L. 1984, Astron. Astrophys., 139, 263
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The quasi-linear relaxation of thick-target electron beams in solar flares

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