The fundamental properties and some numerical results of the solution of the diffusion equation of an impulsive cosmic-ray point source in an uniform, unbounded and spherically symmetrical moving medium is presented. The diffusion-convection(D-C) function is an elementary composite function of the solution of the D-C equation for the particles injected impulsively from a diffusive point source into the medium. It is the analytic solution derived by the dimensional method for the propagation equation of solar cosmic rays in the heliosphere, i.e. the interplanetary space. Because of the introduction of convection effect of solar wind, a nonhomogeneous term appears in the propagation equation, it is difficult to express its solution in terms of the ordinary special functions. The research made so far has led to a solution containing only the first order approximation of the convection effect

Yang, G.

Zhang, G.

English

NASA Technical Reports Server

360
SH 3.2-i0
DI_USION-CONVECTION FUNCTION OF COSMIC RAYS
Gongliang Zhang and Guang Yang *
Institute of Space Physics, Academia Sinica
Beijing, China
ABSTRACT
This paper presents the fundamental properties
and some numerical results of the solution of the
diffusion equation of an impulsive cosmic-ray point
source in an uniform, unbounded and spherically
symmetrical moving medium.
!. INTRODUCTION The diffusion-convection(D-C_ function is
an elementary composite function of the solution of the D-G
equation for the particles injected impulsively from a diffu-
sive point source into a uniform, unbounded and spherically
symmetrical medium. It is the analytic solution derived by
the dimensional method for the propagation equation of solar_e
cosmic rays in the heliosphere, i.e. the interplanetary space.
Because of the introduction of convection effect of solar
wind, a nonhomogeneous term appears in the propagation equa-
tion, it is difficult to exuress its solution in terms of
the ordinary special functions. The research made so far has
led to a solution containing only the first order approxi-
mation of the convection effect. Undoubtedly it is insuffi-
cient to study only t_le propagation of the particle wi_1 not
too high energy. The solution we get with the dimensional
method, up to date, is the most general solution of propa-
gation equation in a uniform, unbounded and spherically sym-
metrical medim,,. It includes the higher orders of approxima-
tion for the convection effect, and has been used in discus-
sing various kinds of propagation effects of solar cosmic
ray, making propagation corrections and evaluating the equi-
valent diffusion coefficients$ '_ The solution raay also have
its value for reference in the discussion of the diffusion
in _he ordlnary moving medium.
It is necessary to ooint out that this solution is
valid only in the case of a constant K. For the low energy
particles, the energy of particles and also their diffusion
coefficients change as a result of the adiabatic expension
loss and other processes. In this case, such limitations
can be retrieved by the equivalen_ diffusion coefficient
reflecting the average nature of pl.opagation space.
Th_ diffusion equation in a uniform, unbounded and
spherically sy_maetrical medium is
m-_U r_U __ Vr_aU ,+ 2CVr
where U is the number density of diffusing particle, C is a
constant relating to the energy spectrum index. In the space
far from the source, the solution satisfyin£ condition
* Presently at Department of Physics, University of New
Hampshire, Durham, NH 03824, U.S.A.
0 
H .2-1  
F1FUSION-CONVECTION O     
onglia  hang and uang ang" 
I tu    si , i   
eij , i  
CT 
is a er resents the funda ental r perties 
 so  erical lts  th  ti n  th  
diffusion equation of an i pulsive cos ic-ray point 
source in an unifor , unbounded and spherically 
metrical oving edi . 
1  N UCTION  -convection(D-C' tI n  
 l entary posite cti n  t  l ti n  t  -C 
uation f r t  rticles i j ted i pulsively fro  a iff -
sive point source into a unifor , unbounded and spherically 
s metrical edi . t i  t  alytic s l ti n ri ed  
the dimensional method for the propagation equation of solar~Q 
cos ic rays in the heliosphere, i.e. the interplanetary space. 
ecause   uction f vection ct  l r 
ind, a nonhomogeneous tor  a ears i  the r pagation e a-
,   ifficult  pre s  l ti n  s  
t  r i ary ecial ctions. 'l  arch ade   s 
l  t  a s l ti n ntaining l  t e fir t r er roxi-
ation f t e nvectiun ff ct. ndoubtodly it i  i uffi-
i nt t  t  l  the r pagation f t  article ith t 
t  i h orgy. e l ti n e t it  t  i ensional 
ethod, up t  ate, i  t e ost eneral s l ti n f r pa-
gation equation i  a unifor , unbounded and s herically sy -
etrical ediulll. It includes the higher orders of approxi a-
ti  r t o nvection f ct,  as en s d  i s-
si  ri us I ds f r pagation ff cts f l r c s ic 
r , aking r pagation c rl'ections and e al ati u; t e e ui-
alent iff sion c efficients.tll 'rhe s l ti n may als  have 
 al e r r nce   i u sion f  iff sion 
i  ~he r inary ovine; ediu . 
t i  ecessary t  point ut t t t is l ti n i  
valid only in tho case of a constant K. For the lo  energy 
articles, t  e ergy f articles a  ls  t eir iff sion 
oefficients cnange as a r sult f the a iabatic e pension 
l ss d t er roce ses.  t is se, ch li itations 
n e r tri ved  t  uivalent iff sion efficient 
reflec ting the average nature of pl'opaga tion space. 
he iff sion equation i  a nif r , unbounde(1 and 
herically YllMetrical ediu   2. cV ~ .. U t:lU r'.}U _ V r 2 au r Ll ( I. , ) 
r
1 ~'f1 -+:ar ar - " a-ti -. "" ~r ,+ K 
hore  is tHe nu ber density of diffusinG article,  is a 
constant relating to tho energy spectru  index. In the space 
far fro  the source. the solution sutisfyinp, condition 
{~ Presently at Department of Physics, University f~ 
ampshire, urham,  3824, .S.A. 
https://ntrs.nasa.gov/search.jsp?R=19850026512 2020-03-20T17:01:11+00:00Z
361
SH 3.2-1o
U(r,t=o)=,_A/-__--_..(r-r._and Eq. (I.i_ist'_
U (r,%)- (_, ),& 4 _%
where € is tdiffusion-convection function I which the paper
is going to discuss in detail. It can be expressed as a func-
tion of two dimensionless parameters as follows: Diffusion
parameter Z- rZ_(4Kt), Convection parameter for time
x=(2c-l_V_t_K):_.Disregarding_,solutio_(1.2_isllk_
a kind of the diffusion caused by a source moving outward
with the convective velocity V. The equation of 4 is
-- r ++ p, +" _.<I,. ')_ 4 )
In case of no convection, V=O and_-----l,solutlon(l.2_ is a
function of a single parameter Z. When convective effect is
not too strong, $ can be expanded into a cower series of
power series of parameter X: _(X,Z_ _X_F(I,Z _, (t4)
F(1,Z_ is called the 1-th order influence functlon of con-
vection, and its equation is
Seeing that the solution must be finite at the origin, Z_O,
expression of F(1,Z_ can be taken as
_ t,__.£._-,.7_._,..._ _ ' ,.,,,, ,,-,,,, _,.+,>
If. MAIN HETHODS A(n,m,k) is a multiple series relating
to m,and k, the lowest recurrence values for m and k are
all zero. For the derivation of the general recurrence
formula of A(n,m,k), it is instructive to evaluate first
from the lower-order coefficient A(n,m-l,k_ the next one
_(n,m_k_ byrecurring k. Then, the general formula of A(n,m,k_
can be yielded as: [++L,IA_.,,, ,_) - A(.,) a.(,, - ) -_[.}.[_,
• a(o,,oj-.-I
S(,..._)-Z _+,3)
S(n,m.,_)=l,_:% .,_,--.,_--o, n_o, _o--I,n-o,(_-4)
whore oooffioien_ A(n) is de_orminod by the initial con-
dition.
The multiple series s(n,m,k% can be transformed into
a series composed of the higher-order I" function (function
_snd G).
From the initial condition at t_ o and the as]nnptotic
value (± _ ) of function as Z_- , the recurrence formula of
coefficient A(1) can be yielded :
_' "_z'..._._-_)_')C_ " "_+.",-) _+A¢+_(-_)_'' s(o, i,) --o, ,:_--,, A(,.I (."_)
361 
sa 3.2-10 
U(r,t=o)= /'/b(r-r.' and Eq. (1.1' ist '1 
~ L 
U(r,t)=(:"lJ/Z lJ/;telC,r-~~~) J '*(1,1), (1.2,) 
where ~ is 'diffusion-convection function' which the paper 
is going to discuss in detail. It can be expressed as a func~ 
tion of two 'dimensionless parameters as follows: Diffusion 
parameter Z = rZ ,<4Kt).. Convection parameter for time 
X=(2C - l')V(tiK): • Disregarding4>, solution (1.2) is likA 
a kind of the diffusion caused by a source moving outward 
with the convective velocity V. The equation of ~ is 
r2'a
1
';' +r~·(2_L)_L~4- == (2C_ 1) Vr 4 ~ ,,' ~ yo let K -:at K 
In case of no convection, V = 0 and4> = 1, solution(1.2\ is a 
function of a single parameter Z. When convective effect is 
not too strong, 4> can be expanded into a nower series 01' 
power series of parameter X: c:P(X, 7,\ = !X"F(l,Z', (1.4) 
F{l,Z) is called the I-th order influence function of COD-
vection, and its equation is ~ 
f " ( 1,) l ) "" [ - I l' -i r'J 1=' (l, 'l ) - ii-If( l. i) == i r H L-I J ~ ) (t. 5" ) 
Seeing that the solution must be finite at the origin, Z - 0, 
expression of F(l,Z' can be taken as 
f(l,'i),.t.'l ... ,·flt<. ••• ,\C.)l'" n."~-"". 
Ma. t--- ' 
II. MAIN l1ETHODS A(n,m,k) is a multiple series relating 
to m,and k, the lowest recurrence values for m and k are 
all zero. For the derivation of the general recurrence 
formula of A(n,m,k), it is instructive to evaluate first 
from the lower-order coefficient A(n,m-l,k\ the next one 
~(n,m,k) byrecurring k. Then, the general formula of A(n,m,k) 
can be yielded as: ..q.. "" 
A(ft, .. ,~) _A(II)(t<'II,,,,q[ "~l, ~11< ~<,I\~ 'M,,\<) ('2.1) 
t~)", a(\\\-I) \ 
') • n~Oj 1.\.(0,"'1= (~,,",) J"'~O; ('1.1) (l"Cl,'" = (M+I)! ll(o,O) =, 
k 'fa' II, ... -I J "-d 
S ( 1\, ... , ~) ,. ~ WI 
"' .. (-\ ~Io\-I""~')('"'i'+"") 
s(n",\t\,.I¥.) 1:' J 1<= 0" IJ 3. ..... , ,"0: 0 ) n~oJ ,"0.:1./ r\:.o, (2.4) 
where coefficient A(n) is determined by the initial con-
dition. 
The multiple series s(n,m,k) can be tl->ansformed into 
a series composed of the higher-order~ function (function 
'" and G). From the ini tial condition at t - 0 and the asymptotic 
value (1 T ) of function as z~ - I the recurrence forLnula of 
coefficient A(l) can be yielded! 
h' A <. " ) .. I L-t ) ) 6', .. 1: -'L (t) ':)<'''J __ ) .... 1\(0)(') S(OI ~ =0 n:l.-""J A(6)_, (2." 
.... l'(,;;) } 
362
8H 3.2-1o
The hig_B(_rorder coefficient A is difficult to express
in a sim_le form. floweret, it is quite convenient to calcu-
late A(1 _ with the recurr(Bnce formulae (a._ and (z.5) by
computer. It can be show_ that A(1) is an alternative seq-
uence, and its absolute value decreases when 1 increases.
Formula ( I._ ) is a nonhomogeneous equation. Its non-
homogeneous terms are composed of the same functions of lower
order. The formula of function Ftl,Z) can be derived from
formulae (t 6 ) and (2.t):
The asymi_totic value of f(n,m,Z) when t = o or Z_-is
Therefrom the recurrence formula ( :2.5 _ of coefficient A(l_
can be formulated.
III. DIFFUSION-_0NVECTION FUNCTION _(X.Z)
For convenience of discussion, function @ can be re-
written as
where Y is another dimensionless parameter, namely, the con-
vection parameter for space: y_ (2C - 1)-_-Vr _ XZ_ (_ _
Fig. 1. illustrates the value of n for calculation of
as a function of Y and T, when the relative magnitude of
the last term os less than l0 . It can be seen that the
term number of converEence increases with the value of Y
and T. _unctlon _ is probably divergent as X > 3._oo. So, we
often restrict our discussion to propagation problem within
the range X _ 3.oo.
30 2S
Abee_s_a is the d_mensionless time r, ordinate _ 30
is the number o_ terms summing up the funvt_on
o os Io 1._ o _5
D_memmnle_ 11m€ T
IV, VARIATION OF D_NSI_ U Variation of _-C function
_ith 'the 'convection p_eter for space Y and thedimenston-
less time _(-----4Kt/r*) have already been shovm in Fig. i.
ofEz] . Plotted here is the variation of diffusing particles
density U with T under the different conditions of solar
wind convections (Fig. 2.). A set of curves on the left
362 
SH 3.2-10 
'rho hir;llcr order coefficient A is difficul t to express in a simple form. HOl-leVer, it is qui te convenient to calcu-late A(l \ wi th the rOCUl~ronce formulae (2.~) and (2,..5 ) by computer. It can be shown that A(l) is an alternative seq-u nce, und its absolute value decreases when 1 increases. Formula ( ,.S ) is a nonhomogeneous equation. Its non-homogeneous terms are composed of the same functions of lower order. The formula of function Fil,Z) can be derived from formulae (,., ) and ('2.' ): 
F ( L, 'l ) = Eo l "'I. G\ ( n, 10\) A {'" f ( r\, "', ~ ) 
The asyml)totic value of f'(n,m,Z) when t ::: 0 or Z ..... is 
L ]..!\:;! <.10\+1)1 (.1.)"'S<. J T(n,""t)i ..... ""1'(2:) t ,. ~r(i) T'll\llt-'IO\) \~ ".MII 
~he efrom the recurrence formula ( 2·5 ) of coefficient can be formulated. 
II. DIFFU3ION-"ONVBCTION FUNCTION p(X.Z) 
<20.~ ) 
A(l) 
For conv i nce of discussion, function ¢ can be re-written as 00 ~= t A<.n)£l(I'I.M)Xnyl\ff(n/,,,/~) (~.I) 
where Y is another dimensionless parameter, namely, the C08-
vection parameter for space: Y =- (2C _ 1) ~~ = XZ~ (1.1 \ 
Fig. 1. illustrates the value of n for calculation of ¢ as a runction of Y and T, when the relative magnitude of the last term as less than 10 • It ca.n be seen tha.t the term number of convergence increases with the value of Y and T. functIon 4> is pro ably divergent as 1C > 3.-00. So, we. often restrict our discussion to propagation problem within the range X" 3.00. 
A bSC1S8Il is the ihmensionless time T. ordinate 
is the number of terms Bumming up the funebon 
", accurate to 10-', 
" ~ 
t 
'e j 
E 
i! 
5 
50 
40 
30 
20 
10 
y 
0 os 10 1.5 20 
D,m.lUl/on'" lime T 
IV. VARIArrrON OF DE:HSITY U Variation of n-c function 
25 
with the convection parameter for space Y and thedimension-le s ti e rr(== 4Kt/r&) have already been shovm in Fig. 1. 01'(21 • Plotted here is the variation of diffusing particles density U with T under the different conditions of solar wind convec~ions (Fig. 2.). A set of curves on the left 
S6S
S_ 3.2-1o
side are the results calculated by Formula (t.Z), taking
2C - 1- 4, or the differential-momentbu_-spectrum index of
oarticles to be 5.5. quch case corresponds to that of the
isotro_)ic propagation. But in fact, propagation of cosmic
rays particles in interplanetary space is anisotropic,the
diffusion coefficient along the magnetic line K,/ is diffe-
rent from the transverse one K_ . In this case, propa_a-
tlon equabion becomes
(€'0
where is the magnetic azimuth of observation point from
Vr (4,)
the source, and H is the modulation parameter, H-- 2N_
Substitute k,, for K when Z is calculated.
Solution (4.1) of the anisotroDic propagation equa-
tion is drawn by a set of curves of the right side of Fig2
when e = o'. It can be shown that the intensity of diffu-
sing particles decreases and the maximum time moves uo as
the convection effect intensifies. Besides, the time scale
of the isotropic D-_ prooagation is longer than that of
the anlsotropic one._Hence, it is possible to form an over
estimation of equivalent diffusion coefficient if obser-
vational data is fitted merely by the former. However, bet-
ter results can be got by using the model of anisotropic
propa_ation, oo _o 2o 30 oo os _o _.s 20
ISOtTOpIC dt|_USlOlt-cohvec(lo- [ Anmotropl¢ dl_u41on-€ol[IveCtlOn
10° 1 10'],
°° !! 1'
€
i.:Z ,o-, ,o-,
o
•_ 10"_ 20 '2.0 10-2
°
O0 )0 20 30 O0 05 10 I._ 20
Dimenstonlem t|me
(a) Iso_pie D-C (b)aniso_opM D.-C
[ 11 Zhang,Gongliang, (1978), HTGI{ _TERGY =HYSICS AND NUC-
LEA PHYSICS, 200.
[2_ Zhm_g, Gongliang, S_,IENTI_ STNICA, 22(1979), 934.
363 
sa . -10 
si e are t e r sults alculated by or ula ("2. ), ta i g 
2  - l:= , r tho differential- omentwn-spectrum index f 
particles t  e .5 • .5. ~uch ase rresponds t  at f t  
isotro;>ic ropagation. 'But i  fact, r pagation f cos ic 
rays articles i  interylanetary space is anisotropic, the 
iff sion oefficient al g t e agnetic li  ~ i  iffe-
l'ent fr  t  tr svel'se e J. •  t is se, r paga-
tl  uation eco es 
U - N i !i/. "" {~)t ~'A' C l (I. ~ ~' ~ . e'lbt 11-.ti J ci'>('(,~) '-I) 
- (41ff' )5/1 1< ... \:Sw,8 I L 1c..L'! I U 
here is tho asnetic azi uth of observation oint fro  
the source, and H is the modulation parameter, H = yr (4.1) 
K." 
ubstitute q r  hen   lculated. 
olution (4.1) of the anisvtropic propagation equa-
tion is drawn by a set of curves of the rieht side of Fig2 
hen   0·. I t n  s n t t t  i t nsity f iff -
si  artlcles creases and t e ax.i um ti e oves up as 
t  nvection ff ct i tensifies. esides, t  ti  l  
f tae is tr pic -r: r oagation i  l ger t a  t t f 
the anisotropic one.J ence, it is possible to for  an over 
sti ation f uivalent iff sion efficient i  bser-
ational data is fitte  erely by the for er. owAver, et-
t r r sults ca  be t by si g t e odel f a isotropic 
propae;a tion. 00 os 20 
IsotropIC 
10' 
I 
Y I 00 
I 
J 0 
, I 
AnllOtropoc d,H .. lOn-con •• cl,on 
10' 
J' 
00 
10"' 
2.0 10-' 
(a) Isotropic D-C 
RE~'811d.'NCSS 
00 
, so l .. ti  T 
(b) anisotropic D-C 
t 11 Zhang,Gongliang, (1971:3), HTGH ~NillGY P SICS' AND NUC-
LEAR. PHYSICS, b. 200. 
C2l ~hru1g, ongliang, S0Ib~TIA ST I , 22(1979), 934. 


Diffusion-convection function of cosmic rays

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