Making use of the perturbation method based on the nonlinear differential equation theory, the author investigates the classical motion of a relativistic electron in a class of curved magnetic fields which may be written as B=B(O,B sub phi, O) in cylindrical coordinates (R. phi, Z). Under general astrophysical conditions the author derives the analytical expressions of the motion orbit, pitch angle, etc., of the electron in their dependence upon parameters characterizing the magnetic field and electron. The effects of non-zero curvature of magnetic field lines on the motion of electrons and applicabilities of these results to astrophysics are also discussed

An, S.

English

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RELATIVISTIC ELECTRON I N  CURVED MAGNE2IC FIELDS 
Physics Department, Bei jing Teachers College 
china 
ABSTRACT 
Making m e  of the perturbation method baaed on the 
wnlinear differential  equation theory, the present work 
imert i&er the classical mt ion  of o re la t iv is t ic  elea- 
tron i n  a class of curved magnetic f ie lds  which may be 
a a 
written u ~ ( 0 ,  ~ y . 0 )  i n  cylirdrical ooordinates (R,?, 2). 
Under &atneral astrophysical condition6 the author derives 
the molyt iaal  expressions of the motion orbit ,  pitch angle 
etcr. of the electron i n  the i r  dependence upon parameters 
oharrroterising the magnetic f ie ld and electron. The effeat8 
of nsn-zero curvature of magnetic f ie ld l ines on the motion 
of eleotronrr and applicabilitiea s f  there result8 t o  aatro- 
PhJraicr are also discussed. 
Ia ar t rephpics ,  some ourved a~gnet iu  f ie lds  with suffiaiently 
small f ie ld  gradients m y  approximately be written i n  a local eoordi- 
mate r p t e ~  u
where Do is  a constant, and (IZ.tp.2) denote the cylilvirical coordinatee. 
The a l u r i u a l  motion of a re la t iv is t ic  electron of c b r g e  -e i n  the 
f ie ld  (1) aan be exactly solved by the tapologioal method, which wa8 
i m a r t i g t e d  by the author i n  sou detail.' The purpose of the preunt  
paper i 8  to  find further the analytioal solution for  the motion, and 
then t a  extend the results t o  more general magnetic fields. In  the 
following treatment, the influence of the radiation damping v i l l  be 
neglected. 
BESULTS 
By virtue of the assumed ( without loss of ganerolity ) i n i t i a l  
position and velacrity 
A a a a 
r It& ro(%vO,O)* v It& = ~ o ~ ~ ~ ~ ~ p ~ ~ ~ ~ o ~ ~  (2) 
the f i m t  integral of the equation of motion f o r  a mla t iv i s t io  eleo- 
tron i n  the f ie ld  (1) may be put into the for r  
398 OG 6.2-11 
I I I  reTR  AGNm'I EL  
Sbuyuan An 
~Bics , !e  
C
 ua   
nonlin , v  
DYestis-tes .o  a c-
  a   II&  ay 
.a. .a. 
as B-B(O,B"O)  n  c ( ,f,Z). 
D general s iY  
anal c  
tc     
c acteriz  c s 
 oD-zero   
 ctrons s o s s  s
J 7sics  d. 
INTRODUCTION 
n s o ysic a c ma etic  vi Ci
aal  a     c i-
nat system as
.a. .a. 
B .B(O,B",O), B., .. BO> 0, (1) 
BO  , R,«f,Z) nd t s. 
c assic    y  har    
 c  o c  v s 
1 nves s-  .1  se t il.  .e
r s  c   
o  
v at-  wi  
t . 
R
:81 Tirta  &Ssu  vi t   e al! ) ! 
y oci
:/t.o ·:O(RO' ,O), (2) 
rst    c   rel c c-
    01"ll 
, 
• 
https://ntrs.nasa.gov/search.jsp?R=19850025797 2020-03-20T17:59:31+00:00Z
i n  -ah wb= e ~ d m c  is the relat ivis t ic  oyclotmn frequency. 7 the 
Lorents frotor, nnd I$, the curvature radium of the f ie ld l ine p s s i n g  
thro- the i n i t i a l  point. Usiog tho porturbetion method braad on 
Foioo.ne t h e e 2  to  solve the nonlinear a ~ t ~ m m o m  equation8 (3)-(5). 
with the imit ia l  oonditione (2) and 
a aoaditien that is adequately met i n  sstrophJrsice, we get the eqor- 
t i o m  of eleotron trajectory whiah, up t o  a d  includiag of the aeaond 
order i n  p. axe 
On differentiating (3). (4) and (5) with rempect t o  the time, one can 
obtain further the analytical expression of the electron velmity. It 
is apparent f ror  (7 ) - (9 )  that the motion of the electron i n  the f ie ld 
(1) may be represented as the superposition of both the helical rotion 
with gyrrtien radius 
a d  the ourvuttuu d r i f t  motion with d r i f t  veloaity uc. 
Tho pitoh -10 of an electron .roving i n  a curved magnetio field,  
p , and its mean value $, definad as 
are ou8torurily aaloulated i n  the refemme f r e ~  where the d r i f t  relo- 
c i ty  ef the alectron vanishes. Following this convention, the pitah 
angle for  the eleotron i n  the f ie ld (1) is found t o  be 
399 
OG 6.2-11 
(:5) 
(4) 
(5) 
 v ioh 'b- Bohao a ro ency, -y
z a t r, a Ro dius  assi  
roqh m U i t. a n e e tebati .eth  ase  
P n ar ' heor;y  utono ous ll )-(5), 
vi  U i ti a  
vbere )A.. ~ 01 wbRO« 1 , ~.1.0:6' ~H.O I ' ~ - v/o, ~.J.D • ~io+ ~~O) 1/2, 
(6) 
/3/10 • A,o, 
 o n a 8  a t Ph7 os v  t t   __ 
na f o v io   n  n  c
 ., r
(R/Ro)-1.JA.E~zo/~)+)\( ~lIoIlS )2J (1-oos wt )+)4(~Ro/1S )Binwt, (7) 
(z-.o t )/Ro.)A~~zcllS )+p.(~IIc1f, )2]sinwt-p.(~RO/1S )(1-o08"" t), (8) 
f -(1- 2p..fJ'&oIfJ)~/ot:.2;(""cI~) ~~zcllS )Sillwt-(f,RO/~){1-oos wt)], (9) 
vbere 
 ti  ,   spe  ll
 ocity. 
.    
sa    moti  
a o
A2 A A2 2 4 2 2 2 2 
P • [t'J.O 2'u""zO t"110-"J.0 2(~0 fJzo f3J.0- (3110)]1/2 
-jJ. -+ r-' +)A. -+-' fJ2 f, ~2 ~4 fJ2 1S2 ' 
n  .ature   o  • 
c 
e angle  avi   c 
""  D  a .ea  1f' e
2~ / 11'- sin-1 (~iI' Jo sin21/' d(wt)J1 2, 
c stoma c c  erenc  ram   v l
o  t  e o  s. Oll o
t  c   l      
In  partionlor, for  an electron i n  typical curvature motion, corresgon- 
dini3 to 1 I&o 1 , the eqa. (10) and (11) reduce to  
DISCUSSION 
In  topies oenoerned with the propertier of the motion and radiation 
of a r e h t i v i s t i c  electron i n  a magnetic f ie ld what is taken in te  
acrcount is mually the influence of radiation damping on the motion and 
3 pitch angle (the so-called wradiation crompressionW) , and somotires 
other effect8 l ike the magnetic lens. However, our resul ts  uhov that 
fo r  a re la t iv is t ic  electron i n  typical curvature motion the influeme 
of the olrrreture of magnetic f ie ld l ines is  also important. 
The l a t t e r  ie luence  w i l l  become quite clear i n  the specrial w e  
vL04 assooiated with primary particle@ flying out along magnetic! f ie ld  
l ines f ro r  the surface of pulsars. In  th is  case, from (12). t+@p. Thir 
indieatem that owing t o  the effect of the non-zero curvature of f ie ld  
lines, the i n i t i a l  motion of the electron, even i f  the initial tranr- 
verse velocity vanishes, can not generally be maintained, but b e  t o  
develsp into the helical protion with the pitoh angle Y i f i a d  the mnr- 
f 
t ion ndlu pLp?, pld the ourvature dr i f t .  Another special case e o c a  
2 i n  vLo=vso~ -FBadB2& +v fo r  vhich the pitch angle of the electron is 
r t r i e t l y  equal ts sero. r 
It aan be expected that these results should be codwive  t o  aalau- 
latin# o r  predicrting aynchro-curvature radiations from some nontherral 
soumes, and osuld exert an influence on the p m e s s  of the elea$.ron 
momentum distribution wone-dimensionalizationn along curved ragnatio 
f ie ld l ines due t o  radiation damping. F'urthennore, h e n  the effect of 
d i r t i o n  damping is taken into account, it may be easily deduced 3 
that, as the result  of the nradiation compressionw, a re la t iv is t la  
electron i n  the f ie ld  (1) should move approximately along Corm sp i ra l  
i n  the guiding center frame, and tend finally tovarda the limiting 
motion aomespsading t o  the l a t t e r  special mse  mentioned above. 
400 OG 6.2-11 
(10) 
(11 ) 
 ti v.la r  tr   l T p
ng . ~~o~I~Hol'  8 )   
11';,.1f.;. ( )).2 +2)J.~zO/ ~ + (3~0/ tl )1/2. (12) 
 io  co c r  vit  s   
 lat T e     aagneti l   to 
c .nt  ua. l    i   aoti
l   8 lled "radiation ooapres ion")3  aetia
s   ve  8  s
r  ti       fluenc  
 cu Ta  l   8   
 t r nflu  l a it     ial cae  
~O.o ci t  s i g  l   i ld 
s roa   8   i8  2),~~~  is 
c tes i    t   t r  i ld 
 tial t   l a
T  T  t haT   
elo  t    m ti  vi t  c l  1f-)). an   gr&-
2 . 
i  ra i .. ra)J., lus  c r t r  t i l  ooour  
  J,O.T sO. JAY/3;o/ {32 .. -JAV r w   l   t  ia 
s riot17 l o z
o   8  t  l  onduci   o lo.
ti g r i ti 8 t re   80a t era l 
rc co l      roce   o.r
 .. " ensi al tion" l  magne c 
i ld li s   urtherm  whe   f t 
radiati  i  8 t  t  . 8   
 t  lt t  "radiatio  o ",  l tiT tio 
l tr  n t  i ld ) 8 l  i t l  ornu iral 
 t  i i   a  t i ll  w s t  aiting 
ti o rrespon    tter ial caS  ti  T
( 
, 
SQ f a r  we have found the analytical solutions fo r  a r e l a t i v i s t i c  
electron moviag i n  the f ie ld  (1 ). Let \u consider now the elms of 
curved magnetic f ie lds  
(where I is a real  constant), which is  more gbneral than the f i e ld  (1) 
and n d w e s  t o  (1) at Nd). Most of cooen c u m d  magnetic f ie lds  with 
axial  myretry i n  astrophysics ouch a8 the digole magnetic f i e ld  may be 
expressed by (13) i n  a local f r e w  of refrerace i f  only the Pognitude of 
the f i a ld  gndien t  along magnetia f ie ld  l iner  is negligibly smull. To 
maintain the f ie lds  (13). there must be e lec t r ic  currenta flowing i n  
the direction pr ra l le l  t o  8 ,  the density 3 of which is  given by 
when j0 denotes the current demity at the i n i t i a l  point. I n  applica- 
tion, one aan select  a confiquration of magnetic f i e ld  from (13). (14) 
so that it i s  appropriate fo r  the considered astronomical objeat. It 
m a ~ r  ba verified that under the condition 
preceding results baed  on the f ie ld  (1 ), provided a& %, w i l l  continue 
t o  be valid f o r  the f ie lds  (13). 
A C I O J O W L E M ; m  
'Ph. author is  grateful t o  Prof. Liaofu Luo f o r  useful disawoione. 
REFERrnCES 
1. Am Shuyuan, (1984). Proc. National Cosmic Ray and Hi&-Energy 
Ast r@pby8i~s  Conference, Chongqing, Chinrr, unpublished. 
2. Hayashi, C., (1964) , Nonlinear Oscillations i n  PAyeical Systems, 
M C G ~ ~ V I E ~ ~ ~ ,  Raw York. 
3. Suvorov, E. V. et.  al., (1973), Astrophys. Space Sci. 23, P189. 
401 
OG 6.2-11 
o   Y  iY
aOT n   . us  c  ..   
.agneti  
• • N B • B(O, B" 0), Br,/BO • (R/Ro) (n) 
 N   aor e   
reduc   . o.  . urve   
sy.a  s s p  __  
. 1»7   aae  ren ma  
e  radi c  s a l. 
s ),  s  
1\ ... a  z  j   ,
(14) 
ere o ns  t.  -
c   aagnet l  a ), 
   ai ct. 
y e nrifie
INl({)A-1, for i3J.0 '" 1~lIo" 
I N I « ()AI~zol/~ +)i r1, (15) for f3J.0« J A,o I 
aae     to':' wb' l 
 Yal  l s 
KNOWLEDGEJmn' 
The  F f.  cussions. 
ENC~
n 1l1'll& ), F . al gh-Energy 
ophysics , , ina, . 
. ~ashi ., 64)   h si  
cGraw-Hill, Ne  . 
. y, . . ta ., , t ys.  i. ~, . 


Relativistic electron in curved magnetic fields

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