The potential drop in the polar cap region of a rotating, magnetized neutron star is found assuming that the magnetic field is dipolar, with the field aligned (or anti-aligned) with the rotation axis. The curvature of the field lines is of critical importance. Charge flow is assumed to be along magnetic field lines. The electric field has a maximum at radius 1.5 R and the magnitude and functional form of the current is determined

Baker, K. B.

Sturrock, P. A.

English

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https://ntrs.nasa.gov/search.jsp?R=19770019100 2020-03-22T10:13:53+00:00Z
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PARTICLE ACCELERATION IN
AXISYMMETRIC, MAGNETIZED
NEUTRON STARS
by
K.B. Baker and P.A. Sturrock
National Aeronautics and Space Administration
Grant NGR 05-020-668
SUIPR Report No. 695
S
f
May 1977
S
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INSTITUTE FOR PLASMA RESEARCH
STANFORD UNIVERSITY, STANFORD, CALIFORNIA
PARTICLE ACCELERATION IN AXISYMMETRIC,
MAGNETIZED NEUTRON STARS
by
*
K.B. Baker and P.A. Sturrock
National Aeronautics and Space Administration
Grant NGR 05-020-668
May 1977
* Also Department of Physics
t Also Department of Applied Physics
PARTICLE ACCELERATION IN AXISYMMETRIC,
MAGNETIZED NEUTRON STARS
K.B. Baker and P.A. Sturrock
Institute for Plasma Research
Stanford University
Stanford, California
ABSTRACT
The potential drop in the polar cap region of a rotating,
magnetized neutron star can be found assuming that the magnetic
field is dipolar, with the fielO aligned (or anti-aligned) with
the rotation axis. The curvature of the field lines is of critical
importance. Charge flow is assumed to be along magnetic field
lines. The electric field has a maximum at radius 1.5R *
 and the
magnitude and functional form of the current can be determined.
* Also Department of Physics
t Also Department of Applied Physics
sI. Introduction
The source of the accelerating potential in pulsars continues to
be a significant problem in our understanding of these oc l-ects. It is
clear that particles are accelerated but the source of the accelerating
electric field is not yet understood. The most widely accepted picture
of a pulsar is one in which particles are accelerated in the region of
the magnetic polar cap, resulting (via some kind of plasma or MHD
instability ) in a pencil beam of radio emission ( Radhakrishnan and
Cooke, 1969). Specific models along these lines have been proposed
by a number of investigators ( Pacini and Rees (1970), Sturrock .'1971),
Roberts and Sturrock 1)72), Ruderman and Sutherland (1975)).
In this paper we will investigate the form of the electric
potential in the polar cap region of an aligned magnetic star.
Clearly this cannot be a realistic model of a pulsar, but it can
give us some insight into the physics of a real pulsar. We shall
assume that the magnetic field is dipolar and the polar cap region
is bounded by the first closed field line ( i.e. the first line that
does not connect to the surrounding nebula). The field lines are
defined by the equation
Sint A
	
= const.	 (1.1)
r
In particular, the polar cap angle is defined as
Sing a	 1
P	 n.
R4	 Rmax
where R^ is the radius of the neutron star and R
max 
is a free
parameter which is fixed by a more detailed model. In the work
of Sturrock `1971) and Ruderman and Sutherland (1975), R
	 is
max
1
the radius of the light cylinder, whereas the work of Roberts and
F	 Sturrock (1972) it is the "force-balance" radius.
II. The Polar Cap Region
In the region of the polar cap, the magnetic field is much larger
than any expected electric field, so that it is reasonable to assume
that charged particles will be tied to the magnetic field lines. In
this case the current density is approximately proportional to the
magnetic field strength
R3
J(r) 1j	 3	 (2.I)
r
If the stream flows in one direction only, e.g. outward, the charge
density is given by p=j/v. We will consider the situtation where v,c
and thus p.j/c. Clearly this is not true in the region immediately
above the polar cap, but we will show that it is a good approximation
when the distance above the cap is small compared with the overall
dimensions of the system. We also note that in this situation the
component: of j rotating with the star is negligible until the particles
approach the light cylinder. In the Roberts and SLurrock model (1972),
the polar-cap region extends only to the force-balance radius and hence
the co-rotating component of j is always negligible.
Poisson's equation can now be written as
R3
r
2
At this point we introduce orthonormal dipole coordinates defined by
•	 (cos 8)
	
n =
si
-- ng	 (2.3)
r
r
The curves of constant n are the magnetic field lines and curves of
constant & are the lines of constant magnetic potential. The remaining
coordinate is the azimuthal angle ?. Since the system is azimuthally
symmetric, we can now write Poisson ' s equation as
	
8	 1	 x	 a^V I 'D 	 [ a  (—D - 	 ) +	 In a	 3(r n
	
(2.4)
4r (cos GA L a& & a&	 an	 an J
where r and 8 are the usual spherical co-ordinates and are implicitly
defined by ^, n.
We now make a major simplifying assumption, that the contribution
of the divergence of the electric field due to the field component
transverse to the magnetic field is small compared with that due to the
field component parallel to the magnetic field. This is opposed to the
assumption of Goldreich and Julian (1969), according to which the parallel
component vanishes. We are implicitly assuming that the assumption of
current outflow from the polar cap necessitates the development of a
parallel electric field, and we are studying the effect of this component,
which is associated with current outflow, independently of the transverse
electric field component, which is associated with corotation. A complete
solution of the electric field configuration of pulsar magnetospheres
will require the simultaneous incorporation of both effects. However,
since much work has been based on the Goldreich -Julian simplification,
it seems not unreasonable to explore briefly the implications of the
converse simplification.
We therefore now assume that the n derivatives can be neglected
with respect to the C derivatives:
3
	Q I
,
 -. 1+3cos 28 	 am = - 4^j* It,((2.5)
4r 5 (cos GA a& 
( .1
	
C a&^	 c	 r3
From equation (2.5), it immediately follows that
^ a ^ _ - 
4Tj* 
R3	
4r2(cos 0)	
d&	 (2.6)*
*	 c	 1+3cos 8
The integration is along a field line n -const., and we can therefore
express d^ in terms of dr. Since
d^ _ - (cos Odr _	 sin 0 d9
r 	 2r (cos 8)
and
	
(2.7)
do - sin 
	
dr + Cos 9 d8 -=0 ,j
we see that
dE _ -
	 1+3cos2o	
dr	 (2.8)
4r 2 (cos e)3/2
so that equation (2.6) becomes
rr4nj*
1 0 
1 
=
R 
3	 dr
	df,	 c	 *	 cos 9
	
(2.9)
*	 R
The equation of a field line is then used to write
cos 0 = (1-qr)^	 (2.10)
where u is a constant for each field line, given by
4
a - sin  8*/R*	 (2.11)
The integration is now trivial:
dO.[8nj* R*	c 	 a L(1-(xR*)	 d T
	
(2.12)
*
We may integrate equation (2.10) again, using the condition that we are
integrating along a field line, and so obtain
	
8n j *R*	 2- 2
^ = ca [
L_
22r 1
	
- r, - 1 (31
 R
+ (1-aR*) & 
2&
2R*
2- 2
+ C 2L* d I + (Wd
At this point we note that the dependence of (D on n (or angle 8 * ) is
completely determined by specifying ^ and 0/dE atE _ ^* as functions of i.
We also note that the value of C * is itself a function of n (i.e. of
the particular field line chosen). However, one may find a surface
f =* which is very close to the surface of the star in the polar-cap
region. Since a neutron star is a nearly perfect conductor, the force
on a charged particle rotating with the star must be zero. This means
that, on the surface of the star,
E* _ -(?: X r)/c X B,	 (2.14)
which in turn implies that on the stellar surface
,iB*R*
,,* _ -2c	
^c0829* - c08 2 9p )	 (2.15)
5
We have used our freedom to change 0 by a constant to set 0 *00 on the
closed field lines that bound the polar-cap regions.
If we now require that the charge flow be space-charge limited
(this condition has been used in a number of models, e.g. Sturrock (1971),
we require that the electric field parallel to the magnetic field lines
-:,	 be zero at the stellar surface. This implies
do	
0	 (2.16)
Finally, if we can determine either 0 or dt jd& at an outer boundary
C - Cmax, 'we can determine j * . Specifically, we note that dO/d& 0 0
for any finite distance above the stellar surface.
To see more clearly what these results mean, we can look at equation
(2.12) in the limit 8 * << 1:
8nj * 	l	 1	 R*
1	 c	
R 3 C 2r	 4R* 	 2]	 {2.17}4r
Thus we see that in this limit 40 appears to behave in the same way as
the free-space electric field produced by a monopole and a dipole. We
note also that the maximum electric field occurs at r - 3 12 R*.
III. Two-Dimensional Approximation
In this section we shall attempt to improve the model by considering
the two-dimensional case. The potential 4^ can be expanded in a power
series in n. The symmetry of the problem guarantees that the series has
only even powers of n. Since 
nmax - 
1 / Rn	 , n may be treated as
small and we Ehall consider the first two terms only in the expansion.
The boundary condition on the field lines n - n
max 
is 4^ - 0, which
6
determines the expansion:
2
ti max'
On substituting this expression into equation 2.4, we obtain
	
©20 - 1+3`os39
	
-a .- ( .L 	 DO(C,0) 	 D ( U	 (1+ 3cos 2@)
	
Ors cos@)	 a& 	 at,
r r,max
(3.2)
—4n j * R,
	
c	 9r
We may now choose the field line n-0 on which to solve equation (3.2).
On this field line 0=0 and r=1/& so that
V 2 (V . ^5	 L(V) _ 4&3,t	 4Trj * 11 3 ^ 3
	(3.3)
	
0	 71-7- -
	max	 c
A particular solution to the inhomogeneous equation is
	
r# p(^,0)
	 R^* R* nix	(3.4)
c
The homogeneous equation can be solved by making the substitution
U =	 z and k(^,0) = u 2 B(u,0)	 (3.5)
The function B(u) must then satisfy the equation
d"t3 + l- dB	 4 + 16	 B = 0 (3.6)
du`	 a du	 u`'	 n
max
7
This to the modified Bessel equation of order 2 and parameter 4/n max,
We now have
rj O 2	 2n
* * max] (I- -D2- ) (3.7)
•W.0
	'ICl2	 + c2&K2 	 + -	 n 2 x
	
^ nmax	 man	
c	 ma)x
We require that it - 0 as r, - 0 (r - -). The 12 term automatically satisfies
this condition. As z ►0, K2 (z)- 2- so that we have
z2
C2	
R3 n 2
	
2 
x 
+ 
C	
max
	
1(0,0)	
8 ma	
(3.8)
which gives
81tj
C-)	
R3	
(3.9)
c
At	
/R9 
we want
* 
xll, n R3
(3.10)
	
2c	 max
Since 4Z*/n x>>I, we can use the asymptotic expansions of 12 and K2ma
to determine the constant Cl.
it	 -Z
12	 K	 with z
(27Tz)	 n max
Hence
.IB*
C	 R n
- 
+	
(--I)
—2	.. m"
	
TiJ R n
	
**	 87t j *
4* 1 X	 max	 2z
R
3(3.12)
(2Tz) '2
 e-Z
c
8
iIf we now require the flow to be space-charge limited, we can determine
J * .
 
We write df/d& in terms of z (as defined in equation 3.11):
dt	 , 8c 1 	 d	 z2
dC	
zn 
2 dz t 16 nmax 1 
{z)^ +
max
(3.13)
8c22 dz^d6 1max K2(z)^
Znmax
Using
dz Cz2 12 (_)1 ' s2 I i (z)	 (3.14)
we find
Lo U10) (	 2Z Z2 T j (z) - 2z z2 K^(z) - Q	 (3.15)
Thus
cl M +7re
-2z c2 8ir2j*	 R*e 2 t (3.16)
c
Combining ( 3.16) with ( 3.12) then. gives us the following condition
on j*:
j * [16n - (2nz) ezR'
*nmax] QB*n 2 ezl2a/	 *R max (3.17)
Since z' 3 1, we can approximate j * by
- 12?r )	 QB*R*n m2 ez nB**
(2trz)
	 ez naw2
	
R* 2n (3.18)
which is the value estimated by Sturrock (1971).
9
We would also like to know how j * varies across the surface of the
star.	 To determine this, we return to equation (2.4) and write it as an
E. expansion in powers of n.
	 From the lefinition of C and n we can write
€ 4
 r4 +	 r n2 . 1 (3.19)
We then solve for r implicitly, giving us
s:
r	 1 I + ('x.20)
11
r3 4
0 
	
This becomes (to ord
	 n2)
	
o	 q
r •	
11 , n, + 
0 ^rl / E 2)	 (3.21)
a
This expansion is valid as long as n2 < 4& (i.e. as long as 0 !^ 85").
We also must expand the pot•-vrial to order n 2 giving us
2
	
^(4' n) • W,0)l - n 2 + u(&)n 4,	 + 0(n 6 )	 (3.22)
	
1
	 nMax
and we expand the current density to order n 2 giving
	
WO • j *(0) 0 + d l n2 ) + 0(n 4 )	 {3.23)
Equation (2.4) then becomes
	
V2 D a 1 + 3 n2 F ,	 1 a^t,0) 1 _ 
n24	 ,ar Q 
	
nm2x^
+
r2 
xt^+ o)
 ^` 
nmax +1 En 
x + lba( )
1 n
2]+	 (3.24)
4w j *(0) 2
c	 * 
	
(	 4 C ) 
(I + di n
10
fV.e zeroth order equation is equation ( 3.3) while the second-order
equation is
5 3 1_ 1	 a	 1	 3&2
^a(1 
a ^ 3	
=+^	 6r n2	 +16a)^4^ — n2n2	
1	 at \\max	 max max
(3.25)
_ 47rj *
}R* 3(4 d l)C	 t
Using the zeroth order equation to simplify equation (3.25), we obtain
)	 41rj * (0)	 1*- 4n ( t, 0
4 +	 c	 n 2	 + ^En62 + 16a(^)^	 ^( ^0)
max max	 max
(3.26)
_ 4'rj*(0) 3
R* d1c
Since the variation of j with respect to r has already been included to
all orders (j-1/0) we know that d l is independent of ^.
	 Equation
(3.26)	 is satisfied for any form of t if
a{^)	 4n12 r	 nl2	 2	 1
l
(3.27)
max max
and
1
d 1 
= - n 22
max
(3.28)
Thus
n 
2
j * (n) = — ^B* ^i - n 2	 (3.24)
2n	 max )
and we have also detcrmined the corrections to (P of order n`'.
11
-	 = -
S
The fact that j * has the same n dependence (to second order) as
0 suggests that we may be able to find other solutions of Poisson's
equation for which this is true. We therefore write equation (2.4)
in the small angle approximation under this assumption. We write 0
and j *
 as follows:
t(E.n)	 *Q)H(n)
(3.30)j * (n) _ j*H(n)
In the small angle approximation Poisson's equation now becomes
EH(n)	 HWO'M + *(E)H"n + n *(&)Hi(n)
(3.31)
_ 4nj*
s s	 c	 R* H (n)
Dividing by *H, we obtain two ordinary differential equations
H"(n) +-!H 1 (n) + A 2H(n) - 0	 (3.32)
E 0W	 1(E) - A2gE)	 4j* R* (3.33)c 
The solution of equation (3.32) is
H(n) a J O (Xn;	 (3.34)
while the solution of equation (3.33) is
47rj*
	
c i EI 2 (^) + c 2 E K2 (*) +	 R*	 (3.35)
J12c
The boundary condition on the closed field lines implies that
	
(An x)	 0	 (3.36)ma
12
To complete the solution, the constants c  and c 2 must be determined
by the boundary conditions at the stellar surface and at infinity. This
requires that * be a sum of the form:
^+()	 E c iiEI2 (J^i ) + c2i^R2 (^i ) + ^^2*^ R*	 (3.37)
where the X  are determined by the zeros of equation (3.36) and
J * (n) = E 3 *iJ O (A in)	 (3.38)
i
13
IV. Conclusion
i
We have determined the analytic behavior of 4 , by several different
methods. The firm conclusion of these various approaches is that the
current density is of the order  QB* at the surface. The one
dimensional approximation indicates that the electric field has a maximum
on the order of one star radius above the surface. We have shown that
it is possible to find a two-dimensional solution which satisfies the
boundary condition on the star, goes to zero at large distances and
results in space-charge limited current flow.
We are currently preparing a numerical analysis of the two-dimensional
solution in order to plot the surfaces of constant electric potential
and determine the location of the maximum electric field for various
values of nmax, B* , and 0.
r
14
El i
Acknowledgements
M I
This work was supported by the National Aeronautics and Space
Administration under grant NGR 05-020-668.
a
0.
1 A
15
References
Goldriech, P. and Julian, W.H. 1969, Ap.J. 157, 869.
Pacini, F. and Rees, M.J. 1970, Nature, 226, 622.
Radhakrishnan, V. and Cooke, D.J. 1969, Astrophys. Let., 3, 225.
Roberts, D.H. and Sturrock, P.A. 1972, Ap. J., 172, 435.
Ruderman, M.A. and Sutherland, P.G. 1975, Ap.J., 196, 51.
Sturrock, P.A. 1971, Ap. J., 164, 529.
16


Particle acceleration in axisymmetric, magnetized neutron stars

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