Pulsars with torque aligning magnetic moment with spin axi

Goldwire, H. C., Jr.

Michel, F. C.

English

NASA Technical Reports Server

ALIGNMENT OF OBLIQUE ROTATORS 
F. Curtis Michel and H. C ,  Goldwire, Jr, 
Space Science Department 
Rice University, - Houston, Texas 77001 
Abstract 
Pulsars may be slowed down by radiation field torques. 
If so, there is also an accompanying torque that aligns the 
magnetic moment with the spin axis. The general implication 
of the alignment effect is that most pulsars would have to 
have been formed either with periods not too different from 
what are now observed or with magnetic mQments almost pre- 
cisely orthogonal to the spin axis. Alternatively, neutral 
sheet formation in the pulsar magnetosphere, or other plasma 
effects, may suppress the alignment torque, 
https://ntrs.nasa.gov/search.jsp?R=19700011287 2020-03-12T01:21:45+00:00Z
-2- 
I. INTRODUCTION 
The discovery of  t h e  pu l sa r  phenomenon has  generated 
considerable  i n t e r e s t  i n  the p r o p e r t i e s  of  r a p i d l y  r o t a t i n g ,  
s t rong ly  magnetized, and highly condensed ob jec t s .  W e  w i l l  
c a l l  such o b j e c t s  " sp ina r s " ,  following Morrison (1969), and 
we w i l l  be concerned w i t h  those s p i n a r s  tha t  a r e  obl ique 
r o t a t o r s  (i.e.,  that  have a sensible angle  between their  sp in  
ax i s  and magnetic moment). Exact ly  why sp ina r s  might also be 
pu l sa r s ,  presumably puls ing  a t  their  frequency of r o t a t i o n ,  
remains a t o p i c  of w i d e  specula t ion .  What we wish t o  r e p o r t  
here i s  t h e  effect  of magnetic d ipo le  r a d i a t i o n  from s p i n a r s  
i n  decreasing t h e  angle  ("magnetic o b l i q u i t y " )  between the 
sp in  a x i s  and the magnetic moments. In  o t h e r  words, the 
torque exer ted  on a sp ina r  no t  on ly  decreases i ts  r o t a t i o n a l  
frequency bu t  a l s o  a l i g n s  the magnetic moment and sp in  a x i s .  
T h i s  alignment effect  has  important consequences f o r  any theory 
t h a t  relies on d ipole  r a d i a t i o n  t o  account f o r  the r o t a t i o n a l  
h i s t o r y  of  the p u l s a r s ,  and may be an e s s e n t i a l  cons idera t ion  
t o  o t h e r  t h e o r i e s  a s  w e l l .  
-3- 
11. TORQUES 
T rad i t iona l ly ,  t h e  torque on a r o t a t i n g  d ipole  i s  
ca l cu la t ed  from the energy l o s t  t o  the r a d i a t i o n  f i e l d ;  
however, only the torque component p a r a l l e l  t o  the 
sp in  a x i s  does work, To c a l c u l a t e  t h e  ne t  torque t h a t  
acts on a r o t a t i n g  d ipo le ,  it i s  p re fe rab le  t o  i n t e g r a t e  the 
Maxwell stresses exe r t ed  a t  the su r face  of the rotator ,  The  
ith component of  the torque i s  then given by 
Ti r 1 “ i j k  j Tkn dSn 
w h e r e  T 
of r o t a t o r  sur face  a rea ,  the normal component being n. Writing 
the torque i n  Car tes ian  coord ina tes ,  b u t  the stress i n  s p h e r i c a l  
coordinates ,  g ives  
i s  the M a x w e l l  Stress t e n s o r  and dSn i s  the element kn 
T 
Y S r coscp T cos r cp 8 s i n  cp ]  dS 
TZ = r[Trq s i n  8 1  dS 
where r o t a t i o n  is  about the z a x i s ,  the magnetic moment l i e s  
i n  the x-z plane,  and the i n t e g r a t i o n  i s  over  the su r face  of 
the sphere,  
Since we are c a l c u l a t i n g  t h e  stress a t  t h e  su r face  of 
the s tar ,  where the magnetic f i e l d  is  f ixed  i n  t h e  h igh ly  
-4- 
conducting s t e l l a r  material ,  the stresses are 
3 
= HrBe T r e  
T = H B  
r r p  r Y  ' 
( e l e c t r o s t a t i c  terms g ive  only a small  con t r ibu t ion ,  which 
we w i l l  n e g l e c t ) .  
Using t h e  exac t  expressions for  t h e  magnetic f i e l d  of 
an obl ique r o t a t o r  i n  a vacuum (Deutsch, 1955) we f i n d  
Tx = T s i n  x cos x , 
and 
2 
= - T s i n  x , 
TZ 
(4) 
where 
T = 2rr a 6 2 3  B R /3 p0c 3 . 
The torque along t h e  sp in  a x i s  ( z )  reproduces t h e  s tandard 
r e s u l t .  H e r e  B is  t h e  sur face  magnetic f i e l d  a t  t h e  poles .  
I t  i s  easy t o  understand t h e  o r i g i n  of  these two 
torque components. Figure 1 i l l u s t r a t e s  the sp ina r  i n  t h e  
x-z plane.  Rotat ion causes t h e  f i e l d  l i n e s ,  w h i c h  emanate 
mainly near  t h e  magnetic po les ,  t o  be s w e p t  back: consequently, 
t h e  tens ion  on t h e  f i e l d  l i n e s  has a component oppos i te  t h e  
d i r e c t i o n  of r o t a t i o n .  There i s  t h e r e f o r e  a n e t  force  appl ied  
-5- 
near the poles  tha t  g ives  a torque component opposi te  t o  L ,  
w h i c h  reduces t h e  angular  momentum, and towards t h e  closest 
N 
magnetic po le ,  w h i c h  reduces the magnetic o b l i q u i t y .  
The equat ions of motion are 
and (5 )  
thus  we have immediately the cons tan t  of motion 
n cos x = n cos xo . ( 6 )  0 
Thus i f  the slow p u l s a r s  ( P  N 1 sec) have evolved from fas t  
pu l sa r s  (P - t o  sec) ,  predominantly a s  a r e s u l t  of 
such electromagnet ic  torques,  t hen  they  must have been formed 
with their  magnetic moments almost exac t ly  orthogonal t o  the 
s p i n  axes. Note t h a t  equation (6)  i s  v a l i d  even i f  the d ipole  
f i e l d  changes w i t h  t i m e .  
-6- 
I11 a DECELERATION PARAMETER 
The time variation of the pulsar frequency can be 
described by (Goldwire and Michel 1969) 
fl = - K'nn , (8) 
where the deceleration parameter (n) is observationally given 
Including the effects of changing x , the torque equations (5) 
give (here assuming constancy of the magnetic field strength) 
(10) 
2 n = 3-t-2cot x , 
The minimum value being n = 3, a result sometimes quoted without 
restriction on x . Determination of n would then determine x 
on the vacuum radiation hypothesis. For NP0532, the preliminary 
value of n = 5 f 3 only requires 32' < x < 90" , but the value 
of n should eventually be known with much higher precision. 
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I V .  AGE/L I €?E TIME RELATIONSHIP 
The slowing down can be parameterized by a c h a r a c t e r i s i t i c  
t i m e  ( " l i f e t i m e " )  def ined by 
7 = - n/n , 
and i f  t h e  a c t u a l  age (t) of t h e  p u l s a r  i s  known, t h e  
re l a t i o n s h i p  
t 2 s i n  X/sin xo = exp(- 7 c o t  x) 
These q u a n t i t i e s  are a l s o  i n t e r r e l a t e d  0 "  i n t e r r e l a t e s  x and x 
by t h e  sp in  r a t i o  (OO/R)  from equat ion (6 )  and can thereby be 
solved f o r .  The C r a b  pu l sa r  i s  presumably 915 years  o ld ,  
whereas the l i f e t i m e  i s  observed t o  be 2484-12 years ,  t hus  we 
can g ive  t h e  magnetic o b l i q u i t i e s  and dece le ra t ion  parameter 
as a funct ion of s p i n  r a t i o  f o r  NP0532, a s  has  been p l o t t e d  i n  
Figure 2 ,  Note t h a t  (1) t h e  dece le ra t ion  parameter n v a r i e s  
between 3 and 4.58, ( 2 )  the minimum i n i t i a l  o b l i q u i t y  i s  78O 
(a t  i-2 /n = 2-55), and ( 3 )  t h e  minimum magnetic o b l i q u i t y  a t  0 
present  would be 48". N o  s o l u t i o n  e x i s t s  f o r  a sp in  r a t i o  less 
than 1 -95  ( i n  general :  The su r face  
magnetic f i e l d  requi red  t o  g ive  a f ixed  value of T v a r i e s  a s  
n,/n 2 (1 - 2 t / 7 )  -'I2). 
B - csc x 
and v a r i e s  only by about a f a c t o r  of 1.34 over t h e  e n t i r e  range 
of poss ib l e  values  f o r  n0/n a 
-8- 
v, LARGE SPIN RATIOS 
If one supposes t h a t  pu l sa r s  are sp ina r s  formed with 
nea r ly  90' magnetic o b l i q u i t i e s ,  then  t h e  age / l i fe t ime 
r e l a t i o n s h i p  can be w r i t t e n  
2 
t / T  = - & ( s i n  x) t a n  x 
thus  
T h e  l i f e t i m e  would then  be determined e n t i r e l y  from observable 
parameters,  as shown i n  Table 1. For s m a l l  sp in  r a t i o s ,  the 
r i g h t  hand s i d e  of the above equat ions must be mul t ip l i ed  by 
e For a l l  condi t ions ,  we have 
-9- 
V I .  NEUTRAL, SHEET MODIFICATION 
The ex i s t ence  of plasma surrounding the sp ina r  (c.f .  
Michel 1969) leads one t o  ques t ion  the v a l i d i t y  of the vacuum 
electromagnet ic  f i e l d  so lu t ions .  N o  d e t a i l e d  s o l u t i o n s  have 
been worked out  tha t  include the plasma effects i n  a se l f -  
c o n s i s t e n t  way. One specula t ion  (Michel and Tucker 1969) i s  
t h a t  t h e  f i e l d  s t r u c t u r e  w i l l  be s u b s t a n t i a l l y  uniforrq, w i t h  
only l o c a l i z e d  r ap id  changes f r o m  inward t o  outward r a d i a l  
f l u x  ( " n e u t r a l  sheets").  Such a model also e x h i b i t s  a slowing 
down torque (c.f .  Michel 1969) ,  b u t  we see from Figure 1 t h a t  
there would be no alignment torque  (neglec t ing  the f i n i t e  
th ickness  of the n e u t r a l  sheets),  s i n c e  the  su r face  stresses 
are then symmetrical about the z ax i s .  W e  can describe t h e  
effect  o f  suppressing the alignment torque  by introducing a 
phenomenological parameter q(< 1) mult ip ly ing  the upper equat ion 
( 4 ) .  Then the cons tan t  of motion becomes 
RT cos x = cons tan t  
hence 
For example, t o  produce a s p i n  r a t i o  of  lo2 with y, 
r equ i r e  q 0-15,  etc.  In  the n e u t r a l  sheet p i c t u r e ,  r) would 
- 60' would 0 
be of t h e  o rde r  of the shee t  th ickness  divided by the s t e l l a r  
rad ius ,  which i s  argued (Michel and Tucker 1969) t o  be much 
less than  lo-'* 
-10- 
The po in t  t o  be emphasized i s  t h a t ,  i f  sp ina r s  a r e  
pu l sa r s ,  t hey  must ei ther have a s i g n i f i c a n t l y  d i f f e r e n t  
f i e l d  s t r u c t u r e  than suggested b y  the vacuum dipole  a n a l y s i s  
o r  be cons t ra ined  i n  one of two e s s e n t i a l  ways: t h e i r  sp in  
r a t i o s  are e i t h e r  small  o r  t he  i n i t i a l  magnetic o b l i q u i t i e s  
w e r e  p r e f e r e n t i a l l y  q u i t e  near 90'. 
This work w a s  supported i n  p a r t  by NASA Grant 
-11- 
Age/Lifetime and Magnetic Obliqui ty  as a Function 
of Decelerat ion Parameter 
n 
3 
4 
5 
6 
7 
8 
9 
10  
30 
i o  0 
1000 
500 
.406 
- 347 
,305 
.275 
,251 
- 2 3 1  
,215 
.099 
., 040 
.006 
x (degrees) 
~ 
90 
54.74 
45 
39.23 
35.26 
32.31 
30  
28.13 
15.23 
8.17 
2.56 
-12- 
CAPTIONS 
Figure 1 
p 
Figure 2 
Origin of  the alignment and slowing down 
torques ,  Sweeping back of  f i e l d  l i n e s  by 
r e t a r d a t i o n  effects  g ives  a force  component 
oppos i te  t o  r o t a t i o n a l  v e l o c i t y  and concen- 
t r a t e d  near magnetic pole .  Resul tant  torque 
vec tor  ( T )  N has component opposi te  angular 
momentum vec tor  (k) and towards nea res t  pole  e 
I n i t i a l  and f i n a l  magnetic o b l i q u i t y  angles  
f o r  NP0532, as a func t ion  of t h e  s p i n  r a t i o  
(n,/n), requi red  t o  g ive  t h e  observed age/ 
l i f e t i m e  r a t i o .  The r i g h t  hand s c a l e  app l i e s  
t o  n . Asymptotic va lues  ( i n f i n i t e  s p i n  
r a t i o )  a r e  ind ica t ed  by the arrow nea res t  
the curve i n  ques t ion .  

i
 
0
 
b IC 
0
 
1 
9V
 
-13- 
REFERENCES 
Goldwire, H, C. ,  Jr., and Michel, F. C., 1969, Astrophys. J. 
(Let ters) ,  156, L113. 
Deutsch, A. J., 1955, Ann. D'Phys., 18, 1. 
Morrison, P , ,  1969, Astrophvs, J., - 157, L73. 
Michel, F. C . ,  1969, Phys. R e v .  Letters, 23, 2 4 7  
Michel, F. C . ,  and Tucker, W. E., 1969, Nature, 223, 277 
(1969). 


Alignment of oblique rotators

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