We analyze a solar flare observed simultaneously in soft and hard X-rays by instruments onboard Yohkoh and the Compton Gamma Ray Observatory. Assuming a simple one-dimensional coronal loop that is heated by field-aligned currents, we solve the energy balance equation to derive the DC-electric field strength necessary to explain the observed soft X-ray emission by current-dissipation. We use the derived DC-electric field to predict the number flux of electrons accelerated by thermal runaway and compare this prediction with the number flux of nonthermal thick-target electrons implied by impulsive phase hard X-ray observations. We find that runaway acceleration can account for the large flux (approx. greater than 10(exp 36 1/s) of nonthermal electrons provided the loop filling factor is approx. less than 10(exp -3) such that heating and acceleration occur in filamented structures within the loop

Schwartz, Richard A.

Zarro, Dominic M.

English

NASA Technical Reports Server

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Energetic Consequences of the DC-Electric Field Model
NASA-CR-204581
Dominic M. Zarro
Applied Research Corporation, NASA/GSFC Code 682.3, Greenbelt,
MD 20771, USA
Richard A. Schwartz
Hughes/STX, NASA/GSFC Code 682.3, Greenbelt, MD 20771, USA
Abstract. We analyze a solar flare observed simultaneously in soft and
hard X-rays by instruments onboard Yohkoh and the Compton Gamma
Ray Observatory. Assuming a simple one-dimensional coronal loop that
is heated by field-aligned currents, we solve the energy balance equa-
tion to derive the DC-electric field strength necessary to explain the ob-
served soft X-ray emission by current-dissipation. We use the derived
DC-electric field to predict the number flux of electrons accelerated by
thermal runaway and compare this prediction with the number flux of
nonthermal thick-target electrons implied by impulsive phase hard X-ray
observations. We find that runaway acceleration can account for the large
flux (_ 1036 s -l) of nonthermal electrons provided the loop filling factor is
10 -s such that heating and acceleration occur in filamented structures
within the loop.
1. Introduction
Magnetic field line reconnection models predict DC-electric field components
parallel to the loop magnetic field (Tsuneta 1995). It has been proposed that
such DC-electric fields can produce nonthermal electrons in solar flares via ther-
mal runaway acceleration (Tsuneta 1985, Holman 1985). In particular, a DC-
electric field will accelerate thermal electrons until a steady state current is
established. Since the collisional drag on the electrons decreases with increas-
ing velocity, those electrons with velocities above a critical velocity will undergo
runaway acceleration producing nonthermal electrons that emit hard X-ray radi-
ation via thick-target interactions (Holman, Kundu, and Kane 1989). Electrons
below the runaway threshold continue to heat the loop plasma producing soft
X-ray emission. This paper investigates the energetics of the DC-electric field
model by examining whether the DC-electric field strength implied by soft X-
ray heating is sufficient to account for the number flux of nonthermal electrons
implied by hard X-ray observations.
In the following section, we derive the DC-electric field strength by solving
the energy balance equation in a single loop with uniform cross-sectional area.
As input to this equation, we use simultaneous soft and hard X-ray observa-
tions from Yohkoh and the Compton Gamma Ray Observatory (CGRO) of a
209
https://ntrs.nasa.gov/search.jsp?R=19970019907 2020-06-16T01:59:24+00:00Z
?.11) D.M. Znrro nnd R.A. Schwartz
_imple loop flare that occurred on 1992 September 6. From the Yohkoh Soft
X-ray Telescope (SXT), we infer the geometry of the flaring loop. From tile
Yohkoh Bragg Crystal Spectrometer (BCS), we derive the temperature "1' anti
emission measure EM of the flare source. From the CGRO Burst and Transient
Spectrometer Experiment (BATSE), we deduce the number flux of accelerated
,,.lcctrons.
2. Method
For a oue-dimensional loop, the energy balance equation is expressed as
dU dec 5nkT _-: ,d---i= Q - n- d--7- (1)
where U = 3nkT is the thermal energy per unit volume, Q is the total flare heat-
iHg rate, fc -_ -lO-6TS/2dT/dz is the Spitzer conductive heat flux, R _- an'ZT --t/2
i._ t.he optically thin cooling rate, and the velocity gradient term is the enthalpy
tlux of convective motions within the loop. For the radiative cooling coeffi-
c:ient, we use a -- 2.2 × 10 -19 which is based on calculations by Raymond, Cox,
;tnd Smith (1976). We shall assume that the density n and temperature T are
uniform along the loop length. This assumption is valid about 20-30 s after
heating onset when thermal conduction will have redistributed the heat energy
throughout the loop, and hydrodynamic motions will have restored approximate
pressure balance within the loop (Fisher and Hawley 1990).
We simplify equation (1) by spatially averaging it with respect to the total
loop volume such that
(IV = QVc - RV + [5nkTv(O) + Fc(O)]A, (2)
where V is the total volume of the loop and Vc is the volume of the current-
heated region. The enthalpy and conductive fluxes vanish at the loop apex and
_re negligible in the chromosphere (z = 0) relative to the heating and radiative
cooling terms. Note that the volume of the current-heated region is assumed to
be less than the total observed loop volume V = 2AL, where A is the observed
loop area and L is the loop half-length. From the ratio of these two volumes, we
define a filling factor f = Vc/V. Following Holman (1985), we assume that the
heat energy generated within each current channel is distributed into the larger
loop volume by conductive (or convective) transport processes on a ti,nescale
less than the heating timescale.
For a loop that is heated uniformly by current-dissipation, the Joule heating
rate is given by
Qo, r_ = nkTu_(E/ED) 2 ergs cm -3 s-', (3)
where t_e _ 3.2 x 102nT -a/_ s-1 is the thermal collision frequency (for classical
r,,sistivity), E is the electric field strength (assumed uniform along the loop
length), and ED = 7 x 10-anT -! volts cm -t is the Dreicer field. The Dreicer
ticld is the field strength at which all the electrons in the plasma undergo thermal
xuaaway. Substituting Qcu_r into equation (2), we derive the following analytic
,_xpression for E
EnergeticConsequencesof theDC-Eh.ctrict"iehlModel 211
E = v
Given n and T, we solve equation (4) for the variation of E during the flare
fi)r different values of tile filling factor. Given E, we can subsequently compute
the rate of runaway electrons assuming that runaway acceleration occurs within
the same current-heated channels. The runaway rate is given by the formula
Nrun _- .35nve(ED/E)a/Sexp[-2t/2(ED/E) t/2 - (II4)(EDIE)]Vc s -t, (5)
which includes electrons that are accelerated out of the thermal distribution
as well as electrons that are scattered into the runaway regime by collisions
(Kruskal and Bernstein 1964). The runaway electrons that propagate along the
loop will heat the plasma by Coulomb collisions, producing thick-target hard X-
ray emission as they impact the chromosphere. Assuming a power-law electron
energy spectrum with spectral index 5 above a low-energy cutoff Ec, the total
number flux of hard X-ray-producing electrons is given by
l_lthicl¢ _ 3 x 1033a1(7 - 1)2B(7 - 1/2, 1/2)Ec -'r s -t, (6)
where B(x, y) is the beta function, "r = _- 1 is the power-law spectral index, and
at is the power-law amplitude, respectively, of the emitted hard X-ray spectrum
(Lin and Hudson 1976). The low-energy cutoff is determined by the critical
energy above which thermal electrons exceed the frictional force and undergo
thermal runaway. The critical energy is given by Ecrit = me(ED/E)ve2/2, where
ve is the electron thermal velocity (Holman 1985).
3. Results
We apply equation (4) to an M3.3 flare that was observed by Yohkoh and CGRO
at 09:00 on 1992 September 6 (Zarro, Mariska, and Dennis 1995). From least-
squares fits of synthetic spectra to the BCS Ca XIX (A 3.177 A) resonance line,
we obtain the variations of T and EM. From preflare SXT images, we infer a
loop half-length L _- 3 x l0 s cm, a loop cross-sectional area A "_ 3 × 10 t7 cm 2,
and a total observed volume V -_ 2 x 102T cm 3. From the emission measure and
filling factor, we derive the loop density n _ (EM/fV) t/2. Substituting into
equation (4), we compute E for different values of st.
Figure 1 shows the time variation of E for f = 0.I, 0.01, and 0.001. The
DC-electric field strength increases with decreasing filling factor. This effect
occurs because the density within the current channels is inversely proportional
to f. Consequently, since _/,-+ n, a higher electric field strength is required to
sustain the enhanced heating rate within the channels. Similarly, the electric
tiekl strength increases with time during the flare rise phase because the density
within the current channels increases as plasma is heated and evaporated into
tim loop.
We use the BATSE LAD Continuous data to derive the nonth,,rmal param-
,'t,.rs of the flare. This data type produces hard X-ray spectra in 16 channels
21"2 D.M.ZairoatJd II.A. Schwartz
0.0008
E 0.0006
0.0004
0
0.0002
w
0.0000
.. ..._."
09:02 09:03 09:04 09:05 09:06
0.8. i i:
0.6 i_
0.4-
0.2 '-
o.oi
09:02 09:03 09:04 09:05 09:06
Start Time (06-Sep-92 09:02:00)
Figure 1. Upper Panel: Variation of DC-electric field strength for
filling factors f = 0.1 (solid), 0.01 (dash), and 0.001 (dash-dot). Lower "_
Panel: Fitted power-law amplitude of nonthermal hard X-ray emission. ,I_i_
_,_tween 10 and > 1000 keV at 2.048 sec temporal resolution. The spectra show
L well-defined power-law distribution of the form ! = ale-7 in the 20-100 keV ,_
•auge. From a least-squares deconvolution of the LAD response function, we 1,brain the temporal variations of the amplitude al and spectral index 7 of the
,ower-law component. The variation of al is compared with E in Figure 1. The
_'mporal variations of these quantities are correlated such that the maximum _
)C-electric field coincides with the peak of hard X-ray burst emission.
For each filling factor, we use the derived values of E to compute the number
hax of nonthermal electrons from the thick-target equation (6) and compare it
vith the number flux predicted by the runaway formula (5). Figure 2 shows
his comparison for f = 0.001. The latter filling factor yields the optimum
tgreement between the measured and predicted fluxes durin6 the rise phase of
repulsive hard X-rays. For this filling factor, the peak DC-field strength is
; × I0 -4 volts crn -l giving a peak runaway flux of 3 x 1037 s -l. Such a high
lit× is a direct consequence of the low cutoff energy (E_ _ 10 keV) implied by
I,' critical energy for thermal runaway. ! ;.
For f > 0.001, runaway acceleration alone does not produce sufficient non- .-4 "
l=,,rmal electrons to match the observed flux. In this case, the density within _i]
t,, _'urrent-heated region is too low to provide a large enough population of/h,'rmal electrons to undergo runaway. For f < 0.001, runaway acceleration also_ils to match the observed number flux. Examination of equation (5), shows
Energetic Consequences of the DC-Electric Field Model 21.3
t_
rr
¢..
o
*d
t_
38
37
36 ....
09:02
...... ' ...... ' 1 I I
', ..
\..*'
• ,J t = = I * , I i ,
09:03 09:04 09:05 09:06
Start Time (06-Sep-92 09:02:00)
Figure 2. Comparison between the number flux of nonthermal elec-
trons implied by hard X-ray observations (solid) and the number flux
predicted by runaway acceleration for a volume filling factor of 0.001
(dashed).
that/_'ru,_ " exp(-ED/E). Since ED "_ n, the runaway rate drops exponentially
with increasing density. Physically, the runaway electrons become thermalized
by collisions when the density in the current-heated region becomes very large.
The predicted and observed electron fluxes show a large discrepancy after
hard X-ray maximum (.._ 09:04 UT) and cannot be reconciled with any value of
f. This discrepancy reflects a breakdown in many of the simplifying assump-
tions that underly the energy-balance analysis. First, the assumption of a filling
factor that is constant in time is likely to become invalid as the flare energy
is distributed throughout the loop system and the heating extends to possibly
multiple loops. Second, the assumption that the hard X-ray emission is pre-
dominantly nonthermal (and thick-target in origin) becomes questionable when
the temperature within the heated region becomes very high and the apparent
power-law spectrum is actually the result of a strong thermal bremsstrahlung
component. The present soft and hard X-ray observations cannot distinguish
this case.
4. Conclusions
We conclude from energy balance that DC-electric field heatitlg an, t acceh'.ra-
ti.tl can self-consistently explain thermal soft X-ray emission and tl(mthermal
n;u'ceh.rationi solarflaresprovidedthat heatingand acceleration occur
,mented subregions with a coronal loop. The filamentation is nece,_.sary
,,,, _mure a sufficiently high density of thermal electrons to undergo rumLw_y
_u:celeration. Such filamentation of loop plasma is in fact necessitated by elec-
trodynamic arguments. In particular, for a loop current system to remain stable,
the induction magnetic field of the current-carrying electrons (including accel-
erated electrons) within each channel must be less than the ambient nmgnetic
field strength. For a cylindrical geometry, this constraint enforces the following
upper limit on the volume of the acceleration region (Holman 1985)
LB 2 ( C_2[ED_ 2
 -ffj , (7)
where B is the loop magnetic field strength. To satisfy this constraint, the accel-
eration region must be fragmented into n¢ -- Vc/_ multiple filaments. Adopting
a typical loop field strength of B -- 100 Gauss, and using the value of E corn-
puled at hard X-ray maximum for J : 0.001, we deduce a pea]( nc --- 1012. The
e×istence of such a large number of current filaments introduces serious prob--
Ictus for maintaining charge-neutrality in the loop plasma. One possible solution
is that the current systems are closed by cross-field drifting of protons at the
chromospheric footpoints (E,mslie and Henoux 1995).
References
Fisher, G., & Hawley, S. 1990, ApJ, 357, 243
Holman, G. 1985, ApJ, 293, 584
Holman, G., Kundu, M., _: Kane, S. 1989, ApJ, 345, 1050
Emslie, A. G., & Henoux, J.-C. 1995, ApJ, 446, 371
Kruskal, M., & Bernstein, I. 1964, Phys. Fluids, 7, 407
Lin, R., & Hudson, H. 1976, Solar Phys. 50, 153
Raymond, J.C., Cox, D.P., & Smith, B.W. 1976, ApJ, 204, 290
Tsuneta, S. 1985, ApJ, 290, 353
Tsuneta, S. 1995, PASJ, 47, 691
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