It is proposed that particles during the second phase of solar flares are accelerated by stochastic resonant scattering off hydromagnetic waves and first order Fermi acceleration in shock waves generated in the impulsive phase of the flare. Solutions allow arbitrary power law momentum dependences of the momentum diffusion coefficient as well as the momentum diffusion coefficient as well as the momentum loss time. The acceleration time scale to a characteristic energy approximately 100 keV for protons can be as short as 5s. The resulting electron spectra show a characteristic double power law with a transition around 200 keV and are correlated to the proton spectra evaluated under equal boundary conditions, indicating that electrons and protons are accelerated by the same mechanism. The correlation between the different spectral indices in the electron double power law and between electron and proton spectra are governed by the ratio of first to second order acceleration and therefore allow a determination of the Alfven Mach number of the shock wave

Droege, W.

Schlickeiser, R.

English

NASA Technical Reports Server

2 SH I.I-4
STOCHASTIC PARTICLE ACCELERATION IN SOLAR FLARES
W. DrSge, R. Schlickeiser
Max-Planck-lnstitut f_r Radioastronomie
Auf dem HHgel 69, 5300 Bonn I, FRG
ABSTRACT
We propose that particles during the second phase of solar
flares are accelerated by stochastic resonant scattering
off hydromagnetic waves and first-order Fermi acceleration
in shock waves generated in the impulsive phase of the
flare. Our solutions allow arbitrary power law momentum
dependences of the momentum diffusion coefficient as well
as the momentum loss time. The acceleration time scale to
a characteristic energy _ 100 keV for protons can be as short
as 5 s. The resulting electron spectra show a characteristic
double power law with a transition around 200 keV and are
correlated to the proton spectra evaluated under equal
boundary conditions, indicating that electrons and protons
are accelerated by the same mechanism. The correlation
between the different spectral indices in the electron
double power law and between electron and proton spectra
are governed by the ratio of first-to second-order accel-
eration and therefore allow a determination of the Alfv_n
Mach number of the shock wave.
I. Introduction. We propose that the combination of first- and second-
order Fermi acceleration is the mechanism responsible for second-phase
acceleration in solar flares. This model has a number of distinct advan-
tages over previous models, in that it gives a natural explanation for
the various observed time delays between first and second phase as well
as for the dependence of the particle spectra to the strength of the flare
and the correlation between electron and proton spectral indices.
2. Theory. Parker and Tidman (1958) have pointed out that Fermi accel-
eration is an zntrinsic property of any sufficiently agitated plasma of
energetic particles. The effects of second-order Fermi acceleration due
to irregularly moving magnetlzed fluid elements, first-order Fermi ac-
celeration off strong shocks as well as loss and escape processes can be
incorporated into a transport equation in phase space (Ramaty 1979)
_f I _ _f I _ f
[p2 D(p) _] -_-_ _p [p2 (_G+ _L) f] + T--_ = Q(p,t) (I),_t p2
where p is the particle momentum, N(p) = 4_ p2 f(p,t) the number of par-
ticles per unit momentum and unit volume, and Q(p,t) represents sources
and sinks of particles. Using the concept of the age distribution
(Schlickeiser and Lerche 1985) the effects of spatial diffusion, convec-
tion and catastrophic losses have been combined in a momentum-dependent
escape time T(p) = To p-b.
In a plasma with a strong MHD turbulence both the momentum diffusion
coefficient D(p) and the spatial diffusion coefficient along the mean
magnetic field K,(p) are governed by the turbulence simultaneously. By
using quasi-linear theory K.(p) can be related rigorously to the spectrum
2 
T STI   TI  IN SO  L  
. ro e, . li eiser 
ax-Planck-Institut fU  adi ast ie 
f e  Ugel ,5300  1,  
 
SH 1.1-4 
e r se t t rti l s ri  the second ase f s l r 
flares are accelerated by stochastic resonant scattering 
 agnetic a es  rst-o r i l r t  
i  ~  a es e erate  i  the i ulsi e ase f th  
l . r l tio  llow r itr r  er law o ent  
ences  h  ent   ffi i t  ell 
  o ent   im .  l r t  im   o 
a aracteristic r y'"  0 eV f r r t s ca   s s rt 
  .  lti  l tr  tr  sho   racteristic 
bl  er law it   tr iti  ro     r  
rr l t  t  t e r t  s ectra e al ate  under e al 
boundary conditions, indicating that electrons and protons 
are accelerate  by t e sa e echanis . he rr l ti  
t ee  th  i t ctr l i i  i  th  l t  
double po er la  and bet een electr  and proton s ectra 
r  er e   t  r ti  f fir t-t  sec d-order el-
  f re o   t inati n   l e  
ac  ber    ave. 
1. cti n. e se t  binati n f  -
order Fer i acceleration is the echanis  responsible for second-phase 
celerati n  l r . his odel   ber  i t an-
t s er r i s odels, i  t t it i s a at ral l ati  f r 
the various observed time delays bet een first and second phase as ell 
as f r t e dependence of the article s ectra to the stre t  of the flare 
a  t  rr l ti  et een l tr  a  r t  s ctral i i s. 
2. heory. arker and id an (1958) have i te  t t at er i accel-
eration is an ~ntrinsic property of any sufficiently agitated plas a of 
ergetic articles.  t   d-order i celerati n  
t  irr l rl  oving t~zed fl i  l ents, fir t- r er er i -
celeration off strong shocks as ell as loss and escape processes can be 
incorporated into a transport equation in phase space (Ra aty 1979) 
of 0 2 of 1 0 2··  
at - p2 op [p D(p) op] -{;Tap [p (PG+PL) f] +T(p') Q(p,t) (0, 
where p is the particle momentum, N(p) = 4np 2 f (p,t) the number of par-
ticles per unit omentu  and unit volu e, and (p,t) represents sources 
and si s f arti l s. sing t e concept f t e age istri ti n 
( chlickeiser and erche 1985) t e effects f s atial iff si n, convec-
tion and catastrophic losses have been co bined in a omentum-dependent 
escape time T(p) = To p-b. 
I  a plas a it  a str  HD t r le ce oth t e o entu  iff si n 
coefficient (p) and the spatial diffusion coefficient along the ean 
agnetic field K .. (p) are governed by the turbulence si ultaneously. By 
using quasi-linear theory II(p) can be related rigorously to the spectru  
https://ntrs.nasa.gov/search.jsp?R=19850026418 2020-03-20T16:53:55+00:00Z
3
SH 1. I-4
of the magnetic field fluctuations (Jokipii 1977). Assuming a power law
spectrum for the magnetic irregularities W(k) = Wo k-q we may write
K,,(p) = 6 v(p) p2-q = Kpq (2)
(6,K = const.) and
V_ p2 (3)D(p) = _2
(VA: Alfv&n speed, _2: constant). As galn process, we consider quasi-
continuous momentum gain by acceleration at a shock wave moving through
the plasma with speed VS. The momentum gain by first-order Fermi accel-
eration at a single shock wave has been determined by Drury (1983)
PG = _I _ p (4)
(_z: constant). Mbbius et al. (1982) pointed out that for condltions in
the flare site the rate of systematic acceleration exceeds the rate of
momentum loss. Thus we assume that PG >> PL at least at all momenta of
interest. Using (2), (3) and (4) in equation (I) and consldering the
steady-state case (_f/_t = 0) yields
I d [p4-q df 3-q] b
7 _ d-_- a p - Xp = Q(p) (5)
(a_ =_.V_/K, a_ =_V_/K, a=a_/a_, % = I/a2T_). We assume that the in-
jection _akes piace_a_ some momentum Q(p) = qo _(P-Po ) and we find for the
steady-state particle number density N(p) =4_p2 f(p) (see DrSge and
Schlickeiser (1985) for details):
q-a+1 q+a+l K ]q+b I ) I -'_"_ p
P<Po
8_ qo 2 2
a2 [q+b[ 2_112
I lq+bI Po )K_;_lq+bl p 2 P>Po
_)= l(3+a-q)/(q+b)l. In the limit q+b+0 solution (6) approaches a power
law distribution
)_ (3+a_Q)2 I4 +XQ-a+1 Q+a+l (P/Po P < Po
2_ qo 2 2
N(p) = 41 ! (7)
_ (3+a_N) 2 , Po P (3+ q)2--'_ _---- +_a2 4 + % (P/Po) P > Po
Equations (6) and (7) are generalizations of the solutions of Ramaty
(1979; b =0, a =0, q =0,1), Barbosa (1979; a =0), Pikel'ner and Tsytovich
(1976; a =0, b =0). The solutions (6) and (7) allow arbitrary momentum
power law dependences of the spatial diffusion coefficient (q) and the
escape time (b). The parameter a = (_l/_2)(V$/V_) = (_l/_2)M_ is of the
order of the square of the shock's Mach number and describes the ratio of
first- to second-order Fermi acceleration. In the case of no Fermi ac-
3 
S  1.1-  
of t e agnetic fiel  fl ct ati ns (J kipii 1977). 
spectrum for the magnetic irregularities W(k) = Wo 
ssuming a power l  
k-q we may write 
KII(p) <5 v(p) p2-q Kpn (2) 
(O,  = const.) ann 
2 VA 2 
a 2 -- p K" 
D(p) (3) 
(VA: Alfven speed, a 2 : constant). As ga1n process, we consider quasi-
conti uous omentu  ai   acceleration t a s ck ave oving t r h 
the plasma with speed VS' The momentum gain by first-order Fermi accel-
r ti  t a si gle s ck ave has een eter ined  rury ( ) 
V§ 
P  = a - P (4) 1 KII 
(al : constant). Mobius et al. (1982) pointed out that for cond1tions 1n 
t e fl r  sit  t e r t  f s ste atic celerati n e ceeds t e r t  f 
momentum loss. Thus we assume that PG » PL at least at all momenta of 
i terest. sing ( ), (3) and (4) i  equation (1) and cons1dering the 
steady-state case (af/at = 0) yields 
1 d [4-n df 3-n] , b 
--r- - p --ap -I\P P dp dp Q(p) (5) 
(a l =alv~/K, a 2 =a2 vl/K, a=al /a2 , jection takes place at some momentum 
steady-state particle number density 
chlickeiser ( ) f r tail ): 
A = 1/ a  T ). We assume that the in-Q(p) = qo g (p-po) and we find for the 
N(p) = 4'TT p2 f(p) (see Droge and 
N(p) 
8'TT q n-a+1 n+a+1 
o -2- -2-
------- Po p 
a2ln+b I 
(
21..1/2 n;b) (21.. 1/2 n;b) K --p I --p 
V In+bl 0 V In+bl 
(
1.. /  n;b) (21.. 112 n;b) I --p K --p 
V In+bl 0 V In+bl 
p<p 
o 
p>p 
o 
(6) 
V = 1 (3+a-n) / (n+b) I. In the li it n+b + 0 solution (6) approaches a power 
la  i tri ti  
j(3+a- n)2 +1..1 
p<p 
2'TT q n-a  n+a+1 
(p/po  0 
N(p) 0 
-2- -2-
(7) = Po p 
_1 (3+~-n) 2 + AI 
a 21 (3+a-n) 2 + A I 4 (p/po) p>p 
0 
Equations (6) and (7) are generalizations of the solutions of Ra aty 
(1979; b=O, a=O, n=O,1), Barbosa (1979; a=O), Pikel'ner and Tsytovich 
(1976; a=O, b=O). The solutions (6) and (7) allow arbitrary momentum 
po er law dependences of the spatial diffusion coefficient (n) and the 
escape time (b). The parameter a = (al /a ) (V~/Vl) = (al /a?) Ml is of the 
order of the square of the shock's ach nu ber and descr1bes the ratio of 
to - r er i l r t . In th     i 
4SH 1.1-4
E0
o ] I I ] ] I I I1 --a=3 '_' 1 ---a=3_
"_ -- a=30 _" -- a=30
_ 10-1 "',.,,z lo- !----I.a-I
z
z u.a
o _ 10-2
10-2 oI.i..I I----
--J 0c=ln-3 _
C3_
i J J if\
101 E0 102 ET 103 101 102 103 104
KINETICENERGY(KEY) KINETICENERGY(KEY)
Fig. 1: Normalized steady-state Fig. 2: Normalized steady-state
electron number density. Parti- proton number density. Particles
cles are injected at Eo =50keV. are injected at Eo = 10 keV.
At ET=200 keV a transition
from solution (6) to (7) occurs.
celeration at shock waves (a =0) (6) and (7) reduce to the solutions of
previous models as quoted above.
3. Discussion. The energy spectra of electrons accelerated in large
flares exhibit a characteristic double power law with a break around ET
200keV (Lin et al. 1982). From this we conclude that for nonrelativistic
kinetic energies (E<E T) the Bessel function solution (6) holds, which
for small arguments is approximately a power law, whereas for E > ET the
particles are relativistzc (p =mc _E=200 keV for electrons) and the
distribution function approaches the power law solution (7) (cf. Fig. I).
We take spatial diffusion along the mean magnetic field as the relevant
escape process. The escape time then is T(p) =L2/K,, = (LZ/K) p-n, whereL
is the length scale of the system and q=b =0 in the relativistic and
n =b = I in the nonrelativistic case. Thus we may consider the evolution
of the particle spectra in the hard-sphere approximation (cf. MSbius et
al. 1982), which gives us
_ I
K,,(p) 3 v(p) _ (8)
v(p) is the velocity and _ the momentum independent mean free path of the
particles. For comparison with data we transform (6) and (7) into energy
space where N(E) =N(p) (I/v) and E= (pZc2+mac_)ZZZ-mc 2 is the particle
kinetic energy. An "effective power law" exponent can be calculated (E>Eo)
a a+2 _I 4xz 2E |
d IgN(E) - _ + _ Vl + (a+2)2 mc 2 p < mc (9)
Yeff(E) d Ig E
a+l a+3 _1 4x 2 |
--2"-+T V I + (a+3)2 p > mc (10)
4 
S  .1-4 
- -a:3 
-a=30 
01 o 02 T 03 
INETIC NERGY (KeV) 
ig. 1: ormalized steady-state 
electr  nu ber e sit . arti-
c les are inj ected at Eo = 50 keV. 
t  = 2 0   sit  
om lution )  ) curs. 
--- a=3 
a=30 
101 02 03 04 
INETIC NERGY (KeV) 
Fig. 2: ormalized steady-state 
proton nu ber e sit . articles 
are inj ected at Eo = 10 keV. 
l t    a es   0)       l t   
r i s odels  ot  e. 
3. iscussion. The energy spectra of electrons accelerated in large 
flares exhibit a characteristic double power law with a break around ET~ 
 ke   t l  2).   e cl e   onrelativistic 
kinetic energies (E < ET) the Bessel function solution (6) holds, which 
for small arguments is approximately a power law, whereas for E > ET the 
arti l s r  l ti i tl.c (   mc ~ E = 200   l t s)  t  
distribution function approaches the power law solution (7) (cf. Fig. 1). 
e take spatial diffusion along the ean agnetic field as the relevant 
escape process. The escape ti e then is (p) 2 /K II =(L2/ ) p-n, hereL 
i  t  l t  s al  f t  s ste  and n = b  0 i  t e r l ti istic and 
n = b = 1 in the nonrelativistic case. Thus we ay consider the evolution 
of the article spectra in the hard-sphere approxi ation (cf. obius et 
l. 82), hich i s s 
1 
"3 v(p) ~ (8) 
v(p) is the velocity and ~ the momentum independent mean free path of the 
particles. For comparison with data we transform (6) and (7) into energy 
space where N(E) =N(p) (l/v) and E = (p 2c 2+m2c'+)1/2 -mc 2 is the particle 
kinetic energy. An "effective power law" exponent can be calculated (E >Eo) 
 I (E) 
  
a a  1/1 2  I _ (9) 
- 4 + -4- Y + (a+2)2 mc 2 p < mc 
a+1 a+3 
---+--2 2 ~1 + x2 (a+3)2 p > mc (10) 
5SH 1.1-4
_ I _ I , ,
5
3 - -
2
3 .
1 -  j-";zo
2
I = J I
1 2 _ 2 3 _ lp
Fig. 3: Correlation between elec- Fig. 4: Correlation between YI
tron spectral exponent below 200 and proton spectral exponent
keV YI and exponent above 200 keV above 10 MeV as a function of a
YII as a function of a
x = %c/L VA is a free parameter which may have different values for elec-
trons and protons and for different flares. Equations (9) and (10) allow
us to determine the correlation between the mean spectral index below 200
keV YI and above 200 keV YII for electrons (Fig. 3) as well as the cor-
relation between YI and the proton spectral exponent above 10 MeV YD
(Fig. 4) as a function of a. The curves obtained from (9) and (10)-are
in excellent agreement with the measurements of Lin et al. (1982), indi-
cating that the strongest shock waves have an Alfv_n Math number MA _ 4,
corresponding to a_30 (see DrSge and Schlickeiser (1985) for details).
4. Conclusions. Combzning first- and second-order Fermi mechanism in
solar flare second-phase acceleration successfully explains the observed
ion and electron energy spectra. The model correctly accounts for the
sometimes very short delay times, and reproduces quantitatively the cor-
relations of nonrelativistic with relativistic electron spectral indices,
and the correlation of nonrelativistic electron with proton indices.
References
Barbosa, D.D., (1979), Astrophys. J. 233, 383
DrSge, W., Schlickeiser, R., (1985), in preparation
Drury, L.O.C., (1983), Rept. Progr. Phys. 46, 973
Jokipii, J.R., (1977), Proc. 15th Int. Cosmi---cRay Conf. (Plovdiv), Vol.
2, 429
Lin, R.P. et al., (1982), Astrophys. J. 253, 949
MSbius, E. et al., (1982), Astrophys. J. 259, 397
Parker, E.N., Tidman, D.A., (1958), Phys.--_v. 11___!I, 1206
Pikel'ner, S.B., Tsytovich, V.N., (1976), Sov. Astron. 19, 450
Ramaty, R., (1979), Particle Acceleration Mechanisms in Astrophysics,
eds. J. Arons, C. Max, C. _cKee, American Institute of Physics, New
York, p. 135
Schlickeiser, R., Lerche, I., (1985), this conference, OG 7.2-8
5 
SH 1.1-4 
1][ 5 II 5 
3 
4 
2 
3 
2 
2 
Fig. 3: Correlation between elec-
tron spectral exponent below 200 
keV YI and exponent above 200 keV 
YII as a function of a 
Fig. 4: Correlation between YI 
and proton spectral exponent 
above 10 MeV as a function of a 
x = 9vc /L VA is a free parameter which may have different values for elec-
trons and protons and for different flares. Equations (9) and (10) allow 
us to determine the correlation between the mean spectral index below 200 
keV YI and above 200 keV YII for electrons (Fig. 3) as well as the cor-
relation between YI and the proton spectral exponent above 10 MeV Yp 
(Fig. 4) as a function of a. The curves obtained from (9) and (10) are 
in excellent agreement with the measurements of Lin etal. (1982), indi-
cating that the strongest shock waves have an Alfven Mach number MA ~ 4, 
corresponding to a ~ 30 (see Droge and Schlickeiser (1985) for details). 
4. Conclusions. CombLning first- and second-order Fermi mechanism in 
solar flare second-phase acceleration successfully explains the observed 
ion and electron energy spectra. The model correctly accounts for the 
sometimes very short delay times, and reproduces quantitatively the cor-
relations of nonrelativistic with relativistic electron spectral indices, 
and the correlation of nonrelativistic electron with proton indices. 
References 
Barbosa, D.O., (1979), Astrophys. J. 233, 383 
Droge, W., Schlickeiser, R., (1985), rn-preparation 
Drury, L.O.C., (1983), Rept. Progr. Phys. 46, 973 
Jokipii, J.R., (1977), Proc. 15th Int. Cosmic Ray Conf. (Plovdiv), Vol. 
2, 429 
Lin, R.P. et al., (1982), Astrophys. J. 253, 949 
Mobius, E. et al., (1982), Astrophys. J.-z59, 397 
Parker, E.N., Tidman, D.A., (1958), Phys. Rev. 111,1206 
Pikel'ner, S.B., Tsytovich, V.N., (1976), Sov. Astron. 19, 450 
Ramaty, R., (1979), Particle Acceleration Mechanisms in-Xstrophysics, 
eds. J. Arons, C. Max, C. l'1cKee, American Institute of Physics, New 
York, p. 135 
Schlickeiser, R., Lerche, I., (1985), this conference, OG 7.2-8 


Stochastic particle acceleration in solar flares

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