c-1 A STEADY STATE SELF-CONSISTENT MODEL FOR PULSAR MAGNETOSPHERES by DENIS ALAN LEAHY B.A.Sc., University of Waterloo, 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS tROR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OR GRADUATE STUDIES Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September,1976 (c) Denis Alan Leahy, 1976
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c-1
A STEADY STATE SELF-CONSISTENT MODEL
FOR PULSAR MAGNETOSPHERES
by
DENIS ALAN LEAHY
B.A.Sc., U n i v e r s i t y of Waterloo, 1975
A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE
REQUIREMENTS tROR THE DEGREE OF
MASTER OF SCIENCE
i n
THE FACULTY OR GRADUATE STUDIES
Department of Physics
We accept t h i s t h e s i s as conforming
to the r e q u i r e d standard
THE UNIVERSITY OF BRITISH COLUMBIA
September,1976
(c) Denis Alan Leahy, 1976
In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements for
an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree that
the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.
I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s
f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head of my Department or
by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t copying or pub l i ca t ion
of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my
w r i t t e n p e r m i s s i o n .
Depa r tment
The U n i v e r s i t y o f B r i t i s h Co l umb i a 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5
Date O C T . -6 , r37<b
ABSTRACT
A steady s t a t e s e l f - c o n s i s t e n t model f o r a p u l s a r magnetosphere
i s developed. I t i s shown that the c e n t r a l neutron s t a r o f a p u l s a r
should possess a magnetosphere. In the f i r s t approximation, the
i n e r t i a of the magnetospheric p a r t i c l e s i s n e g l e c t e d . Steady s t a t e
c o r o t a t i n g models are developed t o c a l c u l a t e the s t r u c t u r e of the
magnetosphere f o r the axisymmetric case and the case of the a r b i
t r a r i l y o r i e n t e d d i p o l e . Two r e s u l t s are t h a t charge d e n s i t y i s
p r o p o r t i o n a l t o the z component of the magnetic f i e l d and t h a t
the z component of the magnetic f i e l d vanishes at the l i g h t c y l i n d e r .
The l i g h t c y l i n d e r i s where the c o r o t a t i o n v e l o c i t y reaches the
speed of l i g h t . The p u l s a r s p i n a x i s i s a l i g n e d w i t h the z a x i s .
I l l u s t r a t i o n s o f the f i e l d s are presented f o r the cases of magnetic
d i p o l e a x i s p a r a l l e l and p e r p e n d i c u l a r t o the s p i n a x i s .
Next these models are a l t e r e d t o take i n t o account the non
zero mass of the p a r t i c l e s i n the magnetosphere. An e x t r a e l e c t r i c
f i e l d i s r e q u i r e d t o h o l d the p a r t i c l e s i n c o r o t a t i o n . Charge
se p a r a t i o n i s assumed. The f o l l o w i n g r e s u l t s are found: 1) F i e l d
l i n e s which p r e v i o u s l y were h o r i z o n t a l i n s i d e the l i g h t c y l i n d e r ,
now have a cusp i n them where they were h o r i z o n t a l . T his cusp
inc r e a s e s i n s i z e as i t s l o c a t i o n approaches the l i g h t c y l i n d e r .
2) F i e l d l i n e s no longer are h o r i z o n t a l at the l i g h t c y l i n d e r but
e r t i c a l , and d i v i d e i n t o two groups those w i t h p o s i t i v e B^
( c a r r y i n g negative charge) and those w i t h negative ( c a r r y i n g
p o s i t i v e charge).
We next cease t o r e q u i r e t h a t the p a r t i c l e s be f i x e d i n the
c o r o t a t i n g frame. F i r s t the s i n g l e p a r t i c l e motion i s c a l c u l a t e d
f o r a r b i t r a r y f i e l d s assuming small v e l o c i t i e s i n the r o t a t i n g
frame. We f i n d that the motion can be separated i n t o a slow d r i f t
along streamlines(which very n e a r l y f o l l o w magnetic f i e l d l i n e s )
and a s p i r a l i n g about these s t r e a m l i n e s . The energy i s conserved,
and can be separated i n t o a l o n g i t u d i n a l and a t r a n s v e r s e energy
a s s o c i a t e d w i t h the two types of motion. The t r a n s v e r s e energy
d i v i d e d by the frequency of s p i r a l i n g i s an a d i a b a t i c i n v a r i a n t .
For the axisymmetric case, a model i s developed from which the
f i e l d s , charge d e n s i t y , and v e l o c i t i e s can be computed. With
a r e s t r i c t i o n on the boundary c o n d i t i o n s , an a n a l y t i c a l model i s
o u t l i n e d f o r the case of a r b i t r a r y magnetic m u l t i p o l e s .
TABLE OF CONTENTS i i i .
page
L i s t of Tables i v .
L i s t of Figures v.
Acknowledgement v i .
I n t r o d u c t i o n 1.
Magnetosphere Ex i s t e n c e 3.
The Force Free Assumption 6.
The Axisymmetric ForceFree Magnetosphere 7.
Covariant Formulation 12.
The Force Free Magnetic P o t e n t i a l 14.
R e l a x a t i o n of the Force Free Assumption 23.
R e l a x a t i o n of C o r o t a t i o n 28.
S i n g l e P a r t i c l e Motion 29.
C o l l e c t i v e Motion 31.
Axisymmetric Case 34.
The General Case 37.
D i s c u s s i o n 42.
B i b l i o g r a p h y . 44.
Appendix 1 46.
Appendix 2 49.
Appendix 3 51.
Appendix 4 56.
LIST OF TABLES
page *
lO.Table I. Numerical E v a l u a t i o n of V ( x , y ) , Where
V(x,y)=(u/Tft (n/c)V(x,y)
* 18.Table 2. Numerical E v a l u a t i o n of x (x,y), Where
2 * <XC x>y) = 0J/7r) (n/<0 X Cx,y) coscb
V . LIST OF FIGURES
page •
10. F i g . 1. Magnetic F i e l d L i n e s , Axisymmetric Force Free Case
11. F i g . 2. E l e c t r i c F i e l d L i n e s , Axisymmetric Force Free Case
11.Fig. 3. Charge Density, Axisymmetric Force Free Case
18. F i g . 4. Lines of Constant Magnetic P o t e n t i a l , x , Axisymmetric .
_ Force Free Case
19. F i g . 5. Lines o f Constant Magnetic P o t e n t i a l ^ , m=l Force Free Case
20. F i g . 6. E l e c t r i c F i e l d L i n e s , m=l Force Free Case
21. F i g . 7. Magnetic F i e l d L i n e s , m=l Force Free Case
22. F i g . 8. Charge D e n s i t y , m=l Force Free Case
27.Fig. 9. Magnetic F i e l d Lines For The Axisymmetric C o r o t a t i o n Case
With I n e r t i a
33.Fig. 10. Streaming and C o r o t a t i n g 'Dead' Zones: Axisymmetric Case
v i . ACKNOWLEDGEMENT
I am indebted t o Dr. M. H. L. Pryce, who has supervised my
t h e s i s work. In a d d i t i o n to p r o v i d i n g the t o p i c of t h i s t h e s i s ,
he has guided my research to enable me to produce t h i s work. I
would l i k e to thank Dr. Pryce f o r h i s h e l p , f o r many u s e f u l
d i s c u s s i o n s , and e s p e c i a l l y '"-''.for appendix 4 , which i s based
on h i s work.
1. INTRODUCTION
Pulsars are astronomical objects which r e g u l a r l y give put
r a d i o pulses w i t h a p e r i o d , which v a r i e s from p u l s a r t o p u l s a r ,
of about a second. The pulses have v a r i a b l e amplitude but a p r e c i s e
p e r i o d . Over 100 p u l s a r s are now catalogued, w i t h p e r i o d s ranging
from 33 m i l l i s e c o n d s to 4 seconds. D. t e r Haar* summarizes the
o b s e r v a t i o n a l r e s u l t s . «
The f i r s t p u l s a r was discovered l a t e i n 1967 by Hewisti et a l
at Cambridge. They used a new r a d i o telescope b u i l t to study
s c i n t i l l a t i o n s , c a u s e d by plasma clouds i n the s o l a r wind,of r a d i o
sources of small angular size.>-Since.-the--time -scale of the s c i n
t i l l a t i o n s i s a f r a c t i o n of a second, the instrument was i d e a l
f o r r e c o r d i n g p u l s a r s i g n a l s .
Some ideas about p u l s a r s are f i r m l y e s t a b l i s h e d . Because o f
the short p e r i o d and i t s slow increase w i t h the passage of time
due to l o s s of energy, a r o t a t i n g neutron s t a r i s most c e r t a i n l y
the c e n t r a l o b j e c t . I t s rotational energy i s coupled to i t s sur
roundings by a huge magnetic f i e l d , having a value at the surface of
12
Bs - 10 Gauss. The reason f o r the p u l s a t i o n i s the asymmetry
introduced by non-alignment of the magnetic d i p o l e and s p i n axes.
A coherent emission process i s necessary f o r the intense r a d i o
emission. However, the emission mechanism and the s t r u c t u r e o f
the magnetic f i e l d s and of matter surrounding the neutron s t a r
i s p o o r l y understood. Many d i f f e r e n t t h e o r e t i c a l models have
been proposed but no c o n c l u s i v e model has yet emerged.
In the f o l l o w i n g work, a p a r t i c u l a r , s e l f - c o n s i s t e n t , steady-
s t a t e model f o r the p u l s a r magnetosphere i s presented. The emission
mechanism i s not considered here.
3. MAGNETOSPHERE EXISTENCE
.The neutron s t a r i s a r o t a t i n g , conducting body and thus w i l l
have an i n t e r i o r e l e c t r i c f i e l d :
(1) FT=-V7C x if
where V<=PJ-<$ i n c y l i n d r i c a l p o l a r c o o r d i n a t e s u n i t v e c t o r s )
centred on the neutron s t a r , w i t h the z-axis along the r o t a t i o n a x i s •
Q i s ' the angular frequency of r o t a t i o n . For an e x t e r i o r
d i p o l e magnetic f i e l d , an i n t e r i o r f i e l d i s 2" w i t h B q a
constant. Thus from (1) one ob t a i n s :
(2) E*=-(arB_/c) £
Th i s can be obtained from the i n t e r i o r e l e c t r o s t a t i c p o t e n t i a l
(3) -$ =ftr 2B o/2c + $ Q
$o a constant, by E=-V$ . For a t y p i c a l p u l s a r B q i s of the order 12
of 10 gauss, the r a d i u s R i s approximately 7 to 10 k i l o m e t e r s and ft i s of the order of 20 sec *. This gives a p o t e n t i a l d i f f e r e n c e
2 17
between poles and equator of A$ =fi R Bq/2C- 10 v o l t s . More
preciselyA<J> = 3.1K10 B 1 2 R 6 /P v o l t s , where B 1 2 = B q /10 gauss,
R^=R/10^cm and P-2v/Q i s the p e r i o d of r o t a t i o n .
E a r l y p u l s a r models assumed a vacuum surrounding the magnetic
neutron s t a r . The e x t e r i o r f i e l d i n t h i s case is' given by s o l v i n g 2
Laplace's equation: V $=0, and f i t t i n g t h i s s o l u t i o n to the boundary
c o n d i t i o n s a t the neutron s t a r s u r f a c e . The t a n g e n t i a l component of
E i s continuous across the su r f a c e . In s p h e r i c a l p o l a r coordinates i t i s , from (2) :
-+ (4) = -(ffcB /c) s i n 6 cos 9 9 ' tan v o . One obtains f o r the e x t e r i o r e l e c t r o s t a t i c p o t e n t i a l (5) _(B_m 5/3cr 3) P 2(cos9)+ (C QR+W 3B Q/3c) 1/r
where ^ s t n e second Legendre polynomial and C Q i s a constant.
T h i s r e s u l t s i n a s u r f a c e charge on the neutron s t a r ^ due-to the
d i s c o n t i n u i t y i n the normal component of the e l e c t r i c f i e l d , o f :
The second term vanishes i f the net charge on the neutron s t a r i s .
zero( Q= C QR+QR B o/3c ). The r e s u l t i n g e l e c t r i c field(Q=0) i s :
(7) £=-v$ = - ( B o ^ 5 / c r 4 ) P (cos6) r -(B o«t^/2cr 4) sin26 6
The magnitude of the surface component p a r a l l e l to the magnetic
f i e l d B i s 10
(8) R/c- 6x10 B 1 2 R 6 / P volts/cm
To understand what e f f e c t t h i s enormous e l e c t r i c f i e l d would
have on the surface m a t e r i a l of the neutron s t a r , one must examine
other f a c t o r s . The g r a v i t a t i o n a l f i e l d on a neutron s t a r i s l a r g e .
The- a t t r a c t i v e f o r c e at the surface i s :
(9) F g= 1 . 6 x l 0 " 1 3 M/R 62 dynes
f o r an e l e c t r o n mass(M i s the neutron s t a r mass i n s o l a r mass
u n i t s ) and correspondingly greater f o r i o n s ( about 10^ times,
assuming n u c l e i near Fe*'*'). The r a t i o of electromagnetic to g r a v i
t a t i o n a l f o r c e i s :
(10) eE/F = 7 . 5 X 1 0 1 1 B 1 0R. 3/PM g 12. 6
4 Thus g r a v i t a t i o n a l b i n d i n g i s i n s i g n i f i c a n t . I t has been shown
56 12 that Fe n u c l e i i n a magnetic f i e l d of 2*10 Gauss w i l l form an
a n i s o t r o p i c , very t i g h t l y - b o u n d l a t t i c e w i t h a b i n d i n g energy of
14 kev per i o n and e =750 eV per e l e c t r o n . L a t t i c e spacing i s
of the order of 10 cm. To remove ions r e q u i r e s a f i e l d o f
(11) E * e j / Z e l = 5 x l O 1 1 v o l t s / c m
The nuclear charge,Z, of F e i > b i s 2 6 ,and 1 i s the l a t t i c e spacing.
Compare t h i s t o the maximum e l e c t r i c f i e l d a v a i l a b l e , c a l c u l a t e d
f o r vacuum i n equation (3). Probably only the f a s t e s t p u l s a r
(the Crab p u l s a r , w i t h P=.033 sec) w i l l be able to p u l l ions from
i t s s u r f a ce. E l e c t r o n s are removed more e a s i l y than ions but the
d e t a i l s have not yet been c a l c u l a t e d . In any case, i t i s assumed
tha t a source of charged p a r t i c l e s f o r the magnetosphere e x i s t s
and serves to reduce the huge value of i n (8) .A p o s s i b l e source
i s the r e g i o n oivtside the magnetosphere. More l i k e l y , p a r t i c l e s
trapped i n the magnetosphere during the formation of the p u l s a r i n
a supernova e x p l o s i o n , so t h a t a huge E«S never develops.
THE FORCE FREE ASSUMPTION 6.
Another ' f o r c e ' enters the problem. Due to the enormous
r o t a t i n g magnetic f i e l d and the p r o p e n s i t y of charged p a r t i c l e s
to f o l l o w f i e l d l i n e s , an i n e r t i a l f o r c e due to the a c c e l e r a t i o n
of the p a r t i c l e s a r i s e s . For a p a r t i c l e i n c o r o t a t i o n w i t h the
c e n t r a l neutron s t a r at d i s t a n c e from the a x i s r , the energy i s
v.mc , wi t h :
(12) Y ( j ) = ( l - n 2 r 2 / c 2 ) - J s
T h i s becomes s i n g u l a r at the ' l i g h t c y l i n d e r ' where the c o r o t a t i o n
v e l o c i t y i l r i s the speed o f l i g h t c. The magnetic energy d e n s i t y i s 2 2 2 much grea t e r than J^THC : .-•> B /8ir >>Y^mc , except f o r very c l o s e
2 2 to r. =c/H . For an e l e c t r o n these are equal f o r B /8ffmc =
5 2 6 2x10 B, , where B, i s B i n u n i t s of 10 gauss. Thus fir i s very D O
n e a r l y c at t h i s p o i n t . Estimates f o r B at the l i g h t c y l i n d e r ( f r o m
energy l o s s by a r a d i a t i o n d i p o l e ) vary from lO^gauss f o r the
Crab p u l s a r downwards.
The f o r c e f r e e assumption i s :
(13) E+v/c XB =0
i . e . that the electromagnetic f o r c e s on a p a r t i c l e v a n i s h . We
neglect non-electromagnetic f o r c e s and assume zero mass p a r t i c l e s .
T h i s i s a good approximation everywhere i n the magnetosphere (
which i s here considered as being l i m i t e d to the r e g i o n between
the neutron s t a r and the l i g h t c y l i n d e r ) except very near the
l i g h t c y l i n d e r .
t
7. THE AXISYMMETRIC FORCE FREE MAGNETOSPHERE
The b a s i c equations are t h e f o r c e f r e e assumption (13), and Maxwell's equations:
(14) V- t=0 Vx £=0 VX&=4IT/C J V»l:=4irp
Note t h a t because o n l y steady s t a t e (3/3t - -fi3/3<{> i n the s t a t i o n a r y
. r e f e r e n c e frame) i s being c o n s i d e r e d , i n combination w i t h a x i a l
symmetry (3/3<j>=0) , Maxwell's equations take the form ( 1 4 ) , i . e .
3/3t=0. In a d d i t i o n we assume o n l y t o r o i d a l p a r t i c l e v e l o c i t i e s
and t h a t the v e l o c i t t a o f a l l p a r t i c l e s at one p o i n t are the same
(15) v=w(r,z)r <j>
The c u r r e n t i s g i v e n by
(16) j=pv .
C y l i n d r i c a l c o o r d i n a t e s are used. Since VxE=0,one w r i t e s E=-W o r :
(17) E r=-8V/3r E z=-3V/3z
V«B=0 gi v e s 3 ( r B z ) / 3 z +3 ( r B r ) / 3 r =D . Thus one can d e f i n e i> by:
(18) B =l/r 3*/3z Bz
= ~ 1 / r ^ / 9 r
The f l u x through a c i r c l e p e r p e n d i c u l a r t o , and cen t r e d on
the z-axis i s t*it = B z 2irr dr = -2ir f^dii /3r d r =-2n^(R,z)-\|>(0,z
The f l u x through the s i d e s o f a c y l i n d e r about the z- a x i s from
z=0 t o z=Z i s f'0;• B 2irr dz = 2n^)(r,Z)-if) ( r , 0 ) ) . Thus the stream f u n c t i o n
ty i s d i r e c t l y r e l a t e d t o the magnetic f l u x .
Greek indices range from 0 to 3, Latin indices from 1 to.3.
The f i e l d variables F^ v are expressed i n terms of the f i e l d
E and B i n the stationary frame, in appendix 2. The covariant **a
current density has the components, in an i n e r t i a l frame, J:=
(p,^) and satisfies the equation of continuity (conservation of
charge):
(29)
The comma(semicolon) denotes ordinary(covariant) differentiation
Maxwell's equations are then:
(30) F ^ J ^ T T / C ) ^
J a ; =0 a
e a y v X F ,=0 uv X (31)
(30) can also be written: (32) (WcX-gljV =((-g)V),v
2 2
where g=determinant g = -c r for the metric (27). aB
14. THE FORCE FREE MAGNETIC POTENTIAL 1 3
We now assume c o r o t a t i o n so that the current i n . t h e r o t a t i n g ' f r a m e
i s zero: J =0. This assumption has some j u s t i f i c a t i o n : 1) i t
holds i n the axisymmetric case, of which t h i s i s a g e n e r a l i z a t i o n t
2) A h i g h l y conducting magnetosphere w i l l have the magnetic f l u x
f r o z e n i n t o i t , thus w i l l be dragged by the f i e l d . The f i e l d i s so
lar g e that i t w i l l be locked i n t o c o r o t a t i o n .
Since;we have time independence i n the r o t a t i n g frame (the
steady s t a t e assumption), (32) reads:
(33) ( r F k l ) 5 1 =0 k l klm
Since F i s an antisymmetric t e n s o r , w r i t e rF = E A m- Then (33)becomes A ,,-A,, =0 so th a t A i s d e r i v a b l e from a p o t e n t i a l : A =X,m Thus we have: m 1 1 m • r m . 111
(34) F k l = ^ l m ( l / r ) X , m
X i s c a l l e d the magnetic p o t e n t i a l .
Now we use the f o r c e f r e e assumption, i . e . the LOrentz f o r c e a
f on each p a r t i c l e vanishes: f =^/cj F v au v=0 where u i s the p a r t i c l e ' s
v e l o c i t y . As pu i s J t h i s g i v e s : v v 6
. ctv (35) F J v =0
We use t h e metric (27) to r e l a t e J to J v : J ^ r g v ^ 0 , so th a t we have: v °
(36) J _ = ( c 2 - f t 2 r 2 ) J 0 J x=J 2=0 J 3=-Or 2J°
One gets from (35): ( F a 0 ( c 2 - ^ 2 r 2 ) -P0^ r 2)J°=0 or s i n c e J°= p(in
the r e s t frame) • i s not i d e n t i c a l l y zero, we have: r-zi-i caOro 2,, 2 2 2 ^ c
a 3 u 2 , 2-. 2_«3
(37) F u = pr / ( c -QT ) J F =\ftr /c JY^F
The c o v a r i a n t form of (35): F j V J V =0 g i v e s :
(38) F ^ O
Thus the homogeneous Maxwell's equations are just:
<39> • F 1 2 ' 3 + F 2 3 ' l + F 3 r 2= 0
ct 3 We know F in terms of x from (34) and (37). To get the
desired equation for X from (39) use:
W V g i « g j s F a B
The results are:
(41) f1 2
= f 1 2
o 2^10 2^13 -ff.2 2, 2» 2 ... 2r13 2 2r13 F = ^ r F +r F =l(fi r /c )y, +l)r F = r y. F. „ 2„20 2„23 r„2 2.2-, 2 2^23 2 2r23 F 2 3=^r F + r. F =[(fi r /c JY^ +l)r F = r F
Thus (39) becomes ( ( / r ) x , 3 ) , 3 + ( n ^ X ) 1), 1+(r Y J X . ^ . J ^ or:
(42) ' S^/a? 2 +r 2Y ( ) )2(9 2x/3z 2 +8 2
X/3r 2) +y 2 (2y 2 - l ) 3 X /3 r =0
The solution to this equation is obtained in appendix 3 for
a dipole magnetic f i e l d ,with axis oriented at angle 9 Q with respect
to the rotation axis(z-axis) , at the neutron star. The second
boundary condition was the requirement of f i n i t e fields at the light
cylinder. No attempt was made in matching the solution within the
magnetosphere to an outgoing wave solution in the region outside
the magnetosphere. This is a p e s s i b i l i t y for further study. The
magnetic potential obtained for the stated boundary conditions i s :
(43) X (z,r,9)<„R o(v)e i a>* * ( e * + e " * O ^ e ^ d x
= - ^ ( P / c ) 2 { c o s V " > / ^ b ) ) % /a V ^ e ^ d x
.' + sin ecosfr/"Jj/(~c )\zk J9. ^ v n + 2 e i x y d a }
where R Q(v) and R^(v) are given e x p l i c i t l y in appendix 3: by equations
(23)5(31) and (23)5(36) respectively, where v=l-fi 2 r 2 / c 2 , y=« z/c •
Vi is the magnetic dipole moment of the neutron star.
16.
From F U V , i n terms of E andB i n the s t a t i o n a r y frame from
appendix 2, we have: •
(44) E ^ r / c ^ 2 3x/ 9r r/c) y^ 2 3 X/ & =0
(45) B = Y^ 23x / a z B =Y^ 23x/3r B 4/rJ3x/3*
(44)gives E"*Vx=0 : l i n e s of constant X are e l e c t r i c f i e l d l i n e s .
A lso i n the r,z plane^B i s p r o p o r t i o n a l to V x . The charge d e n s i t y
p can be c a l c u l a t e d u s i n g : d / d r ( r 2 y ^ 2 ) = 2ry^ to get:
These give, respectively, the equations for \JJ and p when inertia
is included. When the mass m is zero, these reduce to (23) and
(24), using (21).. To find the solution to (57) write:
(58) ty =tyh + ty^
where ty^ is the solution to the homogeneous equation(m=0) as given
by (21)and (25). Since the inhomogeneous term is a function of r 2 2 2 2t alone, we have: d/dr((l/ Y r) d ty^/dr +(mc9/q^fl r /c^Y^O . Setting
the constant of integration to 0,we get; 4JjP={-mcfyqXfi2r2/c2)rY,3
25.
=(-mcfi/q)r( YJ-Y^). With d/dr( y^)=ny^n+2ii2?:/c2 we o b t a i n :
(59) V(-m cVq)(Y^l/Y^) For the magnetic p o t e n t i a l , F1-1 are known from (34) i n terms
of X but F ^ are not. Wc c a l c u l a t e F ^ as f o l l o w s :
F 0 1 4 l / c 2 ) F ( ) 1 < n / c 2 ) F 3 1 = 0 ^ / c 2 j ( n r 2 F 0 1+ r 2 F 3 1 ) g i v e s F 0 1 = Y fcr/c2j 3 X / 8 r
S i m i l a r l y F 0 2= - Y ^ c / q ^ m ^ r / c 2 ) Y ^ frr/c2) 3 X/9z) and F° 3=0 are found.
The only Maxwell's equation that.changes i s f o r u=0, i n (56):
(4Trr/c)p=-mn 2(r/qc)Y ( i )3(Y (j )
2-l) - (P/c2) 2 r Y ( ( )4 3 X/3z
The equation f o r x g i v e n by the V=3 equation o f (56) i s unchanged
so t h a t F1-* are unchanged. Thus o n l y P and F^ 1 are m o d i f i e d .
However, r e f e r i n g to appendix 2, one sees that both E and B are •
a l t e r e d .
In both cases, f o r and X , one has the r e s u l t :
(60) . 4Trp(qc/ft)Y^ - m c ^ (Y^ 2 + l ) -2qB z
•'The l e f t hand s i d e of (60) i s never negative (Pq>Q) . Thus we have 2
2qB^ <-mcftY(j) (Y^ +1) < 0 so tha t q and B^ are always of op p o s i t e sign.Thus one
has |B z >(mcft/2| q|) Y^ ( Y ^ + l ) . In t h i s model q may have d i f f e r e n t
v a l u e s i n d i f f e r e n t r e g i o n s . F i e l d l i n e s along which q does not
change s i g n clo not have a change i n the s i g n o f IV . I f
q changes s i g n , B^ must change d i s c o n t i n u o u s l y , r e s u l t i n g i n a
cusp i n the f i e l d l i n e . Since P=0 at the p o i n t o f change o f q , ( 6 0 ) y i e l d s
(61) LZz= cft(m 2/2q 2 - m ^ q ^ ( Y ^ 2+ l )
For a change from e l e c t r o n s t o protons i n the i n t e r m e d i a t e magneto
sphere, one'has : AB^~ cft(m p+m e)/e~ 2* 10~ 3gauss ( f o r ft-20 s e c " 1 ) .
26.
Thus the cusp i n the f i e l d l i n e s i s n e g l i g i b l e u n t i l one gets
c l o s e to the l i g h t c y l i n d e r . From (61) and the estimate of B - - 3 3 3
on page 17 : B^/B^ 10 y /10 . This i s of the order o f u n i t y 6 2 wheny^-10 . At t h i s p o i n t the p a r t i c l e energy,Y^roc > exceeds
the magnetic energy d e n s i t y . C o r o t a t i o n i s no longer v a l i d , so
t h i s model i s no longer v a l i d . N e g l e c t i n g the f a c t t h a t the mode
i s no longer p h y s i c a l f o r such l a r g e values of y ^ j t h e change i n
f i e l d l i n e s . i s i l l u s t r a t e d i n f i g u r e 9 f o r the axisymmetric case.
Compare t h i s : t o f i g u r e 1. Now the f i e l d l i n e s are not h o r i z o n t a l
at the l i g h t c y l i n d e r but v e r t i c a l and thus do not penetrate the
l i g h t c y l i n d e r .
27.
FIGURE 9: MAGNETIC FIELD LITISS FCR MISTOMETRIC CCROTATION
CASE WITH INERTIA
RELAXATION OF COROTATION 28.
I f one no longer assumes c o r o t a t i o n , what p a r t i c l e motions
can occur and how w i l l the f i e l d s be a l t e r e d ? To answer t h i s
question one procedes as f o l l o w s : C a l c u l a t e the s i n g l e p a r t i c l e
motion i n an a r b i t r a r i l y given f i e l d assuming small departures
from c o r o t a t i o n . I.e. p a r t i c l e v e l o c i t i e s i n the r o t a t i n g frame
are assumed to be of order e where e<<l. Next make assumptions
which give the c o l l e c t i v e motion i n terms of the s i n g l e p a r t i c l e
motion and use t h i s to c a l c u l a t e a l t e r e d f i e l d s . The a l t e r e d
f i e l d s can be used to r e c a l c u l a t e the s i n g l e p a r t i c l e motion. '
One repeats t h i s procedure u n t i l the d e s i r e d accuracy of r e s u l t s
i n orders of e i s a t t a i n e d . The model i s then s e l f - c o n s i s t e n t
to that order.
29. SINGLE PARTICLE MOTION
For s i n g l e p a r t i c l e motion we assume the f i e l d s are such that
c o r o t a t i o n i s p o s s i b l e .I.e. we imagine that a l l the p a r t i c l e s
except the t e s t p a r t i c l e i n question are i n c o r o t a t i o n . Then by 2
(53) we have (q/c)A Q= -mc /y^ s i n c e a l l p a r t i c l e s at any p a r t i c u l a r p l a c e i n the magnetosphere are i d e n t i c a l , w i t h mass m and charge q.
We now w r i t e the s i n g l e p a r t i c l e Lagrangian (49):
2 2 (62) L= -mc ly +mc /y -(q/c) (A 1 z+ A 2f+ A^)
= (m/2XY 9(z 2+ * V Y ^ r V ^ q / c ) A ^ q / c j A ^ 2
-((q/c) A^- my^1- ft)<J>+ terms cubic i n the v e l o c i t i e s
Since we are assuming the v e l o c i t i e s are of order e, e<<l, we
keep terms t o second order and w r i t e L . i n the form:
(63) L=%( Ax 2+ By 2+ C z 2 ) - S^x- <J>2y- * z 3 2
where (x,y,z) = (z ,r ,<j>), A=B=mY(j), C=mY^ r , $ 1=^/c)A ] L, $ 2=^/c)A 2, 2
and $3=^/c) A^- my^r Lagrangians of t h i s form are t r e a t e d i n
appendix 4. The main r e s u l t s a r e :
1) The motion c o n s i s t s of a slow d r i f t tangent to s t r e a m l i n e s ,
s p e c i f i e d by dx/X = dy/Y. = dz/Z where:
(64) X= Cq/c)rBz ^ Y ^ Y ^ l )
Y=(q/c\rB r Z ^ q / c ^
and a s p i r a l i n g motion around these s t r e a m l i n e s .
2) The energy E i s conserved.
3) The energy can be separated i n t o a l o n g i t u d i n a l energy 2
a s s o c i a t e d w i t h the d r i f t — % P J l ' , and a t r a n s v e r s e energy assoc-
30.
i a t e d w i t h the s p i r a l i n g — Noo .
4) An a d i a b a t i c i n v a r i a n t e x i s t s f o r the motion, namely M= *
N a a /IM> the t r a n s v e r s e energy d i v i d e d by the angular frequency
of the s p i r a l i n g .
t P, J2jN and a are d e f i n e d by equations (8), (9), (10), and (16)
2 o f appendix 4. E s s e n t i a l l y , %P£ i s the l o n g i t u d i n a l k i n e t i c
* 2 * 2 2 2*2 ^ energy- h^Y^t z H+ r u+ r <£M ) and Naa i s the t r a n s v e r s e k i n e t i c . 2 • 2 2 2 • 2
energy- ^ Y ^ t ZjJ- r A+ r y^ <J>) . The s u b s c r i p t s r e f e r t o motion
p a r a l l e l t o , and p e r p e n d i c u l a r to s t r e a m l i n e s .
•I
COLLECTIVE MOTION 3 1 *
If,instead of allowing just one particle to move,one allows a l l to
move,then the fields will be altered. In the formalism of the
single particle treatment this can be taken into account since
the four potential, except for A q , was- not assumed to be of a
particular form. Thus one on l y needs t o all o w f o r a different A o.
We write:
(65) A q= (-c/q>c 2/Y 9 + A Q
This modifies the Lagrangian L and thus the energy, which becomes:
(66) E = %P*2 + Cq/c) A Q (for M=0)
2 Thus one sees that A is second order in e since both E and 1
o 2
are of order e (see (67) below).
To determine the fields we must make assumptions about how
the currents derive from the single particle motion. Since we
have steady state*all particles which move along a given streamline
must be of the same type. The Coulomb interaction between streaming
particles results in no spiraling motion (M=0) and ensures that
a l l particles along the same streamline have the same energy E.
If M were not zero, collisions would occur between particles on
neighboring streamlines which damp out the transverse motion.
Thus we have convection current only: j=pv or: (67) J =p(l,x,y,z) = p(l, JtX, £Y, 11 )
using (16) of appendix 4(with c=0). X,Y,Z are given by (64). The equation
of continuity (yJ^),^ =0 reads:
(68) y(P*X),x + (yP*Y),y + y (p£Z) , z = 0
32.
From the d e f i n i t i o n (2) o f appendix 4,one has the i d e n t i t y
(corresponding to v-ko) : x y + Z = 0., thus (68) becomes:
(69) X(p£y), x + Y(P£y), y + Z ( P * / ) , Z = 0
Therefore P£y i s constant along the streamlines s p e c i f i e d by
dx/X = dy/Y = dz/Z . We w r i t e :
(70) p£y =(c 2/4Ttq) K
where K i s constant along a s t r e a m l i n e , the v a l u e of which i s
determined by the boundary c o n d i t i o n s . Write p = P Q + P J where
PQ i s zero order ..in e and p ^ i n c l u d e s a l l higher orders. (70)
•gives £ t o f i r s t order i n e as a f u n c t i o n of p o s i t i o n along a
s t r e a m l i n e s i n c e p ^ i s known from the c o r o t a t i o n case as given
by (60). Maxwell's equations(56) w r i t t e n out are:
(71) ( r F ? 1 ) ^ =(4Trr/c) J V = (4Trrp/c)(l, *X, £Y, IZ)
= (4Trrp/c, Cc/q)KX, fc/q) KY, Cc/q) KZ)
Equation (70) r e q u i r e s t h a t K and £ be zero along a s t r e a m l i n e
where p changes s i g n . In steady s t a t e , and w i t h charge s e p a r a t i o n ,
no streaming can occur along any s t r e a m l i n e along which the charge
changes s i g n . From (60) p Q changes s i g n , to a good approx
imation except very near the l i g h t c y l i n d e r , when B z changes s i g n .
Thus the p u l s a r must have a c o r o t a t i n g 'dead' zone, where no streaming
occurs, which very n e a r l y c o i n c i d e s w i t h the r e g i o n c o n t a i n i n g
f i e l d l i n e s which go h o r i z o n t a l i n s i d e the l i g h t c y l i n d e r . This
s i t u a t i o n i s i l l u s t r a t e d f o r the axisymmetric case i n f i g u r e 10.
33.
FIGURE 10. Streaming and C o r o t a t i n g 'Dead' Zones
Axisymmetric Case
LEGEND -
- C o r o t a t i o n Region
1 Streaming Region
v
0 1 Note: the s i g n of charge i s i n d i c a t e d by + and - above.
34. AX I: SYMMETRIC CASE
Write (72): A^= (fc-c/ql mc2/y^, 0, 0,* ) + ( A Q, A j , A 2 , Ap
so t h a t the zero order p a r t i s i d e n t i c a l t o (54) . The • terms
higher order i n e are contained i n the second set o f b r a c k e t s .
For a x i a l symmetry the p a r t i a l d e r i v a t i v e of any q u a n t i t y , w i t h r e s p e c t
t o 9 , i s zero. Thus A 1 and A 2 always appear i n the
combination 8A ] L/3r - 3A 2/3z = F j 2 = . Now from (64) we have:
(73) X= Q-q/c) (3A 3/3r +3i|//3r ) + nSlry ( Y ^ + l ) "
Y=(q/c)(3A 3/3z +3i|>/3z) Z= Cq/c) B^
F a 3=3A a/3x 3 -3A e/3x a i s c a l c u l a t e d from (72) and F a 3 c a n be found
u s i n g g from (28). The r e s u l t i s :
(74) F 0 1 = ( - l / c 2 ) 3 A 0 / 3 z + (Q/c 2) (3<J//3 z + 3A 3/3z) . 02 ? 9 9 9 F = ( - l / c ) 3 A Q / 3 r - (c/q^mQ r/ c .) y +(«/c ) (3^/3r + 3A 3/3r)
F 2 3 = (-mcA/q^r/c^Y - ( ^ / c 2 j 3 A Q / 3 r 4/Y^ V ^ ^ r +3A 3/3r)
F 3 1=(fi/c 2)3A 0/3z + ( l / Y ( J )2 r 2 ) ( 3 ^ / 3 z + 3A 3/3z)
The zero order p a r t s o f (74) are i d e n t i c a l t o (55), as necessary
f o r c o n s i s t e n c y .
Maxwell's equations equations (56) w i t h JV g i v e n by (67) a r e :
(75) y = 0: (4irr/c> (pQ+p j)=3/3 z (C-r/c 2) 3A Q/3 z^r/c2) (3^/3 z+9A 3/3 z))
+9 /3 r ((-r/c 2) 9 A Q/3 r - t / q X m f 2 2 r 2 / c 2 ) Y^ - ^ r / c ^ / S r+3 A 3/3r ) )
cmfi Y ( { )46(k ) (5c 2r 2/qrC 2J (iH% ( l ^ 2 ) ^ 2 ^ ( 1 - Y ^ 2 ) )
For the axisymmetric case(m=0) (99) becomes:
(100) d/di((l/ rY ( j ) V x ( )/dr)--k 2X 0 /Y ( j )2r^cmn/qr)2Y ( { )(l - Y ( { )
2)6(k)
The F o u r i e r transform of t h i s i s not the same as the equation
. f o r i|> (57), but i s the same as the equation f o r i> ' = (-cm^/q) r Y^+S'
(from ( 7 9 ) ) . . We have: d / d i { ( l / r 2 Y ^ 2 ) d/dr ((-cm^/q) r 2 Y ^ y ) = 3 3
•: (-m /cq) rY^ (Y 9 - 1 ) . When t h i s i s added t o the r i g h t hand s i d e
of (57) one gets the equation f o r 4": (101) V ^ C l/rY^ 2) W'/tW/yfo 9V/9z2<cm^/q) 2Y<f| (1-Y^ 2)
40.
T h i s i s j u s t the F o u r i e r transform of (100). This r e s u l t i s c o n s i s
t e n t s i n c e i t i s and not ij>which i s constant along s t r e a m l i n e s .
The f i r s t order p a r t of (98) i s :
(102) d / d r f t k ^ / O d A j d r ^ - O ^ / Y ^ r ) A 1 =
(-2KkirY^ 2/0((crafi/q) <5(k) Y <^ k^^/cV
This gives X=AQ+ X l U P t o s e c o n c ^ order i n the streaming v e l o c i t i e s ,
The u=0 equation o f (71) can be w r i t t e n , u s i n g (91), as:
(4irr/c) (p0+p 1)=(c/c 2r)£ ( f(Y 4 )2/c 2)d/dr(r d^ / d r ) - ( m a 2 r / c q ) y^ 3 ( Y ^ + l ) 6 (k)
2 2 2 2 - _ •*§x /c Kf 1 2 - ^ Q / c ) y ( j ) f 2 3 where p Q and p j are the F o u r i e r components o f p Q.andpj. The zero order p a r t of t h i s equation i s :
(103) 4T r rp 0/c=(-ft/c) T ( f )2r((mcft/q)Y ( ( )( Y^ 2+l)6(k) + 2b z) •
The F o u r i e r transform of t h i s i s i d e n t i c a l t o (60) ,as necessary
f o r consistency. The remainder i s :
(104) 4 1 r r p 1/c=(l/c 2;((?a 0/r ) - Y ( j )2d/dr(r d i Q / d r ) Hft/e2) Y ^ r 2 ^
T h i s has f i r s t order and second order order p a r t s , i n c o n t r a s t
to the axisymmetric case. This i s because b^ now has a zero order
p a r t , a^ i s s t i l l second order ,as given by the F o u r i e r transform
of (66).
The f i e l d components given by (94), (95),(96) have zero order
p a r t s :
(105) h w = f u 0 .4mk/c)(dX0/dr ^ m f y q^Y ^ C k ) )
r b r 0 = f 3 1 0 = i k X 0 r b z 0 = f 2 3 0 ^ -k^^OY^dXQ /dr^mJVq^Y^raCk) . ( m ^ - l / Y ^ )
2 2 2 -k r Y^(Y^ +1))
41.
The f i r s t order p a r t s o f the F o u r i e r components of the f i e l d s
are given by:
(106) b + 1= - K k ^ Y ^ / C - t m k / ^ d A j / d r
r b r l = i k X x
r b z l = ia£ry2XQ/^ - ( k ^ V / ^ d X ^ d r
One can c a l c u l a t e the magnetic f l u x through a v e r t i c a l s l i c e ,
width rA<j>, o f a c y l i n d e r w i t h a x i s c o i n c i d i n g w i t h the s p i n a x i s
( z - a x i s ) , extending from z=0 t o Z = Z Q . - : . T h i s i s : z z
(107) rA<j>/00 B rdz =A<|,/0
0 F 3 1 d z =A(f»(A(z0,r,<j»)-A(0,r,<|)))
-a simple r e s u l t . I f one i n t e g r a t e s over <j>, t h i s w i l l v a n i s h except
f o r the m=0 case, f o r which one obtains the same r e s u l t as f o r the
stream f u n c t i o n if>: 2iTjfi ( z Q , r ) - A (0,r))
From the F o u r i e r transform (87) and the formulae f o r f „ i n Ctp
terms of Xand a^ i t i s p o s s i b l e t o c a l c u l a t e a l l the f i e l d s . To
c a l c u l a t e X^ f o r a s p e c i f i c set of boundary c o n d i t i o n s there
are two approaches: i ) C a l c u l a t e the s e r i e s s o l u t i o n to (100), and
compute s e r i e s f o r ^°ZQ>^TQ>^^Q• Match these to the F o u r i e r transforms
of the f i e l d s at the neutron s t a r t o get C-(k) . i i ) Use the r e s u l t
(107) and c a l c u l a t e the c y l i n d r i c a l s l i c e f l u x near the neutron
s t a r ( e . g . f o r an obli q u e d i p o l e f i e l d ) . Then d i f f e r e n t i a t e w i t h
respect to <f> t o get A(z,r,<f>) at the boundary, Inverse F o u r i e r t r a n s
form to get X at the boundary. Match t h i s to the s e r i e s s o l u t i o n
t o (100) to determine C^(k). The f i r s t procedure i s s i m i l a r t o
tha t used i n s o l v i n g e x p l i c i t l y the axisymmetric f o r c e f r e e c o r o t a t i o n
case f o r a d i p o l e f i e l d at the neutron, s t a r , i n appendix 1.
42. D I S C U S S I O N
A f t e r arguing f o r the existence o f a magnetosphere we s t a r t e d
h i t h a simple but r e s t r i c t i v e model u t i l i z i n g the stream f u n c t i o n
IJJ. T h i s was generalized t o f i t any boundary f i e l d at the neutron
s t a r v i a the magnetic p o t e n t i a l x Next the f o r c e f r e e
assumption was r e l a x e d , ivjVich had the e f f e c t of a l t e r i n g the f i e l d s
to hold the p a r t i c l e s in c o r o t a t i o n . F i n a l l y the p a r t i c l e s were
allowed to move. The type o f motion th a t occurs was c a l c u l a t e d 'as1.was
i t s r e s u l t i n g e f f e c t on the f i e l d s .
The two major assumptions are: 1) the model i s steady
state(time independence i n the r o t a t i n g frame); 2) p a r t i c l e motions
are small departures from c o r o t a t i o n . .The f i r s t assumption seems .
q u i t e reasonable. However some p u l s a r models, most notably those
i n v o l v i n g some type of r e l a x a t i o n o s c i l l a t i o n to explain the emissions,
such as the spark gap model of Ruderman and Sutherland 4, abandon assumption
1) e n t i r e l y . In any case, keeping 1) i s reasonable i n order to
c a l c u l a t e the conditions under which non-steady processes occur.
The second assumption i s on l e s s s o l i d f o o t i n g . I t i s q u i t e reason
able well within the l i g h t c y l i n d e r , but as one approaches regions where
i s very large, the p h y s i c a l s i t u a t i o n i s ' u n c l e a r . Many other
processes become important. Others have proposed a shock f r o n t
near the l i g h t c y l i n d e r . Associated with 2 ) i s the assumption
of charge separation, which follows n a t u r a l l y from the steady
9 s t a t e c o r o t a t i o n case. Again some authors assume countering views.
43.
The l a s t models presented here, a l l o w slow p a r t i c l e
motions. The c o n t i n u i t y equation gives r i s e to the parameter K
which i s constant along a s t r e a m l i n e . Except f o r the axisym
metric case; K i s f u r t h e r r e s t r i c t e d t o be piecewise constant.
K must be known before one can proceed w i t h the c a l c u l a t i o n o f a
model. It i s determined by the c o n d i t i o n s at the neutron s t a r sur
f a c e . Since one knows the p o s t i o n of the streamlines from the
zero order c a l c u l a t i o n , K i s - then known everywhere i n s i d e the
l i g h t c y l i n d e r . From (64),(67) and(70) we have:
KCB^cmn/q^CY^.+l), B r, B ^ ^ P / O (z,*,*)
The zero order charge d e n s i t y and f i e l d s are known. The physi c s
of the emission process enters the problem i n determining the
i n i t i a l v e l o c i t i e s and thus K. If the neutron s t a r has a uniform
surface and a pure d i p o l e f i e l d , the surface p r o p e r t i e s and thus
K should be a function only of the angle from the d i p o l e a x i s .
P r e c i s e l y what t h i s f u n c t i o n i s depends on a knowledge of the phy s i c
of the s u r f a c e , which i s outside the scope of t h i s i n v e s t i g a t i o n .
The r e s t r i c t i o n f o ttatise a, piecewise constant K f o r the case of
the general stream f u n c t i o n A ,may or may not be an unreasonable
approximation to the a c t u a l p h y s i c a l s i t u a t i o n .
44.
BIBLIOGRAPHY
1. t e r Haar D, Pulsars,'Phys .Rep .3_, no . 2,57-126, (1972)
2. Ruderman M, P u l s a r s : S t r u c t u r e S Dynamics,A.R.A.A., 427-476(1972)
3. G o l d r e i c h P § J u l i a n PulsarEiectrodynamics,Ap.J,157_,869-880(1969
4. Ruderman M § Sutherland P,Theory of P u l s a r s : P o l a r Gaps, Sparks
and Coherent Microwave Radiation,Ap.J.,196,51-72,(1975)
5. Arfken G, Mathematical Methods f o r P h y s i c i s t s , Academic Press
1970 (2nd e d i t i o n )
6. G o l d s t e i n H, C l a s s i c a l Mechanics, Addison Wesley 1965(7th p r i n t i n g )
7. P r o t t e r M § Morrey C, Modern Mathematical A n a l y s i s , Addison
Wesley 1970
8. Ardavan H, Magnetosphere Shock D i s c o n t i n u i t i e s i n Pu l s a r s . I . '
A n a l y s i s of the I n e r t i a l E f f e c t s at the Lig h t Cylinder,Ap.J,203,226-232(1976)
9. Henriksen R § Rayburn D, Hot Pu l s a r Magnetospheres, M.N.R.A.S.,
166, 409-424,(1974)
10. Henriksen R § Norton J , Obiique R o t a t i n g P u l s a r Magnetospheres
With Wave Zones, Ap.J,201,719-728,(1975) •
11.. Mestel L, P u l s a r Magnetosphere, Nat .Phys.Sci. ,233,149-152, (1971)
12. M i c h e l F, Ro t a t i n g Magnetosphere: A Simple R e l a t i v i . s t i c
Model, Ap.J,180, 207-225,(1973)
13. Cohen J § Rosenblum A, Pu l s a r Magnetosphere, Astr.SSpace S c i . ,
6,130-136,(1972)
45.
14. Mestel L, Force-Free P u l s a r Magnetospheres, Astr.§Space S c i . ,
24,289-298,(1973)
15. Okamoto I, Force-Free P u l s a r Magnetosphere I,M.N.R.A.S.,167,
457-474, (1974)
16. Scharlemann E 5 Wagoner R, A l i g n e d R o t a t i n g Magnetospheres I
General A n a l y s i s , Ap.J,182, 951-960,(1973)
APPENDIX 1
( l - n 2 r 2 / C 2 ) 0 2V / 8 r 2 + 9
2 V / 9 z2 ) - ( 1 / r ) ( l + n 2 r 2 / c 2 ) 9 V / 9 r = 0
(1)
Since t h i s equation i s separable i n r and
V(r,z)=X(x) Y(y)
we write:
where x=9x/c, y=ftz/c. Separation of va r i a b l e s gives:
(11) f v n + 6 ~ 1 ( ( a / 2 ) 2 a T i _ + ((n +3-2)(n +3-3)-(a/2) 2-(m/2) 2)a r» 1 1 - 0
-(n+S-1) (2n+2B-5)a j+Cn+6) (n+3-2)a n)=0
The i n d i c i a l equation i s B(8-2)=0 , so that 8=0,2.. For 8=0 a l = 0 ' a 2 , a 3 e t c ' d i v e r 8 e - Thus we set a Q=3b so that (9R/8B)^
i s a l s o a s o l u t i o n . But we have: co n
(.12) ( 3V3B ) B = 0 = : R m ( v ) log(v) + E b n v "
Here R. (v) i s the solution.for.3=2, as fol l o w s : m
(13) R (v)=fa v n + 2
m ' 5 nm
With a n m = ( ( n + l ) ( 2 n - l ) a n l + ( ( a / 2 ) 2+ ( m / 2 ) 2 - n ( n - l ) ) a n 2
- ( a / 2 ) 2 a n 3 )*/n.(n + 2))
n-2
54,
Both s e r i e s are f i n i t e as v goes to zero. From appendix 2, we have:
F =E /c r Using t h i s and (37) from the main t e x t , we have:
E r=- (f2r 2/e 2) Y (j )
2F 23=--(«r/c) Y ^ x / 3z = - (Or/c) Y ^ i a W c ) x
Since we assume th a t E^ i s w e l l behaved as ftr/c approaches u n i t y ,
X must approach zero as one approaches the l i g h t c y l i n d e r ( v = 0 ) .
Thus the s o l u t i o n (12) i s r e j e c t e d . We are l e f t w i t h , f o r m=0:
(14) R 0 ( v ) = a 0 0 ( v 2 + ( 2 / 3 ) v 3 + ( l / 8 ) ( ( a / 2 ) 2 + 4 ) v 4 + • • • )
and, f o r m=±l:
(15) R 1 ( v ) = a 0 1 ( v 2+ ( 2 / 3 ) v 3
+ ( l / 8 ) ( ( a / 2 ) 2+ ( 1 7 / 4 ) ) v 4 + - - - )
da
The magnetic p o t e n t i a l x i s given by:
(16) X ( v , y ) = / ; R 0 ( v ) e ^ d a + ( e i * + e - i * ) £ R ^ e 1 ^
w i t h RQ and R^ from (13) as given i n ( 1 4 ) 5 ( 1 5 ) . A l s o we have
X"*Xj • i - as x,y ->0 A d i p o l e " , • •, • i_ - i a u .. , - i a u +i<j> , . . . . , , By m u l t i p l y i n g by e , -Uand e e .. a n d - i n t e g r a t i n g over y and <f>
one ge t s , f o r x->-0(v->-l) :
(17) R Q(v+l)+ - ( y / T r ) ( f 2 / c ) 2 c o s 6 o i a l o g ( x )
(18>:Rj(v*l>»- - ( y A ) ( R / c ) 2 s i n 9 o (1/x)
R e c a l l x d/dx=2(v-l)d/dv ((10) i n appendix 1) so th a t (17) i s :
(19) 2(v-l)d/dv R 0 ( v + 1 > - (y /TT) (fl/c) 2cos6 i a
Write the q u a n t i t y on the l e f t hand s i d e as the s e r i e s :
a n r Z b v" with h ~2(n+l)a </ann -2(n+2)a /ann a are the 00$ n n v ' n-1 00 v 1 n 00 n c o e f f i c i e n t s of RQ(V) as given by ( 1 3 ) . Then as v * l , we have:
(20) a 0 ( )|b n= - ( y / 7 r ) ( f i / c ) 2 c o s 6 Q i a
Thi s determines the c o e f f i c i e n t of R r t ( v ) , s i n c e b are known. 0 1 ' n
55.
To determine the c o e f f i c i e n t of Rj(v) from (18), we solve the
d i f f e r e n t i a l equation f o r W(v). We d e f i n e :
(21) W(v)=xR 1(v) = ( l - v ) i s R ( v )
W(v) w i l l be f i n i t e as x-K). With R 1(v) = ( l - v ) 2W(v), we have:
Rj =(1-V)_5SW + JS(1-V)" 3 / 2 W
RJ =(l-v) _ J 2W + ( l - v ) " 3 / 2 W +(3/4) ( l - v ) " 5 / 2 W
Thus (10) (With m=l) becomes,upon m u l t i p l y i n g by (1-v) 2:
(22) v(l-v)W-(l-v)W-(V(a 2/4)v)W=0 0 0 n+8 W r i t i n g W=ar.1Ec v , c n = l , (22) becomes: 01 » n 0
(23) v n + 6 " 1 ( - ( a 2 / 4 ) c N _ 2 - ( J S + ( n + e - ^ ( n + 6 - 3 ) ) c n _ 1 + ( n + 6 ) ( n + 8 - 2 ) c n ) = 0
The i n d i c i a l equation i s 6(B-2)=0. The 8=0 s o l u t i o n i s r e j e c t e d
s i n c e as v-*-0, R (and thus W) must; approach zero.' We are l e f t w i t h :
(24) W(v)=a 0 1Zc nv n w i t h c n = ( ( ( n + l ) ( n - l . ) + % ) c n _ 1 + ( a V 4 ) c n _ 2 )
T((n+2)n) and CQ=1
As x->0, from (18) and (21) we get:
( 2 5 ) a 0 l K =-^/ i r)Cn/c) 2sin8 0
Here a m i s the c o e f f i c i e n t of R 1 ( v ) .
Now x>as given i n (16) , i s completely determined :
S i . APPENDIX 4
»
We consider the Lagrangian:
(1) L=^(Ax 2+By 2+Cz 2) + 01x+$2y+<I>3z
Here A,B,C/!^ >*2»*3 a r e f u n c t i o n s o f x,y,z but not e x p l i c i t l y
o f t or the v e l o c i t i e s . We .define:
(2) X=8*3/Dy -9$ 2/8z and c y c l i c a l l y f o r Y and Z.
Then the Lagrange equations o f motion a r e : (3) d/dt(Ax)=Zy-Yz + i i ( A , x x 2 + B , x y 2
+ C , x z 2 ) d/dt(By)=Xz-Z* + l 2 ( A , y x 2
+ B , y y 2+ C , y z 2 )
d/dt(Cz)=Yx-Xy +3i(A, zx 2+B, zy 2+C, zz 2)
w i t h A,x=8A/8x e t c . The Hamiltonian energy f u n c t i o n i s constant:
(4) E=MAx 2+By 2+Cz 2)
s i n c e L i s not e x p l i c i t l y a f u n c t i o n o f time.
The equations (3) have x=y=z=0 as a p o s s i b l e s o l u t i o n , f o r
a r b i t r a r y x,y,z. We are l o o k i n g f o r sm a l l d e v i a t i o n s from these
p a r t i c u l a r motions. I f the q u a n t i t i e s A,B,C,X,Y and Z were
constant, then the motion would be a s u p e r p o s i t i o n of
s o l u t i o n s of normal mode type, each o f which has a time dependence
o f the form e~ l t 0 t. The c h a r a c t e r i s t i c f r e q uencies are given by
itoA Z -Y
(5) determinant -Z iuB X ' = 0
Y -X iwC
or (6) w(ABCu)2-AX2- 2 2 BY -CZ )=0
The root w=0 corresponds to uniform motion w i t h the v e l o c i t i e s
(x,y,z) • p r o p o r t i o n a l t o (X,Y,Z) . The other r o o t s correspond to
tr a n s v e r s e e l l i p t i c a l motion.
57,
to= ( (AX 2+BY 2+CZ 2)/AEC) J 5
One sees that A,B,C,X,Y and Z, which are functions of •:
p o s i t i o n , w i l l vary to f i r s t order i n the v e l o c i t i e s . They vary
slowly, due to the uniform motion, and rapi.dly.with frequency
w , due to the o s c i l l a t i n g motion. We d e f i n e :
(7)
Thus the roo t s o f (6) are 0,± w. With t h i s value f o r u, we
wr i t e the matrix equation f o r the amplitudes o f the transvei-se motion
K Z -Y (8) -Z . iwB X
Y -X iwC
=0
T h i s has n o n - t r i v i a l s o l u t i o n s (£,n>c) which are determined up to
a n - a r b i t r a r y complex m u l t i p l y i n g f a c t o r .
We s h a l l use the v e c t o r s : (X,Y,Z), (£ ,n ,0 and (£*,ri*^ , as a
ba s i s f o r d e s c r i b i n g the general motion. We introduce the d e f i n i t i o n s
(9) AX 2+BY 2+CZ 2=P •k *k *
(10) AU +Bnri =N
(11) 7 u dt = X
M u l t i p l y i n g (8) on the l e f t by (X,Y,Z) and (£,n,r.), we have:
(12) AX5+BYn+CZC=0
(13) 2 2 2 A£ +Bn +CC =0
From (8) , we can express any one of£,n,r, i n terms of another:
C=-A((XZ-itoBY)/(BY 2+CZ 2)) 5 and c y c l i c r e l a t i o n s
Thus we can express N, from (10), as a f u n c t i o n of only one of
£,n>C and one complex conjugate quantity £,n,r.. Inverting these
r e l a t i o n s g i v e s :
St.
(14) «*=(1/A -X2/P)N/2 and cyclic relations
(15) £*n=(-iwCZ-XY)N/2P and cyclic relations
We now express the velocity vectors as a superposition of
the instantaneous normal modes. These have a real amplitude SL describi
the longitudinal component and a complex amplitude amplitude o*
representing the e l l i p t i c a l component, as follows:
(16) x=£X+o5e~ l x+o*£*e l x " . —iv * * iv y=£Y+ane A+a n e A
z=£Z+a!;e"lx+a*?*e:LX
The factor e~ Aincorporates the rapid time dependence of the traiisvers
e l l i p t i c a l motion.
These equations can be inverted to give £ and ain terms of the •
velocity components. The results are:
(17) ?l= AXx+BYy+CZz
(18) Nae-1*=Ac;*x+Bn*y+(:c*z
Using (16), the energy (4) is given by:
(19) . E=%P£2+Noa*
This shows that the particle energy i s unambiguously expressible
as the sum of a longitudinal partf^PJi') and a transverse part(Noo ).
We can obtain the equation of motion for £ anda by different
iating (17) and (IP.), using the equations of motion (3),and then
using (16) to express the results in terms of % and o. The results
are expressions for I and 6 which contain slowly varying terms,
plus oscillating terms proportional to e ~ l x , e " 2 l x . In considering
59.
the " a d i a b a t i c " aspects of the motion the o s c i l l a t i n g terms can
be ne g l e c t e d , to second order i n the v e l o c i t i e s . When one i n t e g r a t e s
over many c y c l e s the f i r s t order p a r t s of the o s c i l l a t i n g terms
v a n i s h , whereas the f i r s t order p a r t s o f the s l o w l y v a r y i n g terms
do not. The s l o w l y va'rying terms o n l y c o n t a i n £,n, orC as a product * * *
w i t h one of £ ,n , or X, . (14)5(15) are used to e l i m i n a t e these terms.
The r e s u l t of the c a l c u l a t i o n f o r & i s w r i t t e n : (20) PJ--Js.i 2(X3/8x +Y3/3y +Z3/3z)P
*
-Noa (X3/3x +Y3/3y +Z3/3z)logu) + o s c i l l a t i n g terms
The r a t e of change of any f i e l d q u a n t i t y , a l o n g the path o f a
p a r t i c l e , i s gi v e n by a p p l y i n g the operator:
d/dt = x3/3x + y3/3y + z3/3z
The p a r t i c l e ' s v e l o c i t i e s , x,y,z, are given e x p l i c i t l y by (16).
When we average over s e v e r a l c y c l e s , t h i s operator becomes, t o
f i r s t order:
*(X3/3x + Y3/3y + Z3/3z )
Thus from (20), we can w r i t e , f o r a p a r t i c l e moving along a stream
l i n e : P£d£ +%l2dP +No*od(logu>) =0 or
(21) d(%P£ 2) + Nac*d(logu>) = 0
From (19) and the constancy o f E, t h i s y i e i d s : * * *
d(No o)=Na od(logu) o r d(log(No o/u))=0 , i . e . *
(22) Na a/w =M M i s a constant.
T h i s r e s u l t i s a g e n e r a l i z a t i o n of a r e s u l t f a m i l i a r i n many asp e c t s ,
of p a r t i c l e motions i n a magnetic f i e l d . Namely , the magnetic
moment i s an a d i a b a t i c i n v a r i a n t . Combining (19) and (20) g i v e s :