Interaction of plasma discontinuities in pulsar emission mechanis

Michel, F. C.

Tucker, W. H.

English

NASA Technical Reports Server

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A PULSAR EMISSION MECHANISM 
F. C. Michel and W. H. Tucke 
Department of Space Sc ience  
R i c e  U n i v e r s i t y  
Houston, Texas 77001 
February 1969 
A PULSAR EMISSION MECHANISM 
F. C. Michel and W. H. Tucker 
Department of Space Science 
Rice University 
Houston, Texas 77001 
We point out that the interaction between two 
basic types of plasma discontinuities in relative 
motion, namely the tangential and shock disconti- 
nuities, should result in pulsed emission of electro- 
magnetic radiation. These discontinuities are both 
observed in the interplanetary plasma flow. Currents 
must flow on these discontinuities, and the radiation 
is generated by the acceleration of these currents at 
the intersection points of the shock and tangential 
discontinuities. Under conditions thought applicable 
to pulsars, Np 0532 (the pulsar associated with the 
Crab Nebula) 
radio energy 
3 0  would radiate about 4 x 10 erg/sec of 
below about 10 MHz. The efficiency of 3 
-8 this mechanism is quite low (- 10 for NP 0532) for 
the assumed conditions, and direct extention to the much 
slower pulsars would require that they have moments of 
inertia much larger than attributable to neutron stars. 
The basic purpose of this letter is to propose a 
mechanism for pulsar radiation. We also offer a relation- 
ship, based on this mechanism, that appears to be satis- 
fied by the pul-sar associated with the Crab Nebula. It 
is necessary in any very specific discussion to adopt 
some basic model for the pulsar. We have argued (Tucker 
and Michel 1968) that (1) the pulse repetition represents 
rotation of (2) a very condensed object (e.g., neutron 
star), with (3) the radiation being associated with 
neutral surfaces in the pulsar magnetosphere. Gold 
(1968) has independently forwarded the first two sug- 
gestions (c.f. Maran and Cameron 1968) and we wish to 
discuss a possible mechanism (point 3) and its conse- 
quences in more detail here. 
The pulsar associated with the Crab Nebula 
is observed to be slowing down with an e-folding 
of about 2 x 10 years (Craft et al. 1968), and, 3 
(NP 0532) 
period 
even 
for a neutron star, the resultant rotational energy 
loss  rate must be comparable to the luminosity of the 
entire nebula (Finzi and Wolf 1968). The observed 
slowing down implies a large torque,and a well-known 
mechanism for such a torque is mass loss  in the 
presence of a strong magnetic field. In the case of 
the sun, the outward flow of plasma carries out solar 
magnetic field lines (hence the plasma flows faster 
than the local Alfven velocity) to form a predominantly 
radial magnetic field structure. Rotation of the sun 
I 
-2- 
causes t h e  f i e l d  l i n e s  t o  take  t h e  form of Achimedian 
s p i r a l s  ( c . f .  Parker 1963) .  The r a d i a l  extension of 
t h e  s o l a r  magnetic f i e l d  t r a n s f e r s  angular  momentum t o  
t h e  outflowing plasma and thereby r e s u l t s  i n  a s i g n i f -  
i c a n t  angular  momentum loss by t h e  sun  (Dicke 1963; 
Modisette 1967; weber and Davis 1967) .  The angular  
momentum loss i s  a f a c t o r  of 10 g r e a t e r  than t h a t  ex- 3 
pected i f  t h e  e f f e c t s  of t h e  magnetic f i e l d  w e r e  
neglected.  
we have enough information a t  hand t o  estimate 
t h e  f i e l d  and plasma flow f o r  NP 0532 on t h e  basis of 
t h e  above model. Corotation can t ake  p l ace  out  t o  a 
r a d i a l  d i s t ance  of a t  most 
8 2 which f o r  NP 0532 i s  about 1.5 x 10 c m  (w = 2 x 10 
0 
radians/sec) . The 
Torque = 
giv ing ,  f o r  e i t h e r  
torque i s  of t h e  order of 
- dM R 2w 
d t  o o 
t h e  energy loss r a t e  es t imated by 
F inz i  and wolf (1969) o r  t h e  observed luminosity of t h e  
38 Crab of M 10 erg/sec (Haymes e t  a l .  1968) ,  
17 - dM = 10 g/sec,  
d t  
(3) 
which f o r  n e a r - r e l a t i v i s t i c  f low v e l o c i t i e s  corresponds 
t o  a dens i ty  n e a r  t h e  co ro ta t ion  l i m i t  of 
-11 3 
p M 10 g/cm (4) 
-3- 
For co ro ta t ion  t o  a c t u a l l y  be e f f e c t i v e  o u t  t o  R , t h e  
azimuthal component of t h e  Alfven v e l o c i t y  m u s t  be 
comparable there t o  t h e  v e l o c i t y  required f o r  r i g i d  
co ro ta t ion  (Modisette 1967),  o r  
0 
/ 
Thus,  
5 
B M 3 x 10 gauss,  
0 
w h i c h  corresponds t o  a trapped f l u x  i n s i d e  a p u l s a r  
o b j e c t  10 k m  i n  r ad ius  of 2 x gauss-cm e s s e n t i a l l y  
equal  t o  t h a t  for  t h e  sun. The magnetic f i e l d  i s  r a d i a l  
(B a l/r ) for r<<F&and becomes azimuthal ( B  a l/r) f o r  
E->% a s  shown i n  Figure 1, whereas t h e  flow i s  very 
nea r ly  r a d i a l  a t  l a r g e  d i s t ances  ( p  CI l/r ) . Since t h e  
n e t  magnet ic  f l u x  ou t  of any closed su r face  m u s t  be 
zero,  regions of both outward and inward r a d i a l  f l u x  
m u s t  be p resen t ,  which the re fo re  are separated by s u r -  
faces of zero radial  f l u x  ( n e u t r a l  surfaces, o r  tangen- 
t i a l  d i s c o n t i n u i t i e s  i n  g e n e r a l ) .  These d iscont in-  
u i t i e s  are observed i n  t h e  i n t e r p l a n e t a r y  medium (c.f. 
Siscoe e t  a l .  1968) and the minimum th ickness  of these 
shee t s  i s  of t h e  order  of t h e  ion cyclotron diameter.  
The proton ene rg ie s  w i l l  be on t h e  order  of 10 e V  f o r  
t h e  above magnetic f i e l d  s t r e n g t h  ( s t ronger  f i e l d s  
would r e s u l t  i n  more ene rge t i c  p a r t i c l e s  and r equ i r e ,  
f o r  a given torque,  a propor t iona te ly  s m a l l e r  p a r t i c l e  
loss rate) and the re fo re  t he  cyc lo t ron  r ad ius  f o r  e i t h e r  
2 
2 
2 
9 
-4- 
protons  o r  e l e c t r o n s  of t h a t  energy i s  
a -10 c m .  (7) C 
W e  now show t h a t ,  given t h i s  model, a c e r t a i n  
p a r t  of the  energy loss w i l l  be i n  t h e  form of pulsed,  
l i n e a r l y  po la r i zed ,  low frequency (4 10 Hz) e l e c t r o -  
magnetic r a d i a t i o n .  Most of  t h e  energy loss i s  i n i t i a l l y  
i n  t h e  form of  outflowing magnetized plasma; however, 
w e  note  t h a t  (1) t h e  t a n g e n t i a l  d i s c o n t i n u i t i e s  d e l i n e a t e  
r ap id  changes i n  f i e l d  d i r e c t i o n  and (2) t h e  magnetic 
f i e l d  drops o f f  a s  l/r. In  o t h e r  words, we  a l ready  have 
very nea r ly  a/ga&ia$ion f i e l d .  A shock t r a n s i t i o n  m u s t  
e x i s t  a t  some l a r g e  r a d i a l  d i s t ance  t o  s l o w  t h e  flow t o  
submagnetoacoustic v e l o c i t i e s  i n  response t o  the  s u r -  
rounding medium. The i n t e r s e c t i o n  of each n e u t r a l  sheet 
with t h e  shock t r a n s i t i o n  causes t h e  su r face  c u r r e n t s  
flowing on these t w o  d i s c o n t i n u i t i e s  t o  undergo r ap id  
changes i n  v e l o c i t y  and magnitude. The shock-neutral  
shee t  i n t e r a c t i o n  should the re fo re  be a source of 
e lectromagnet ic  r ad ia t ion .  This  mechanism w i l l  be ex- 
amined i n  d e t a i l  elsewhere (Michel and Wolf 1969).  
Figure 2 shows t h e  geometrical  r e l a t i o n s h i p s  among t h e  
var ious d i s c o n t i n u i t i e s  and t h e  r a d i a t i o n  pu l se  
t h a t  emanates from t h e i r  i n t e r a c t i o n .  It  i s  s t r a i g h t -  
forward t o  estimate t h e  r ad ia t ed  i n t e n s i t y  and spectral 
d i s t r i b u t i o n  s i n c e  the  pu l se  amplitudes w i l l  be com- 
parable  t o  t h e  ambient f i e ld  amplitudes bu t ,  f o r  a 
f ixed  observer ,  w i l l  be of dura t ion  A t  - 2 a /c. The 
9 
u se 
C 
-5- 
radiated energy is therefore essentially the fraction 
of the energy flux within neutral sheets, or 
- 2 a /nR = 2 wo/w (9) Lr ad io/Lro tat ion c o  max . 
- 8 For a = 10 cm, R = 1.5 x 10 cm, and Lrotation 
C 0 
erg/sec, we obtain 
(10) 
30 - 4 x 10 erg/sec, Lradio 
in accord with the observed luminosity for NP 0532 (our 
value for a is only an 
We have shown elsewhere 
a /R (the neutral sheet 
between neutral sheets) 
C 
C 
order-of-magnitude estimate). 
(Michel 1969) that the quantity 
thickness divided by the spacing 
remains constant despite expan- 
sion and energy loss from the plasma. The resultant 
constancy of the pulse width together with B - l/r 
means that the pulsed radiation output is insensitive 
to the location of the shock: we would expect such a 
shock to be many A.U. from the pulsar object. The 
frequency spectrum will simply be the Fourier transform 
of a pulse of duration At, which for a smooth (e.9. 
gaussian) shape is constant at very low frequencies and 
drops off sharply at higher frequencies (around 10 Hz), 
as seems to be observed for the pulsars. 
9 
We are only radiating rotational energy in the 
above model, thus we can regard the moment of inertia 
as the only unknown and solve for it, using equation 
(9 ) ,  in terms of observable quantities, Thus, 
-6- 
2 2 3  2 
v P T / <ne> I = I F ( l n  dA) o v  e C 
where I 
dens i ty  a t  low frequency, Jn dR t h e  d i spe r s ion  measure, 
is a p r o p o r t i o n a l i t y  factor, FV t h e  f l u x  
e 
0 
v t h e  cu to f f  frequency (adopted t o  be frequency a t  
which f l u x  d e n s i t y  drops t o  l /e) ,  P the p u l s e  per iod ,  
T t h e  e f f e c t i v e  l i f e t i m e  (P/fi), and <n > i s  t h e  average 
e l e c t r o n  concentrat ion i n  t h e  d i r e c t i o n  of observat ion,  
Table 1 shows t h e  assumed va lues  and r e s u l t a n t  moments 
f o r  those p u l s a r  with measured va lues  of T. The agree- 
ment  among t h e  f irst  three i s  c e r t a i n l y  adequate,  
e s p e c i a l l y  s i n c e  CP 1919 is c l o s e  t o  t h e  g a l a c t i c  p lane  
(4') and i s  perhaps r e l a t i v e l y  c l o s e r ,  The u s u a l  views 
regarding compact s t a r s  argue t h a t  e lectron-degenerate  
stars should have moments of i n e r t i a  exceeding about 
10 I w h i l e  nucleon-degenerate stars should have 
moments less than o r  about I (I  M 1 .2  x 10 g c m  
i n  equation (11) f o r  the  u n i t s  ind ica ted  i n  Table 1). 
Thus, w h i l e  t he  f irst  t h r e e  p u l s a r s  could be regarded 
as electron-degenerate  stars, CP 0950 would either have 
C 
e 
4 
0 
45 2 
0 0  
t o  be 10 t i m e s  c l o s e r  than t h e  rest ( t o  be a neutron 
s tar  of conventional design) or have a mass- rad ius  
r e l a t i o n s h i p  no t  thought t o  be stable. Possibly 
r o t a t i o n a l  effects p l ay  a major r o l e  i n  t h e  s t a b i l i t y  
of t hese  o b j e c t s ,  Any r o t a t i o n a l  model must face up t o  
s u b s t a n t i a l l y  t h e  s a m e  d i f f i c u l t y :  i f  a l l  p u l s a r s  are 
-7- 
substantially the same and the slowing down is pro- 
portional to the radio luminosity, then the Crab pulsar 
is anomalously dim. Furthermore if they are all neutron 
stars, the conversion efficiency into radio pulses must 
be near 100% (except again for NP 0532). 
In the spirit of such short communications, we 
defer the discussion of many important questions, such 
as the morphology of the pulse features and the rele- 
vance of NP 0532  to the Crab Nebula as a whole, although 
it has already suggested that neutral sheet structures 
within the volume of the Nebula are essential to the 
observed synchrotron radiation (Michel 1968, 1969). The 
observed pulse width can be understood in terms of per- 
turbations to the shock position and shape, and so on; 
the only basic difficulty encountered has been the 
moment of inertia question outlined above. 
We wish to acknowledge the interest and encourage- 
ment by A. J. Dessler, and the many valuable comments 
by R. A. Wolf. This work was supported in part by NASA 
under Grant No. NGR-44-006-033. 
-8- 
TABLE 1 
Moment of I n e r t i a  determined from equat ion  (11). 
The u n i t s  are 10 J / m  Hz ( a t  81.5 MHz), e lec t rons -  -26 2 
3 7 parsec/cm , 100 MHz, seconds, and 10 y e a r s  r e spec t ive ly .  
45 2 The q u a n t i t y  I i s  then 1 .2  x 10 g c m  , and we assume 
an average e l e c t r o n  concent ra t ion  of 0 . 1  electron/cm . 
D a t a  i s  f r o m  Ekers and Moffet t  (1968);  S t a e l i n  and 
Rei fens te in  (1968) ; R i c k e t t  and Lyne (1968) ; Maran and 
Cameron (1968). 
0 
3 
C 
P [n d4 v F81.5 e Pu l sa r  P T I/Io 
CP 1919 0.55 12.5 5 1,337 4.2 16 l o 3  
CP 1133 0 0 4  4.9 7 1.188 1.0 7 l o 3  
3 CP 0834 0.4 1 2 , 8  3 1.274 0.7 7 x 10 
CP 0950 0.2 3.0 6 0,253 3.0 124 
NP 0532 0.3" 56.0 lo** 0.033 0,00024 0.3 
* Only ind iv idua l  p u l s e s  b r i g h t e r  than  about  10 i n  these 
u n i t s  are v i s i b l e  above the  C r a b  background. These p u l s e s  
are q u i t e  in f requent ,  and t h e r e f o r e  t h e  average f l u x  should 
be considerably less as ind ica t ed  by t h i s  rough estimate 
f r o m  very  l i m i t e d  da ta .  
** Assumed -- a c t u a l  measurements ei ther unknown or un- 
a v a i l a b l e .  
-9- 
CAPTIONS 
F i g u r e  1. S p i r a l  magnetic f i e l d  s t r u c t u r e  i n  t h e  
plasma flow away from t h e  proposed pulsar 
model as seen i n  t h e  e q u a t o r i a l  ( r o t a t i o n )  
plane.  The plasma flow is  e s s e n t i a l l y  
r a d i a l  everywhere, The shock t r a n s i t i o n  i s  
shown much too  close on t h e  scale of t h i s  
f i g u r e ,  and a neutron star would be smaller  
than t h e  c e n t r a l  do t .  The  shaded a rea  i s  
i l l u s t r a t e d  i n  Figure 2, except t h a t  here  
t h e  inward pu l se  has  been omitted i n  an  
a t tempt  t o  s impl i fy  t h i s  f i gu re .  
Figure 2 .  Passage of a n e u t r a l  shee t  through a shock 
t r a n s i t i o n ,  i l l u s t r a t e d  f o r  a compression 
f a c t o r  of 3 and an inc iden t  flow v e l o c i t y  
of 0.9 c. Sheet c u r r e n t s  f l o w  t o  maintain 
t h e  change i n  s ign  of a c r o s s  t h e  n e u t r a l  
shee t  and t h e  change i n  magnitude of  B 
ac ross  t h e  shock, The a c c e l e r a t i o n  of these 
c u r r e n t s  genera tes  t h e  r a d i a t i o n  p u l s e .  
This  pu l se  i s  po la r i zed  p a r a l l e l  t o  t h e  
su r face  cu r ren t  d i r e c t i o n  (E-vector per- 
pendicular  t o  t h e  p lane  of the  f igu re )  and 
i s  analogous t o  Zerenkov r a d i a t i o n  i n  t h a t  
t h e  phase ve loc i ty  of t h e  i n t e r s e c t i o n  be- 
tween the  n e u t r a l  shee t  and t h e  shock moves 
-10- 
very much f a s t e r  than  c, g iv ing  a shock- 
wave l ike form t o  t h e  r a d i a t i o n  pulse .  
Equivalent ly ,  it could be regarded as co- 
he ren t  bremsstrahlung generated by t h e  change 
i n  d i r e c t i o n  of t h e  c u r r e n t  carriers ( t h e  
c u r r e n t  flowing on t h e  shock su r face  changes 
s i g n ) ,  which a l l o w s  t h e  p o l a r i z a t i o n  sense 
t o  be more r e a d i l y  appreciated.  
-11- 
REFERENCES 
C r a f t ,  H. D., Jr., Lovelace, R. V. E , ,  R i c h a r d s ,  D. W., 
and Sut ton ,  J. E ,  1968, t o  be published. 
D i c k e ,  R. H. 1964, N a t u r e ,  202, 4 3 2 .  
E k e r s ,  R. D. and Moffett ,  A, T .  1968, N a t u r e ,  - 2 2 0 ,  756. 
G o l d ,  T. 1968, N a t u r e ,  - 218, 731, 
H a y m e s ,  R. C.,  Ellis, D. V., Fishman, G. J., K u r f e s s ,  
J, D., and Tucker ,  W. H. 1968, Ap. J. ( L e t t e r s ) ,  
- 151, L9. 
Maran, S .  P. and C a m e r o n ,  A.G.W. A u g u s t  1968, Physics 
Today, - 21 ,  8, 41. 
Michel, F. C. 1968, J. G e o p h y s .  R e s . ,  _73, 4135, 
Michel, F. C. 1969, Ap. J., t o  be publ ished.  
Michel, F. C, and W o l f ,  R. A. 1969, i n  preparation. 
Modisette, J. L. 1967, J. G e o p h y s .  R e s . ,  72, 1521. 
-12- 
Parker, E. N. 1963, Interplanetary Dynamical Processes 
(New York: Interscience Publishers), Ch. 10, 
Rickett, B. J. and Lyne, A, G. 1968, Report presented 
at the Fourth Texas Symposium on Relativistic Astro- 
physics, Dallas, Texas, 16-20 December. 
Siscoe, G. L,, Davis, L., Jr., Coleman, P, J., Jr., 
Smith, E, J., and Jones, Do E. 1968, J. Geophys, Res., 
- 73, 61. 
Staelin, D, H. and Reifenstein, E. C., I11 1968, Science, 
- 162, 1481. 
Tucker, W. H. and Michel, F. C. 1968, Report presented 
at Conference on Pulsars, New York, N. Y,, 20-21 May 
1968, See Maran, S. P. and Cameron, A.G.W. loc, cit, 
Weber, E. J. and Davis, Lo, Jr. 1967, Ap. J., 148, 217. 
FIGURE I 
1 
J 
1 
/ 
/ 
/ 
J 
I 
\ 
RADIATION -1 
PULSE I 
I 
J 
I 
I 
- SHOCK TRANSITION 
L 
I 
L 
\ 
V ( POSTSHOCK) 
_IC 
’\ 
\ 
\ 
I \ 
1- RADIATION 
\ PULSE 
\ 
\ 
\ 
\ 
1- NEUTRAL SHEET 
FIGURE 2 


A pulsar emission mechanism

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