Intensity modulation and deceleration dynamics of solar wind by cosmic ray

Dorman, I. V.

English

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National Aeronautics and S 
Goddard Space Fli 
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ST - CR - NP - 10784 /J69-/c@6 
COSMIC RAYS AND DYNAMICS OF THE SOLAR WIND 
I 
I. V. Dorman 
& 
L. I. Dorman 
(USSR) 
10 DECEMBER 1968 
https://ntrs.nasa.gov/search.jsp?R=19690005674 2020-03-12T08:26:35+00:00Z
ST - CR - NP - 10784 
COSMIC RAYS AND DYNAMICS OF THE SOLAR WIND 
I - 
Geomagnetizm i Aeronomiya 
Tom 8, No.5, 817 - 821, 
Izdatel'stvo "NAUKA", 1968. 
SUMMARY 
by I. V. Dorman 
& L. I. Dorman 
When investigating the propagation of cosmic rays in interplane- 
tary space, solar wind is considered as given and independent of cosmic 
rays. It is shown that, considering rigorously the problem, one must 
take into account the inverse action of cosmic rays on the fluxes of 
solar plasma, which is materialized by way of magnetic fields. The in- 
tegral-differential equation is derived, which describes the interaction 
of solar wind with cosmic rays, taking into account the intensity modu- 
lation and the deceleration of solar wind by cosmic rays. 
* 
* *  
During the investigation of modulation of galactic cosmic rays and the 
propagation of solar cosmic rays in interplanetary space, solar wind is con- 
sidered as given and independent of cosmic ray intensity. In particular, in 
the equation, describing the propagation of cosmic rays in interplanetary spa- 
ce and the modulation of their density - n E13 
the particle fluxes ja with charge Ze ( a  = 1, 2, 3,  4 are the spatial coordi- 
nates r, 8, $I and therfgidity R; 4 is the function of the source) are deter- 
mined by the given solar wind velocity u(r), while the pressure of galactic 
cosmic rays is comparable or even greater than the pressure of the galactic 
magnetic field [ 2 ] .  This is why one must take into account that as the in- 
tensity of cosmic rays varies in time (with the variation of solar activity) 
2 
and in space (with recession from the Sun), the action of cosmic rays on 
solar wind varies also, which leads in its turn to the variation of u(r). 
In the first part, we shall find the equation describing the interaction of 
solar wind with cosmic rays. In the second part [31 ,  its approximate solu-  
tion is found under specific assumptions on the properties of the low-energy 
part of cosmic radiation beyond the limits of the solar system. 
Integral-Differential Equation describing the Solar Wind Interaction 
with Cosmic Rays. 
tially on cosmic rays, but undergoes itself the ingluence of the latter, 
whereupon this influence is so substantial that it may be one of the basic 
causes of variation of solar wind velocity and limitation of its propagation 
in the interstellar space. This is why all the works where the modulation of 
solar wind by cosmic rays is investigated and the solar wind is considered 
as given (see, for example, [l, 4-61 ] ,  are not strict. It is necessary to 
take into account the inverse action of cosmic rays on solar wind, i. e. to 
resolve a self-consistent problem. 
It was noted above that solar wind not only acts substan- 
Let us consider the action of cosmic rays on the radial motion of an 
isolated plasmoid with frozen-in magnetic field, the transverse and longitu- 
dinal dimensions being respectively and L. We shall consider that in the 
first approximation cosmic rays are distributed isotropically in interplane- 
, tary space. Then the pressure exerted by them will be isotropic, and at the 
distance - r from the Sun the pressure will be determined by their energy density 
Q. -
iL(r,12)l?,cdRl 
0 
where Ek is the kinetic energy of particles, n(r, R) is the hard spectrum of 
cosmic ray concentration, The deceleration of the isolated plasmoid will be 
determined by the difference in pressure of cosmic rays on the forward and 
rear surfaces: 
cliz ( r ,  R )  W 
Fdec = P  C [ i 1 ( 1 . , I Z ) - - ( ) . - 1 - L , H ) ] ~ , ~ d R =  -PL\ E,, d R .  
+ dr 
The concentration of plasma, N(r), at the distance - r from the Sun on ac- 
count of continuity law, will be 
. .I.. 
3 
N ( r )  = Iv pf r d  u-'r-2, ( 4 )  
where r6 = 1 a.u. is the distance Earth - Sun; N h  " l V ( 1 . 6 ) ;  ut = b ( r h ) ,  where 
- u is the radial velocity of plasma motion. 
of plasma will be 
Then the mass of the isolated area 
where % is the mass of the hydrogen atom. 
shall find the acceleration of deceleration du/dt = Tdec/M and the velocity 
gradient 
On the basis of (2) and ( 4 )  we 
where it is taken into account that udt = dr. 
The multiplier dn(r, R)/dr, entering into (5), is determined by plasma 
velocity u(r, t) and by transport free path for particle scattering A(r, R, t) 
by means of a nonstationary equation of anisotropic diffusion [l], or, to be 
more precise, by the kinetic equation [ 7 ] .  However, it is then impossible t o  
obtain a simple analytical expression and difficult to investigate Eq. (5). 
This is why we shall make use of the following two cases, substantially alle- 
viating our problem. First, as is shown in [l], when investigating long- 
period variations of cosmic rays, one may use with great precision the sta- 
tionary equation, considering t as a parameter. Secondly, according to [l], 
a rigorous analytical solution of the modulation problem in two simplest 
cases, when A = const and when A % r, may be represented with a sufficient 
precision (to 10%) in the form 
where no(R) is spectrum of cosmic ray concentration beyond the limits of 
solar wind; ro are the dimensions of solar wind assigned in [8] on the basis 
of hysteretic phenomena in cosmic rays; - v is the velocity of particles. 
However, in our case one should spread the integration over r in the right- 
hand part of ( 6 )  from - r to m, inasmuch as the solar wind limitation is 
4 
materialized by the quantity u(r,t) with the aid of Eq.(5). 
noted, we may consider in accord with [l, 91, t in (6)  and (5) as a paramete- 
and in the following we shall drop it. Differentiating (6) with respect to - r
and substituting into (5), we shall obtain for the determination of u(r) the 
integral-differentia1 equation 
As already 
The substitution of velocity u(r), determined with the aid of (71, into (6) 
(as r + a) will allow us to find also u(r,R). 
Transformation of the Integral-Differential Equation into a Nonlinear 
Differential Equation of Second Order. Selection of no(R). - According to [2 ] ,  
for the density spectrum of nucleons beyond. the limits of the modulating volu- 
me, one should choose the expression 
Here y ~ $ , 5 ,  no(n) 
asserted on the basis of experimental data (see [ 2 ] )  that it is knowingly 
smaller than 0.5 Bv (the value &in @ 0.5 Bv corresponds to energy density 
of cosmic rays in the interstellar medium 'L 1 ev/cm3). Subsequently, cal- 
culations will be conducted for Rmin = 0.5, 0.2, 0.1 and 0.05 Bv. Taking into 
account that in the interstellar medium cosmic rays are distributed nearly iso- 
tropically and that the number of nucleons in nuclei and protons is about iden- 
tical, we find 
per nucleon cmm3 'Bv-' . As to the quantity sin, it may be 
( 9 )  
4n 
a x --.1,4 x G.-fo- io  nucleon cm-3 'Bv' 
c 
Approximate Expression for Ek. For protons the kinetic energy is 
--- 
Ek = b( dR2 + 0.88 - 0.94) ergs; 
R is given in Bv, b = 1.6*10'3 is the conversion factor to the CGS system. 
For nucleons in nuclei Ek = b( JR2 + 3 , 5 2  - 1.88). 
nucleons in nuclei and protons is about identical, the mean value of Ek will 
be : 
Inasmuch as the number of 
. e / .  
5 
where t h e  last express ion  is v a l i d  wi th  a relative e r r o r  of no more than  10%: 
R, Bv 0 0 . 5 1 2 4 8 0 0  
Ek ex 
Ek appr  
0.90 0.98 1.07 1.09 1.06 1.00 0.90 
Approximate Expression f o r  v(R) - . Inasmuch as f o r  pro tons  v(R) = c ( R 2  + 
+ 0.88)-1/2 , and f o r  nucleons i n  n u c l e i  v(R) = cR(R2 + 3.52)-’6 
is  
t h e  average 
(11) 
v ( R )  = 0,s (nz + o,sq - t i 2  + (nz + 3,521 -34 = C R ( R ~  + 1,571 -I/?, 
where t h e  l as t  express ion  i s  c o r r e c t  w i th  a re la t ive e r r o r  no t  exceeding 6%: 
R,  B v  0 0.5 1 2 4 8  m 
Ek ex 1.06 1.05 1.04 1.02 1.01 1.00 1.00 
G p r  
Approximate Expression f o r  t h e  Dependence of A on R. According t o  [ l o ] ,  
i n  t h e  r eg ion  (R 6 1 Bv) is almost independent of R. For 3 > R >  1 B ~ ,  A - I’R 
[ l l ] .  
This  is  why i t  i s  p o s s i b l e  t o  approximately r ep resen t  
I n  t h e  r eg ion  R 3 2 - 3 Bv, A rises by Q R through R Q 15 Bv [9]. 
A = O,G3 AoYll’ + 137, (12) 
where A .  i s  t h e  t r a n s p o r t  f r e e  pa th  f o r  s c a t t e r i n g  a t  R % 1 Bv. 
of A dependence on R i s  as fol lows:  
The cha rac t e r  
R,  Bv 0 0.5 1 2 4 8 16 
A /Ao  0.79 0.85 1.00 1 . 4 8  2.63 5 . 1  10 .1  
Note t h a t  t h e  express ion  (12) s a t i s f a c t o r i l y  d e s c r i b e s  t h e  dependence of 
A on R only  f o r  R 4  15  Bv (according t o  [9], f o r  R 3 15 Bv, w e  apparent ly  
have A Q R ; however, t h i s  energy r e g i o n  is no longer  e s s e n t i a l  f o r  our  problem). 
Densi ty  of Cosmic Rays i n  Interstellar Space. S t r i c t l y  speaking, the form 
of t h e  f u n c t i o n  no(R) i n  t h e  reg ion  of s m a l l r i g i d i t i e s  i s  unknown. I n  accord 
wi th  [ 2 ] ,  w e  choose n o  (R) Q R-* ’. 
t h e  i n t e r s t e l l a r  space i s  then  v e r y  f e e b l y  dependent-on t h e  q u a n t i t y  %in. 
The energy d e n s i t y  of cosmic r a y s  Wcr i n  
. . I* .  
6 
where the last expression was obtained by utilizing for Ek(R) formula (101, 
the values of Wcr as a function of &in are as follows: 
%in* *v 10 2 1 0.5 0.2 0.1 0.05 0 
Wcr, ev/cm3 0.39 0.71 0.84 0.96 1.09 1.15 1.20 1.32 
Transformations of Eq.(7). Substituting (8), (10) - (12) into (7), we 
obtain 
where 
By variable substitution we transform Eq.(14)  to the form 
P/R,,l,, 
The results of numerical calculations of Y ( B ,  bin) at y = 1.5, are com- 
piled in Table 1. 
T A B L E  1 - 
I--- \I O W O F F 5  -____ 1 0,7 I 1,0 1 1,5 1 2 1 I 1 
1,15 1,0S3 0,900 0,739 0,590 0,439 0,333 0,lGG 
2,84 7,33 1,SO 1,5!15 1,183 0,895 0,637 0,459 0,339 0,166 
4,GO 3,19 2,32 1,805 1,235 0,912 0,659 0,IGO 0,330 0,166 
7,15 3,80 2,117 1,850 1,243 0,913 0,GGO I 0,460 0,340, u,lGti 1 1.34 1 1.30 1 1 
- - -  
7 
G,X9 0,590 
1,63 2,30 
0,895 0,657 
0,975 1,71 
0,012 0,657 
0,355 1,71 
0,912 0,657 
1.71 -q(y) 0,0175 5,346 0,112 3,216 J.528 0.955 
Assume that Eq,(16) is resolved relative to 13, i.e. that f3 = B(P. hin) 
has been found. Then, inasmuch as 
0,439 
3,99 
0,459 
3,29 
0,459 
3,29 
0,459 
3,29 
and, according to (16) and (17), 
we obtain 
Eq.(21) is written in dimensionless variables 
\I = U / ~ m 9  p = r / ' r a 9  a = A o / r g  
and the denotation cp(Y) d p  / GY. The dimensionless parameter 
is determined only by solar wind concentration near the Earth's orbit and by 
the density of cosmic rays in the interstellar space {coefficient 2). 
Determination of Function $(Y). In dimensionless variables (221, and 
according to (16) and (23), parameter Y will be 
Y = -2,s. 10-2u bN&:hop-2l)/'lt-', 
Functions #(Y) = df3 / dY, which are found by graphical differentiation 
( 2 4 )  
with respect to data of Table 1, are compiled in Table 2( when finding dB / dY 
at small and great 8 ,  asymptotic expressions (17) and (18) were used). Note 
that Y varies within the limits 0 f Y S Ymax, where Ymax 1.34, 2.84, 4.60 
and 7.15, respectively at &in = 0.5, 0.2, 0.1 and 0.05 &. 
.- 
Bv 
Bv 
Bv 
Bv - 
8 
It follows from Table 2 that in the region 0.05 C Y B Y m z  function $(Y) 
may be approximately (with relative precision +lo%) represented in the form 
q ( Y )  = -0,8 Y-2. (25) 
As Y -+ 0 ,  we have in accord with (18), 
little from (25). Besides, the region of very small Y (correspondnng to the 
regions of very great 6) has no essential value for our problem and this is 
why we shall spread the variation of Y over the entire region. 
$(Y) = -0.87 Y- 5'3 . This differs 
Differential Equation for the Determination of I/J. Substituting (25) into 
(21) and taking into account (24) and !25), we shall obtain a nonlinear differ- 
ential equation of second order for the determination of $: 
9" = $/ [ 5.1 i)--13u 2N6 p-2$'$-2 + 2 p.4 + *'j,-i __ &'hO-i]. (26) 
As to boundary conditions, we actually have only one condition: by defi- 
nition $(I) = 1. 
of the differential equation (7) or relations equivalent to it. 
The second condition for $'(l) must be obtained with the aid 
*** T H E E N D 
I Z M I R A N  Manuscript received on 
2 April 1968 
CONTRACT No.NAS-5-12487 Translated by ANDRE L. BRICWT 
VOLT INFORMATION SCIENCES, INC. 
1145 - 19th St.NW 
WASHINGTON D.C. 20036 
Telephone: 223-6700 (X-36) 
on 10 December 1968 
References follow... 
9 
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Cosmic rays and dynamics of the solar wind, 1

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