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, 2007, 33, 1, . 62-71

523.985


c 2007 .
1 2

. . 1* , . .

2**

. .. , . .. ,
25.05.2006 .

, , . , , , . - . , . . : , , , , , -. FORMATION OF POWER-LAW ELECTRON SPECTRA IN COLLAPSING MAGNETIC TRAPS, by S. A. Bogachev and B. V. Somov. 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 in which the betatron mechanism dominates. Nonthermal sources with a power-law spectrum are formed when electrons are accelerated by the Fermi mechanism. PACS numbers : 95.10.Ce Key words: Sun, solar flares, magnetic reconnection, particle acceleration, X-ray emission, gamma-ray emission.

- , , , . (., , , , 1995; , 2001; , 2002), , .
* **

, , - . ""- - , . . - (, , 1997) . 62

: bogachev@sci.lebedev.ru : somov@sai.msu.ru




63

, , . , (, , 2003). (2005) , . RHESSI, (HXR) , HXR- . . , (. ., 2005). , , . , , , -. , . , , - . . . , RHESSI , HXR-, , HXR- . , . (2003), . (2004), ,
33

5-7. 10-30 , , 10-1 -102 2 . , , HXR- RHESSI. . HXR-, , HXR- . . HXR-, - . (, , 1997). , . , (Bm ) , (B0 ). (, 1992, 2000), (T 100 MK). . HXR-, . , HXR-. . , L B0 ( ), Bm ; Bm /B0 . .
1 2007


64

,

B < 100 20 , , . B < 100 Rc 108 . , . . b(t) = B (t)/B0 . b = 1 bm = Bm /B0 , , . L l(t) = = L(t)/L0 , . , - . ( ), . , . , . (2005) , N0 f0 (K), K - , , l b, l bm - b , (1) N = N0 1+ (bm - b)l2 , dN = 2N0 f (K) KdK sin d, l f (K,) = f0 (KA ), b 1+(bl2 - 1) cos2 b - l b . A = (2)

, (1), l 0, , f (K), , dN = 4N f (K) KdK, (3) N - (1). (3) dK f (K,) - : dN = 2N0 K
-esc

f (K,)sin d,
esc

(4)

cos esc = 1 - B/Bm =
/2

1 - b/bm .

(3) (4), (2), N0 l f (K) = Nb
esc

f0 (KA )sin d

, x cos ,
1-b/bm

N0 l f (K) = Nb
0

f0 (KAx )dx,

(5)

1+ x2 (bl2 - 1) (6) b - , A , - , x. (5) f0 (K0 ), . Ax =
- f0 (K0 ) = C0 K0 .

(7)

(7) (5), f (K) = C K
-

.

, , C (b, l) = C з
0 1-b/bm 0

1+ (bm - b)l2 з b bm - b
-

1+ x2 (bl2 - 1) b
33

dx.



1

2007




65

, , , ,
= lim C (b, l = 1) = C0 bm- bbm 1.5

.

, T0 : f0 (K0 ) = 1 4 2 k
3 3 T0

exp -

K0 . kT0

. , , . b bm (11) l = 1, 2 1 lim f (K) = з (12) bbm 4 k3 (bm T0 )3 з exp - K . kbm T0

(8)

(8) (5), f (K) = з
0

N0 l 1 N b 4 exp -

2
3 k3 T0

з

(9)

1-b/bm

KAx dx. kT0

(1) (6) N Ax . (9) f (K) = з з
0

1+ (bm - b)l2 1 з 4 b bm - b K 2 exp - з 3T 3 bkT0 k 0 exp - K bl2 - 1 2 x dx. kT0 b

(10)

1-b/bm

(10) erf(x) (. ., 1968). C 2 +1 1 1 з (11) f (K) = bkT0 C 4 K K erf C з exp - bkT0 C= (1 - b/bm )(bl2 - 1). K bkT0 ,

(12) bm T0 . , bm ( , ) : T0 bm T0 . , , . (1/3)T0 , (2/3)T0 (bm - 1/3)T0 . . , , . (11) b = 1, , C , f (K), , . l 0 1 1 1 з (13) lim f (K) = l 0 kT0 bm - 1 4 K з exp - K erfi kT0 bm - 1 bm K kT0 ,

erfi(x) - ( ., 1968, . 72):
x

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

exp t2 dt.

(11) , , , , .
5 33

, , (, , ) , , :
1 2007


66
-2 a -3 -4 lg f -5 -6 -7 c

,

a -4 b c lg f -5

b

1

10

100 K, ЭЧ

1000

10 000

30

40

50 60 K, ЭЧ

70 80 90 100

. 1. bm = 100: a - T = 108 ; b - - ; ; c - - .

. 2. bm : a - bm = 25, b - bm = 50, c - bm = 100.

T0 Teff = T0 (bm +2)/3 (. , , 2005). Teff , l 0 (1). bm , , bm T0 , , , . . 1. a , b - l 0, (13), c - b bm , (12). , , 20-200 . , : f K- , - , bm . . 2 bm . , , :

Д bm . bm : bm = = 25 "" 130-150 , bm = 100 0.5-1.0 . , , bm 15-20 . HXR- HXR
V

ME =
0

n1 n2 dV .

n1 () n2 () V . ME . (14) ME = Ne np , Ne - , np - . (1), (2005): b bm - b . (15) n = n0 1+(bm - b)l2 (15) (1) (14), bl(bm - b) , ME = ME0 1+(bm - b)l2
33 1 2007

(16)






67

ME0 - , : ME0 = np0 Ne0 . , (16) : l(bm - 1) , ME = ME0 1+(bm - 1)l2 b(bm - b) . ME = ME0 1+(bm - b) (18) (17)
ME/ME0

6 5 4 a 3 b 2 c 1 1.0 0.8 0.6 I (З) 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 ()

100 80

(17), (18) . 3. , . , . HXR- , f (K), . HXR- (X ). [X ,X + dX ] (X )dX X : dN
X

= (X )dX .

. 3. () () . : a - bm = 10, b - bm = 25, c - bm = 100. HXR-.

, , dI
X

= (X )X dX .

(19)

dNK dNX dNX =n dX
p
X

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

20-100 . . : ve (K) = 2K/me . (21)

np - , ve - , - . (X ) (19), f (K) (3), (X ) = 4np Ne з


( ., 2001) (K,X ) =
2 8r0 me c2 1+ ln 3 KX 1-

1 - X /K 1 - X /K

, (22)

(20)

з

X

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

r0 - (2.82 з з 10-13 ), 1/137 - .
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68

,

1.0 () 0.8 шгкгЭ HXR-ЫСЮл~ТЬЫfl, з 10-8 йгкгЬА в-1 ЭЧ-1 0.6 0.4 0.2 0 1.0 8 (З) 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

. 4. : a - 25 , b - 50 , c - 75 , () () .

(21) (22) (20), np Ne з (23) (X ) = CX X


з

X

f (K)ln

1+ 1-
2 8r0

1 - X /K 1 - X /K

dK,

C
X

= 4

3 me /2
14

me c2 =

3 1/2 . (23) "" . f (K) = 1.86 з 10-

, +1. . Ne , f (K) np (1), (5) (15), np0 Ne0 l bm - b з (24) (X ,l,b) = CX X 1+(bm - b)l2
1-b/bm

з

X

f0 (KAx )ln
0

1+ 1-

1 - X /K 1 - X /K

dxdK.

. - f0 - ,
33 1 2007






69

HXR- ( ) , bm = 3 25 50 75 25 bm = 10 50 75 25 bm = 100 50 75

4.9 з 10
-9

5.2 з 10

-10

8.9 з 10

-11

6.1 з 10

-8

1.5 з 10

-8

5.8 з 10

-9

9.0 з 10

-7

3.5 з 10

-7

2.0 з 10

-7

1.6 з 10
-9

1.0 з 10

-10

1.4 з 10

-11

8.7 з 10

-9

1.5 з 10

-9

4.6 з 10

-10

5.3 з 10

-8

1.7 з 10

-8

8.3 з 10

-9

3.1 5.2 6.4 7.0 10.0 12.6 17.0 20.6 24.1

F0 (K0 ) (24) . (24) . 4. T = 108 . , 1036 , 1028 -1029 ( ) 25 , 101 -102 2 . RHESSI . . HXR- . , . HXR-, . . T = = 108 , 75 25 l 0.05, .. 5%
33

. . HXR- , . . (2003) HXR-, RHESSI 23 2002 . 20 7 5 . (24), , 4 з 1029 e = 7, np0 = 109 -3 bm = 20. - , , , , - . 5. 20 . , -, - . . , . . .
1 2007


70

,

02/07/23, 00:21:42.00 10 югкгЬА вЯ-2 в-1 ЭЧ-1 10
2

()

10 10

2

(З)

1

1

1 10 10 10
-1

1 10 10 20 30
-1

-2

-2

-3

10

40 50 10 10 ?ЬТ,,Ыfl, ЭЧ

-3

20

30

40 50

. 5. () () HXR-, 23 2002 . RHESSI ( ., 2003).

: . . . () , . , . ( ) , , . , , , . , . , , . , bm T0 , T0 -

(.. ), bm - . , , . 20-200 . HXR, , , , . HXR- . RHESSI HXR- . ( 04-02-16125).
1. (M.J. Aschwanden), Particle Acceleration and Kinematics in Solar Flares (Dordrecht: Kluwer Acad. Publ., 2002). 2. . (P. Balciunaite, S. Krucker, and R.P. Lin), Am. Astron. Soc. Meet. 204, abstract #54.07 (2004). 3. .., .., .., (.: , 2001).
33 1 2007




4. .., .., . . 31, 137 (2005). 5. . (R.P. Lin, S. Krucker, G.D. Holman, et al.), Proc. of the 28th Int. Cosmic Ray Conf. (Ed. T. Kajita, Y. Asaoka, A. Kawachi, Y. Matsubara and M. Sasaki, 3207, Tokyo: Univers. Acad. Press Inc., 2003). 6. (L.I. Miroshnichenko), Solar Cosmic Rays (Dordrecht: Kluwer Acad. Publ., 2001). 7. (B.V. Somov), Physical Processes in Solar Flares (Dordrecht: Kluwer Acad. Publ., 1992). 8. (B.V. Somov), Cosmic Plasma Physics (Dordrecht: Kluwer Acad. Publ., 2000).

71

9. .., .., . . 29, 621 (2003). 10. , (B.V. Somov and T. Kosugi), Astrophys. J. 485, 859 (1997). 11. . (B.V. Somov, T. Kosugi, I.V. Oreshina, et al.), Adv. Space Res. 35, 1712 (2005). 12. , (H. Hudson and J. Ryan), Ann. Rev. Astron. Astrophys. 33, 239 (1995). 13. ., ., ., : , , (.: , 1968).



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2007