The solar wind deceleration by the interstellar medium may result in the existence of the solar wind terminal shock. In this case a certain fraction of thermal particles after being heated at the shock would obtain enough energy to be injected to the regular acceleration process. An analytical solution for the spectrum in the frame of a simplified model that includes particle acceleration at the shock front and adiabatic cooling inside the stellar wind cavity has been derived. It is shown that the acceleration of the solar wind particles at the solar wind terminal shock is capable of providing the total flux, spectrum and radial gradients of the low-energy protons close to one observed in the interplanetary space

Nikolaev, V. S.

Petukhov, S. I.

Turpanov, A. A.

English

NASA Technical Reports Server

196
SH I.5-17
COSMIC RAY ACCELERATION BY STELLAR WIND.
SIMULATION FOR HELIOSPHERE
S .I.Petukhov, A.A.Turpanov, V.S .Nikolaev
Institute of Cosmophysical Research and Aeronom_,
T,enin Ave., 31, 677891 Yakutsk, USSR
ABSTRACT
It is shown that the acceleration of the
solar wind particles at the solar wind
terminal shock is capable of providing
the total flux, spectrum and radial gra-
dients of the low-energy protons close
to ones observed in the interplanetary
spac e.
q. IntroductiQn_ The solar wind deceleration by the in-
terstellar medium may result in the existence of the solar
wind terminal shock _lJ. In this case a certain fraction of
thermal particles after being heated at the shock would ob-
tain enough energy to be injected to the regular acceleration
process [2-5J. An analytical solution for the spectrum in the
frame efa simplified model that includes particle accelera-
tion at the shock front and adiabatic cooling inside the
stellar wind cavity has been derived in _6J.
A more realistic model is discussed in the present pa-
per in order to estimate whether this process might occur
the main source of the low-energy cosmic rays in the helio-
sphere.
2. tModel. Let the solar wind velocity u(r) in a steady
state spherically symmetric model be equal to u1=const at
and u2(R/r)2 at r > R, where R is the solar wind ter-r < R
minal shock radius, _ = Ul/U 2 - the compression ratio of
the shock. Cosmic ray propagation in the diffusive approxi-
mation is governed by the transport equation:
where n(r,p) is the particle number density per unit inter-
196 
COSMIC RAY A CELERATION BY STELLAR WIND. 
SI ULATION FOR HELIOSPHERE 
S. t hov, A.A.Turpanov, V.S. ikolaev 
Institute of CosmophJsical Research and Aeronomy, 
Lenin Ave., 31, 677891 Yakutsk, U SR 
ABSTRACT 
It is sho n t at t e a celeration of t e 
solar wind particles at t e solar wind 
t r inal shock is capable of providing 
t e t tal fl x, s ectr  and r dial gra-
dients of t e l -energy protons cl se 
t  ones observed i  t e i t r lanetary 
space. 
SH 1.5-17 
1. I tr duction. The solar ind deceleration bJ t e i -
terste lar ediu  may r sult i  t e exist nce of t e solar 
ind t r inal s ock [1]. I  t is case a ertain fr cti n f 
t er al articles after being heated t t  s ock ould b-
t i  e ough energy t  be i j ct d t  t  r gular acceleration 
proce s [ 1. n analytical s luti n f r t e s ectr  i  t  
f e af a si plified odel t at i cl des article ccelera-
t n t t  s ock fr nt a  diabatic cooli g i si e t  
ste lar ind avit  has been eri ed i  [ . 
 ore r alistic odel  is ssed i  t  resent a-
er i  order t  st ate hether t is r ce s ight ccur 
t e ain urce f t  l -energJ ic r s i  t  elio-
s here. 
. odel. et t e s lar ind elocity (r) i  a stea  
t t  s herically sJ ruetric odel e ual t  1=const t 
r   and u2(B/r)2 at r > R, where R is the solar wind ter-
inal shock radius, (/ = u1/u2 - the co pression ratio of 
t  s ock. os ic  r pagation  t  if si e proxi-
ation  verned J  sport uati n: 
(1) 
here (r, )   article ber ensit  er nit t r-
https://ntrs.nasa.gov/search.jsp?R=19850026467 2020-03-20T17:02:31+00:00Z
197
SH 1.5-17
val of moment-_m p, _ (r,p) - cosmic r_ diffusion coeffi-
cient. A_ r < R and r > R (subscripts q and 2, respectively)
it is given by
The matching conditions are that the particles current
and the number density should be contlnious at the shock.
The other conditions are the absence of source or sink at
r=O (r.n(r,p)-* O as r-_ O) and a given spectrum at infinity
(n(r,p_-_ N_(p) as t-*co ).
Integrating (I) with respect to r from 0 to R-0 and
using the analytical solution to (q) at r > R, the matching
condition might be reduced to the following:
3 6" ,_p l J
(2)
0
where Wyp)= _e,xp1- _ R ,/[,.e_)_[p)]
Hence, the problem is to solve the equation (I) at
r _<R with respec_ to the boundary condition (2). The equa-
tion (I) is valid at particle momentum p higher than some
limiting value, sa_ Po" One more condition is necessary to
make the system (I),(2) being completed, namely n(R,po)=no.
This condition relates the injection rate to the spectrum
amplitude p=po.
It has been shown [7-9_, that the shock acceleration
is efficient enough to transfer a significant fraction
( _ 50%) of the inflowing plasma kinetic energy to the acce-
lerated particles. The accelerated particles spectrum takes
on the power law form with the index equal to that one given
by the linear theory at particles velocity v _-3 uI 19_.
The corresponding values for the solar wind particles are
T _ 10 keV or po__3.10 -3 mpc. Particles of energy correspon-
197 
SH '.5-17 
val of momentlim p, de (r ,p) - cosmic ra::J diffusion coeffi-
cient. At r < R ano r > R (subscripts 1 and 2, respectively) 
it is given by 
The matching conditions are that the particles current 
and the number density should be continious at the shock. 
The other conditions are the absence of source or sink at 
r=O (r.n(r,p) ~ 0 as r - 0) and a given spectrum at infinity 
(n(r,p~~ Noo(p) as r- oo ). 
Integrating (1) with respect to r from 0 to R-O and 
using the analytical solution to (1) at r > R, the matching 
condition might be reduced to the following: 
~":L[ p"ncR f)7 + h(R,ej :=. 
36' ?;If ':J 6'{1-WP)] 
R-o 
=~ J...:!.- :L{f'2t(f)~[p.n(fV,f)J,.:J.df" 7-u~R2 3,.2~f" ~p 
o 
where 1f(f) -= ex/> [- u~ R /0.,. ..ft:i) £),t, Cf)] 
eftf) IVOQ (f) 
6' [1- (Nf)] 
Hence, the problem is to solve the equation (1) at 
(2) 
r ~ R with respect to the boundary condition (2). The equa-
tion (1) is valid at particle momentum p higher than some 
limiting value, say Po' One mor~ condition is necessary to 
make the system (1),(2) being completed, namely n(R,po)=n o ' 
This condition relates the injection rate to the spectrum 
amplitude P=Po' 
It has been shown [7-9J, that the shock acceleration 
is efficient enough to transfer a significant fraction 
( ~ 50%) of the inflowing plasma kinetic energy to the acce-
lerated particles. The accelerated particles spectrum takes 
on the power law form with the index equal to that one given 
by the linear theory at particles velocity v ~; u1 [9]. 
The corresponding values for the solar wind particles are 
T ~ 10 keV or Po~ ;.10-; ~c. Particles of energy correspon-
198
8111,5-,17
ding to the cutoff velocity diffuse far upstream and suffer
catastrophic adiabatic cooling.
7. Results and Discussion. The results of the numerical
calculations are presented in Figure in comparison with the
observational data [10_. The curve labelled"1" is the differen-
tial proton flux near the shock, "2"- that one at the orbit of
the Earth. The following parameters were used: u1=5.107 cm/s,
R=7,5.101@ cm, _ =_. The solar wind energy density near the
shock Ek has been taken equal to 5 eV/cm3. The value of n o
has been defined so, that accelerated particles (T _ 10 keV)
contain 10% of the kinetic energy density E_. The diffusion
. 22 0,3 _ 0,2 2. -I
coefficients are • =6,5 10 (p/mpC) (r/R) cm s ,
_<_ _ . The diffusion
coefficient in the out-
side vicinity of the
shock might be signifi- ml
cantl7 reduced due to the
qP
magnetic field compres- '
sion and convection of
Z:
the turbulence excited & III
b_ the accelerated par- __ :1".|@73\ (_)
ticles. This leads to the _ I@4
strong modulation of the
galactic proton flux at _
T 6 10 MeV. Radial gra-
dient n-1 ?n/_ r takes
on a value of 15%/AU at 10_ (2)
T=IO keV and 5%/AU at 1176
T _- (1-10)_eV.
The comparison of I@||
the results with the ob-
,,
servations shows that IO°
under the above asaump- I0' tO° t MeV/NII
tions the acceleration
rate (source _) is ca-
pable of providing the
\,
8 
8B '.5-'7 
i  t  t e c t ff el cit  iff se f r strea  and s ff r 
catastrophio adiabatio cooling. 
3. esults and iscussion. he res lts of the nu erical 
oalculati ns are rese ted i  i re i  co parison it  t e 
observational data [1 OJ • The curve labelled "1" is the differen-
l r t n fl  ar t  ck, - t t  t t  r it f 
the Earth. The following parameters were used: u1=5·107 cm{s, 
=7,5·1014 c , 6 4. e s l r i  e er  nsit  ar t  
shock Ek has been taken equal to 5 eV/cm3 • The value of no 
s  fi  , t t el rat d arti les (  ~1  Y) 
ntain 10  f t  i eti  e er  sit  k • e iff si  
coefficients are £--1 =6,5 10 2(p/~c)O  (r/R)  cm2.s 1 , 
Je,2 <.::. J-i1 •  i i  
e ficient   t-
i  i i it  f t  
shock might be signifi-
tly r ed  t  t  
.. 
agnetic fi l  c pres- I 
  vection f i 
I: 
t  t l ce it  
by the accelerated par-
ti l s. his lea s t  t e 
strong modulation of the 
alactic r t n fl  t 
 ~  e . adial r -
i t -1 (J n/o r t  
  l  f /  t 
:.:10   /  t 
 ~ ( -10)M . 
The comparison of 
 lts it   -
r ati ns s t 
under the above assump-
  celerati n 
t  r  ~   -
pable of providing the 
., 
CIt 
... 
E 
. 
ot 
-
to' 
1014 
101 
lI .tl7S \ 
\ 11 1173 \ 
\ \ 
• \ 
• 
fl13 '. 
174 
-, 
10 
\ , 
• \ 
o 
f  
(0 
fO· H,V IN 
199
registered intensity of the low-energy particles in the in-
terplametamy space. In the energy range T _ 100 KeV the solar
cosmic rays might contribute significantly to the total in-
tensity, especially during the maximum of solar activity (11]
This would cause the complicated variations of the low-ener-
gy cosmic ray intensity: in the energy range T < 100 KeV the
intensity is correlated with the solar activity level _I0_;
at the emergy T=6 MeV/nucl. the anticorrelation takes place
Lll). The relative contribution of the source _ increases
with the heliocemtric distance r.
The cutoff energy for the accelerated particles spectrum
is determined by _ (Tmax) _-Ru I; on the ether hand, Tmax _-
10 MeV/nucl. as is observed for different nuclear species.
That shpuld be consistent with our model if z is a func-
tion of T per nucleon only. The verification of this sugges-
tion should be based on the treatment of the cosmic ra_ che-
mical composition.
_. Acknowled6ements. The authors are grateful to Dr.E.G.Be-
rezhko for useful discussions.
References
1. Baranov, V.B., Krasnobayev, K.V., (1977), Gidrodinamiches-
ka_ Teori_a Kosmicheskoi Plasm_, Moskva, Nau_a. ....
2. Krymsky, G.F., (1977), Dokl. AN SSSR, 254, 1306.
3. Axford, W.I. et al., (19-_, _-_oc._-th----YCR_qC,Plovdiv,
_., 132
Belll A.R., (1978), Mon.Not.Roxal Astren.Soc., 182, 147.
5. Blandford, Rm., Ostr-12e'OT-,J_(TOVD-T, X_T.J.,_221, 29.
6. Webb, G.M. etal., (1983), Proc.18-th IORS_-_-anaa--'_re,2,
26_ .
7. Eichler, D., (1979), AD J., 229, 419.
8. Erymsky,G.F.,(1981),Tzv.i T-SSSR, .r.fiz.,45, 4509. Berezhko, E.G. etal., (19837-,r_.18-th ICRC,_an_alore,
259.
O. Zeldovich, M.A., Logachev, Yu.I., (1983), Kosmicheski_e
Issledovani_a, 21, No.5, 803.
11. Zamow, R., _975), _, I__, 167.
199 
til t.,.'1 
registered intensity of the low-energy particles in the in-' 
terplanetary space. In the energy range T < 100 KeV the solar 
cosmic rays might contribute significantly to the total in-
tensity, especially during the maximum of solar activity [11]. 
This would cause the complicated variations of the low-ener-
gy cosmic ray intensity: in the energy range T < 100 KeV the 
intensity is correlated with the solar activity level [10J; 
at the energy T=6 MeV/nucl. the anticorrelation takes place 
[11]. The relative contribution of the source 01 increases 
wit  the heliocentric distance r. 
The cutoff energy for the accelerated particles spectrum 
is determined by Je., (Tmax) .~ Ru1 ; on the other hand, Tmax ~ 
10 MeV/nucl. as is observed for different nuclear species. 
That shpuld be consistent with our model if ~ is a func-
tion of T per nucleon only. The verification of this sugges-
tion should be based on the treatment of the cosmic ray che-
mical composition. 
4. Acknowledgements. The authors are grateful to Dr.E.G.Be-
rezhko for useful discussions. 
References 
1. Baranov, V.B., Krasnob~ev, K.V., (1977), Gidrodinamiches-
kaYirTeOriya Kosmicheskoi Plasmy, Moskva, Nauka. 
2. ymskY, G.F., (1971), Dokl. AN SSSR, 23f, 1306. 3. Axford, W.I. et al., \1~, Proc:15-th~, Plovdiv, 
11, 132. 
4: Bell, A.R., (1978), Mon.Not.Rpyal Astron.Soc., 182, 147. 
5. Blandford, R.D., OstrikeF;-J. ., (1978), lj)."J. ,~1, 29. 
6. Webb, G.M. et al., (1983), Proc.18-th ICRe, BangiIOre, 2, 
263. ---- ---- - -
7. Eichler, D., (1979), Ap.J., ii9, 419. 
8. Kr~msky, G.F., (1981), Izv. SSSR, Ser.fiz., ~, 450 
9. Berezhko, E.G. et al., mS3),' Froc 1-s=th lCRe, jjangalore, 
2, 259. 
10. Zeldovich, M.A., Logachev, Yu.I., (1983), Kosmicheskiye 
Issledovaniia, 21, No.5, S03. 
11. Zamow,., t1'975) , Ap.J., .122,167. 


Cosmic ray acceleration by stellar wind. Simulation for heliosphere

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