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ISSN 1063-7737, Astronomy Letters, 2007, Vol. 33, No. 1, pp. 54­62. c Pleiades Publishing, Inc., 2007. Original Russian Text c S.A. Bogachev, B.V. Somov, 2007, published in Pis'ma v Astronomicheski Zhurnal, 2007, Vol. 33, No. 1, pp. 62­71. i

Formation of Power-Law Electron Spectra in Collapsing Magnetic Traps
S. A. Bogachev
1

1*

and B. V. Somov

2**

Lebedev Institute of Physics, Russian Academy of Sciences, Leninski i pr. 53, Moscow, 117924 Russia 2 Sternberg Astronomical Institute, Universitetski i pr. 13, Moscow, 119992 Russia
Received May 25, 2006

Abstract--The energy distribution of the fast electrons captured into a collapsing magnetic trap in the solar corona is calculated as a function of the trap length and diameter. It is shown that if the electrons injected into the trap have a power-law spectrum, then their spectrum remains a power-law one with the same slope throughout the acceleration process for both the Fermi and betatron acceleration mechanisms. For electrons with a thermal injection spectrum, the model predicts two types of hard X-ray sources, thermal and nonthermal. Thermal sources are formed in traps dominated by the betatron mechanism. Nonthermal sources with a power-law spectrum are formed when electrons are accelerated by the Fermi mechanism. PACS numbers : 95.10.Ce DOI: 10.1134/S1063773707010070 Key words: Sun, solar flares, magnetic reconnection, particle acceleration, X-ray emission, gammaray emission.

INTRODUCTION The acceleration of particles in a cosmic plasma is a classical problem of astrophysics that, as applied to solar flares, is accessible to a comprehensive study. It is well known from observations (see, e.g., Hudson and Ryan 1995; Miroshnichenko 2001; Aschwanden 2002) that a large number of protons and electrons are accelerated during flares in the solar atmosphere to energies that are many orders of magnitude higher than the thermal particle energies in the solar corona. The deceleration of these particles in the plasma of the solar atmosphere is accompanied by intense hard X-ray, gamma-ray, and radio bursts. The particle acceleration region coincides with the magnetic reconnection region in the solar corona and the adjacent cusps--helmet-like structures located above flare loop arcades. Here, collapsing magnetic traps are produced by reconnection. Two effects increase the energy of the charged particles captured into a collapsing trap--Fermi acceleration as the length of the trap decreases (Somov and Kosugi 1997) and betatron acceleration as it contracts in the transverse direction. The additional increase in energy caused by betatron acceleration is exactly offset by the decrease in the time of particle confinement in the trap. As a result, the particle energy at the time of its escape remains the same as
* **

E-mail: bogachev@sci.lebedev.ru E-mail: somov@sai.msu.ru

it would be during the escape from a collapsing trap without contraction (Somov and Bogachev 2003). Previously (Bogachev and Somov 2005), we developed a trap model in which both acceleration mechanisms were simultaneously taken into account in the collisionless approximation. The model can explain the RHESSI satellite observations that have revealed hard X-ray (HXR) sources in the solar corona whose emission is more intense than the HXR emission from the chromosphere. These observations are indicative of a high efficiency of electron capture and acceleration in collapsing solar flare traps. The radio emission from such traps must also have the distinctive properties that allow the dominant acceleration mechanism in them to be judged (see Somov et al. 2005). The physics of particle acceleration and confinement in rapidly collapsing magnetic traps has a number of important features that distinguish it from the physics of particle confinement in stationary traps, for example, in coronal magnetic loops. In stationary traps, where there are no regular particle acceleration and precipitation mechanisms, such stochastic effects as the Coulomb particle scattering into the loss cone and the particle scattering during the wave­ particle interaction play the dominant role. Models of this type are very popular in solar physics and are commonly used to interpret type IV solar radio bursts and as the stochastic particle acceleration mechanism. Regular acceleration mechanisms,
54


FORMATION OF POWER-LAW SPECTRA

55

more specifically, the Fermi and betatron mechanisms, which govern the particle pitch angle and determine the time the particle falls into the loss cone and the energy of its escape from the trap, dominate in collapsing traps. The stochastic effects play a secondary role in this case. In recent years, reliable data on the intensities and spectra of coronal and chromospheric X-ray sources have been obtained. Of particular interest are the RHESSI satellite measurements during limb flares, when the chromospheric HXR sources at the footpoints of flare loops and the coronal HXR sources above the solar limb can be studied separately. In these studies performed by Lin et al. (2003), Balciunaite et al. (2004), and several other authors, coronal sources with power-law spectra with an index of 5­ 7 were observed. The intensity of a coronal source in the energy range 10­30 keV at the Earth's orbit was 10-1 ­102 photons cm-2 s-1 . Despite this obvious progress in observations, the mechanism that is responsible for the formation of coronal HXR sources and that allows the experimental RHESSI data to be interpreted remains unclear. The model of a collapsing trap offers new possibilities for explaining the results of these observations. Based on this model, we can explain the power-law shape of the HXR spectrum, calculate the emission intensity, and determine the number of electrons required for the formation of an observable HXR flux at the Earth's orbit. The main goal of this paper is to present and substantiate these assertions. Our results can be used to interpret not only HXR observations, but also gamma-ray and radio observations of solar flares. THE FORMATION AND COLLAPSE OF A TRAP The appearance of collapsing magnetic traps in the solar corona is related to magnetic reconnection (Somov and Kosugi 1997). Magnetic traps are formed from flux tubes of reconnected magnetic field lines with their footpoints located in the photosphere. Such a flux tube confines particles, since the magnetic field strength at its footpoints, Bm ,is largerthan that at its top, B0 . A high-temperature turbulent current sheet (Somov 1992, 2000) that accelerates the particles and heats up the plasma to anomalously high temperatures (T 100 MK) is located in the magnetic reconnection region. The preaccelerated particles are injected into the trap near its top and are confined in it by magnetic mirrors. The electrons decelerate appreciably in the trap through Coulomb collisions and produce HXR bremsstrahlung, which is observed as a coronal source. As the electrons escape from the
ASTRONOMY LETTERS Vol. 33 No. 1 2007

trap, they precipitate into the chromosphere, where they also produce HXR bremsstrahlung. Let us simplify the actual configuration. We will assume that the electrons are confined in an axially symmetric tube of finite length L with a field B0 (at the initial time) that is minimal at the center of the trap and increases at its edges to Bm ; Bm /B0 is called the mirror ratio. This simplification is admissible. The Larmor radius of an electron depends on its transverse momentum and the magnetic field strength in the trap. It does not exceed 20 cm for nonrelativistic electrons at B < 100 G, which is much smaller than the characteristic length scale on which the magnetic field varies. The drift gradient displacements of the electrons in their acceleration time in the trap at B < 100 G and a radius of curvature of the magnetic field lines Rc 108 cm are less than one kilometer. This is much smaller than the presumed sizes of coronal magnetic traps. Under these conditions, the curvature of the field lines is unimportant. We will describe the transverse contraction of the tube by the quantity b(t) = B (t)/B0 . Its value changes from b = 1 to bm = Bm /B0 , at which the field at the center of the trap becomes equal to the field in the magnetic mirrors and the trap ceases to confine the particles. In place of the trap length L, we introduce the parameter l(t) = L(t)/L0 that decreases from one to zero. The electron energy is increased by the Fermi mechanism as the length of the trap decreases and by the betatron mechanism as it contracts. If the trap collapses simultaneously in both (longitudinal and transverse) directions, both mechanisms are at work. THE SPECTRUM OF TRAPPED ELECTRONS Let us consider a trap with decreasing length and thickness. We are interested in how the energy distribution of the trapped electrons changes with time. Previously (Bogachev and Somov 2005), we showed that if N0 nonrelativistic electrons with a distribution f0 (K),where K is the particle kinetic energy, fall into the trap at the injection time, then, as the trap contracts to the sizes corresponding to given l and b, the number of electrons in it decreases to l bm - b , (1) N = N0 1+ (bm - b)l2 while their distribution function normalized as dN = 2N0 f (K) KdK sin d takes the form l f (K,) = f0 (KA ), b (2)


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BOGACHEV, SOMOV
2 2

where 1+(bl - 1) cos A = b is a function of the dimensionless parameters l and b and pitch angle . Since, according to Eq. (1), the number of electrons captured into a collapsing magnetic trap approaches zero as l 0, we introduce their energy spectrum, f (K), normalized in such a way that (3) dN = 4N f (K) KdK, where N is the number of particles in the trap (1). Let us write the number of electrons (3) in the energy range dK as an integral of the distribution f (K,) over the pitch angle outside the loss cone: dN = 2N0 K
-esc



1-b/bm

â
0

1+ x2 (bl2 - 1) b

-

dx.

If the trap contracts in the transverse direction, while its length remains constant, then in this special case corresponding to betatron acceleration,
C = lim C (b, l = 1) = C0 bm- bbm 1.5

.

Let us now analyze the spectrum of the trapped electrons if it was a thermal one with temperature T0 at the injection time: f0 (K0 ) = 1 4 2 k
3 3 T0

exp -

K0 . kT0

(8)

f (K,)sin d,
esc

(4)

Substituting (8) in (5) yields N0 l 1 f (K) = N b 4
1-b/bm

2
3 k3 T0

(9)

where the loss cone is defined by cos esc = 1 - B/Bm =
/2

1 - b/bm .

â
0

exp -

KAx dx. kT0

Equating (3) and (4) using (2), we obtain f (K) = N0 l Nb
esc

f0 (KA )sin d

We will take into account Eqs. (1) and (6) for the number of particles N and the function Ax . After their substitution in (9), we obtain f (K) = â (5) â 1+(bm - b)l2 1 4 b bm - b K 2 exp - 3T 3 bkT0 k 0 exp -
0

or, after the change of variable x cos ,
1-b/bm

(10)

N0 l f (K) = Nb
0

f0 (KAx )dx,

1-b/bm

where 1+ x2 (bl2 - 1) (6) b is a function similar to A , but it depends not on the pitch angle , but on the integration variable x. Equation (5) describes the energy distribution of the trapped electrons for any injection spectrum f0 (K0 ), for example, a power-law or Maxwellian spectrum. In the former case, Ax =
- f0 (K0 ) = C0 K0 .

K bl2 - 1 2 x dx. kT0 b

The integral on the right-hand side of Eq. (10) can be reduced to the error integral erf(x) (see Jahnke et al. 1960). Therefore, the sought-for electron distribution function takes the form C 2 +1 1 1 (11) f (K) = bkT0 C 4 K â exp - where C= (1 - b/bm )(bl2 - 1). Equation (11) shows that if the electron energy distribution at the injection time was thermal, then it does not remain Maxwellian, but is modified differently for the betatron and Fermi mechanisms during the acceleration in a collapsing trap.
ASTRONOMY LETTERS Vol. 33 No. 1 2007

(7)

K erf C bkT0

K bkT0

,

Substituting (7) in (5) yields f (K) = C K
-

.

Thus, the power-law electron spectrum in a collapsing traps remains a power-law one with a constant index , while the coefficient C increases as C (b, l) = C0 1+(bm - b)l b bm - b
2


FORMATION OF POWER-LAW SPECTRA

57

Let us again consider the two cases separately. If the trap contracts in the transverse direction, then the electrons are accelerated by the betatron mechanism until the field in the trap becomes equal to the field in the magnetic mirrors. Calculating the corresponding limit b bm for Eq. (11) at l = 1, we obtain an energy distribution of the trapped electrons in the form 2 1 lim f (K) = (12) 3 (b T )3 bbm 4 k m 0 K . â exp - kbm T0 Equation (12) describes a Maxwell distribution with temperature bm T0 . Thus, in a contracting trap with initial mirror ratio bm , the spectrum of the trapped electrons (and, hence, their bremsstrahlung spectrum) changes in the following sequence: thermal with temperature T0 nonthermal thermal with temperature bm T0 . Note that the kinetic energy of a particle under betatron acceleration is distributed between the degrees of freedom not uniformly, but in the following proportions. As the trap contracts, the longitudinal particle temperature does not change and remains equal to (1/3)T0 , while the transverse temperature increases from (2/3)T0 to (bm - 1/3)T0 . This leads to the result obtained above for the angleintegrated energy spectrum of the accelerated electrons. If, however, the length of the trap decreases in the longitudinal direction, but it does not contract transversely, then the particles are accelerated in it by the Fermi mechanism. Setting b = 1 in Eq. (11), we see that the coefficient C in this case becomes imaginary, but the distribution f (K), of course, remains real. As l 0, this distribution of the trapped electrons tends to 1 1 1 (13) lim f (K) = l 0 kT0 bm - 1 4 K â exp - K erfi kT0 bm - 1 bm K kT0 ,

­2 a ­3 ­4 ­5 ­6 ­7 c b

log f

1

10

100 K, keV

1000

10 000

Fig. 1. Energy spectrum of the captured electrons in a trap with mirror ratio bm = 100: a is the initial thermal distribution with temperature T = 108 K; b is the final particle distribution as the trap length decreases--Fermi acceleration; the dashed straight line indicates the slope of the power-law segment of the spectrum; c is the final thermal electron distribution as the trap contracts transversely--betatron acceleration.

where erfi(x) is an imaginary value of the error function (Jahnke et al. 1960):
x

2 erfi(x) = -ierf (ix) =
0

exp t2 dt.

In a trap with the dominant Fermi mechanism, the electron spectrum (and, as a result, their bremsstrahlung spectrum) changes in a different way than in a trap with betatron acceleration. This change
ASTRONOMY LETTERS Vol. 33 No. 1 2007

passes through the following stages: thermal spectrum with initial temperature T0 nonthermal spectrum nonthermal spectrum with effective temperature Teff = T0 (bm +2)/3 (see Bogachev and Somov 2005). The effective temperature Teff characterizes the mean kinetic energy of the trapped electrons whose number, according to Eq. (1), approaches zero as l 0.Since, by definition, the mirror ratio bm is larger than one, the temperature of the confined particles as they are accelerated by the betatron mechanism, bm T0 , is higher than the effective temperature of the particles accelerated by the Fermi mechanism. The results of our calculations for traps of both types are presented in Fig. 1. Curve a indicates the Maxwellian spectrum at the time of electron injection into the trap, curve b indicates the spectrum of the accelerated electrons for l 0 calculated using Eq. (13), and curve c indicates the spectrum for b bm calculated using Eq. (12). The distribution formed when thermal electrons are accelerated by the Fermi mechanism differs greatly from the Maxwellian distribution in the energy range from 20 to -200 keV. Here, as the trap length decreases, a rectilinear segment of the spectrum in which the electrons have a power-law distribution in kinetic energy is formed: f K- , where is the spectral slope that depends on the trap mirror ratio bm . Figure 2 shows the electron spectra under Fermi acceleration in traps with various mirror ratios bm . In all three cases, the spectrum is a power-law one, but with a different slope that depends on the mirror ratio: the spectra are harder in traps with larger bm . As bm increases, the extent of the region of the power-law


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BOGACHEV, SOMOV

a ­4 log f b c

spectrum becomes a power-law one changes only slightly with bm and is approximately equal to 15­ 20 keV. THE EMISSION MEASURE OF HXR BREMSSTRAHLUNG The HXR emissivity of a magnetic trap is characterized by the emission measure

­5

V

30

40

50 60 K, keV

70 80 90 100

ME =
0

n1 n2 dV .

Fig. 2. Shape of the power-law segment of the spectrum as a function of the trap mirror ratio bm : a, bm = 25; b, bm = 50; c, bm = 100.

6 5 4 a 3 b 2 c ME/ME0 1 1.0 0.8 0.6 I (b) a 60 40 20 0 0.2 b c 0.4 0.6 (b ­ 1)/(bm ­ 1) 0.8 1.0 0.4 0.2 0 ()

It depends on the densities of the interacting particles n1 (electrons) and n2 (protons) and on the volume V of the emitting region. Let us define ME as a function of the trap length and its diameter. Integration over the trap volume yields ME = Ne np , (14) where Ne is the total number of electrons in the trap and np is the proton density. The number of particles is defined by Eq. (1), while the formula for their density was derived previously (Bohachev and Somov 2005): b bm - b . (15) n = n0 1+(bm - b)l2 Substituting (15) and (1) in (14), we obtain ME = ME0 bl(bm - b) , 1+(bm - b)l2 (16)

100 80

where ME0 is the initial emission measure equal to the product of the proton density and the number of electrons at the injection time: ME0 = np0 Ne0 . If the trap contracts only in one direction, then Eq. (16) is simplified: for the Fermi acceleration ME = ME0 l(bm - 1) , 1+(bm - 1)l2 (17)

Fig. 3. Emission measure from a collapsing trap as a function of its contraction in the longitudinal (a) and transverse (b) directions. The calculations were performed for three mirror ratios: a, bm = 10; b, bm = 25,and c, bm = 100. The vertical dashed lines indicate the times the HXR flux reaches its maximum.

for the betatron acceleration b(bm - b) . ME = ME0 1+(bm - b)

(18)

spectrum also increases toward the high energies: the spectrum is cut off at energies 130­150 keV for bm = 25 and extends to 0.5­1.0 MeV for bm = 100. The minimum electron energy starting from which the

The results of our calculations using Eqs. (17) and (18) are presented in Fig. 3. They show that the increase in the emission measure in traps with betatron acceleration is larger than that in traps with Fermi acceleration. The electrons inside betatron traps produce more intense bremsstrahlung than the electrons in Fermi traps.
ASTRONOMY LETTERS Vol. 33 No. 1 2007


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59

1.0 () 0.8 0.6 HXR-flux, â 10­8 photons s­1 keV­1 0.4 0.2 0 1.0 8 (b) 6 a b c 0.8 0.6 l 0.4 0.2 0 a

4

2

b c

0

0.2

0.4 0.6 (b ­ 1)/(bm ­ 1)

0.8

1.0

Fig. 4. HXR Flux from the trap in three energy ranges: a,25 keV; b,50 keV; and c, 75 keV, as a function of the trap contraction in the longitudinal (a) and transverse (b) directions.

THE SPECTRUM AND INTENSITY OF THE HXR EMISSION Thus, we have found the spectrum of the electrons f (K) captured into a collapsing trap. Let us calculate the spectrum of their HXR bremsstrahlung (X ).In the energy range [X ,X + dX ], (X )dX photons of energy X are produced in one second: dN
X

where np is the proton density in the emission region, ve is the electron velocity, and is the differential bremsstrahlung cross section. Expressing the number of photons in terms of their emission spectrum (X ) using Eq. (19) and the number of electrons in terms of their distribution f (K) using Eq. (3), we obtain (X ) = 4np Ne


= (X )dX . â


(20)

The total energy emitted from the trap in this energy range is dI
X

ve (K) (K,X )f (K) KdK.

= (X )X dX .

(19)

X

The number of emitting electrons, dNK , is related to the number of photons produced by them, dNX ,by the formula dNX =n dX
p
X

ve (K) (K,X )dNK (K),

The most intense nonthermal X-ray emission from solar flares is observed in the energy range 20­ 100 keV and is produced mostly by electrons of the same energies. Therefore, it will suffice to give nonrelativistic formulas. Let us express the electron velocity in terms of their kinetic energy: ve (K) = 2K/me . (21)

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Vol. 33 No. 1 2007


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BOGACHEV, SOMOV

HXR flux (the number of photons per second) from the trap at the burst peak normalized to the number of injected electrons bm = 3 25 keV 50 keV 75 keV 25 keV bm = 10 50 keV 75 keV 25 keV bm = 100 50 keV 75 keV

Flux from betatron trap 4.9 â 10
-9

5.2 â 10

-10

8.9 â 10

-11

6.1 â 10

-8

1.5 â 10

-8

5.8 â 10

-9

9.0 â 10

-7

3.5 â 10

-7

2.0 â 10

-7

Flux from Fermi trap 1.6 â 10
-9

1.0 â 10

-10

1.4 â 10

-11

8.7 â 10

-9

1.5 â 10

-9

4.6 â 10

-10

5.3 â 10

-8

1.7 â 10

-8

8.3 â 10

-9

Flux ratio 3.1 5.2 6.4 7.0 10.0 12.6 17.0 20.6 24.1

For the bremsstrahlung cross section in the nonrelativistic approximation, we will use the formula (Berestetskii et al. 2001) (K,X ) =
2 8r0 me c2 1+ ln 3 KX 1-

we obtain np0 Ne (X ,l,b) = CX X
1-b/bm 0

l bm - b 1+(bm - b)l 1+ 1-
2

(24)

1 - X /K 1 - X /K

, (22)

where r0 is the classical electron radius (2.82 â 10-13 cm) and 1/137 is the fine-structure constant. Substituting (21) and (22) in Eq. (20) yields (X ) = CX


â

X

f0 (KAx )ln
0

1 - X /K 1 - X /K

dxdK.

np Ne X dK,

(23)

â

X

f (K)ln

1+ 1-

1 - X /K 1 - X /K

where C
X

= 4

2 8r0

3 me /2

me c2
1/2 -1

= 1.86 â 10-14 cm3 keV

s

.

Equation (23) establishes the relationship between the "instantaneous" distribution of the trapped electrons and their bremsstrahlung spectrum. If f (K) is a power-law distribution with index and since the slowly changing logarithmic factor may be considered constant compared to the exponential, the produced bremsstrahlung also has a power-law spectrum with slope +1. This is a well-known prediction of the thin-target theory for the emission of electrons in a tenuous plasma. Expressing Ne , f (K),and np in terms of their values at the injection time using Eqs. (1), (5), and (15),

Thus, the problem of calculating the trap emission spectrum from a given injection spectrum has been solved. The solution of the inverse problem, i.e., reconstructing the injection spectrum f0 from the observed emission spectrum , is more complex, since Eq. (24) for F0 (K0 ) is a second-order integral equation. The results of our calculations using Eq. (24) are presented in Fig. 4. The calculations were performed for a thermal injection spectrum with T = 108 K. According to these calculations, if 1036 electrons have been injected into the trap, then the trap produces 1028 ­1029 (depending on the acceleration mechanism) photons with energy 25 keV per second at the emission maximum, which corresponds to their flux at the Earth's orbit, 101 ­102 photons cm-2 . This value agrees with the RHESSI measurements in this spectral range. At a lower or higher photon flux, the number of trapped electrons would increase or decrease proportionally. The intensity of the HXR bremsstrahlung from a trap is also affected by its mirror ratio. Information about the dependence of the flux on the mirror ratio, energy range, and acceleration mechanism is given in the table. At the same number of injected electrons, traps with betatron acceleration produce several times more intense HXR bursts than traps with Fermi acceleration.
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02/07/23, 00:21:42.00 102 Photons ðm­2 s­1 keV­1 101 1 10 10 10
­1

()

10

2

(b)

101 1 10 10 20 30
­1

­2

­2

­3

10

40 50 10 10 Energy, keV

­3

20

30

40 50

Fig. 5. Experimental (a) and theoretical (b) spectra for the coronal HXR source observed in the July 23, 2002 flare. The experimental data were obtained by the RHESSI satellite (Lin et al. 2003).

The higher-energy emission in a trap can be delayed relative to the lower-energy emission. If the injection spectrum is thermal with T = 108 K, then the delay of the emission in the 75-keV range relative to that in the 25-keV range is l 0.05, i.e., about 5% of the trap lifetime. For betatron acceleration, our calculations yield similar delays. It is of interest to compare the experimental HXR spectrum with the theoretical spectrum calculated in the model of a collapsing trap. Lin et al. (2003) reported the results of their study of the nonthermal HXR source observed by the RHESSI satellite in the solar corona during the July 23, 2002 flare. The HXR spectrum was a powerlaw one with a knee at 20 keV and slopes of 7 and 5 to the right and to the left of the knee, respectively. Using Eq. (24), we can interpret this spectrum as the spectrum of the bremsstrahlung produced by an ensemble of 4 â 1029 electrons with a power-law injection spectrum and slope e = 7 captured and accelerated in a collapsing coronal trap with proton density np0 = 109 cm-3 and initial mirror ratio bm = 20. Both spectra, the theoretical spectrum corresponding to these parameters and the experimental spectrum, are shown in Fig. 5. They do not coincide at energies below 20 keV. The experimental spectrum in this energy range is harder, probably because the low-energy electrons decelerate in the background plasma. This effect is disregarded in the collisionless model. Note that the interpretation of the observations in the model of a collapsing trap is not unequivocal. The model predicts most reliably the slope of the injection spectrum for a power-law emission spectrum. The spectral slopes
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coincide in this case. The intensity of the observed emission depends on three parameters: it is directly proportional to the number of injected electrons and the initial proton density and depends in a complex way on the mirror ratio in the trap. CONCLUSIONS The particles in the solar corona are accelerated during flares in two stages. During the first stage, the electrons and ions are accelerated by the electric field in a high-temperature turbulent current sheet. During the second stage, the particles are captured into collapsing magnetic traps, where their kinetic energy additionally increases through the first-order Fermi mechanism and the betatron mechanism. In this paper, we investigated the second stage of electron acceleration and showed that it is highly efficient. We calculated the (angle-integrated) energy spectrum of the electrons captured into a collapsing trap as a function of its length and diameter. The electrons with a power-law energy distribution at the time of injection from a high-temperature turbulent current sheet into the trap were shown to be additionally accelerated by the two mechanisms. In this case, their spectrum remains a power-law one with the same slope throughout the acceleration, shifting toward the high energies. The electrons with a thermal injection spectrum in traps with the dominant betatron acceleration heat up strongly. The angle-integrated energy spectrum remains Maxwellian, but with temperature bm T0 , where T0 is the initial particle temperature (i.e., the electron temperature inside the highly turbulent current sheet) and bm is the mirror ratio in the trap.


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BOGACHEV, SOMOV 3. V. B. Berestetskii , E. M. Lifshitz, and L. P. Pitaevskii , Quantum Electrodynamics (Fizmatlit, Moscow, 2001) [in Russian]. 4. S. A. Bogachev and B. V. Somov, Astron. Lett. 31, 537 (2005). 5. H. Hudson and J. Ryan, Ann. Rev. Astron. Astrophys. 33, 239 (1995). 6. E. Jahnke, F. Emde, and F. Losch, Tables of Higher Functions (McGraw-Hill, New York, 1960; Nauka, Moscow, 1968). 7. R. P. Lin, S. Krucker, G. D. Holman, et al., in Proceedings of the 28th International Cosmic Ray Conference, Ed. by T. Kajita, Y. Asaoka, A. Kawachi, et al., (Univ. Acad. Press Inc., Tokyo, 2003), 3207. 8. L. I. Miroshnichenko, Solar Cosmic Rays (Kluwer, Dordrecht, 2001). 9. B. V. Somov, Physical Processes in Solar Flares (Kluwer, Dordrecht, 1992). 10. B. V. Somov, Cosmic Plasma Physics (Kluwer, Dordrecht, 2000). 11. B. V. Somov and S. A. Bogachev, Astron. Lett. 29, 621 (2003). 12. B. V. Somov and T. Kosugi, Astrophys. J. 485, 859 (1997). 13. B. V. Somov, T. Kosugi, I. V. Oreshina, et al., Adv. Space Res. 35, 1712 (2005).

In traps with the predominant Fermi acceleration, the thermal electron injection spectrum during the acceleration transforms into a nonthermal one with an effective temperature lower than the electron temperature during betatron acceleration. In the energy range 20­200 keV, the particles have a power-law spectrum. Thus, the existence of two types of coronal HXR sources is predicted, more specifically, sources with a thermal spectrum formed in traps with betatron acceleration and sources with a nonthermal powerlaw spectrum corresponding to Fermi acceleration. We calculated the HXR bremsstrahlung spectra of the electrons captured into collapsing traps. These spectra are of considerable interest in interpreting the RHESSI observations of coronal HXR sources during solar flares. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project no. 04-02-16125a). REFERENCES
1. M. J. Aschwanden, Particle Acceleration and Kinematics in Solar Flares (Kluwer, Dordrecht, 2002). 2. P. Balciunaite, S. Krucker, and R. P. Lin, Am. Astron. Soc. Meet. 204, abstract No. 54.07 (2004).

Translated by V. Astakhov

ASTRONOMY LETTERS

Vol. 33 No. 1

2007