Restoration of original synchrotron radiation formul

Noerdlinger, P. D.

English

NASA Technical Reports Server

U. of Iowa 68-27 
Restoration of the Original Synchrotron 
4 2 2  ZC3 * Radiation Formula P = 2e H, y /bo 
Peter D. Noerdlinger 
Department of Physics and Astronomy 
University of Iowa 
Iowa City, Iowa 
April 1968 
* 
Research supported in part by the National Aeronautics 
and Space Administration under grant %C 23: * AGA- / L  - u ~ / - P o &  
https://ntrs.nasa.gov/search.jsp?R=19680019352 2020-03-12T09:53:21+00:00Z
2 
I 
ABSTRACT 
IJfi&er most ccndi t ions,  such as cases where the r ad ia t ing  
e lec t rons  a r e  confined t o  a f ixed o r  slowly expanding region,, 
recent  r e s u l t s  amending the synchrotron rad ia t ion  r a t e  cquation 
t o  replace H, by H2 are not applicable;  the o r i g i n a l  form of the 
equation p reva i l s .  
2 
c t 
The formula usual ly  given' for the  r a t e  of retdiat,ion from 
an u l t r a r e l a t i v i s t i c  e l ec t ron  of energy ym c2 sp i r a l ing  with p i t c h  
angle a i n  a uniform magnetic f i e l d  8, namely 
0 
4 2 2  2 2 2  p ( 7 , a )  = 2e H 7 sin c 
has recent ly  been cr i t ic ized.? ' j  Spec i f ica l ly ,  i t  w a s  pointed out 
t h a t  t o  see the r ad ia t ion  a t  any appreciable in t ens i ty ,  3 d i s t a n t  
observer must be loca ted  so tha t  a l i n e  jo in ing  him t o  the e l cc t ron  
makes an angle (2 with H". 
the  power spectrum, and the t o t a l  rad ia t ion  r a t e  increased by the 
Doppler f a c t  o r  
Therefore, he sees the gyro frequency, 
where v = e lec t ron  ve loc i ty .  
f a c t o r  s i n  a be omitted from (1) and t h a t  the r e su l t i ng  formula 
A s  a r e s u l t ,  i t  i s  proposed t h a t  the 
2 
( 3 )  4 2 2  ~ ( 7 , a )  = 2e H 7 /ko 2c2 
<> . 
be used i n  a l l  ca lcu la t ions  of synchrotron emission. '-'"' Equation ( 
*--..la :-- 
W U U L U  I l , l & J l j i  a nofizero rad lc t ion  r n t c  vhen cx = 0, an iqcss:i??le r e s u l t  
4 
tha t  suggests f u r t h e r  scrutiny; however, 
s ingular  case of no physical  importance. 
The new calculat ions,  i n  f a c t ,  2, j 
t h i s  could conceivably bc :L 
are  qui te  rigorotis f o r  the 
case t o  which they are  applied, namely an e lec t ron  t h a t  s p i r a l s  
forever along a p e r f e c t l y  uniform mslgnetic f i e l d .  
however, one deals  with electrons t h a t  a re  forced t o  occupy a 
bounded region of slowly changing s ize  and posi t ion,  f o r  o w  reason 
o r  another. For example, the f i e l d  l i n e s  may be r a t h e r  disorderc\d, 
o r  may gradually curve so as to  wind through ct bounded rc3i:ion. 
Sca t te r ing  by magnetic i r r e g u l a r i t i e s ,  mirroring, sca t te r ing  by 
protons, o r  the two-stream i n s t a b i l i t y  may prevent, the electrons 
from proceeding i n d e f i n i t e l y  along H. If there i s  somc' re:sidiinl 
streaming along H, or a bulk motion, or expansion of t.hc e lec t ron  
cloud, i t  w i l l  be c l e a r  presently t h a t  t h i s  should be t r e a t e d  by 
standard not by the introduction of the f a c t o r  
( s i n  a) . The reason i s ,  tha t  the new e f f e c t  being introduced 
through ( 2 )  i s  valid only so long as the e lec t ron  t r a v e l s  toward 
the observer; during any return t r i p ,  one must pay f o r  the time 
borrowed through the time contraction impl ic i t  i n  ( 2 ) .  
t h i s ,  consider a simple model i n  which €I i s  uniform, but t.nd-plates 
which r e f l e c t  the e lec t ron  are placed a t  the o r i g i n  and some point  
a distance L from the o r i g i n  toward the observer. Frac t icn l ly ,  
t h i s  would induce a pulse of rad ia t ion  as the e lec t ron  i s  re f lec ted ,  
In m o c t  C Z G ~ S ,  
4 
4 
-2 
'1'0 sc'c 
4 
5 
but t h i s  could be reduced t o  as small a proportion as desired of the 
t o t a l  rad ia t ion  by increasing L. Actually, the abrupt r e f l e c t i o n  at 
the ends w i l l  be seen t o  be i r re levant ,  and the model serves as w e l l  
f o r  cases where the f i e l d  l i n e s  gradually bend back on themselvc:;, 
o r  where the e lec t ron  i s  mirrored o r  sca t te red  so as t o  makc i t s  
mean vector  veloci ty ,  averaged over a su i tab ly  long  time, much less 
than c i n  magnitude. 
In t h i s  end-wall model, the e lec t ron  t r a v e l s  toward the 
observer a time T = L/v  COS^ and an equal time away, as measured 
i n  a l o c a l  rest frame. However, the observer, due t o  the f i n i t e  . 
veloc i ty  of l i g h t ,  sees the electron approach and recede during 
consecutive t i m e  i n t e r v a l s  T- and T , respect ively,  when> i 
T- = T - L(coSCXj/C (approach) ( h a )  
and 
T+ = T + ~ ( c o s a ) / c  ( recession) .  
The f r a c t i o n  of observer time during which the electron approaches 
i s  
fo r  v xz c. Without the re la t ive  t ime-di la ta t ion and contraction 
given by ( 4 ) ,  one would have j u s t  f -  = - 2 .  Thus, exact ly  the same 
6 
c 
I 
' \  
time transformation given by ( 2 )  leads t o  a reduction of f -  by a 
fac tor  s i n  a. During periods of recession, the u l t r a r e l a t i v i s t i c  
p a r t i c l e  cannot be seen. Therefore, i f  one has an ensemble of siicli 
p a r t i c l e s  rad ia t ing ,  all having s t a r t e d  with the same a but bt-in;: 
a t  d i f f e r e n t  po in ts  along t h e i r  t r a j e c t o r i e s ,  only the f r ac t ion  f-, 
which contains the f a c t o r  s i n  Q, can be seen a t  one timc, nnd thc- 
t o t a l  r ad ia t ion  i s  reduced t o  i t s  former value (1). The spcc t r a l  
2 
2 
2 d i s t r i b u t i o n  P( v ,  y , a )  "for tui tously"  
only by the same f a c t o r  ( 2 )  as  appeared i n  P ( y , a ) ;  hence the fore-  
going argument r e s to re s  P (v ,~ , cx )  t o  i t s  o r i g i n a l  value. 
was supposed t o  be modified 
6 
To be sure t h a t  the same kind of argument can be developed 
f o r  a r a t h e r  d i f f e r e n t  model geometry, consider the mods1 of 
Komesaroff f o r  a non-thermal emit t ing region i n  the Southern Nillqv 7 
Way. I n  t h i s  model, an i so t ropic  d i s t r i b u t i o n  of e lec t rons  i s  
trapped i n  a t o r o i d a l  magnetic f i e l d .  Takakura and Uchida a s s e r t  
t h a t  the f ac to r  (sins)-* inval idates  the analysis  of Komcsaroff. 
8 1 
Yet e lec t rons  t h a t  approach the observer mus t  r ec i r cu la t e  aroiind 
the cyl inder  before they can again approach. Thus, i f  one 
described t h e i r  o r b i t s  i n  terms of the  retarded time a s  secn by 
the observer, they spend only a f r ac t ion  f '  of the time i n  p a r t s  
of t h e i r  t r a j e c t o r y  where they can be seen, where f '  again contains 
the f a c t o r  s i n  0. Rather than ca lcu la t ing  t h i s  i n  d e t a i l ,  one can 2 
'7 
derive i t  i n  a more f'undamental way. The e lec t rons  having the 
cor rec t  p i t c h  angle a t o  be seen have an 2 p a r e n t  ve loc i ty  ( t ak inc  
v % c )  
2 
V = c /s in  QI 
aP 
( 6 )  
which exceeds c,  a s  i s  allowable because i t  i s  only an apparciit, 
ve loc i ty  i n  terms of the observer 's  time. (The l a t t e r  has :I varyin:: 
re ta rda t ion  r e l a t i v e  t o  the time i n  the  r e s t  frame o f  the p a r t i c l e s '  
guiding cen te r . )  
s i t y  f o r  a f l u i d  i n  steady motion 
Now, the usual I-aw of conservation of number den- 
div(n  5) = 0 ( ' I  1 
i s  a kinematic equation t h a t  appl ies  i n  terms of time, number, and 
ve loc i ty  as measured cons is ten t ly  i n  any frame o r  manner whatever. 
In a one-dimensional case, such a s  t h i s  c i r cu la t ion  around R r ing,  
(7) implied t h a t  number density and ve loc i ty  are reciprocal ,  w l i c r c  
these quan t i t i e s  may be defined e i t h e r  a s  seen by l o c a l  Lorentz 
observers - o r  by the d i s t a n t  observer, with h i s  prejudices  due t.o 
re ta rda t ion  e f f e c t s .  Using the l a t t e r  de f in i t i ons  of n and v, wc 
-2 see t h a t  when v i s  increased by ( s i n a )  
go down by the  rec iproca l  factor! These e f f e c t s  then compensate 
so a s  t o  r e s to re  Komesaroff's analysis!  
according t o  (c;), n must 
a 
From the foregoing examples, i t  i s  c l e a r  that  tht\ drily correc- 
t ion  that may be properly applied t o  (1) i s  one f o r  bulk motion, whrn 
an e n t i r e  cloud or e lec t rons  streams toward o r  away from the. observer, 
o r  undergoes some bulk motion. I f  a long time average ovcr the 
p a r t i c l e  o r b i t s  i s  taken, a l l  t ime-di la ta t ion ef'fccts nllist be reduc- 
i b l e  t o  ones t h a t  can be calculated from the mean motion of the 
ma t2 r i a l .  Similarly,  the relaxat ion of the p a r t i c l e  d i s t r i b u t i o n  
must be calculated i n  a Lorentz-invariant manner i n  the‘ Loc,zl w s t  
frame, and then converted t o  the observer 's  frrunc. according t o  ruiy 
possible bulk motion. This contrasts  with the SuGGtIstion of Epstcin 
and Feldman* t h a t  re laxat ion e f f e c t s  should be t.rcated through the 
use of (j). 
9 
WFERENCES 
'L. Landau and E. M. Lifshi tz ,  The C las s i ca l  Theory of Fields  
(Addison-Wesley h b l i s h i n g  Co. , Reading, Mass. , 19!;'?), 
Eq. ('(;,.1,). 
'R. I. Epstein and P. A.  Feldman, Astrophys. J. Le t t c r s  - 150, L109 
'T. Takakura and Y. Uchida, Astrophys. Le t t e r s  1, 14,, (19~;8). 
( 1967 1 * 
Additional references t o  s imi la r  Russianwork a re  givc3n 
i n  t h i s  paper. 
P. D. Noerdlinger, J. R. Jok ip i i ,  and L. Woltjer, Astrophys. J .  4 
- 146, 523 (1966). 
'M. J. Rees, Nature - 211, 468 (1966). 
6 
'M. M. Komesaroff, Australian J. Phys. - 19, '(5 (1966). 
K. C .  Westfold, Astrophys. J. - 130, 241 (1959). 
The l i n e s  run around the torus, which has the cross-section of 3 a 
rectangle  very long along the ax i s  of symmetry. 'rhus, i t  
i s  a cy l ind r i ca l  shel l .  
NOTE: Since preparing th i s  report, the author has found that  a 
very similar discussion reaching the same conclusions has been 
published by P. A. G. Scheuer, Ap. J. Letters - 151, Ll39 (1968). 


Restoration of the original synchrotron radiation formula

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