The generation of rapid fluctuations, or spikes, in hard X-ray and microwave bursts via the disruption of electron heating and acceleration in current sheets is studied. It is found that 20 msec hard X-ray fluctuations can be thermally generated in a current sheet if the resistivity in the sheet is highly anomalous, the plasma density in the emitting region is relatively high, and the volume of the emitting region is greater than that of the current sheet. A specific mechanism for producing the fluctuations, involving heating in the presence of ion acoustic turbulence and a constant driving electric field, and interruption of the heating by a strong two-stream instability, is discussed. Variations upon this mechanism are also discussed. This mechanism also modulates electron acceleration, as required for the microwave spike emission. If the hard X-ray emission at energies less than approx. 1000 keV is nonthermal bremsstrahlung, the coherent modulation of electron acceleration in a large number of current sheets is required

Holman, Gordon D.

English

NASA Technical Reports Server

N87-21824
THE GENERATION OF RAPID SOLAR FLARE HARD X-RAY AND MICROWAVE
FLUCTUATIONS IN CURRENT SHEETS
Gordon D. Hoiman
Laboratory for Astronomy and Solar Physics, Code 682
NASA/Goddard Space Flight Center
Greenbelt, MD 20771
Abstract. The generation of rapid fluctuations, or spikes, in
hard X-ray and microwave bursts via the disruption of electron
heating and acceleration in current sheets is studied. It is
found that 20 msec hard X-ray fluctuations can be thermally
generated in a current sheet if the resistivity in the sheet is
highly anomalous, the plasma density in the emitting region is
relatively high (_i0 II cm-3), and the the volume of the emitting
region is greater than that of the current sheet. A specific
mechanism for producing the fluctuations, involving heating in
the presence of ion acoustic turbulence and a constant driving
electric field, and interruption of the heating by a strong
two-stream instability, is discussed. Variations upon this
mechanism are also discussed. This mechanism also modulates
electron acceleration, as required for the microwave spike
emission. If the hard X-ray emission at energies less than _100
keV is nonthermal bremsstrahlung, the coherent modulation of
electron acceleration in a large number (>104 ) of current sheets
is required.
I. Introduction
Fluctuations in the impulsive hard X-ray emission from solar
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2flares with rise and fall times as short as 20 msec have been
reported (Kiplinger et al. 1983). Spikes in the microwave
emission from solar and stellar flares with time scales as short
as i msec are also observed (Slottje 1978, 1980; Kaufmann et al.
1980, 1985; Zhao and Jin 1982; Gary et al. 1982; Lang et al.
1983). The high brightness temperature and polarization of the
microwave spikes clearly indicate that this emission is
nonthermal, while the hard X-ray emission may be either thermal
or nonthermal. Mechanisms that might be responsible for these
rapid fluctuations are (1) an instability in the
heating/acceleration region, (2) rapid compression and expansion
of the emitting and/or heating/acceleration region, or (3) an
instability in the "beam" of streaming, accelerated particles
when the emission is nonthermal.
Flares are generally understood to be associated with the
formation and instability of current sheets. This paper
therefore addresses mechanisms of the first type for producing
the observed fluctuations or spikes; specifically, unstable
electron heating and acceleration in current sheets. In a
previous study of electron heating and acceleration in current
sheets (Holman 1985; hereafter, H85) it was found that impulsive
phase solar flare bard X-ray and microwave emission can be
generated in a single current sheet if the bulk of the X-ray
emission is thermal, while a large number of oppositely directed
sheets (>104 ) is required if the bulk of the hard X-ray emission
is nonthermal (see also Spicer 1983, Hoyng 1977). Hence, the
following section is concerned with the generation of thermal
hard X-ray fluctuations in a single current sheet. The general
physical conditions required for the fluctuations to be produced
are determined, and a specific mechanism for generating the
fluctuations is discussed. In Section 3 the generation of
microwave and nonthermal hard X-ray spikes is discussed.
362
2. Thermal Hard X-ray Fluctuations
The results of Kiplinger et al. (1983) indicate that the
X-ray flux can increase by a factor of two on a time scale of 20
msec. For one such fluctuation they deduce a peak temperature
and emission measure of 2.4 x 108 K and 6 x 1044 cm -3. Since the
bremsstrahlung emissivity is proportional to Tl/2exp(-Emin/kT),
where Emi n is the minimum observed photon energy (29 keV), if the
emission measure remains constant the temperature must increase
by a factor of 1.4 in 20 msec, and the initial electron
temperature, To, is 1.7 × 108 K. These values will be used in
the following paragraphs.
The characteristic time required to heat the plasma within a
current sheet, _j _ nkT/JE, where J is the current density and E
the electric field strength, can be written as (see H85, Eqns. 4
and 20)
xj = (Ve/Vd)2,e -I, (i)
where v e is the electron thermal speed, v d is the drift speed of
the current-carrying electrons, and _e is the thermal collision
frequency. The precise relationship between this time scale and
the actual rise time, tr, depends upon which quantities vary or
remain constant while the heating is occurring. When the
electric field E and the total current I are both held constant,
and 9e = T -312, this relationship is (H85, Eqn. 24)
tr = (3/2)[(T/To)-I]T J.
Writing, in general, tr = _Tj, T/T o = 1.4 gives _ = 0.6.
T/T o _ i, other conditions give comparable values for _.
(2)
Since
Holding
E and the volume of the current sheet constant gives a smaller
value (0.3), while holding J and I constant gives a slightly
363
4larger value (0.8) for _.
In addition to obtaining the 20 msec rise time, the emission
measure, EM = n2Vj, must also be correct. The current sheet
volume can be written as Vj = wL_r, where w, L, and _r are the
width, length, and thickness of the sheet. All three dimensions
are constrained to be less than Ctr, where c is the speed of
light, and 6r is further constrained by the requirement that the
induction magnetic field associated with the current sheet not be
unacceptably large. Taking w and L to be !Ctr, using Eqn. 8 from
H85 for _r, and combining this with tr = _j and Eqn. 1 for Tj
gives the following expression for _e:
_e I 1.89 × 1011 _(EM44)2T8(B 2)
-2(nli}-2 ( -5 -itr(_2)) s , (3)
where all parameters are in cgs units and the numerical
subscripts indicate the exponent of the value assumed for each
parameter (nll= n/i011). With B = 300 gauss and n = 1011 cm -3,
a collision frequency of 3 x 1010 s-1 is obtained. This is well
above the classical collision frequency. A plasma density of
1012 cm -3 brings this down to a value that is comparable to the
highest anomalous collision frequency expected from the ion
acoustic instability, but the corresponding value of vd is, from
10-4Ve, much too small to drive such an instability.Eqn. 1, 3 X
Electron acceleration in such a sheet would also be negligible.
The reason for the low value of v d (and the correspondingly large
value of _e ) is the need for the thickness of the sheet, _r, to
be large enough for the required emission measure to be obtained.
Therefore, in the following an emission volume that exceeds the
current sheet volume is considered.
The collision frequency required for the Joule heating of a
volume V > Vj in time t r is found from Eqn. 22 of H85 to be
364
5_e _ 4.36 x 106 _(EM)44(To(8)) I/2(B2)-1 (nll)-l(tr(_2)) -3.
• (Ve/V d) s -1. (4)
With B = 300 gauss, n = 1011 cm -3, and (Ve/Vd) = 16, the
collision frequency _e = 1.4 x 107 s-1 is obtained. This is well
above the classical collision frequency, but an order of
magnitude smaller than the maximum effective collision frequency
expected from the ion acoustic instability. The ion acoustic
instability will be driven by this current when Te/Ti _ 10, and
this current and resistivity will result in significant electron
acceleration. The length and width of the X-ray emitting region
have been taken to be 6,000 km (Ctr) , and the thickness of the
emitting region is found from the emission measure to be 1.7 km.
Hence, the rise time of the X-ray fluctuations can be attained
with Joule heating if the resistivity in the current sheet is
highly anomalous and the volume of the X-ray emitting region
exceeds that of the current sheet. It is interesting, however,
that, because of the strong dependence of Ue upon tr, rise times
shorter than i0 msec rapidly become more difficult to obtain.
Since V > Vj, for the 20 msec rise time to actually be
achieved the heat generated in the current sheet must be
transported to the larger emission volume in 20 msec or less.
Both classical and anomalous (Bohm) cross-field conduction are
too slow to do this. This could easily be achieved with
classical conduction along field lines, however. Hence, a bent
or tangled field structure would allow the source volume to be
achieved. Alternatively, if the heat is transported convectively
or behind a shock or conduction front, a propagation speed of at
least 40 km/sec is required.
Producing the X-ray fluctuations requires a mechanism for
interrupting the enhanced heating of the flare plasma. If the
365
6current is driven so that E remains constant, and the plasma
resistivity decreases with increasing temperature, as for
classical resistivity, the fluctuations can be produced as
follows. The Joule heating rate is E2/_, where _ is the
resistivity in the current sheet, and, hence, is regulated by _.
The current density J = nev d in the sheet is J = E/_ and,
therefore, is also regulated by _. Hence, if the resistivity
decreases as the temperature in the sheet increases, both the
heating rate and J increase as T increases. The number of
electrons accelerated out of the thermal plasma increases as
well. If a new instability sets in because of the increased
current of either thermal or accelerated electrons, _ will
increase, causing both the heating rate and J to decrease. The
fall time of the fluctuation will be determined by either (a) the
growth time of the new instability, (b) the time required for
heat to be transported out of the emitting volume, or (c) the
size of the emitting region (tf _ L/c). The longest of these
time scales will dominate. After the enhanced turbulence
dissipates, heating will be occurring at the inltial lower level
and, as T increases again, the process can repeat itself.
It should be noted that the rise time of the fluctuation can
also be determined by either (b) or (c) if the heating time is
shorter than tr. If the rise is dominated by heating and the
fall by heat transport, then the X-ray spectrum is expected to
harden during the rise and soften during the falling phase. If
heat transport dominates both the rise and fall, the spike will
be fairly symmetric (t r _ tf) with the X-ray spectrum becoming
softer throughout the spike. The spike (0.25 sec rise time)
studied by Kiplinger et al. (1984) may be of this type.
A likely source of the anomalous resistivity during the rise
phase of the fluctuation is ion acoustic turbulence, since it can
provide the required collision frequency, and the current drift
366
7speed is likely to be near the threshold for the ion acoustic
instability. Whether or not the effective resistivity has the
required temperature dependence will most likely be determined by
how the instability saturates. The instability required to
interrupt the heating must also be strong, since the turbulence
level required in the sheet to attain the 20 msec rise time is
already high. The oscillating two-stream instability, associated
with the accelerated electrons, is a likely instability that can
generate the required level of turbulence. Determining when the
oscillating two-stream instability will occur is difficult, since
it depends upon the details of the electron distribution. Since
the instability grows rapidly compared to the msec time scale of
the X-ray bursts, the growth time of the instability is not
likely to be the time scale that determines the fall time of the
fluctuations.
A variation upon this scenario might be to have the initial
level of turbulence determined by the oscillating two-stream
instability, and the heating interrupted by the ion acoustic
instability when J reaches the instability threshold. It is not
apparent that the oscillating two-stream effective resistivity
can have the required temperature dependence, however, since it
is proportional to the number density of accelerated electrons.
Other likely sources of anomalous resistivity in the current
sheet are the electrostatic ion cyclotron instability and the
lower hybrid drift instability. One of these may be able to
supply the required initial level of turbulence if the plasma
density is higher and B is larger or Ve/V d is smaller than
assumed above. In the extreme case of ve/v d _ i, the Buneman
instability could be the interrupting instability.
An alternative mechanism for interrupting the plasma heating
has been studied by Krishan and Kundu (1985; see also Spicer
1977, 1981). In direct analogy to the sawtooth oscillations seen
367
8in Tokamaks, the current is interrupted by the onset and
nonlinear growth of the m=l tearing mode. A difficulty with this
analogy is that the tearing mode is driven by an internal kink
instability, requiring that the system oscillate about the
threshold for this instability, q = i, where
q = rBz/LB 8 = CBz/21LJ z = (CBz/2_enLve)(Ve/V d) (5)
is the system safety factor. A cylindrical current channel of
radius r with uniform current density, Jz' is assumed here, as
for the Tokamak. B z is the magnetic field strength in the
direction of the current, and B8 is the azimuthal magnetic field
associated with the current. For solar parameters, however, q is
found to be orders of magnitude smaller than one and, therefore,
the oscillations cannot occur (the system is very unstable,
however). This mechanism may still be able to operate as
required if, instead of being distributed throughout a
cylindrical volume, the current is concentrated in a thin sheet
at the surface of the cylinder. The right side of Eqn. 5 then
becomes multiplied by the factor _(r/_r), where _r is the
thickness of the sheet, and, for the parameters used above, r
106_r _ 103 km is obtained for q = I. Hence, although the formal
analogy with the Tokamak disruption is no longer valid because of
the different current geometry, this mechanism may also be able
to provide the required interruption of the current heating.
3. Nonthermal Microwave and Hard X-ray Spikes
In the above mechanism for generating thermal hard X-ray
fluctuations the acceleration of electrons in the current sheet
is modulated as well. The rate at which runaway electrons is
produced is sensitive to the value of Ve/V _ = ED/E , where ED is
the Dreicer electric field. With _ = T-3-2, ED/E = T -I and thee
flux of accelerated electrons increases as the temperature in the
368
current sheet increases. Hhen the heating is interrupted by a
sudden increase in resistivity, the rate of runaway production
suddenly drops. For the parameters obtained above (§2),
electrons can be accelerated to energies of up to 1 GeV. If the
particles are pitch angle scattered within the current sheet,
however (e. g., Holman, Kundu, and Papadopoulos 1982, Moghaddam-
Taaheri et al. 1985), the maximum energy will be smaller, _a few
MeV, since electrons with a perpendicular energy greater than a
few MeV have gyroradii that exceed the thickness of the sheet.
Microwave emission generated by the accelerated electrons is
not likely to directly mimic the X-ray fluctuations, since the
rise and fall of the microwave spikes is likely to be determined
by the growth and saturation of the emission mechanism. If the
emission mechanism is gyrosynchrotron masering (Holman, Eichler,
and Kundu 1980, Melrose and Dulk 1982, Sharma, Vlahos, and
Papadopoulos 1982; for a review of possible emission mechanisms,
see Holman 1983) a delay is also involved in the time required
for the accelerated electrons to propagate to a mirroring point
within the flaring loop. 0nly when conditions are right for both
the X-ray fluctuations and the masering will both be seen
simultaneously.
Generating the hard X-ray fluctuations at photon energies
below _i00 keV through nonthermal bremsstrahlung requires
modulating simultaneously, to within the time scale of the
fluctuations, electron acceleration within a large number of
current sheets. For a typical electron flux requirement of 1035
electrons/sec for !25 keV X-rays, at least 104 sheets are
required. For a typical sheet thickness of 100 cm, the total
thickness of the sheets is _i0 ks if the sheets are arranged
adjacently, well below ct r = 6,000 km (and on the order of vAt r,
where vA is the Alfven speed). Hence the sheets can in principal
be driven with the required degree of coherency. The
369
I0
interruption of the Joule heating within the sheets must keep the
temperature (or emission measure) of the thermal X-ray source low
enough so that it does not dominate the nonthermal emission.
Acknowledgements
This work was supported in part by NASA Grant 188-38-53-01.
The author thanks Dr. Daniel Spicer for a helpful comment.
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371


The generation of rapid solar flare hard X-ray and microwave fluctuations in current sheets

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