High energy cosmic ray propagation of the energy region 10 to the 14.5 power - 10 to the 18th power eV is simulated in the inter steller circumstances. In conclusion, the diffusion process by turbulent magnetic fields is classified into several regions by ratio of the gyro-radius and the scale of turbulence. When the ratio becomes larger then 10 to the minus 0.5 power, the analysis with the assumption of point scattering can be applied with the mean free path E sup 2. However, when the ratio is smaller than 10 to the minus 0.5 power, we need a more complicated analysis or simulation. Assuming the turbulence scale of magnetic fields of the Galaxy is 10-30pc and the mean magnetic field strength is 3 micro gauss, the energy of cosmic ray with that gyro-radius is about 10 to the 16.5 power eV

Honda, M.

Kifune, T.

Matsubara, Y.

Mori, M.

Nishijima, K.

Teshima, M.

English

NASA Technical Reports Server

A Simulat ion of High Energy Cosmic Ray Propagation I 
* * ** t 
M. Honda, T.Kifune, Y. Matsubara , M. Mori , K .  Nishij ima and M. Teshima 
I n s t i t u t e  f o r  Cosmic Ray Research, Univers i ty  o f  Tokyo, Tokyo, 188 Japan 
'Department of Physics ,  Kyoto Univers i ty ,  Kyoto, 606 Japan 
**The Graduate School o f  Science and Technology, Kobe Univers i ty ,  Kobe, 657 
Japan 
Abs t rac t  
We ~Amula te  high energy cosmic ray  propagat ion o f  t h e  energy region 
1 0 ~ ~ * ~ - 1 0  eV i n  the  i n t e r  s t e l l e r  circumstances.  In  conclus ion,  t h e  
d i f f u s i o n  process  by t u r b u l e n t  magnetic f i e l d s  is c l a s s i f i e d  i n t o  s e v e r a l  
region by r a t i o  o f  the  gyro r d 'us  and the  s c a l e  o f  turbulence .  When t h e  
--b .4 
r a t i o  becomes l a r g e r  then 10  , the  a n a l y s i s  with 3he assumption o f  p o i n t  
s c a t t e r i n g  can be app l i ed  w ' t h  t h e  mean f r e e  path  E . However, when the 
-0.5 
r a t i o  is smal le r  than 10  , we need a more complicated a n a l y s i s  o r  
s imulat ion.  Assuming t h e  turbulence  s c a l e  o f  magnetic f i e l d s  o f  t h e  Galaxy 
is 10-30pc and the  mean magnetic f i e l d s  s t r e n g t h  is 3 ~ 8 ~ 5 0  gauss ,  t h e  
energy of  cosmic ray  wi th  t h a t  gyro-radius is about 10  eV. 
1 In t roduc t ion  
au thors  have s t u d i e d  cosmic r a y  propagation i n  magnetic 
f i e l d s  . Espec ia l ly ,  Gleeson and Axford had solved a d i f f u s i o n  equat ion 
wi th  t h e  assumption t h a t  cosmic ray  a r e  s c a t t e r e d ( 9 y  s c a t t e r i n g  c e n t e r s .  
The r e c e n t  observat ion o f  Ga lac t i c  magnetic f i e l d  shows t h e  average 
f i e l d  s t r e n g t h  is about 3 micro gauss and t h e  t u r b u l e n t  f i e l d s  a r e  about 
1 .5  micro gauss .  The gyro-radius o f  cosmic r a y  i n  the  energy reg ion ,  i n  
which w e  a r e  i n t e r e s t e d ,  becomes 0.1-300pc. The random magnetic f i e l d s  
cannot d e f l e c t  t h e  cosmic r a y s  s o  hardly ,  b u t  an amount o f  smal l  d e f l e c t i o n  
may cause a l a r g e  s c a t t e r i n g  angle  as a r e s u l t .  We know t h a t ,  i n  some 
c a s e s ,  we can sum up t h e  e f f e c t s  i n  terms o f  p o i n t  s c a t t e r i n g  and mean- 
free-path(mfp1. However, we a r e  n o t  s u r e  t h i s  is t h e  case  i n  t h e  cosmic r a y  
propagat ion i n  the  t u r b u l e n t  magnetic f i e l d s ,  e s p e c i a l l y  when the  gyro- 
r a d i u s  o f  t h e  cosmic ray  becomes near  to t h e  s c a l e  of  i r r e g u l a r i t i e s .  The - 
assumption o f  p o i n t  s c a t t e r i n g  can be an over  s i m p l i f i c a t i o n  i n  t h e  s tudy  
of cosmic ray  propagation i n  t h a t  energy region.  I t  should be checked by 
the  numerical  c a l c u l a t i o n  o r  s imulat ion.  
I n  t h i s  paper,  we p resen t  a Monte Carlo s imulat ion o f  cosmic r a y  - 
propagat ion i n  the  Ga lac t i c  space and show a r e s u l t  i n  terms o f  d i f f u s i o n  
t ensor .  Gleeson and Axford showed the  d i f f u s i o n  t ensor  have t h r e e  
independent element (K ,KL ,  K T ) .  The a n a l y t i c  form o f  these  element as a 
funct ion of mean f r e e  LLth a r e  given by 
1 
K//= g Ac 9 (1 1 
K 
K = // and ( 2  1 
l +  ( w + I 2  - 
A (W K//  
K = ( 3 )  A 2 1 + (WE) 
9 
6 
where \ i s  t h e  mfp and [r! is t h e  gyro frequency o f  cosmic r a y s  i n  t h a t  
magnetic f i e l d .  When we l eave  from t h e  i d e a  of p o i n t  s c a t t e r i n g  of cosmic 
272 
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https://ntrs.nasa.gov/search.jsp?R=19850025765 2020-03-20T18:01:16+00:00Z
ray ,  we should not  use mfp a t  the  beginning o f  the  study.  Rather it should 
be given a s  t h e  r e s u l t  of d i f f u s i o n  process.  In our s imula t ion ,  we 
c a l c u l a t e  the  two diagonal elements K and KL . We check t h e  consis tency / L  
comparing these  formula. When the  consistency is maintained,  we t r y  t o  
c a l c u l a t e  t h e  mfp from the  formula. 
2  Generation o f  random magnetic f i e l d s  
I t  seems t o  be a  d i f f i c u l t y  i n  the  s imulat ion with i r r e g u l a r  magnetic 
f i e l d s  t o  make the  f l u x  l i n e  conserved. This d i f f i c u l t y  is avoided i n  our  
case  by genera t ing  t h e  vec to r  p o t e n t i a l s  by random numbers. The s t e p s  t o  
genera te  the  random magnetic f i e l d s  a r e  following.  F i r s t  we embed a  l a t t i c e  
i n  the  s imulat ion space ,  whose l a t t i c e  constant  is chosen a s  the  s c a l e  of  
the  i r r e g u l a r i t y  of  the  magnetic f i e l d .  We genera te  the  vec to r  p o t e n t i a l  
A = ( A x , A y , A Z )  f o r  each l a t t i c e  p o i n t  with the  p r o b a b i l i t y  given by 
exp(-Ai/t) dAi ( i = x , y , z )  , ( 4 )  
where t is a  parameter t o  a d j u s t  the  i r r e g u l a r  magnetic f i e l d s  s t r e n g t h .  
The vec to r  p o t e n t i a l  i n  t h e  p o i n t  o t h e r  than the  l a t t i c e  p o i n t s  a r e  g iven 
by l i n e a r  i n t e r p o l a t i o n .  The i r r e g u l a r  magnetic f i e l d s  a r e  given by 
3 + 
DH = r o t (  A ) ,  ( 5 )  
with the  random vec to r  p o t e n t i a l  A.  The t o t a l  f i e l d s  a r e  given a s  t h e  sum 
of i r r e g u l a r  magnetic f i e l d  and average f i e l d .  When we wish t o  s imula te  t h e  
tu rbu len t  magnetic f i e l d s  with some spectrum l i k e  the  Kolmogorov's one, we 
prepare a  s e r i e s  of l a t t i c e s  wi th  d i f f e r e n t  l a t t i c e  cons tan t s .  And sum up 
the  i r r e g u l a r  magnetic f i e l d s  with the  weight g iven by the  spectrum. 
The motion of the  cosmic ray  p a r t i c l e s  is determined by equat ion o f  
motion; 
+ dv - 9 3 +  
m - d t  = q(E + v x  H ) .  
However, i n  our  s imulat ion we neg lec t  the e f f e c t s  from E .  I n  o t h e r  words, 
we assumed the  magnetic f i e l d s  is s t a t i c a l .  This assumption can be 
j u s t i f i e d  because t h e  v e l o c i t y  o f  cosmic ray  is e s s e n t i a l l y  l i g h t  v  l o c i t y ,  
-9 but  t h e  v e l o c i t y  of t h e  magneto-hydrodynamical waves a r e  around 1 0  t imes 
of l i g h t  v e l o c i t y  i n  t h e  i n t e r s t e l l a r  circumstances. This a l s o  means t h a t  
i n  our  s imula t ion ,  we neg lec t  t h e  e f f e c t  o f  Fermi type a c c e l e r a t i o n  o r  the  
d i f f u s i o n  process  i n  energy a x i s .  
3 Simulation 
Our s imulat ion is c a r r i e d  o u t  as follows.  We assumed t h a t  the  average 
magnetic f i e ld (Ho)  is a l i n e  with z-axis,  and the  average s t r e n g t h  o f  t h e  
magnetic f i e l d  is 3 micro gauss and t h a t  of the  t u r b u l e n t  magnetic f i e l d s  
a r e  1 . 5  micro gauss i n  average. (We note  t h a t  what is r e a l l y  meaningful is 
the  r a t i o  ( D H ) / H ~ ,  which is 1/2  i n  our  case.  The abso lu te  s t r e n g t h  is 
needed when we t r a n s l a t e  the  gyro-radius t o  the energy i n  t h e  r e l a t i o n  t o  
the  s c a l e  o f  the  turbulence  i n  r e l a t i o n  t o  the  (maximum) t u r b u l e n t  s c a l e  
( = L  ) ) .  We f i r s t  pu t  a  number of cosmic ray  p a r t i c l e s  a t  the  o r i g i n  and 
folyow the  motion numerically determined by the  equation.  The s t e p  time f o r  
so lv ing  ( 6 )  numerically is chosen a s  1/10 o f  the  gyro frequency and is 
changed with gyro frequency. Th i s  s t e p  time is chosen s o  t h a t  t h e  e r r o r  
caused by it is small  i n  our  numerical c a l c u l a t i o n  and t h a t  the  s imula t ion  
is executed wi thin  t h e  reasonable  cpu- k ing the  a t i s t i c s  of t h e  
p o s i t i o n  o f  cosmic r a y  i n  terms of bs =m and G//= f l  z$ we can c a l c u l a t e  
273 OG5.4-1 
  
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      (jl j~ +y >  0/ :j\z?   t  
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conversion of d i f f u s i o n  cons tan t s  t o  average value. 
The random magnetic f i e l d  is 
4 Resul ts  
In  f i g .  1, we show a r e s u l t  of Fig. 1 
our s imulat ion i n  t h e  form of m3 
5 Summary and Discussions 
If we assume t h a t  t h e  t u r b u l e n t  s c a l e (  L,) is 30pc and the  average 
dimension-less quan t i ty ;  Ki/(c x Lo), 
where i s tands  f o r  // and L, c r i g h t  2- 
ve loc i ty  and Lo f o r  maximum turbu len t  
s c a i ~ ? .  The s imulat ion of f i g . 1  is 1 
c a r r i e d  o u t  by assuming t h a t  t h e  h 0 
magnetic f i e l d s  ' turbulence follows 
t h e  Kolmogorov spectrum. The c rosses  d 
and c l o s e  ones show K and KL 1 0 -  25 
respec t ive ly .  We a l s o  kLde a ~ 7 ,  
0 s imulat ion by assuming o f  t h e  only  
one t u r b u l e n t  s c a l e .  We found a -1 
l i t t l e  d i f f e r e n c e  between them. I n  
magnetic f i e l d  s t reeggh is 3 m i 5 6 0  gauss ,  theuenergy o f  the  cosmic ray  with 
the  gyro-radius 10- xLO is 10 05ev f o r  the  protons.  For the  higher 
energy cosmic rays ,  the  assumption of po in t  s c a t t e r i n g  i n  the  study of its 
propag9tion i n  Galact ic  space is a f a i r l y  good approximation with the  
mfpwE . However t h e  f a c t e r  i n  f r o n t  o f  the  power o f  E is a rap id ly  varying 
both cases ,  however, we can say t h a t  
t h e  q u a l i t a t i v e  tendencies  a r e  t h e  
same. Theref o re  t h e  f a r t h e r  
s imulat ion is executed assuming t h e  0 1 -2 -3 -2 -1 
only one s c a l e  i n  turbulence of log(RL/Lo) 
magnetic f i e l d .  
The whole region can be divided i n t o  severa l  regions  by the  f i g f u s i o n  
tensor-gegendences on t h e  RL. When R is i n  the  region between 10- ' xLo L 
and 10 xLo, t h e  r t ' o  K /KL is almost cons tan t  and b8tg a r e  
p ropor t iona l  t o  R ~*'- '*~./hhen R becomes l a g e r  than 10'. xLo, K is L p ropor t iona l  t o  R ~ '  and KL is almost constant .  we a l s o  gave a & a l e  i n  
f i g . 1  t o  read t h e  mfp d i r e c t l y  6rgm K us ing ( 1 ) .  We f i n d  i n  the  region 
where t h e  RL is l a r g e r  than 10- xL0!'(2) g ives  c o n s i s t e n t  r e l a t i o n  
between K L  and K a s s  m'ng t h e  mfp. However, t h e  r e l a t i o n  ( 2 )  is not  f o l d  
f o r  R smal ler  t h i k  10 L -'*'xL . We may-gee t h a t  the re  is an another region,  
where t h e  r a d i u s  is smal ler  ?hp 10- xLo, t h e  s lope  o f  K dependences on 
t h e  R changes t o  s t e e p e r ( w R  ) . However, t h e  ambiguity o p t h e  c a l c u l a t i o n  L L becomes l a r g e  i n  t h i s  region and the  r e l i a b i l i t y  of s imulat ion may be 
small .  
We a l s o  executed s imulat ions  with t h e  condi t ion < D H ) / H ~  = 1 and 1 /4  
assuming of t h e  only one (maximum) tu rbu len t  s c a l e .  These r e s u l t s  a r e  shown 
i n  f i g . 2  and i n  f ig .3 .  The simbols a r e  s e with f i g . 1  T e whole region is 10Y. 5 divided i n t o  t h r e e  regions  a t  t h e  R L ; xLO and 10 'O*'xLo. The dependences of K and KL on R a r e  t h e  same with t h e  case  o f  <DH)/HO = // 1/2. However, t h e r e  is a quan t ika t ive  d i f f e r e n c e  i n  the  value of K KL 
and i n  t h e  r a t i o  K / . They a r e  r a t h e r  r a p i d l y  varying functionb'of t h e  
<DH). 
1 
SH/H=0.5 
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  is i   it  ti  
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 ar    
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tic ield ~8 5h   irSo5 ,  r y  e ic  t  
t e  .  • O  0 • eV r t e  t e i  
r  s ic , e ti n t tt ring  e t  t  
ag~t  i     irly  ti    
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Fig. 3 
I I 
function of <DH>. We note,  however, t ha t  t h i s  is a r e s u l t  of the 
accumulation of  small def lec t ion  by the magnetic f i e l d s  and the  f ac t e r  
should be obtained by d i f f e r e n t  ~ g q o d ;  f o r  example simulation. For the 
lower energy cosmic rays than 10 ' eV, the d i f fus ion  constants  do not 
follow the  formula (1)-(3) .  To study the cosmic ray propagation i n  t h i s  
region, w e  need a simulation o r  more sophis t icated ana ly t ic  methods. I t  is 
noted t h a t  i n  t h i s  region the  r a t i o  of K and KI is almost constant.  I f  
we assume the  average s t rength  of tu rbulMt  magnetic f i e l d  is 1.5 micro $Yes the  r a t i o  is about 10. For the smaller energy cosmic ray than 
eV, the  r e s u l t  o f  the simulation show a r e l a t i v e l y  la rge  difference 
between d i f f e r e n t  random number series. This f a c t  might ind ica te  t ha t  the 
other  mechanism than the  d i f fus ion  process l i k e  convectionshold be 
considred i n  the  study of the  propagation of cosmic ray with t h i s  energy i n  
the  Galaxy. 
Acknowledgements 
The authors thank the other  member of Akeno group f o r  t h e i r  
discussions espec ia l ly  from the  experimental view point.  They a l so  a r e  
g r a t e f u l l  t o  Prof. K.Murakami f o r  the  advice i n  the s t a r t  of  t h i s  work. The 
simulation were made by the  use of FACOM M380 of the computer room i n  the 
I n s t i t u t e  fo r  Nuclear Study. 
References 
1, Gleeson,L.J. and Axford,W.I., Ap. J .  149, L115 (1967), 
J o k i p i i  , J.R. , Rev. Geophys. and spacePhys.  9, p27 (1971) , 
Barge,P., Mi l le t , J . ,  and P e l l a t ,  R. ,  Ap. J .  584, p817 (1984), 
2 ,  Lee, L.C. and Jok ip i i ,  J . R . ,  Ap. J. 206, p 7 3 5 1 9 7 6 ) ,  
Verschuur, G. L. Fundamentals of ~ o s m i ~ h ~ s i c s ,  5 p113 ( 1979) . 
Simard-Normandim, M. and Kronberg, P .P., ~ s t r o ~ b s .  J. 242, p74(1980) . 
-
275 OG5.4-1 
Fig. 2  
· 3 3 
2 cSH/H=1.0 2 cSH/H=0.25 )( 
2 2 
....... 
....... )C ....... 0 ....... 0 0 
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...J K" ...J u ...J u )( "-
"- "- ~O :l )C ~ .... 0 .-< ......., ~ 
• - ~ 
01 
....., )( 01 ....., t 0.3 01 K,.,. • 0 0 01 0 X -' 0 
-' )( 
• 
..J 
-1 x -1 KJ. • 
• KJ. x • • 
-1 
• -1 • • 
-2 -2 t 
-2 -2 
-3 -2 -1 0 1 -3 L2 -1 0 1 
log(RL/Lo) log(RL/Lo) 
ti  >.  , t is   t  
l ti  ll ti   ti  i lds   t r 
l  i  f rent Tstgod; r l   
l  r  ic s  0 • ,  f sion t ts  t 
ll   l   t    i   is 
, e  ti    i    
t  is   tio  /  l.    
   t  leh  i ld   
ga~gs~  tio  t .   i    
10 • ,  t  i l ti    ti l  r  re ce 
t  . i  t t    
   i   i e l  
      i   is   
 
l t  
     r  
i s l      
ll  i r     t is  
       
 r 
s 
, ,L.J. r , .I.,  , l ), 
, ., S ce P  , , 
., t, .,  .,  2 , ), 
, . .  . ., ~1976), 
 Cosmic Physics,   . 
i , . ., Astrophys.  . 


A simulation of high energy cosmic ray propagation 1

The cosmic ray propagation in the Galactic arm is simulated. The Galactic magnetic fields are known to go along with so called Galactic arms as a main structure with turbulences of the scale about 30pc. The distribution of cosmic ray in Galactic arm is studied. The escape time and the possible anisotropies caused by the arm structure are discussed

Kamata, K.

A Simulation of High Energy Cosmic Ray Propagation I1 
* * *t 
M.Honda, K.Kamata, T.Kifun$, Y.Matsubara , M.Mori , K.Nishijima , 
G.Tanahashi, and M.Teshima 
I n s t i t u t e  fo r  Cosmic Ray Research, University o f  Tokyo, Tokyo, 188 Japan 
'Department of Physics, Kyoto University,  Kyoto, 606 Japan 
**The Graduate School of Science and Technology, Kobe Universi ty ,  Kobe, 657 
Japan 
Abstract 
The cosmic ray propagation i n  the Galactic arm is simulated. The 
Galactic magnetic f i e l d s  a r e  known t o  go along with so ca l led  Galact ic  arms 
a s  a main s t r u c t u r e  with turbulences o f  the s ca l e  about 30pc. W e  s tudy the  
d i s t r i bu t ion  of cosmic ray i n  Galact ic  arm and discuss  the  escape time and 
the possible  an iso t rop ies  caused by the a r m  s t ruc tu re .  
1 Introduction 
In the previous paper''), we have reported a method t o  s imulate  the  
. - 
propagation of the  cosmic ray i n  the turbulent  magnetic f i e l d s .  (We r e f e r  
t ha t  paper as  I here a f t e r ) .  In t h i s  paper, we make a l i t t l e  extension of  
the work in  I. The magnetic f i e l d  i n  Galaxy is now considered t o  be a l ined  
along the Galact ic  a r m .  We simulate the propagation o f  the  cosmic ray  i n  the 
Galact ic  a r m .  It is expected t h a t  the cosmic ray is trapped i n  magnetic 
f i e l d  of  a r m  j u s t  l i k e  i n  TOKAMAK. We a r e  i n t e r e s t ed  i n  the d i s t r i b u t i o n s  of  
cosmic rays i n  the Galact ic  arm by following reasons. A s  the  o r i g i n  of  
anisotropy of  the a r r i v a l  d i r ec t i on  o f  cosmic ray,  one may consider two 
reasons; the source d i s t r i bu t ion  and the propagation i n  the Galaxy 
( including the  leakage from i t ) .  I f  the d i s t r i bu t ion  o f  the cosmic ray  is 
not uniform, w e  can expect the anisotropy from the l a t t e r  reason. A s  many 
authors suggested, the main reason the acce le ra t ion  of  cosmic ray is by 
the shock waves from the supernova(qf, the both reasons f o r  anisotropy can 
be considered as  the same thing. The study of the d i s t r i bu t ion  of cosmic ray 
becomes important. We w i l l  s tudy the possible  anisotropy caused by the a r m  
s t ruc tu re  ( o r  by the d i s t r i bu t ion  of cosmic ray i n  the arm) and the  escape 
of cosmic ray from the Galact ic  arm. w 
2 Magnetic f i e l d  i n  Galact ic  A r m  and Model 
The method of Rotation Measure a s  well as  o thers  ( 3-4 ) gave us the 
informations of the Galact ic  magnetic, f i e lds .  I t  can be summarized a s  a 
follows; the main s t ruc tu re  of  magnetic f i e l d s  is a l i n e  along the Galact ic  
arms and its average s t rength  is 3.0 micro gauss. About the turbulence of  
magnetic f i e l d s ,  the s t rength  is about 1 .5  micro gauss and the s c a l e  o f  the 
turbulences is 10-30pc i n  average. 
A s  a model of the Galact ic  arm, we consider a r i g h t  cyl inder  with the  
radius  of 300pc, which is comparable t o  the thickness of Galact ic  d i s c  
determined by the ro t a t i on  measure. We ident i fy  the a x i s  of  the cy l inder  
with the z-axis. The 'average' magnetic f i e l d  is assumed t o  be al ined along 
the cyl inder  ax is .  The magnetic f i e l d  is assumed a s  the sum of  'average'  one 
* 
and the turbulent  ones. The 'average'  magnetic f i e l d s  var ies  with radius  by 
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where r is taken t o  be the  rad ius  of  g a l a c t i c  arm; 300pc. H is determined 0  
so  t h a t  the  average of  the  magnetic f i e l d  i n s i d e  the  arm is 8 micro gauss.  
(The average o f  t u r b u l e n t  magnetic f i e l d s  is 0 . )  The genera t ion  o f  tu rbu len t  
magnetic f i e l d s  is e s s e n t i a l l y  same with I. However, t h e  s t r e n g t h  of the  
t u r b u l e n t  magnetic f i e l d s  with same r is generated p ropor t iona l  t o  ' averaget  
magnetic f i e l d  i n  average. The s c a l e  of magnetic f i e l d s  turbulence  a r e  
taken 30pc. 
3 Simulation 
The s imulat ion is executed almost same way with I. However, i n  t h i s  
case  the  i n i t i a l  d i s t r i b u t i o n  of cosmic ray is taken a s  the  uniform 
d i s t r i b u t i o n  i n  t h e  Ga lac t i c  arm. The d i r e c t i o n s  o f  v e l o c i t y  a r e  assume 
uniform. When a  cosmic ray  goes beyond r from the  arm a x i s ,  i t  is 1 
considered t o  escape from t h e  arm. In t h i s  s imulat ion r is taken 600pc, 1 
with which d i s t a n c e ,  t h e  gyro rad ius  i n  the  magnetic f i e l d  given by (1) 
comparable t o  o r  l a r g e r  than r f o r  protons with energy g r e a t e r  than 10 1BS5 1 
eV. ( I n  t h i s  paper ,  we assume i m p l i c i t l y  t h a t  the  cosmic rays  a r e  p ro tons . )  
We observe t h e  d i s t r i b u t i o n  o f  cosmic rays  and c a l c u l a t e  the  escape 
p r o b a b i l i t y  of  cosmic ray  from the  arm. To save the  cpu t ime, we made a  
t r i c k  t h a t  t h e  escaped p a r t i c l e s  a r e  resumed i n t o  the  a r m  with the  p o s i t i o n  
determined by the  p r o b a b i l i t y  propor t ional  t o  the  d i s t r i b u t i o n  of o t h e r  
cosmic rays .  This resumption is executed f o r  the  every time s t e p  
corresponding t o  250 years .  With t h i s  t r i c k  the  number of  cosmic rays  i n  
s imulat ion is kept cons tan t .  The escape p r o b a b i l i t y  is c a l c u l a t e d  from the  
resumed p a r t i c l e  numbe$ a f t e r  the  d i s t r i b u t i o n  of cosmic ray  is considered 
s t a t i o n a l  f o r  a few 10  yea rs .  The s t e p  t i ~ g  f o r  t h e  f g l g u l a t i o n  o f  cosmic 
ray p a r t i c l e  motion isl+,aken 1 year  f o r  10 eV and 10 ' eV and 10  yea rs  f o r  
g r e a t e r  energy than 10 eV. The s t e p  time is same o r  smal le r  than t h a t  o f  I. 
4 Resul t  
In  f i g .  1, we show the  escape 
-lo5 L 
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0, 
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0, 
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I I . 
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- 
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time c a l c u l a t e d  by the  escape 
p robab i l i ty .  I t  is i n t e r e s t i n g  t h a t  
the  escape t i m  fpg  ghe cosmic ray  E6- . 
with energy 10  eV is c o n s i s t e n t  
with the  e x t r a p o l a t i o n  of e  l ' f  
time o f  lower energy by E -FB.3-6.S) 
The s loyy gf the  escape time l a r g e r  
than 10  becomes s t e e p e r ,  which can 
be considered a s  the  r e f l e c t i o n  of the  
s t r u c t u r e  o f  the  Ga lac t i c  arm. W note  
O 1 0 3 .  I I 
the escape time of  the  energy 10B6eV 
16 17 may show a  smal ler  value ,  because the  l8 r used i n  t h i s  s imulat ion is too 
log E 1 Fig. 1 small  t o  consider  t h a t  the  cosmic ray  
a t  r s u r e l y  escape from the  arm. They 1 
can be returnl&igol+,h~ arm with  the  gyro motion by the  magnetic 
- 
the  region 10  eV, t h e  escape time is propor t iona l  t o  E -(Gg%: 6fn 
In f i g . 2 ,  we show t h e  r - d i s t r i b u t i o n s  of  the  cosmic ray  f o r  var ious  5  
energy a f t e r  10  yea r s  when the  d i s t r i b u t i o n  o f  cosmic r a y  is considered 
a l r e a d y l ~ t ~ t i o n a l .  The r - d i s t r i b u t i o n  of cosmic ray with the  energy lower 
than 10  ' eV show a  similar f e a t u r e .  In  t h i s  energy region the  cosmic ray  
dens i ty  decrease  exponent ia l ly .  We can consider  t h a t  t h e  cosmic r a  ~8 a r e  
trapped i n  the  Ga lac t i c  arm. The r - d i s t r i b u t i o n  with t h e  energy 10 eV show 
 
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a  l a r g e  d i f f e r e n c e  from the  lower 
one, which shows a slower decrease  
with r. This d i s t r i b u t i o n  suggests  
t h a t  t h e  cosmic ray is no t  trapped 
i n  the  arm s o  long time and e a s i l y  g- 
escape from t h e  arm. This  is a l s o  2 c 
g 5 r shown i n  t h e  escape time of t h i s  
energy. I t  a l s o  noted t h a t  the  =-2 0 o , 
L L d i s t r i b u t f g n  of cosmic ray  wi th  the  r 
encj?g~3-lq eV is d i f f e r e n t  from the  3 10  ' 5 e ~  i n  the  shape of  the  z s  2 0 r 
s lope.  This  may i n d i c a t e  the  4 
d i s t r i b u t i o n  of  cosmic ray wi th  3. 
t h a t  energy is still i n  t h e  course  
of the  formation of  s t a t i o n a l  form 
wi th i8  the  s imulat ion corresponding 2. 
t o  10 yea r s .  Af te r  t h e  formation r 
of  the  s t a t i o n a l  form, t h e  escape 1. 
p r o b a b i l i t y  can be smal le r .  
In  f i g .  3, we show t h e  V6 i n  
average over  a l l  cosmic r a y s  f o r  . 
var ious  energ ies .  V g  is def ined by; 
x. v - Y. vx I O ' ~ - ~ ~ V  
Y VO = --2- --~T 
J x  + Y 
- Fig. 2 
r 
We can s e e  t h e  Vg become 0 300 600 (PC) 
considefgbky l a r g e  f o r  l a r g e r  energy 
than 10 ' eV. I t  takes  its maximum a t  10 17'5 and i n  the  energy 1018ev it 
becomes smal le r  value.  This value have 
d i r e c t  r e l a t i o n  t o  the  anisot ropy o f  
the  a r r i v a l  d i ref&+gn of cosmic rays .  
In  the energy 10  eV, we can expect  a  
few % of excess  of the  number of cosmic 
r a y s  coming from the  r o t a t i o n a l  
d i r e c t i o n  a r o  d  the  arm a x i s  and i n  - 
the  energy 1 "5eV a  few 10%. In the P6 
energy of 10  eV, t h i s  value  a l s o  show 
a  non-zero V e  value.  However, t h i s  
- 
value  f l u c t u a t e s  with time and a l s o  
v a r i e s  with r. We a r e  not  s u r e  the  we 
can expect the  anisot ropy i n  t h i s  
energy. 
In  f i g  .4 we show the  average o f  
I v ~ / v I  i n  var ious  energies .  I f  the 
d i r e c t i o n  of  cosmic ray v e l o c i t y  is 
i s o t r o p i c  t h i s  value  is 0.5. In the  
energy 10I8ev, it  shows a  l a r g e  
1 -  
A lo-'- - 
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-
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1 0 ' ~  16 17 18 devia t ion  from 0.5 t o  a  l a r g e r  value:  - 
log E 0.6. The d i r e c t i o n  o f  v e l o c i t y  o f  Fig. 3 cosmic r a y s ,  which s t a y  i n  the  arm f o r  
long time, a r e  not  uniform i n  t h i s  
energy. In  the  lower energy, the  value  - 
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is very near  t o  0.5.  In  the  context  of  
anisot ropy , 1 V ~ / V  1) 0.5  means t h e r e  is 
an excess i n  the  numbers of cosmic rays  
coming from the  d i r e c t i o n  p a r a l l e l  t o  
the  arm a x i s .  In t h i s  case  the  
anisot ropy is observed a s  the second 
harmonics. 
> 5 Summary and Discussions We have shown t h a t  the  cpgmic r a y s  
" .48 with the  energy lower than 10  eV a r e  
trapped i n  the  Ga lac t i c  arm. A s  t he  
log E physical  consequences, we can expect  a 
I few % ( f i r s t  harmonics) anisot ropy of  
Fig, 4 the  a r r i v a l  d i r e c t i o  y6:g-fqyic r a y s  i n  
the  energy region 10  eV t o  the  
r o t a t i o n a l  d i r e c t i o n  around the  arm 
a x i s .  We note  t h i s  is the  d i r e c t  consequence of the  d i s t r i b u t i o n  of  cosmic 
r a p p d  t h e  gyro-rota t ion o f  them i n  the  magnetic f i e l d .  I n  the  energy 
10 eV, t h e  cosmic ray  become easy t o  escape from t h e  G a l a c t i c  arm. The 
cosmic ray ,  which s t a y  long time i n  the  arm have smal ler  p i t c h  angles .  
Therefore we can expect t h e  anisot ropy o f  second harmonics t o  the  d i r e c t i o n  
of arm a x i s .  (Of course ,  if the re  is t y p i c a l  source nea r  t o  the  Ear th ,  the  
anisot rpgy i n  t h i s  energy region is a f f e c t e d  by it. ) For the  lower energy 
than 10  eV, t h e  e f f e c t  o f  a r m  s t r u c t u r e  i n  anisot ropy o f  the  a r r i v a l  
d i r e c t i o n  is uncer ta in .  The anisot ropy observed i n  t h i s  energy may be t h e  
r e f l e c t i o n  o f  t h e  l o c a l  s t r u c t u r e  o f  the  Ga lac t i c  magnetic f i e l d .  I t  should 
be noted t h e  ma ude of t h e  anisot ropy p red ic ted  he re  is c o n s i s t e n t  with 
the  o b s e r v a t i o n F S f .  However, the  d i r e c t i o n  determined by the  observat ion is 
d i f f e r e n t  from p g ~ ~ r e s u l t .  The observation of t h e  an i so t ropy  i n  the  energy 
region above 10  eV have some ambiguity y e t  caused by poor s t a t i s t i c s .  The 
more e l a b o r a t e  experiments a r e  expected. 
Acknowledgements 
The au thors  thank t h e  o t h e r  members o f  Akeno group f o r  t h e i r  
- 
discuss ions  and f o r  showing t h e i r  experimental r e s u l t s .  They a l s o  a r e  
g r a t e f u l  t o  Prof .  K.Murakami f o r  the  d i scuss ion  from t h e  experimental  po in t  
of view and h i s  encouragements. The s imulat ion were made by the  use of  
FACOM M380 o f  t h e  computer room i n  the  I n s t i t u t e  f o r  Nuclear Study. 
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B e l l ,  A.R. Mon. Not. R. A s t r .  Soc. 182 p147(1978) -  
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, .I.  i  i  ays, U  981) 
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A simulation of high energy cosmic ray propagation 1

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