Radiation retardation effect on angular and energy distribution of charged particle flow across magnetic fiel

Gernet, G. E.

English

NASA Technical Reports Server

~ 
19 APRIL 1967 
National Aeronautics and Space Administration 
Goddard Space Flight Center 
Contract ?;a J S - 5 - 1 2 4 g 7  
ST-PF-10 589 
VARIATION OF THE ANGULAR AND ENERGY DISTRIBUTION IN A 
CHARGED PARTICLE FLOW ACROSS A MAGNETIC FIELD 
bY 
G. E. Gernet 
(USSR) 
I 
I 
I ITHRUI 
W 6 7 -  2 5 8 2 5 
P 
https://ntrs.nasa.gov/search.jsp?R=19670016496 2020-03-16T18:18:15+00:00Z
. L 
S T - P F  -10589 
VARIATION OF THE ANGULAR AND ENERGY DISTRIBUTION IN A 
CHARGED PARTICLE FLOW ACROSS A MAGNETIC FIELD 
("1 
Doklady A.N. SSSR, Fizika, 
Izd-vo '"AUKA", Moskva, 1967 
Tom 173, NO.3, 544-545, 
by G. E. Gernet 
The influence is determined of the radiation retardation on the distribu- 
The results obtained may have a significance 
tion by angles and energies in a flux of charged particles across a magnetic 
field layer of finit? thickness. 
in astrophysics when considering the motion of fast electron fluxes through 
regions of stellar atmospheres and nebulae with intense magnetic fields. 
* 
* * 
The influence of radiation retardation on the motion of a charged particle 
We shall now determine in a uniform magnetic field was considered in ref. [l]. 
how the radiation retardation influences the distribution by angles and energies 
in a flow of charged particles across a magnetic field layer of finite thickness. 
We shall consider that the magnetic field is concentrated in a plane-paral- 
Assume that a flux of par- 
le1 layer of thickness L ,  that it has inside the layer a constant intensity H 
and is directed along the normal to layer's surface. 
ticles with a specific energy and a density i~(&), 0 < €lo< n / 2 ,  is incident upon 
it, 0, being the angle between the particle'% velocity direction and the axis OZ 
of the coordinate system and the direction H coinciding with OZ. The character 
of the distribution by angles and energies of particles having emerged from the 
layer is easy to represent. 
Let us denote by e the angle between the direction of the velocity and the 
axis 
we shall also have e = 0. 
e a =  x / S  the particle will be moving in the layer a long time, losing nearly 
all of its tranverse velocity, so that e g i .  
OZ at egress from the magnetic field layer. It is evident that ate0= 0 
On the other hand, for As e, increases, so does e. 
Therefore, one may expect, 
-- 
(*> TZMENENIYE UGLOVOGO I ENERGETICHESKOGO RASRREDELENIYA V POTOKE ZRYAZHEN- 
NYKH CHASTITS PRI PROKHOZHDENII CKEKEZ MAGNITNOYE POLE. 
u . 
2. 
and t h i s  is corroborated by d i rec t  calculation, tha t  a l l  egress angles e are 
comprised within a cone 0 < 0 < 0,,, of which the aperture e,, depends on the 
thickness of the layer and the intensity of the magnetic f ie ld .  To every e 
inside the cone correspond two values o f e , ;  inasmuch as the radiation energy 
losses depend on $,particles with two energy values move i n  the flow of par- 
ticles having emerged from the layer along each direction e. 
Passing to  the computation, we notethat,  according t o  [ l ] ,  i n  the extreme 
r e l a t i v i s t i c  case the transverse component L*Idecreases with time according t o  
the l a w  
where 
The longitudinal velocity L ’ ~  =vu cos Oa remains invariable. 
the t i m e  t of motion through the layer may be expressed through vz 
layer t h i h e s s  L 
This is why 
and the 
t = L / ut =, L f vo cos eo. (3) 
Let us introduce tg 13 = uI / cz and denote 
then (4) 
It  stems from t h i s  formula that  t o  one value of e correspond two values 
1,2, and is determined by the 
of Bo , provided 8 does not exceed the value e,, ( O u ’ < j u  ,Oo”> e,,). 
value of em is reached a t  0,) = 6, where h. tg<,, 
re la t ion  tg On,  = 2 / 3k. 
The maximum 
The energy of the pa r t i c l e  is determined by the formula 
th (k tg e,,). I 1 sine0 E E,, me2 ---=- 
which is obtained from formula (12) of r e f .  [ l ]  a t  subst i tut ion of expression 
(3) fo r  the time - t by the angle 0, *. 
metric form (by the angle % ) .  
of t h i s  dependence a re  given i n  Fig.1 f o r  
several values of K .  The flux of par t ic les  
emerging from the iayer  through one element 
of so l id  angle, 
Formulas (6) and (5) express the dependence of 1 / E  - 1 / E  on e in  a para- 
The graphs 
dN=2;rj(O) sin O do, 
is  equal t o  tile sum of fluxes of par t ic les  
entering the layer a t  angles e, ’ and e o 1 ’ .  
We have 
t g e  - 
j ( Q )  = lo (6,’) cl C O S  0 / CZ cos Oo’ + io (e,“) cl cos 0 I cl cos 00”. 
* see next page. 
(7) 
Fig.1 
3.  
c 
Utilizing (S), we may obtain 
d COS 0 / d COS e o  = COS' 80 / cos3 0 X ch? (X t g  On) / [i - 1; t& th (k tg e o ) ] .  (8) 
In conclusion we shall bring forth estimates for the values of the coef- 
ficient k for some of the cosmic objects. 
H - 1 gauss, we have IC - 5-10-8. 
H - 103 gauss, 1; - 5.104. For some of the giant stars of later spectral clas- 
ses [3] L - 1013 cir, H - 105 gauss,k - 0,5. 
For the Sun, assuming L - 10'2 CJI, 
For a magnetic spot of the Sun, L - ioio CJI, 
I wish to extend my thanks to Prof. L. E. Gurevich for the discussion. 
Received on 6 April 1966. 
REFERENCES 
1. G. E. GERNET. Dokl. AN.SSSR, 168, 1, 1966. (ST-PF-10490) 
2. I .  YA. p(=MERANCHUK. ZhETF, 9, 916, 1939. 
3. A. B. SEVEFWYY. UFN, 88, v.1, 1966 
CONTRAfl N~.NAS-5-12487 
VOLT TECHNICAL CORPORATION 
1145 19th St. NW 
WASHINGTON D.C. 20036 
Translated by ANDE L. BRICHANT 
on 19 April 1967 
DISTRIBUTION 
GSFC 611 (lo), 612 (6), 613 (8), 614 ( 6 ) ,  615 (6), 640 ( 8 ) ;  rest as usual 
* (from the preceding page) It follows from Eq. (6) the result, obtained 
by Pomeranchuk [2] as early as 1939, 
approaching a constant limit, not dependent on Eo. 
we do not replace here the true trajectory by a straight line. 
that as Eo+ - , the final energy is 
However, contrary to [2], 


Variation of the angular and energy distribution in a charged particle flow across a magnetic field

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