THE

MAGNETIC D. M.

FIELD

OF

ROTATING

NEUTRON

STARS

Sedrakyan

The e l e c t r o m a g n e t i c p r o p e r t i e s of neutron s t a r s (pulsars) a r e studied. Equations for m a c r o scopic p l a s m a movement in the p r e s e n c e of strong gravitational fields a r e obtained. It is shown that calculation of the f i r s t c o r r e c t i o n s of the general t h e o r y of relativity in the p l a s m a hydrodynamic equilibrium equations and the Maxwell equations f o r the e l e c t r o m a g n e t i c fields leads to the generation of s t r o n g toroidal magnetic fields. The o r d e r of magnitude of the r i s e time and m a x i m u m value of t h e s e fields is estimated. F o r a model with central density of 8 910 i9 g/cm 9 the growth of the magnetic field continues up to 109 y e a r s , reaching a value of 1012 G, while for a c e n t r a l density of 5 9 1014 g/era 9 the maximum value of the magnetic field is on the o r d e r of 10 ~4 G, and is attained in a time period on the o r d e r of 105 y e a r s .

1. Introduction. The m a j o r i t y of t h e o r i e s which explain the nature of p u l s a r radiation a s s u m e the existence of magnetic fields on the o r d e r of 1012 G [1-3]. On the other hand, it is well known that the most probable p u l s a r model is a rotating neutron s t a r . Thus, to provide a sound base for these t h e o r i e s , a m o r e detailed study of the possibility of intense magnetic field f o r m a t i o n in p u l s a r s is needed. The goal of this study is to show that calculation of c o r r e c t i o n s demanded by the general t h e o r y of relativity in the Newtonjan equations for the m a c r o s c o p i c p l a s m a movement and the Maxwell equations for the e l e c t r o m a g n e t i c fields leads to the generation of strong magnetic fields. The m a x i m u m value of t h e s e fields is dependent on the central density of the configuration and the initial value of the angular velocity. F o r neutron s t a r s with a c e n t r a l density on the o r d e r of 8" 1013 g/cm 3 the toroidal magnetic field can grow o v e r the c o u r s e of 109 y e a r s , attaining a value on the o r d e r of 10 l~ G. F o r c e n t r a l densities on the o r d e r of 5 ~10 I4 g/cm 9 the growth of the t o r o i d a l field continues f o r 2 "105 y e a r s and its m a x i m u m value r e a c h e s 1014 G. We will a s s u m e that the s t a r consists solely of neutrons, protons, and electrons and that the entire s y s t e m is in a state of complete degeneration. Such a neutron s t a r model was examined in [4], and it was shown that the s t a t i o n a r y magnetic fields could be no g r e a t e r than 1 G. To examine the nonstationary m a g netic fields, we will f i r s t derive the basic equations. Then we will show the possibility of the formation of a toroidal magnetic field which i n c r e a s e s with time, and evaluate its m a x i m u m magnitude. 2. Fundamental Equations. In [5,6] it has been shown that rotation leads to insignificant d e p a r t u r e s f r o m s p h e r i c a l s y m m e t r y , even when the p a r t i c l e velocity at the equator approaches the velocity of light. T h e r e f o r e , we will not c o n s i d e r the effect of rotation on the distribution of m a t t e r and the f o r m of the configuration s u r f a c e and will a s s u m e that the objects examined a r e s p h e r i c a l l y s y m m e t r i c . M o r e o v e r , in describing the gravitational field we will limit o u r s e l v e s to the post-Newtonian approximation; i.e., we will a s s u m e that ~/c 2 (e is the gravitational field potential) is s m a l l c o m p a r e d to unity, and we will include only t e r m s linear in e / c 2 in our examination. We will derive the fundamental equations for the c a s e of strong gravitational fields, for the present not setting any limitations. The time and space m e t r i c s have the f o r m [8] ds 2 = gtkdx ~dx ~.

(1)

E r e v a n State University. T r a n s l a t e d f r o m Astrofizika, Vol. 6, No. 4, pp. 6 1 5 - 6 2 4 , O c t o b e r - D e c e m b e r , 1970. Original a r t i c l e submitted April 17, 1970.

9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

339

w h e r e gik a r e the s p a c e - t i m e components of the m e t r i c t e n s o r . We introduce the e l e c t r i c E , D and m a g n e t i c - ~ , g field intensities ~ and g a r e dual to the a n t i s y m m e t r i c t e n s o r s H a 5 and B a i l ) in the following manner: = B~ = F~,

G0 F ,

(2)

H 4 = ] gooF ,

(3)

where a , fl =1, 2, 3 and F a i l a r e the spatial components of the e l e c t r o m a g n e t i c field t e n s o r . Considering the s t a t i o n a r y c h a r a c t e r of the gravitational field, we w r i t e the Maxwell equations L~ the f o r m [8] div/) = 4~n.

(4)

div/} = 0

(5)

l~

o~-

c

----

(~)

c Ot

lh

T 0-7

(7)

(9)

: - - -!- [ g E l .

w h e r e g - ~ - - g 0 a / g 0 0 , n is the e l e c t r i c c h a r g e density, and 7 is the e l e c t r i c c u r r e n t v e c t o r with components s a =en(dxa/dt), whereby the c u r l and d i v e r g e n c e have the f o r m

(ou,~a)~= 2 V---7= diva--

1

\~

O~~ /

(to)

0 (l/Ta~)

gaff = -gaff +g00gagfl is the three-dimensional

space metric, g is the corresponding earl y is the unit a n t i s y m m e t r i c t e n s o r with e123e123=1.

determinant,

and

To go f u r t h e r , it is n e c e s s a r y to derive the m a c r o s c o p i c motion equation f o r the e l e m e n t a r y p a r t i c l e gas in the s p a c e d e s c r i b e d by Eq. (1). The hydrodynamic equilibrium equations can be written in the following form"

T,~ = o,

(ll)

w h e r e Tik is the sum of the e n e r g y - i m p u l s e t e n s o r s of the m a c r o s c o p i c bodies and the e l e c t r o m a g n e t i c field [8]

T/'=(P@[')uiuk--~P+

7

5tl*lmP --FizF~l"

(Fik is the electromagnetic field tensor, p and P are the energy density and elementary particle pressure , u i is the four-dimensional velocity vector). Substituting Eq. (12) in Eq. (11) and assuming that the particle system undergoes a rotary motion with constant velocity ~2 =d~o/dt and u 3 =~2u 0 [12], we obtain

ljOE~ ~, I ] ~ B ~ ~ c)P O In u~ = O, c c -- dx" d- (P q- P) -c)x--~

(I3)

where f

-

]/g0oe'&-7'Cd'd

( "~ =

2o

goo

The e x p r e s s i o n f o r u ~ follows f r o m the fact that u i "u i =1.

340

9~

+-7--G~+ 77&~

)- ",-/2

"

We will examine the gaseous s p h e r e consisting of neutrons, protons, and e l e c t r o n s . Having designated the e n e r g y density and p r e s s u r e of the n e u t r o n s , protons, and e l e c t r o n s r e s p e c t i v e l y by pl, P2, P3 and Pi, P2, P~ and the c u r r e n t s c r e a t e d by the protons and e l e c t r o n s by j~, Ji, we will r e w r i t e Eq. (13) f o r each s o r t of p a r t i c l e :

Oxi + ( P x 4z',,x)d d x ~ lnu ~ ~ ~ C

--~

+ Z I~,___

OP~

+ (p~ +

c

Ox ~

]~E~-- c

dx--:-

(14)

,~ ) ~ ,2

~

C)x~

~. ,o = p~ .

~3 dx ~ lnu~ -: -- P~

(15) (16)

In writing Eq. (13) f o r the protons and e l e c t r o n s , we calculated with the a s s i s t a n c e of the v e c t o r pie the i m p u l s e t r a n s f e r for collisions of t h e s e p a r t i c l e s [9]. F o r convenience, as was done in [9], we will r e p l a c e E q s . (14-16) with o t h e r equations. We will take the sum of Eqs. (14-16), the difference of Eqs. (15, 16) and an equation obtained f r o m c o n s i d e r a t i o n of the conditions of c h e m i c a l equilibrium between the neutrons, protons, and e l e c t r o n s : gl =t~2 + g3 (tq, #2, ~3 a r e the c o r r e s p o n d i n g c h e m i c a l potentials). Simple m a t h e m a t i c a l calculations lead to the following.

e[sB] C

V-gTovP§ g~ooo(P-~,) -~lnu ~ = 0 .

s~ =E-q- 1s: ._ l/-gT~ [vP3--~(n)~-P~] 1 o cn en 1 + ~ (n)

[sB] ecn (1 + ~ (n))

~1 = ~2 + ~3 '

(17)

(18) (19)

w h e r e ~s is the e l e c t r i c c u r r e n t with c o m p o n e n t s - ~ a = n ( 3 x ~ / O t - O x ~ / 3 t ) , p (n)= (P3 + P3)/(P2 +P2), e is the e l e c t r o n i c c h a r g e , and n is the e l e c t r o n or proton density (we a s s u m e the two to be equal), whereby the v e c t o r product has the f o r m

e ~ s , B = [sB]~.

:=

In analogy with n o n r e l a t i v i s t i c t h e o r y , we a s s u m e the v e c t o r p~e, which d e s c r i b e s collisions between protons and e l e c t r o n s , is p r o p o r t i o n a l to the c u r r e n t [9] "

z ] / ~ 0 0 /3;e en

(cr is the e l e c t r i c a l conductivity of the neutron s t a r ) . Equations (17-19), the Maxwell equations (4-9), the equations d e s c r i b i n g the g r a v i t a t i o n a l field, and the equation of s t a t e f o r a f r e e gas of d e g e n e r a t e p a r t i c l e s c o m p r i s e a s e l f - c o n g r u e n t s y s t e m of d i f f e r e n tial equations f o r the solution of the p r o b l e m we seek~ 3. The G e n e r a t i o n Effect. F o r f u r t h e r study it will be n e c e s s a r y to choose a model f o r the neutron s t a r . By limiting the study to a m o d e l with c e n t r a I density not exceeding 5 - 101~ g / c m 3 we can r e g a r d the e l e m e n t a r y p a r t i c l e gas as a l m o s t n o n r e l a t i v i s t i c , and include in the m e t r i c t e n s o r only t e r m s l i n e a r in

~/c 2 [sl goo =]/ 1 @ 27/C ,

g:~ ' ~ "~ ~ ~ ( 1 - - ~q~/c~

go~ = O.

(20)

H e r e it is n e c e s s a r y to note the following: g03 is p r o p o r t i o n a l to ~/c 2 . s w h e r e r is the distance f r o m the axis of rotation. F o r the models to be examined, the m a x i m u m value of f2r/c does not exceed 0.1, and we m a y t h e r e f o r e r e g a r d g03 to be equal to z e r o e v e r y w h e r e . We then w r i t e the t i m e component of the f o u r dimensional velocity u i in the f o r m

u~ =

goo + 2~-' c go~_+_ a ~176

~ ~- 2c 2

Hence, the last t e r m of Eq. (17) is equal to c~

(21)

341

As an equation of state, we choose the equation of state of a n o n r e l a t i v i s t i c d e g e n e r a t e gas [4]: 5/3

ni mz

Pl =

a

a ~ ,

=

1 -~-

( 3 r 1 6 2~

) 2/a1~2, i : 1, 2, 3.

(22)

Then, in a c c o r d a n c e with Eq. (22), we obtain

P3 + P'~ ~ m:,c2n _ m3 ~ << 1, (n)-~- Po + P2 :rncen - - - m = where ~ is the ratio of the m a s s e s of the electron and proton. The m a s s e s of proton and neutron a r e taken as equal and indicated by m. Since a << 1, we r e p l a c e l + f l ( n ) ~ 1 and, in a c c o r d a n c e with Eq. (22), we wilt have

;P~--~ (n) vP2 = (1--~t2) vP3 ~ - V P 3. The indicated modifications in Eqs. (17-19) produce the following equations:

[sB] CR

|/~ooovP 4- mnl (V~-- 2~5 ]/goo : O.

Z=#+__t

+V

ell.

(23)

[;#].

ip _ en

(24)

eBC

3.~ /2 ~

(25)

~ ' Jl 1

(n I is the neutron density). Eliminating P3 f r o m Eq. (24) with the aid of Eqs. (23) and {25) we finally have

[s2B}c_Z__ ~ -d"

~=~+

goo(V~- ~ r )

~nc

~6~

Employing the curl o p e r a t o r on Eq. (26),

(ourtE)~ -

1

21"T

=

-

-

--

~ ]/ g00] "

(27)

Working out the v e c t o r product on the right of Eq. (27) and substituting the Maxwell equation (6) on the left, we obtain --

at

= --c~ R e

OR

sin 0 cos 0~,

(28)

where R, O, r a r e s p h e r i c a l c o o r d i n a t e s , a n d 7 is the unit v e c t o r in the direction of the s t a r t s rotation. Equation (28) is obtained f o r the initial m o m e n t of t i m e when the e l e c t r i c c u r r e n t s =0 and the magnetic field-B =0. If we substitute in Eq. (28) the e x p r e s s i o n g00 f r o m Eq. (20) and r e p l a c e d~/d1~ =Gu/R 2 (u(R) is the m a s s contained in a s p h e r e of radius R), then we finally obtain the following: 6B _

at

_

mc Q~ Gu sin~cos0~. e c2R

(29)

(c is the velocity of light; G is the gravitational constant). F r o m Eq. (29) it follows that a toroidaL magnetic field a r i s e s in the s t a r , i n c r e a s i n g with t i m e . The p r e s e n c e of rotation with calculation of the Einstein c o r r e c t i o n s leads to the r e s u l t that the e l e c t r o n v e l o c ity b e c o m e s v a r i a b l e with r e s p e c t to the proton velocity, being dependent on the z coordinate. And it is this which leads to the production of a magnetic field [11]. It is e a s y to see f r o m Eq. (18) that, ha those regions of the s t a r where the state of the p a r t i c l e s is d e s c r i b e d by the u l t r a r e l a t i v i s t i c equation of s t a t e for a d e g e n e r a t e g a s , ~ (n)=1, Pl =P2, and the b a t t e r y effect mentioned is absent. Since the magnetic field f o r m e d is p a r a l l e l to s2, it does not change the gradient of the magnetic field o v e r t i m e . During the t i m e the c u r r e n t is not too large, the magnetic field grows linearly with time:

B = I 'race ~ c2R(--Tusin 0 cos ~ l.t"

(30)

The question of how long the linear growth of the magnetic field will continue m a y be a n s w e r e d by e x a m i n ing Eq. (24) for an a r b i t r a r y m o m e n t of t i m e . Applying the curl o p e r a t o r to Eq. (24), we obtain

342

O Ot

mc

~2 .Gu sin0cos0m . . . e

.ccurl. s . ccurl~[S .

c2 R

:

(31)

enc

The f i r s t t e r m of Eq. (31) appears as a constant b a t t e r y (if, of c o u r s e , ~ =const) for production of the m a g netic field, while the second and t h i r d t e r m s act in the opposite direction, tending to reduce the field. Initially, s =0, and t h e s e t e r m s a r e absent; but with i n c r e a s e in the magnetic field, s i n c r e a s e s , as c o n s e quently do t h e s e t e r m s . In the c o u r s e of t i m e the rate of growth of the magnetic field d e c r e a s e s , and finally, when the dissipative t e r m s c o m p e n s a t e for the battery effect, the field tends to its maz.imum value. To estimate the m a x i m u m attainable value for the toroidal magnetic field, it is n e c e s s a r y to know the t i m e o v e r which the second o r t h i r d t e r m of Eq. (31) becomes c o m p a r a b l e to the f i r s t . The magnetic field changes according to Eq..(30), so that the c u r r e n t i n c r e a s e s : cA.t.

J'~ D

(32)

To obtain Eq. (32), Eq. (7) was employed. A has the sense of the growth rate of the magnetic field, while D is the order of distance at which the field changes significantly. In the evaluations, we will take D on the order of the starts radius. The time over which the second term of Eq. (31) becomes comparable with the f i r s t is D~z tl "~ c-2- sec.

(33)

In [10] the conductivity of a neutron s t a r was evaluated, being on the o r d e r of 1029 sec -1. The time o v e r which the t h i r d t e r m will compensate for the battery effect can be calculated f r o m tz~ \-~c/

'

(34)

F o r a neutron s t a r with c e n t r a l density of Pc =8" 10 i3 g/cm 2, the m a s s is equal to M =0.12 MQ, the radius R=107 cm, and ~ =40 sec - I [7]. Then, f r o m Eq. (30), the growth rate of the magnetic field is A =10 -4 G/sec. The time o v e r which the ohmic l o s s e s c o m p e n s a t e the b a t t e r y effect is on the o r d e r o f t I ~1015 y e a r s , while t2~109 y e a r s . To evaluate the m a x i m u m value of the magnetic field, it may be considered that over the c o u r s e of approximately 109 y e a r s the field grew almost linearly and attained a value of ~1012 G. F o r a neutron s t a r with c e n t r a l density Pc =5 9 10 it g/cm 3, the m a s s , radius, and ~ m a x a r e equal to 0.637 MQ, 10 ~ cm, and 6" 103 sec - l . The growth r a t e of the field is on the o r d e r of 3 9 102 G/see, and t i m e s tl and t 2 a r e equal to 1012 and 2 9105 y e a r s . Over a t i m e of the o r d e r of 10 ~ y e a r s linear growth will lead to a m a g netic field on the o r d e r of 1014 G. If we identify the p u l s a r in the Crab nebula with the model of a neutron s t a r with Pc = 5 910 it g/cm 3 and a s s u m e that o v e r its lifetime (which has been estimated as being on the o r d e r of thousands of y e a r s ) its angular velocity has d e c r e a s e d linearly f r o m a m a x i m u m value of ~ m a x =6 9103 see -i to the presently obs e r v e d value of ~2~30 sec -/, then the toroidal magnetic field generated by the battery effect should be on the o r d e r of 1012 G. To solve the p r o b l e m of the distribution of the magnetic intensity within the neutron s t a r after the d i s a p p e a r a n c e of the battery effect, it is n e c e s s a r y to solve the s y s t e m of equations consisting of Eq. (23), the equation for the gravitational potential ~ in the post-Newtonian approximation, the equations of state f o r the m a t t e r , and equations e x p r e s s i n g the decay of the battery effect in the f o r m curl

e

-~

ecn I

=

(35)

The author e x p r e s s e s his gratitude to P r o f e s s o r Roxbourgh, f o r his evaluation of the r e s u l t s . It was a discussion with the p r o f e s s o r which led the author to this examination of the b a t t e r y effect in neutron s t a r s . I am grateful to P r o f e s s o r G. S. Saakyan for his evaluation of the study, as well as to E. V. Chubaryan and V. V. Papoyan for evaluating the r e s u l t s and p r e p a r i n g the m a n u s c r i p t for publication. This study was done in Cambridge, at the Institute for T h e o r e t i c a l A s t r o n o m y .

LITERATURE I. 2.

H.Y. Chiu, V. Canuto, and L. Fassio-Canuto, T. Gold, Nature, 221, 25 (1969).

CITED Nature, 22___~I,529 (1969).

343

3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

344

V. Canuto, H. Y. Chiu, C. Chiuderi, and H. J. Lee, Nature, 225, 47 (1970). D.M. Sedrakyan (Sedrakian), M. N. (in press). J . J . Monaghan and J. W. Roxburgh, M. N., 131, 13 (1965). V . V . Papoyan, D. M. Sedrakyan, and E. V. Chubaryan, Soobshch, Byur. Obs.~ 40, $6 (1969); Astro~ fizika, 3, 41 (1967). G . G . Arumyunyan, D. M. Sedrakyan, and E. Vo Chubaryan, Astrofizika {in press). L~ D. Landau and E. M. Lifshits, Field Theory [in Russian], Fizmatgiz, Moscow (1967)~ po 330. L. Jo Spitzer, Physics of Fully Ionized Gases, Interscience, New York (1967), p. 23~ G. Baym, Ch. Pethiek, and D. Pines, Nature, 224, 674 (1970). J . W . Roxbourgh and P~ A. Strittmatter, M. N., 133, 1 (1966). J . B . Hartle and D. H. Sharp, Ap. J., 147, 317 (1967).