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Flare heating by energetic nonthermal electrons
J Quant. Spec/rose. Radial. Trans!er. Vol. 17. pp. 711-720. Pergamon Press 1977. Printed in Great Britain
FLARE HEATING BY ENERGETIC NONTHERMAL ELECTRONS J. DAVIS, P. C. KEPPLE and D. J. STRICKLANDt Naval Research Laboratory, Washington, IX 20375, U.S.A. (Received 2 November 1976)
Abstract-A model has been developed which includes, in a self-consistent fashion, the energy degradation and attendant bremsstrahlung emission of a high energy electron beam, the heating of the plasma by the beam and the subsequent cooling by thermal conduction and radiation. To assist in the interpretation of experimental diagnostics the model also characterizes the radiative behavior of iron ions present in the flare plasma. The electron deposition is described by the Fokker-Planck equation for an initial power law particle distribution. Results are presented for the bremsstrahlung radiation emitted by the incident beam as it impinges on the disturbed atmosphere. A comparison is made between the direct beam heating and thermal conduction heating of the flare plasma. Finally, the radiation emitted by several selected spectral lines from the Fe ions are shown as a function of time during and after the deposition. I. INTRODUCTION
OVER the years, there has been considerable speculation on the various mechanisms responsible for heating solar flares. One process that has received a good deal of attention is the heating generated by the deposition of energetic electrons. How the electrons acquire energies in the hundreds of keV range is still a controversial subject, but that they exist is confirmed experimentally by an enhancement of the bremsstrahlung radiation. In this investigation, the existence of energetic electrons is assumed. The primary emphasis in this paper is on the role they play in heating the plasma and in characterizing the radiative behavior of the Fe-ions in the flare plasma. In the absence of flare activity, the X-ray emission from the quiet sun is characteristic of thermal emission from a coronal plasma with temperatures on the order of three million degrees. However, during flare activity both the soft and hard components of the X-ray emission are enhanced. For the purpose of this discussion, we assume the soft component consists of X-rays in the OJ-JOkeY range while all X-rays with energy above JOkeV are considered hard. It is the hard component that we will be particularly interested in, since there is speculation that they may be due to non-thermal electrons. CHENd 1) showed that it is possible to heat the region of the Corona and upper chromosphere by Coulomb collisions to temperatures on the order of millions of degrees by a stream of nonthermal electrons. STRAUSS and P APAGIANNIS(2) have developed a constant density model heated by energetic electrons and cooled by thermal conduction and radiation. CULHANE et a/.(3) have concluded that the cooling of the high temperature region could be explained by thermal conduction. For the chromosphere, BROWN(4) argues in favor of direct beam heating while SHMELEVA and SVROVATSKU(SJ find that the energy in the optical flare is transported by thermal conduction from hotter regions of the flare heated by nonthermals. Each of these authors placed special emphasis on one aspect of the problem at the expense of the others. We have incorporated all these features in a self-consistent fashion. In an effort to understand some of the observed iron line emissions we developed a simple but illustrative nonhydrodynamic model that includes the effects of particle deposition, thermal conduction, and optical radiation. 2. MODEL
The particle deposition is described by the Fokker-Planck equation (FPE). Since multiple small angle scattering dominates over large angle scattering in a single encounter and the amount of energy lost per mean free path is a small fraction of the total energy, the FPE is particularly well suited to describe the transport of energetic electrons in the flare plasma. We tScience Applications. Inc.. McLean, VA 22101. U.S.A. 711
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adopt a form of the FPE that is appropriate to describe the evolution of the electron flux in a predominantly ionized hydrogen plasma with trace amounts of iron ions. Since the iron exists in small amounts, it does not alter the energetics of the deposition or the overall energetics of the plasma and is thus ignored in the deposition dynamics. However, they play a significant role in providing diagnostic information on the internal state of the plasma and will be discussed later. The form of the FPE adopted here is given by a4> = Q(E) ~ [(1 _ 2) a4>] + ~ [L(E)4>] JL a( 2 aIL JL aJL aE '
(1)
where 4> = 4>«(, E, JL) is the flux of electrons, ( is the column density, IL is the cosine of the pitch angle referenced to the direction of the magnetic field, E is the energy in keV, Q is the momentum transfer cross section, and L(E) is the loss function. The column density ( in differential form is d( = n(z) dz, where n is the total density (sum of the hydrogen and proton densities). The momentum transfer cross section can be expressed as (2)
where X is the fraction of ionization and Qp(QH) is the momentum transfer cross section for protons (hydrogen atoms). QH is represented by(6)
1
2 4
1]
27TZ e [ + QH = 27T Jof" d9 cos 9(1- cos 9)u(9) = 4£2 In ~-1 + Tf
'
(3)
where u(9) is the differential cross section for elastic scattering and Tf is the atomic screening parameter which appears in u(9) as follows:
Z2 e4 1 u(9) = 4E2 (1- cos 9 + 2Tft
(4)
Our choice for Tf is the Dalitz screening parameter (see JACOB(6». For protons, we assume (5)
where Dmin , the scattering angle at one Debye length, AD, is e2IEAn, with AD = 7.43 x Ilf(Tlnp )ll2. (T is the temperature in eV and np is the proton density.) The loss function is expressed as L(E) = X[L.(E) + LB(E)] + (1- X)LH(E)
(6)
where L., L B and L H are, respectively, the loss functions for energy loss to the ambient electrons, by bremsstrahlung emission, and to hydrogen atoms. The expression for L.(E) is
(7) L B is 2
L B (E)
16 137 r0 = 3" me 2 = 1.6 x 10- eV-cm,2 21
which is obtained from the following differential form:
[1
L = fE dE'5£, (E E') = ~ !02 me (E dE' In + VI - Ell§]. B Jo B, 3137 E Jo I-VI-E'lm 2
(8)
Flare heating by energetic nonthermal electrons
713
For hydrogen, the loss function is
(9) which is valid for the energy range considered, i.e. 1- 250 keY. From BETHE,(7) I has the value 15 for hydrogen. In addition to the bremsstrahlung emitted by the precipitating electrons, the electrons have their energy partitioned between the systems degrees of freedom (neglecting convection): electron thermal energy, ion internal and thermal energies, and line and continuum radiation. In order to account for this energy partition, a set of coupled equations must be solved that determine the ion population {N..}, and electron and ion temperatures, T. and Til respectively. For the ion populations, a system of rate equations of the following form must be solved:
az.. = ~
W...Nv
(IL
= 1, ..., s),
(10)
where S is the number of energy levels. The rates W..v for IL -t II correspond to transitions from the 11th to the ILth level, and W.... = -I Wv ... The total ion density is N i = I N.. and is a constant of the motion. The rates depend on the energy density of the plasma through the electron temperature and the electron density N. = I Z..N.., where {Z..} are the ionic charges. The internal
..
energy of the ILth ionic level is obtained by multiplying N.. by E... On summing over IL one obtains the plasma internal energy, E i = I N..E...
..
A complete description of the energy flow in a volume element !1 V of the plasma requires knowledge of the electron and ion thermal energies. If one neglects convective terms these equations take the form(8) (11)
and (12)
where
3m
Q'i = M
T.
N.k(T. - T.),
(13)
M is the ion mass and T. is the electron-electron 71'/2 collision time. The first term on the RHS of eqn (11) is the beam energy deposition rate, q. is the electron heat flow vector and represents thermal conduction losses; q. = -K. VT•. The third term describes the exchange of thermal energy between electrons and ions, Qi is the net rate of energy transfer from electrons to ions via collisional excitation and ionization less recombination processes, and R b represents radiative energy loss due to bremsstrahlung and is approximated as lI
R b = 1.2 x 1O-19(kT.) 1/2 N.
L Z..2N.., ..
(14)
assuming unit Gaunt factor for this initial estimate, qi is the ion heat flow vector, which is negligible compared to q., and is ignored in the calculations. 3. RESULTS AND INTERPRETATION
We have studied the behavior of a fully ionized hydrogen plasma, embedded with an abundance of 10-4 iron impurities, subjected to a stream of energetic nonthermal electrons. The
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beam of electrons has an isotropic pitch angle distribution and is assumed incident on a column of plasma 3 x 104 km long by 104 km in radius. The upper portion of the column is in the lower corona. A magnetic field of 100 gauss is assumed parallel to the column axis, which defines the z-axis. The presence of the field inhibits diffusion across the field lines and mainly serves in the present model to define direction. Proper treatment of the cross field diffusion processes requires a two dimensional treatment. The plasma atmosphere is described by an exponential density dependence. The temperature at 2 x 104 km from the top of the column is taken to be 30 eV. This is a mathematical boundary condition and is not to be misconstrued as the temperature of the photosphere. Also, the calculation was terminated at 50 sec so that the "heat wave" did not reach the lower boundary. Thus, the particular temperature imposed at this boundary does not influence the calculated temperature profiles during the time of the calculations. The calculations were performed using a power-law flux distribution function with a 10 keY break for the incident beam. The form of this function is given by F(E) - E-a, with a = 2.5 and 4.5, corresponding to a hard and soft spectrum, respectively. The incident spectra are shown in Fig. 1. The bremsstrahlung radiation emitted by the precipitating particles, observed at the earth, is shown in Fig. 2. The total energy content is approximately the same for both incident fluxes and is taken as 2-6 x 1022 eV/cm2 sec. The harder flux generates a harder bremsstrahlung spectra, particularly at higher energies. The correspondence between the functional form of the incident flux and the emitted bremsstrahlung results do not exhibit a simple relationship over the entire range. The bremsstrahlung spectrum is usually assumed to be of the form dJldE - E-Y. For photons in the energy range 10 «: hv"; 100 keY, KANE and ANDERSON(9) find, after investigating 13 bursts, that the spectral index 'Y of the burst lies in the range 2.7"; 'Y"; 4.5. LIN and HUDSON(\O) have observed five relatively soft nonthermal bursts with a spectral index between 4.4 and 5.1. The results of our calculations show that simple fits of the type obtained for thick and thin targets are valid only over limited portions of the spectra. The form of the emitted 1DIS ,..------,--------,,..--------,
10'4
~
10 13
x
:3
10' 2
\ \
U.
\ \
10' ,
\ \ 10' 0..,..10 3
\ 4-:-
--'-'-::--_ _'----'
10 4
10 5
106
E (eV)
Fig. 1. Incident particle flux as a function of electron energy.
715
Flare heating by energetic nonthermal electrons 10",-----,-----,--------.--------,
;; ~10"
E
~
~104 o
.=
..e-
""'" 310' :I:
«
a:
f-
~ :;; ~w co
0
1O- .'"'01.----'O"'.1,.---------.-'c1.0.-----.i 10'.----..-.!100 E (keV)
Fig. 2. Beam bremsstrahlung as a function of photon energy.
bremsstrahlung is dependent on several quantities including the kind of atmosphere the incident stream encounters. In the present model an exponential atmosphere with a constant scale height was selected and is shown in Fig. 3. The electron density at the top of the atmosphere is taken as 109 cm-3 and at 2 x 104 km from the top of the column as 1013 cm-3 • Also shown in Fig. 3 is the deposition rate, i.e. the energy per unit volume and time deposited by the beam into the flaring region as a function of altitude. As seen on Fig. 3 the deposition rate peaks for a = 4.5 10 14
10 13 a= 2.5 \
\
/
\ \
/
/
E
/\<"e
..e
/
UJ
f-
/
c:(
a:
/
'"
0
j::
",/
;:;; 0
CL.
~ 10
~
"'"
\
\
/
1]
10 10
\
/ /
/
~
JOlI]
\
/
101 1'-:-
10 12
~
\
~/ " (:1/
12
"
~
'y
-!: :>
.........
/
-;; 10 13 ~
/-
--'-:
"-
2]
-----'-
2J
-----'" 109
3.0
Z (10 9 em)
Fig. 3. Deposition rate and electron density profile as a function of altitude.
1. DAVIS et al.
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(2.5) at an electron density of about 5 x WO (lOll) electrons/cm3 • The collisional range of these electrons appears to be dependent only on the distribution function to the extent that it is a measure of the hardness of the flux. The range for the cases considered here can be estimated by 2 x 10 17E 2 (keV) electrons/cm2 • Due to its softer nature the deposition rate for a = 4.5 decays rapidly after it peaks. The a = 2.5 case is fairly broad and at the lower boundary (2 x W km from the top) had decreased by about a factor of 5 from its peak value. These two curves demonstrate the characteristic behavior of the fluxes considered here. Figures 1-3 summarize the results of the deposition as described by the FPE in an exponential atmosphere. Another problem of considerable interest is to determine to what extent the incident beam is responsible directly for the bulk of plasma heating. An incident beam of 10 sec duration for a = 4.5, in the absence of thermal conduction, locally heats the plasma to excessively high temperatures in contradiction to the observed X-ray data. These results are shown in Fig. 4 along with the temporal evolution of the heating. These calculations show that one can achieve temperatures on the order of 104 eV during the deposition. The plasma is definitely in an over-heated state. Longer deposition times would only further maintain elevated temperatures. The only mechanism by which the plasma can rapidly dissipate this energy in our model is through bremsstrahlung radiation. However, the bremsstrahlung emission is proportional to n.nz~T\i2, i.e. it has a weak temperature dependence but varies as the square of the charge state, which for totally stripped iron would be Z = 26. For bremsstrahlung radiation to be an effective cooling mechanism over relatively short time periods would require a greater abundance of Fe ions than the 10-4 considered here. Therefore, although the bremsstrahlung losses may, on the whole, be large numerically relative to laboratory plasmas, it is insufficient to significantly reduce the plasma temperature. For the same set of initial conditions but with thermal conduction included, we note a gradual departure from the results due to direct beam heating. At 0.5 sec the effects of thermal conduction are noticeable. At about 4 sec after the onset of the beam the influence of thermal conduction is rather apparent. There is an overall smoothing and lowering of the peak temperature accompanied by the heating of a much larger region of the plasma. At 50 sec, well after the deposition ceases, the total region has been heated to about 800 eV. These results are shown in Fig. 5 along with a more complete time history of the heating. This type of behavior is characteristic of heat transport by thermal conduction. Since the conduction loss rate goes as TS/2, conduction losses also increase as the plasma becomes hotter. It is clear from these calculations that thermal conduction influences the shapes of the temperature profiles throughout the column.
103
-~ '"
~
10 2
I o \'""0------,L1.3;--------,1'7. 7-----;f-2.;:;"0----;;2':;-.4---2;;';.80----,'3.1 Z (10 9 em)
Fig. 4. Temperature profile as a function of altitude in the absence of thermal conduction.
Flare heating by energetic nonthermal electrons
717
----
10 3
1/ I{!
~
{ II III
~
10'
11/
/II
10;'-=-.0------,1-1.=-3-----,1-'-=,7----:!-2.""""0-----::2:'-:.4----:!-2.=-a----='3.1 Z (10' em)
Fig. 5. Temperature profile as a funclion of altitude with thermal conduction.
The high temperatures associated with the various heating mechanisms are also responsible for the appearance of high ionization stages of the Fe ions. The appearance and behavior of the line and continuum radiation characteristic of these keV plasmas is directly observable. For illustrative purposes we studied the temporal behavior of several selected lines integrated over the column and at a fixed point on the column located at Z = 1.8 x 104 km (2.2 x 104 km from the top). The heated electrons excite and ionize the different iron ionization stages and produce thermal bremsstrahlung, radiative and dielectronic recombination and line emission losses from the column. The ion dynamics of the iron plasma was modeled as a corona plasma. The neighboring ground ionization stages were connected to each other by the upward process of collisional ionization and by the downward processes of radiative and dielectronic recombination. Selected excited states of Fe(XXV), (XXIV), (XXIII), (XV) and (XIV) were included in the model. Each could be formed by collisional excitation from their respective ground states and by radiative recombination from the ground state ionization stage above it. They were depopulated by radiative decay to its own ground state. A more complete discussion of the ion dynamics and the rate coefficients is discussed fully elsewhere (JACOBS et al.(ll)). Since the ratio of spectral lines are often used to estimate the electron temperature, we will adopt this convention and present results for the behavior of the following line ratios: Fe(XXV) (300)/Fe(XXIV) (192), Fe(XXIV) (192)/Fe(XXIII) (133) and Fe(XV) (233)/Fe(XIV) (219). The quantity in parenthesis is the (approximate) wavelength of the transition in Angstrom units. The population densities of these levels were obtained as solutions to the system of rate equations given by eqn (10). Thus, line radiation is generated both in the ionizing (or heating) and recombining (or cooling) stages of the plasma's time evolution. The temporal evolution of the line ratios at Z = 1.8 X 104 km is shown on Fig. 6. As the beam impinges on the column, the lower ionization stages are rapidly depleted. This is seen on Fig. 6 by the broken line curve for the Fe(XV)/Fe(XIV) ratio. The higher ionization stages peak in the time interval between 5 to 10 sec as shown by the solid curves. This is in accord with the temperature profile as a function of time. As the beam is depositing energy, it is directly heating the region while thermal conduction is indirectly heating the surrounding region, as previously discussed. The temperature profile exhibits a peak value of about 1.75 keY at 10 sec, which is just at the end of the deposition, and then slowly decays to a value of about 800 eV at 50 sec, at which time the calculation was terminated. In order to interpret the results shown on Fig. 6 we must first understand the behavior of the individual lines. The relative abundance of Fe(XXV) at peak temperature is about three times greater than Fe(XXIV) and about eight times greater
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Fe XXIV 192 A/Fe XXIII 133 A
Fe XXV 300 ~/ Fe XXIV 192 A
"\ \
10-4
\ \
I l--I
Fe XV 233 A. Fe XIV 219 A-
I
do
25.5
51
TIME (sec.)
Fig. 6. Temporal evolution of spectral line ratios.
than Fe(XXIII). Hower the Is2p 3P21evel is not significantly populated from IS2S 3S level of Fe(XXV), i.e. the 300 'A transition. On the other hand, the temperature is sufficiently high to populate significantly the 2p 2P3/21evel of Fe(XXIV) and the 2S2p Ip level of Fe(XXIII) at 192 and 133 'A, respectively. In fact, the 192 and 133 'A transitions track the temporal evolution of the temperature, i.e. the plasma remains hot enough over the duration of the calculation to maintain the population in these levels. Once the deposition terminates, the plasma maintains itself for a brief duration and then enters a cooling phase. During this phase the plasma begins to cool and recombine, producing, at first, fractional reductions in the higher stages of ionization. As the plasma continues to cool the lower ionization stages are populated. This is seen on Fig. 6 from the behavior of the Fe(XV)/Fe(XIV) ratio. The production rate for these lines begins around 20 sec into the run (10 sec after the deposition has terminated) and steadily increases over the remainder of the calculation. This behavior is characteristic of recombining plasmas. Monitoring the radiative behavior of selected spectral lines for a fixed point on the flare is useful for qualitatively understanding the behavior of the flare plasma. In reality, however, observers cannot resolve fixed points on the flare, but rather make measurements that are spatially integrated over the entire flare or, at least, large parts of it. The integrated behavior of these line ratios is shown on Fig. 7. These results are easily interpretable with the aid of the temperature profiles and our previous remarks.
4. SUMMARY
The primary purpose of this investigation was to study the behavior of a fully ionized hydrogen plasma embedded with iron ions subjected to a stream of energetic nonthermal electrons. In the absence of convection the energy of the incident beam is partitioned among the system's degrees of freedom; electron thermal energy, ion'internal and thermal energies, line and continuum radiation. It has been shown that direct beam heating without thermal conduction would provide localized flare regions with very high temperatures, in contradiction to the observed facts. The inclusion of thermal conduction reduces the peak temperature to values more in agreement with the observations and heats a larger region of the flare. For a hydrostatic atmosphere the deposition, for both incident fluxes, peaks at an electron density in the neighborhood of 1011 electrons per cm3. It has also been shown that, for two kinds of initial power law distributions in a hydrostatic atmosphere, the emitted bremsstrahlung is not describable simply in terms of thick or thin target results but falls somewhere in between. Finally, we considered the variation of spectral line intensities as a function of time at a fixed position and integrated along the flare column. It was shown that the peak temperature was
719
Flare heating by energetic nonthermal electrons
0
( 1)
XXTIZ: (192 A) 0
XXllI(133A)
10- 2 0
(2)
XXJZ: (300 A)
A)
XXIil ( 192
(f)
~
I-
cr
(3)
lJ.J Z
A) A)
:x::x:I'i[
(192
xxnz:
(255
...J
10- 3
10- 5 L...-_L----<_-...J._--l._---'-_--'---""-'-_--'-_-'-------'
o
10
20
30
40
50
TIME (SEC)
Fig. 7. Temporal evolution of space integrated spectral line ratios.
sufficient to significantly populate some of the higher stages of ionization, In particular, Fe(XXV), Fe(XXIV) and Fe(XXIII). The temporal history of several spectral line ratios was presented for a fixed spatial position and spatially integrated along the plasma column, during and well after the deposition. During both the heating and cooling phases the plasma was emitting line radiation. The X-ray line emission peaked at about 10 sec, which is just the duration of the deposition. After that its behavior was typical of any high temperature recombining plasma. The results presented here are indicative and illustrative of the behavior of the flare plasma due to the deposition of energetic nonthermal electrons in the absence of hydrodynamics. Our remarks were confined to the deposition, thermal conduction heating and radiative losses from the Fe ions due to electrons injected into the plasma with a power law distribution function. We could just as easily have considered the injected electrons to have a Maxwellian, double-peaked Maxwellian or various combinations of Maxwellian and power law distribution function. The effects due to these types of distribution will be the subject of a forthcoming paper. There are several other variables that are essential to any realistic model of flare heating. They are the
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total energy content and duration of the burst and whether there are several short or one long burst or a combination of these. Questions of this nature are best answered by appealing to the data and then incorporated into our model as known parameters. Another major unknown in our understanding is the role of collective interactions of the electron beam with the flare plasma. The principal effect of most collective plasma processes is to drive the system to a state of marginal stability. That is, the system is at the threshhold from which it may be stabilized, excited, restabilized, and then reexcited both in space and time. This behavior will also affect the anomalous transport resulting from the collective effects. For a discussion of these processes the reader is referred to the work of P APAOOPOULOS.o 2 ) It is clear that plasma instabilities may play a more important role than has hitherto been considered. The collective plasma effects may also play an important role in the overall energy balance of the source. However, we must turn first to our most pressing problem: the incorporation into our model of the gas kinetic effects resulting from the deposition. Preliminary calculations show that the hydrodynamics has a profound influence on the overall behavior of the flare plasma. There is considerable mass redistribution, shock wave heating, and changes in the equation of state, to mention a few. The details of these calculations will appear in a subsequent paper. Acknowledgements-The authors would like to thank Dr. JOHN ROGERSON for computational assistance and Dr. V. L. JACOBS for reviewing the manuscript. This work has been supported by the E. O. HULBERT Center for Space Research, NRL, as part of the ATM Data Analysis Program funded by NASA DPR-60404-G.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
REFERENCES C. C. CHENG, Solar Phys. 22, 178 (1972). F. M. STRAUSS and M. D. PAPAGGIANIS, Appl. J. 164,369 (1971). J. L. CULHANE, J. F. VESECKY and K. 1. H. PHILLIPS, Solar Phys. IS, 394 (1970). J. C. BROWN, Solar Phys. 29, 421 (1973). S. I. SYROVATSKII and O. P. SHMELEVA, Sov. Astron. 16, 273 (1972). 1. H. JACOB, Phys. Rev. 8, 226 (1973). H. A. BETHE, Handbuch der Physik, Vol. 24(2), p. 273. Springer, Berlin (1930). S. I. BRAGINSKll, Reviews of Plasma Physics, Vol. I, p. 205. Consultants Bureau, New York (1965). S. R. KANE and K. A. ANDERSON, Appl. J. 162, 1003 (1970). R. P. LIN and H. S. HUDSON, Solar Phys. 17,412 (1971). V. L. JACOBS, 1. DAVIS, P. C. KEPPLE and M. BLAHA, Appl. 1. (Jan. 1977). K. PAPADOPOULOUS, Phys. Fluids 18, 1769 (1975); 1. Geophys. Res. 79, 674 (1974).
FLARE HEATING BY ENERGETIC NONTHERMAL ELECTRONS J. DAVIS, P. C. KEPPLE and D. J. STRICKLANDt Naval Research Laboratory, Washington, IX 20375, U.S.A. (Received 2 November 1976)
Abstract-A model has been developed which includes, in a self-consistent fashion, the energy degradation and attendant bremsstrahlung emission of a high energy electron beam, the heating of the plasma by the beam and the subsequent cooling by thermal conduction and radiation. To assist in the interpretation of experimental diagnostics the model also characterizes the radiative behavior of iron ions present in the flare plasma. The electron deposition is described by the Fokker-Planck equation for an initial power law particle distribution. Results are presented for the bremsstrahlung radiation emitted by the incident beam as it impinges on the disturbed atmosphere. A comparison is made between the direct beam heating and thermal conduction heating of the flare plasma. Finally, the radiation emitted by several selected spectral lines from the Fe ions are shown as a function of time during and after the deposition. I. INTRODUCTION
OVER the years, there has been considerable speculation on the various mechanisms responsible for heating solar flares. One process that has received a good deal of attention is the heating generated by the deposition of energetic electrons. How the electrons acquire energies in the hundreds of keV range is still a controversial subject, but that they exist is confirmed experimentally by an enhancement of the bremsstrahlung radiation. In this investigation, the existence of energetic electrons is assumed. The primary emphasis in this paper is on the role they play in heating the plasma and in characterizing the radiative behavior of the Fe-ions in the flare plasma. In the absence of flare activity, the X-ray emission from the quiet sun is characteristic of thermal emission from a coronal plasma with temperatures on the order of three million degrees. However, during flare activity both the soft and hard components of the X-ray emission are enhanced. For the purpose of this discussion, we assume the soft component consists of X-rays in the OJ-JOkeY range while all X-rays with energy above JOkeV are considered hard. It is the hard component that we will be particularly interested in, since there is speculation that they may be due to non-thermal electrons. CHENd 1) showed that it is possible to heat the region of the Corona and upper chromosphere by Coulomb collisions to temperatures on the order of millions of degrees by a stream of nonthermal electrons. STRAUSS and P APAGIANNIS(2) have developed a constant density model heated by energetic electrons and cooled by thermal conduction and radiation. CULHANE et a/.(3) have concluded that the cooling of the high temperature region could be explained by thermal conduction. For the chromosphere, BROWN(4) argues in favor of direct beam heating while SHMELEVA and SVROVATSKU(SJ find that the energy in the optical flare is transported by thermal conduction from hotter regions of the flare heated by nonthermals. Each of these authors placed special emphasis on one aspect of the problem at the expense of the others. We have incorporated all these features in a self-consistent fashion. In an effort to understand some of the observed iron line emissions we developed a simple but illustrative nonhydrodynamic model that includes the effects of particle deposition, thermal conduction, and optical radiation. 2. MODEL
The particle deposition is described by the Fokker-Planck equation (FPE). Since multiple small angle scattering dominates over large angle scattering in a single encounter and the amount of energy lost per mean free path is a small fraction of the total energy, the FPE is particularly well suited to describe the transport of energetic electrons in the flare plasma. We tScience Applications. Inc.. McLean, VA 22101. U.S.A. 711
712
1.
DAVIS
et al.
adopt a form of the FPE that is appropriate to describe the evolution of the electron flux in a predominantly ionized hydrogen plasma with trace amounts of iron ions. Since the iron exists in small amounts, it does not alter the energetics of the deposition or the overall energetics of the plasma and is thus ignored in the deposition dynamics. However, they play a significant role in providing diagnostic information on the internal state of the plasma and will be discussed later. The form of the FPE adopted here is given by a4> = Q(E) ~ [(1 _ 2) a4>] + ~ [L(E)4>] JL a( 2 aIL JL aJL aE '
(1)
where 4> = 4>«(, E, JL) is the flux of electrons, ( is the column density, IL is the cosine of the pitch angle referenced to the direction of the magnetic field, E is the energy in keV, Q is the momentum transfer cross section, and L(E) is the loss function. The column density ( in differential form is d( = n(z) dz, where n is the total density (sum of the hydrogen and proton densities). The momentum transfer cross section can be expressed as (2)
where X is the fraction of ionization and Qp(QH) is the momentum transfer cross section for protons (hydrogen atoms). QH is represented by(6)
1
2 4
1]
27TZ e [ + QH = 27T Jof" d9 cos 9(1- cos 9)u(9) = 4£2 In ~-1 + Tf
'
(3)
where u(9) is the differential cross section for elastic scattering and Tf is the atomic screening parameter which appears in u(9) as follows:
Z2 e4 1 u(9) = 4E2 (1- cos 9 + 2Tft
(4)
Our choice for Tf is the Dalitz screening parameter (see JACOB(6». For protons, we assume (5)
where Dmin , the scattering angle at one Debye length, AD, is e2IEAn, with AD = 7.43 x Ilf(Tlnp )ll2. (T is the temperature in eV and np is the proton density.) The loss function is expressed as L(E) = X[L.(E) + LB(E)] + (1- X)LH(E)
(6)
where L., L B and L H are, respectively, the loss functions for energy loss to the ambient electrons, by bremsstrahlung emission, and to hydrogen atoms. The expression for L.(E) is
(7) L B is 2
L B (E)
16 137 r0 = 3" me 2 = 1.6 x 10- eV-cm,2 21
which is obtained from the following differential form:
[1
L = fE dE'5£, (E E') = ~ !02 me (E dE' In + VI - Ell§]. B Jo B, 3137 E Jo I-VI-E'lm 2
(8)
Flare heating by energetic nonthermal electrons
713
For hydrogen, the loss function is
(9) which is valid for the energy range considered, i.e. 1- 250 keY. From BETHE,(7) I has the value 15 for hydrogen. In addition to the bremsstrahlung emitted by the precipitating electrons, the electrons have their energy partitioned between the systems degrees of freedom (neglecting convection): electron thermal energy, ion internal and thermal energies, and line and continuum radiation. In order to account for this energy partition, a set of coupled equations must be solved that determine the ion population {N..}, and electron and ion temperatures, T. and Til respectively. For the ion populations, a system of rate equations of the following form must be solved:
az.. = ~
W...Nv
(IL
= 1, ..., s),
(10)
where S is the number of energy levels. The rates W..v for IL -t II correspond to transitions from the 11th to the ILth level, and W.... = -I Wv ... The total ion density is N i = I N.. and is a constant of the motion. The rates depend on the energy density of the plasma through the electron temperature and the electron density N. = I Z..N.., where {Z..} are the ionic charges. The internal
..
energy of the ILth ionic level is obtained by multiplying N.. by E... On summing over IL one obtains the plasma internal energy, E i = I N..E...
..
A complete description of the energy flow in a volume element !1 V of the plasma requires knowledge of the electron and ion thermal energies. If one neglects convective terms these equations take the form(8) (11)
and (12)
where
3m
Q'i = M
T.
N.k(T. - T.),
(13)
M is the ion mass and T. is the electron-electron 71'/2 collision time. The first term on the RHS of eqn (11) is the beam energy deposition rate, q. is the electron heat flow vector and represents thermal conduction losses; q. = -K. VT•. The third term describes the exchange of thermal energy between electrons and ions, Qi is the net rate of energy transfer from electrons to ions via collisional excitation and ionization less recombination processes, and R b represents radiative energy loss due to bremsstrahlung and is approximated as lI
R b = 1.2 x 1O-19(kT.) 1/2 N.
L Z..2N.., ..
(14)
assuming unit Gaunt factor for this initial estimate, qi is the ion heat flow vector, which is negligible compared to q., and is ignored in the calculations. 3. RESULTS AND INTERPRETATION
We have studied the behavior of a fully ionized hydrogen plasma, embedded with an abundance of 10-4 iron impurities, subjected to a stream of energetic nonthermal electrons. The
714
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beam of electrons has an isotropic pitch angle distribution and is assumed incident on a column of plasma 3 x 104 km long by 104 km in radius. The upper portion of the column is in the lower corona. A magnetic field of 100 gauss is assumed parallel to the column axis, which defines the z-axis. The presence of the field inhibits diffusion across the field lines and mainly serves in the present model to define direction. Proper treatment of the cross field diffusion processes requires a two dimensional treatment. The plasma atmosphere is described by an exponential density dependence. The temperature at 2 x 104 km from the top of the column is taken to be 30 eV. This is a mathematical boundary condition and is not to be misconstrued as the temperature of the photosphere. Also, the calculation was terminated at 50 sec so that the "heat wave" did not reach the lower boundary. Thus, the particular temperature imposed at this boundary does not influence the calculated temperature profiles during the time of the calculations. The calculations were performed using a power-law flux distribution function with a 10 keY break for the incident beam. The form of this function is given by F(E) - E-a, with a = 2.5 and 4.5, corresponding to a hard and soft spectrum, respectively. The incident spectra are shown in Fig. 1. The bremsstrahlung radiation emitted by the precipitating particles, observed at the earth, is shown in Fig. 2. The total energy content is approximately the same for both incident fluxes and is taken as 2-6 x 1022 eV/cm2 sec. The harder flux generates a harder bremsstrahlung spectra, particularly at higher energies. The correspondence between the functional form of the incident flux and the emitted bremsstrahlung results do not exhibit a simple relationship over the entire range. The bremsstrahlung spectrum is usually assumed to be of the form dJldE - E-Y. For photons in the energy range 10 «: hv"; 100 keY, KANE and ANDERSON(9) find, after investigating 13 bursts, that the spectral index 'Y of the burst lies in the range 2.7"; 'Y"; 4.5. LIN and HUDSON(\O) have observed five relatively soft nonthermal bursts with a spectral index between 4.4 and 5.1. The results of our calculations show that simple fits of the type obtained for thick and thin targets are valid only over limited portions of the spectra. The form of the emitted 1DIS ,..------,--------,,..--------,
10'4
~
10 13
x
:3
10' 2
\ \
U.
\ \
10' ,
\ \ 10' 0..,..10 3
\ 4-:-
--'-'-::--_ _'----'
10 4
10 5
106
E (eV)
Fig. 1. Incident particle flux as a function of electron energy.
715
Flare heating by energetic nonthermal electrons 10",-----,-----,--------.--------,
;; ~10"
E
~
~104 o
.=
..e-
""'" 310' :I:
«
a:
f-
~ :;; ~w co
0
1O- .'"'01.----'O"'.1,.---------.-'c1.0.-----.i 10'.----..-.!100 E (keV)
Fig. 2. Beam bremsstrahlung as a function of photon energy.
bremsstrahlung is dependent on several quantities including the kind of atmosphere the incident stream encounters. In the present model an exponential atmosphere with a constant scale height was selected and is shown in Fig. 3. The electron density at the top of the atmosphere is taken as 109 cm-3 and at 2 x 104 km from the top of the column as 1013 cm-3 • Also shown in Fig. 3 is the deposition rate, i.e. the energy per unit volume and time deposited by the beam into the flaring region as a function of altitude. As seen on Fig. 3 the deposition rate peaks for a = 4.5 10 14
10 13 a= 2.5 \
\
/
\ \
/
/
E
/\<"e
..e
/
UJ
f-
/
c:(
a:
/
'"
0
j::
",/
;:;; 0
CL.
~ 10
~
"'"
\
\
/
1]
10 10
\
/ /
/
~
JOlI]
\
/
101 1'-:-
10 12
~
\
~/ " (:1/
12
"
~
'y
-!: :>
.........
/
-;; 10 13 ~
/-
--'-:
"-
2]
-----'-
2J
-----'" 109
3.0
Z (10 9 em)
Fig. 3. Deposition rate and electron density profile as a function of altitude.
1. DAVIS et al.
716
(2.5) at an electron density of about 5 x WO (lOll) electrons/cm3 • The collisional range of these electrons appears to be dependent only on the distribution function to the extent that it is a measure of the hardness of the flux. The range for the cases considered here can be estimated by 2 x 10 17E 2 (keV) electrons/cm2 • Due to its softer nature the deposition rate for a = 4.5 decays rapidly after it peaks. The a = 2.5 case is fairly broad and at the lower boundary (2 x W km from the top) had decreased by about a factor of 5 from its peak value. These two curves demonstrate the characteristic behavior of the fluxes considered here. Figures 1-3 summarize the results of the deposition as described by the FPE in an exponential atmosphere. Another problem of considerable interest is to determine to what extent the incident beam is responsible directly for the bulk of plasma heating. An incident beam of 10 sec duration for a = 4.5, in the absence of thermal conduction, locally heats the plasma to excessively high temperatures in contradiction to the observed X-ray data. These results are shown in Fig. 4 along with the temporal evolution of the heating. These calculations show that one can achieve temperatures on the order of 104 eV during the deposition. The plasma is definitely in an over-heated state. Longer deposition times would only further maintain elevated temperatures. The only mechanism by which the plasma can rapidly dissipate this energy in our model is through bremsstrahlung radiation. However, the bremsstrahlung emission is proportional to n.nz~T\i2, i.e. it has a weak temperature dependence but varies as the square of the charge state, which for totally stripped iron would be Z = 26. For bremsstrahlung radiation to be an effective cooling mechanism over relatively short time periods would require a greater abundance of Fe ions than the 10-4 considered here. Therefore, although the bremsstrahlung losses may, on the whole, be large numerically relative to laboratory plasmas, it is insufficient to significantly reduce the plasma temperature. For the same set of initial conditions but with thermal conduction included, we note a gradual departure from the results due to direct beam heating. At 0.5 sec the effects of thermal conduction are noticeable. At about 4 sec after the onset of the beam the influence of thermal conduction is rather apparent. There is an overall smoothing and lowering of the peak temperature accompanied by the heating of a much larger region of the plasma. At 50 sec, well after the deposition ceases, the total region has been heated to about 800 eV. These results are shown in Fig. 5 along with a more complete time history of the heating. This type of behavior is characteristic of heat transport by thermal conduction. Since the conduction loss rate goes as TS/2, conduction losses also increase as the plasma becomes hotter. It is clear from these calculations that thermal conduction influences the shapes of the temperature profiles throughout the column.
103
-~ '"
~
10 2
I o \'""0------,L1.3;--------,1'7. 7-----;f-2.;:;"0----;;2':;-.4---2;;';.80----,'3.1 Z (10 9 em)
Fig. 4. Temperature profile as a function of altitude in the absence of thermal conduction.
Flare heating by energetic nonthermal electrons
717
----
10 3
1/ I{!
~
{ II III
~
10'
11/
/II
10;'-=-.0------,1-1.=-3-----,1-'-=,7----:!-2.""""0-----::2:'-:.4----:!-2.=-a----='3.1 Z (10' em)
Fig. 5. Temperature profile as a funclion of altitude with thermal conduction.
The high temperatures associated with the various heating mechanisms are also responsible for the appearance of high ionization stages of the Fe ions. The appearance and behavior of the line and continuum radiation characteristic of these keV plasmas is directly observable. For illustrative purposes we studied the temporal behavior of several selected lines integrated over the column and at a fixed point on the column located at Z = 1.8 x 104 km (2.2 x 104 km from the top). The heated electrons excite and ionize the different iron ionization stages and produce thermal bremsstrahlung, radiative and dielectronic recombination and line emission losses from the column. The ion dynamics of the iron plasma was modeled as a corona plasma. The neighboring ground ionization stages were connected to each other by the upward process of collisional ionization and by the downward processes of radiative and dielectronic recombination. Selected excited states of Fe(XXV), (XXIV), (XXIII), (XV) and (XIV) were included in the model. Each could be formed by collisional excitation from their respective ground states and by radiative recombination from the ground state ionization stage above it. They were depopulated by radiative decay to its own ground state. A more complete discussion of the ion dynamics and the rate coefficients is discussed fully elsewhere (JACOBS et al.(ll)). Since the ratio of spectral lines are often used to estimate the electron temperature, we will adopt this convention and present results for the behavior of the following line ratios: Fe(XXV) (300)/Fe(XXIV) (192), Fe(XXIV) (192)/Fe(XXIII) (133) and Fe(XV) (233)/Fe(XIV) (219). The quantity in parenthesis is the (approximate) wavelength of the transition in Angstrom units. The population densities of these levels were obtained as solutions to the system of rate equations given by eqn (10). Thus, line radiation is generated both in the ionizing (or heating) and recombining (or cooling) stages of the plasma's time evolution. The temporal evolution of the line ratios at Z = 1.8 X 104 km is shown on Fig. 6. As the beam impinges on the column, the lower ionization stages are rapidly depleted. This is seen on Fig. 6 by the broken line curve for the Fe(XV)/Fe(XIV) ratio. The higher ionization stages peak in the time interval between 5 to 10 sec as shown by the solid curves. This is in accord with the temperature profile as a function of time. As the beam is depositing energy, it is directly heating the region while thermal conduction is indirectly heating the surrounding region, as previously discussed. The temperature profile exhibits a peak value of about 1.75 keY at 10 sec, which is just at the end of the deposition, and then slowly decays to a value of about 800 eV at 50 sec, at which time the calculation was terminated. In order to interpret the results shown on Fig. 6 we must first understand the behavior of the individual lines. The relative abundance of Fe(XXV) at peak temperature is about three times greater than Fe(XXIV) and about eight times greater
QSRT Vol. 17. No. 6---C
J.
718
100
DAVIS
et al.
Fe XXIV 192 A/Fe XXIII 133 A
Fe XXV 300 ~/ Fe XXIV 192 A
"\ \
10-4
\ \
I l--I
Fe XV 233 A. Fe XIV 219 A-
I
do
25.5
51
TIME (sec.)
Fig. 6. Temporal evolution of spectral line ratios.
than Fe(XXIII). Hower the Is2p 3P21evel is not significantly populated from IS2S 3S level of Fe(XXV), i.e. the 300 'A transition. On the other hand, the temperature is sufficiently high to populate significantly the 2p 2P3/21evel of Fe(XXIV) and the 2S2p Ip level of Fe(XXIII) at 192 and 133 'A, respectively. In fact, the 192 and 133 'A transitions track the temporal evolution of the temperature, i.e. the plasma remains hot enough over the duration of the calculation to maintain the population in these levels. Once the deposition terminates, the plasma maintains itself for a brief duration and then enters a cooling phase. During this phase the plasma begins to cool and recombine, producing, at first, fractional reductions in the higher stages of ionization. As the plasma continues to cool the lower ionization stages are populated. This is seen on Fig. 6 from the behavior of the Fe(XV)/Fe(XIV) ratio. The production rate for these lines begins around 20 sec into the run (10 sec after the deposition has terminated) and steadily increases over the remainder of the calculation. This behavior is characteristic of recombining plasmas. Monitoring the radiative behavior of selected spectral lines for a fixed point on the flare is useful for qualitatively understanding the behavior of the flare plasma. In reality, however, observers cannot resolve fixed points on the flare, but rather make measurements that are spatially integrated over the entire flare or, at least, large parts of it. The integrated behavior of these line ratios is shown on Fig. 7. These results are easily interpretable with the aid of the temperature profiles and our previous remarks.
4. SUMMARY
The primary purpose of this investigation was to study the behavior of a fully ionized hydrogen plasma embedded with iron ions subjected to a stream of energetic nonthermal electrons. In the absence of convection the energy of the incident beam is partitioned among the system's degrees of freedom; electron thermal energy, ion'internal and thermal energies, line and continuum radiation. It has been shown that direct beam heating without thermal conduction would provide localized flare regions with very high temperatures, in contradiction to the observed facts. The inclusion of thermal conduction reduces the peak temperature to values more in agreement with the observations and heats a larger region of the flare. For a hydrostatic atmosphere the deposition, for both incident fluxes, peaks at an electron density in the neighborhood of 1011 electrons per cm3. It has also been shown that, for two kinds of initial power law distributions in a hydrostatic atmosphere, the emitted bremsstrahlung is not describable simply in terms of thick or thin target results but falls somewhere in between. Finally, we considered the variation of spectral line intensities as a function of time at a fixed position and integrated along the flare column. It was shown that the peak temperature was
719
Flare heating by energetic nonthermal electrons
0
( 1)
XXTIZ: (192 A) 0
XXllI(133A)
10- 2 0
(2)
XXJZ: (300 A)
A)
XXIil ( 192
(f)
~
I-
cr
(3)
lJ.J Z
A) A)
:x::x:I'i[
(192
xxnz:
(255
...J
10- 3
10- 5 L...-_L----<_-...J._--l._---'-_--'---""-'-_--'-_-'-------'
o
10
20
30
40
50
TIME (SEC)
Fig. 7. Temporal evolution of space integrated spectral line ratios.
sufficient to significantly populate some of the higher stages of ionization, In particular, Fe(XXV), Fe(XXIV) and Fe(XXIII). The temporal history of several spectral line ratios was presented for a fixed spatial position and spatially integrated along the plasma column, during and well after the deposition. During both the heating and cooling phases the plasma was emitting line radiation. The X-ray line emission peaked at about 10 sec, which is just the duration of the deposition. After that its behavior was typical of any high temperature recombining plasma. The results presented here are indicative and illustrative of the behavior of the flare plasma due to the deposition of energetic nonthermal electrons in the absence of hydrodynamics. Our remarks were confined to the deposition, thermal conduction heating and radiative losses from the Fe ions due to electrons injected into the plasma with a power law distribution function. We could just as easily have considered the injected electrons to have a Maxwellian, double-peaked Maxwellian or various combinations of Maxwellian and power law distribution function. The effects due to these types of distribution will be the subject of a forthcoming paper. There are several other variables that are essential to any realistic model of flare heating. They are the
720
1. DAVIS et al.
total energy content and duration of the burst and whether there are several short or one long burst or a combination of these. Questions of this nature are best answered by appealing to the data and then incorporated into our model as known parameters. Another major unknown in our understanding is the role of collective interactions of the electron beam with the flare plasma. The principal effect of most collective plasma processes is to drive the system to a state of marginal stability. That is, the system is at the threshhold from which it may be stabilized, excited, restabilized, and then reexcited both in space and time. This behavior will also affect the anomalous transport resulting from the collective effects. For a discussion of these processes the reader is referred to the work of P APAOOPOULOS.o 2 ) It is clear that plasma instabilities may play a more important role than has hitherto been considered. The collective plasma effects may also play an important role in the overall energy balance of the source. However, we must turn first to our most pressing problem: the incorporation into our model of the gas kinetic effects resulting from the deposition. Preliminary calculations show that the hydrodynamics has a profound influence on the overall behavior of the flare plasma. There is considerable mass redistribution, shock wave heating, and changes in the equation of state, to mention a few. The details of these calculations will appear in a subsequent paper. Acknowledgements-The authors would like to thank Dr. JOHN ROGERSON for computational assistance and Dr. V. L. JACOBS for reviewing the manuscript. This work has been supported by the E. O. HULBERT Center for Space Research, NRL, as part of the ATM Data Analysis Program funded by NASA DPR-60404-G.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
REFERENCES C. C. CHENG, Solar Phys. 22, 178 (1972). F. M. STRAUSS and M. D. PAPAGGIANIS, Appl. J. 164,369 (1971). J. L. CULHANE, J. F. VESECKY and K. 1. H. PHILLIPS, Solar Phys. IS, 394 (1970). J. C. BROWN, Solar Phys. 29, 421 (1973). S. I. SYROVATSKII and O. P. SHMELEVA, Sov. Astron. 16, 273 (1972). 1. H. JACOB, Phys. Rev. 8, 226 (1973). H. A. BETHE, Handbuch der Physik, Vol. 24(2), p. 273. Springer, Berlin (1930). S. I. BRAGINSKll, Reviews of Plasma Physics, Vol. I, p. 205. Consultants Bureau, New York (1965). S. R. KANE and K. A. ANDERSON, Appl. J. 162, 1003 (1970). R. P. LIN and H. S. HUDSON, Solar Phys. 17,412 (1971). V. L. JACOBS, 1. DAVIS, P. C. KEPPLE and M. BLAHA, Appl. 1. (Jan. 1977). K. PAPADOPOULOUS, Phys. Fluids 18, 1769 (1975); 1. Geophys. Res. 79, 674 (1974).