overestimation.

For

"true"

Te ~

12 000~,

the

error

may

reach

1500~

When allowance is made for the "patchy" distribution of the anomalies over the surface of the Ap stars, the above effects of an increased silicon abundance are, of course, smoothed, but, judging from the results of [15], are still readily observable, We

are sincerely grateful We are also grateful of the State University

method. center

to V. V. Tsymbal, to the mathematical at Rostov-on-Don.

who assisted support group

us in mastering the at the computational

SAM-I

LITERATURE CITED 1. 2.

3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

PAIR

B. P i k v l ~ n e r and V. L. K h o k h l o v a , U s p . F i z . Nauk, 107, 3 ( 1 9 7 2 ) . S w i h a r t , A s t r o p h y s . J . , 123, 143 ( 1 9 5 5 ) . S W r i g h t a n d J . A r g y r o s , Comm. U n i v . L o n d o n O b s . , No. 76 ( 1 9 7 5 ) o R S. K u r u c z , SAO S p e c i a l R e p o r t , No. 309 ( 1 9 7 0 ) . V V. L e u s h i n , A s t r o f i z . Issled. ( I z v . SAO), 3 , 36 ( 1 9 7 1 ) o C W. A l l e n , A s t r o p h y s i c a l Quantities, A t h l o n e P r e s s , London (1972). D F . G r a y , The O b s e r v a t i o n a n d A n a l y s i s o f S t e l l a r Photospheres, Wiley-Interscience, New York ( 1 9 7 6 ) . K. R. L a n g , A s t r o p h y s i c a l Formulae, Springer, Berlin (1974). Yu. V. G l a g o l e v s k i i , K. I . K o z l o v a , a n d N. ~{. C h u n a k o v a , A s t r o f i z . Issled. ( I z v . SAO), 5 , 52 ( 1 9 7 3 ) . K. S t e p i ~ n a n d H. Muthsam, A s t r o n . A s t r o p h y s . , 92, 171 ( 1 9 8 0 ) . S. E. S t r o m a n d K. M. S t r o m , A s t r o p h y s . J . , 155, 17 ( 1 9 6 9 ) . Yu. B. G l a g o l e v s k i i , A s t r o n . Z h . , 43, 1 ( 1 9 6 6 ) . O. G i n g e r i c h a n d J . C. R i c h , A s t r o n . J . , 71, 161 ( 1 9 6 6 ) . D. S. L e c k r o n e , A s t r o p h y s . J . , 185, 577 ( 1 9 7 3 ) . L. I . S n e z h k o , A s t r o f i z . Issled. ( I z v . SAO), 8 , 14 ( 1 9 7 6 ) . S

R

CREATION V.

S.

IN

A

STRONG

MAGNETIC

FIELD

Beskin

The creation of electron--positron pairs in an inhomogeneous magnetic field is considered. It is shown that for B > 5"1012 G and y-ray energy Ey < i04 MeV the particles are created in the zeroth Landau levels (n = n' = 0). In the case of weaker fields, the d i s t r i b u t i o n of the p r o d u c e d particles with respect to the numbers of the Landau levels is investigated. For Ey < 108 MeV, the d i s t r i b u t i o n has a sharp peak at n = n', but for Ey > 108 MeV the case when one of the particles acquires the entire energy is the most probable. Some astrophysical consequences of the results are discussed~

I. Introduction It is now clear that allowance for the creation of electron--positron pairs in a strong m a g n e t i c field is the key to an u n d e r s t a n d i n g of the processes taking place in t h e m a g n e t o s p h e r e s of pulsars [i, 2]. Hard y rays p r o d u c i n g pairs are emitted by r e l a t i v i s t i c particles either by virtue of motion along curved m a g n e t i c lines of force or by virtue of s y n c h r o t r o n losses. The p r o b a b i l i t y of p r o d u c t i o n of a pair by a p h o t o n of energy Ey in a m a g n e t i c field BI is [3, 4]

IV:

3 m eaB~ 29[~ ~mc3

e-2~;

q =-

4

Bo m c 2

--

3 B ~ E~,

>> 1,

(1)

P. N. L e b e d e v Physics Institute. T r a n s l a t e d from Astrofizika, Vol, 439-449, July-September, 1982. Original article submitted July 14, 1981; p u b l i c a t i o n May 3, 1982.

266

0571-7132/82/1803-0266507.50

9 1983 Plenum Publishing

18, No~ 3~ pp. accepted for

Corporation

W:

where

B0 = m 2 c 3 / e h

= 4.4"!013

5

~/i/ 3

eaB ~

(2)

G.

Nevertheless, for investigation of particle production in pulsar magnetospheres the expressions given above are inadequate. Above all, Eqs. (I) and (2) are obtained under the assumption that the electron and positron are created in fairly high Landau levels, so that the created particles are relativistic. However, there exists a region of T-ray energies and magnetic fields for which the production occurs in low and even the zeroth Landau levels. In this region, Eqs. (i) and (2) are naturally inapplicable. Second, in the relativistic case the particles are produced with nonzero pitch angles, so that the photons emitted as a result of synchrotron losses will also be absorbed in the magnetic field and give new electron--positron pairs. And to investigate such processes, it is necessary to know the distribution of the produced particles with respect to the pitch angles and energies (or, which is the same thing, with respect to the numbers n and n' of the Landau levels). In the first part of the paper, we determine the probability of pair production in the "nonrelativistic" case, when the production takes place in low Landau levels. This case corresponds to fields B > 5"1012 G (which are possible on the surface of neutron stars). In the second part of the paper, we investigate the distribution of the particles with respect to the numbers of the Landau levels in the "relativistic" case. It is found at low y-ray energies (corresponding to Eq. (i)) the distribution function fn,n' has a sharp peak at n = n'. In contrast, at high y-ray energies the case when one of the particles takes virtually all the energy is most probable. In the third part, we discuss the astrophysical consequences of our results. 2.

Nonrelativistic

Case

We note first one further feature characteristic of pair production in pulsar magnetospheres. This is that irrespective of the emission mechanism the y ray is emitted effectively along the magnetic field and a transverse component B 1 arises because of the curvature of the magnetic lines of force. This is due to the high magnetic field on the surface of neutron stars (B ~ 1012 G), as a result of which particles can move only along the field. In all that follows, we shall assume B• = x/pB, where x is the mean free path of the T ray, and p is the radius of curvature of the magnetic line of force (p ~ 107 cm for pulsars). Suppose a photon and W(ET, B, 0) is the a frame of reference in Then in this system 0' factor of the relative obvious equality

of energy ET moves at angle 0 = x/p to the magnetic field B, probability of pair creation per unit length. We go over to which the photon moves at right angles to the magnetic field. = 7/2, B' = B, and ET = Ey/TI], where Tl] = 1/@ is the Lorentz motion of the frames of reference. On the basis of the

~(&, and

bearing

in

mind

that

dl'/d2

= I/T[[

B, ~i) d! = = 0,

we

W(EI, B,

=12) d r

obtain '

W(E~, B, ~))=oW(I~E~, B, We now w r i t e down a n e x a c t e x p r e s s i o n electron in level n and a positron in level unpolarized photon)

IV,,,', , ( x ) Here, K == I//--~ k 0 ~ k23 ~ 4 n ~ E~,mce-mc/l~ is the e n e r g y

1

2

for n'.

the probability Using Eq. (3),

(3)

of we

e ~ 1 mc ~ B S d k a ( l ~i ' - i t ,~ - t - i ~, a ~ I ~ ) ; ( K * ~ _ '

hc }~ ~ ao

production of have [4] (for

an an

(4)

K---z)

are the e n e r g i e s of the e l e c t r o n and the p o s i t r o n , of the photon. The matrix elements are

and

~ = x/~,.

267

i~,

--

~ ( A i A~ - A;A3}

=

(E~B.~ ~

,

Cn,

-

n

:

BIB/o

....

o i

-1,

.

.

9

<5) . ~ 113

~4!

(A~ A z

--

A'4A3)(B~B3L,_~ o . . , '

--

"

B~B41~

,r),

where

ko

] k

-

mc ~

-

~

;

Kd=

K ~-k-,:

,

", =

-eH -; 2ch

= il is t h e s p i n p r o j e c t i o n , ~ = 1 f o r t h e e l e c t r o n , ~' = --! for t h e p o s i t r o n , p r i m e corresponds to t h e p o s i t r o n , a n d In, n, a r e L a g u e r r e f u n c t i o n s . A f t e r s u m m a t i o n o v e r t h e s p i n p r o j e c t i o n in t h e e a s e w h e n (which means that we are nonrelativistic)., we have

! % 1~ =

'~n,o' -- I ~: I" +

k 3 << ko~

the

4 n 7 << k~0 2

,,,.'-~ (a) + I Z" I , . ' ~a ) + / , 2 ~_ 1 , . ' - l ( a" ) . ],,~, .' (~) + ]~-

(6)

The a r g u m e n t i n t h e L a g u e r r e f u n c t i o n s i s a=2(Bo/B)(x/lo) ~, w h e r e ~ = 2 7 (mc~iEj i s t h e threshold ranffe of the photon at which the "transverse e n e r g y " b e c o m e s e q u a l t o 2me 2. I f any i n d e x i n t h e L a g u e r r e f u n c t i o n s is negative, the Laguerre f~n~tion vanishes; thus 90,0 d I~,0" After

integration

o f Eq.

(4)

over

W., ., (x) = ~

e2

2 v V ~

lows

Since the argument from the condition

in the of being

k3,

we o b t a i n

1 mc 2 B

~ E~

~o ~/-Co~

Laguerre functions nonrelativistic),

for

the

total

probability

B - ~ o (. + .')

satisfies a >> n, a >> n' we can use the asymptotic

,l~,,~.(a) ~.12 .... '(a)=~e As a r e s u l t ,

(7)

'-P.... ' (a)

1 n!n'!

. . . . .a

(which fol= behavior

~,,,

(8)

W,,,,,'(x) we h a v e

DY(x]=~ n) n'

1

e"

t~.'(~)-- 2 I ---Y hc

1 m c ~ B X~8

x--l o-

lop

>~

Bo

>-, E., Bo ~

/ ~

2B

T

\-~: / i "

V p

1

H e r e Zp :=

(p

!; p = n -~ n I .

k)

F i g u r e 1 shows t h e can t a k e o n l y t h e f i r s t

d e p e n d e n c e o f W(x) on x . few numbers p. Indeed,

interested in the present case are dition 2n(B/B O) << 1 is valid only For

such

.....

small

p,

the

large (see for p 5 i0.

distribution

Note however that to g o o d accuracy one the magnetic fields in which we are below), so that the nonrelativistic con-

function

with

1

respect

to

n

and

n'

is

1

',n+n' n[ n ; !

and

the

value

p : n + n'

itself

can

be

determined

by

integrating

ltp) From t h e

condition

~ W,/(x)dx : 1 we o b t a i n 0

268

approximately

the

expression

(9).

10la I

345

20

50 100

7"10~:

/ ~

5"10~

/

/ 2

4 log Ey

6 (MeV)

\ \ co

a3 §

\ co e0

F i g . 2. M e a n v a l u e of p = n + n' as a f u n c t i o n of the 7 - r a y e n e r g y Ey and the m a g n e t i c f i e l d B. The curves corresponding to p = 20, 50, i00 w e r e o b t a i n e d u s i n g the r e l a t i v i s t i c f o r m u l a (19).

cn c~

x/Lo

Fig. W on

1 . Dependence the mean free

of

the

path

probability

x of

a y

ray.

-

B

I

P \2mc 2 I

('!1)

Inl2B~ I--2 g i The

photon

range

can

be

simply

determined

as

l~ ~-; [n-= 2[, (rnc2/Ey ).

Figure 2 shows the dependence of p on the photon energy Ey and the magnetic field B. We see that for fields B > 5-1012 G and energies Ey < 10 4 MeV production takes place in the zeroth Landau levels. For y-ray energies greater than 10 4 MeV, the boundary goes over to the region B ~ BO, where the single-particle approximation breaks down altogether. We recall that at energies Ey < 10 2 •eV pulsar magnetospheres are transparent, and pair production does not occur at all. 3.

Relativistic

Landau emitted

Case

We n o w c o n s i d e r the distribution function /n,., with respect to the numbers of levels in the relativistic case. We s h a l l again assume that the photon is along the magnetic field. It is clear that

the

/ .... ' =,I' W,,. o, ( x ) F ( x ) d x ,

(12)

0

0

At

the

same

time

2/,

,,,

=1, since

n, n

n, R' 0

Integrating we o b t a i n

Eq.

(4)

over

k 3 and

0

going

over

to

the

dimensionless

variable

y~-1-~-~'E-r B x 2 rnc 2 Bo F'

269

rnc ~ =~dyep,,

?

Here y,,=(1/V~T)(B/Bo)aa(Vn+ over the spin p r o j e c t i o n by

q)

J

t~c ?:

""

I-]),

4---

the

(14)

t 24' ~ ~t" n' -t // y,,,.v

'

.%,

and

F (y)

element

matrix

~ ....

is defined after summation

[4]

Bo ..... " - 2 B

nn'

""' ~' -k nnl" (n -~ n') ( W F + ], n ) , .... . + - - ] / n--~

\ Y,

/

(15)

'

Note that in [4] the f u n c t i o n [~'n..' did not arise in the i n t e r m e d i a t e calculations, the i n t e g r a t i o n was p e r f o r m e d first over n and n' and only then over k 3. In fact, shown that

the L a g u e r r e functions

are of interest only for y = Yn + O.

'

*J"vv+, :)::'(~176176v~ 't t "

(VT+V~-#)="%.') ~ ] '= ( y . -} O) = ....

where Ai(

(I/n-;- I 2)2:a(nn'/;6

(nn')~/6

'

(V7+

1w

} is an A i r y function,

~

it can be

Bo ,,

1.~ 2B~;--~--')--~n.. 1 ~ - ~

since

1,

li

(16)

-~ ~. l'

"

and the prime denotes the derivative.

In the general case, it is not p o s s i b l e to c a l c u l a t e the integral (14). However in the two l i m i t i n g cases c o r r e s p o n d i n g to the a s y m p t o t i c b e h a v i o r s (1) and (2) the i n t e g r a t i o n can be completed. We note first that the p h o t o n range 1y has the d i s t r i b u t i o n function W(x)F(x). We define the m e a n as

range

>= ('xW(x)F(x)dx--

4 3

3

(17)

,9o m e ~ 1 ~, B E~ q

0

where

q--

Writing

the

4

Bo mc 2

r,

3

B E~



energy

Bo mc 2 3 B= E~ 4

conservation

E~ ':' < l~ "\ P

----

mc ~ k

law

is

in

the

the

quantum

parameter

in

(i) and

Eqs.

(2).

form

B + mc~ ] / 1+2# ~ / 1 4- 2n -~o

~ ]/' ~-/~; B me 2(],n + ]/n'), "

-

-

we have l~ n ~ |

.~=

-

If is

we

assume

that

n

=

n'

=

N,

argument

in

the

Airy

functions

n',

then for

q

3~2 q"

(z8)

(19)

9\BY can

(nn

==

If n ~ Therefore,

(0)

then the classical value for the n u m b e r of the L a n d a u level

N: The

2k2 3

d ~ (B0/2b)I/NI/3 >> 1 we can use

be

)

- q2/3, and the asymptotic

A ~ ( d + bt) -

,

q~ written

as

d

+

b(y

-- yn)~

where

d = ( V-~-+ 1 2 ):"~ & (nn'/is for large behavior

q

the

(2o)

2*9 value

of

d

is

also

large~

1 ---.~-(d bt)m2], 4=(b + bt) in exp [ + (2~)

270

exp 1~__3_(4 +

4= In the and we

case have

q << I,

we

can

simply

set

d =

In both e a s e s t h e i n t e g r a t i o n

0.

is elementary,

exp I - (4/3),d ~1

3 ~ / " Y I Bo C ' ~' ~' (mc'~ ~

(22)

fn,,---C1--~.~C

(ll...~_~V2)Sj3tTlll,)213

~ t g ] /]

{~njffr+;

q<
Here

1 -=

1 D In the calculation increasing y -- Yn, Going

over

we so to

have that

the

F(5/'6)

1

,

,o.~r

._

_

"""

used the fact that the Airy function decreases the function F(y) can be taken in front of the

rapidly integral.

with

variables 4 . - r = 1,t I ' ~ '2~ - (B. I B 0) " ? 2 (.l -n- -}- l ' n 7) = g,,: w = a r c tgV" nln',

we o b t a i n f(r, ~ ) - 2 l ' ~

tic

[10 !(,-, ,~)= 2 C , ~ -e2- , ~ .r' l/ m d "~2 ~F(r) C t7 \ ,/ ~ / B After the

integration

over

the

angle

3 rsine2 ,, ; q;,'l,

~ \Er/.

~,

form

we

obtain

(23)

rgm(! =- ~o~ -~2T)(sin2~)'/3; qgs 1. in

both

limiting

cases

a distribution

of

~i2

(24)

f(r) = If(r, ?)d'~:= llF(rlY(r), Y and the expressions W(r) (after return to the variable x) are equal to the asymptotic expressions (I) and (2). Therefore, r is distributed in the same way as Iy. And it follows directly from this that the sum ~nn + nr T is equal to its classical value (18). Incidentally, this is no surprise, since at large n it is natural to go over to the co

classical

case.

~loreover,

it

follows

automatically

from

Eq.

(13)

that

~f(r) dr~--l. e) 0

As we have already respect to the angle ~. value from Eq. (18), we

said, we are primarily interested Replacing r in the first of Eqs. finally obtain

f(~)~exp 4.

Discussion

of

the

I

in the (23) by

sin~2~2q] ; q ~ 1, / ( ? ) ~ (1 + cose2"~) (sin

distribution with its most probable

2~)~;3; q ~

l.

(25)

Results

Figure 3 shows the function f(~) for q << I. Whereas for q ~ 1 the distribution has a sharp peak at n = n' (9 = 7/4), for q << 1 the situation is quite different. Thus, with probability of only 10% we have 1/3 < n/n' < 3. With probability 90% either the electron or the positron takes the entire "transverse" energy. Note also that in this case the distribution with respect to the angle ~ does not depend on the y-ray energy. However, the case q << 1 is impossible in a pulsar magnetosphere. that the energy of the radiating particles is bounded by the radiative so that the most energetic photons have energy of order 106-107 MeV. these energies we have q = 2-4, and with decreasing energy the value

The point is reaction force, Corresponding to ol q increases.

271

!

V---

cp Fig. 3. Universal distribution The region in which 1/3 < n/n' is hatched.

f(~). < 3

Therefore, it can be c o n c l u d e d that in fields B < 5.1012 G the energies of the p a r t i c l e s p r o d u c e d in a p u l s a r m a g n e t o s p h e r e will be approximately equal, and the class:ical ~ formula (19) can be used. The c o n d i t i o n q > 1 makes it p o s s i b l e to c a l c u l a t e using the c ! a s s $ C a l e x p r e s s i o n s and synchrotron r a d i a t i o n of the particles. A c c o r d i n g to the e x i s t i n g theories [5, 6] there exists a v a c u u m cavity, i n w h i c h the p a r t i c l e s are accelerated, in the m a g n e t o s p h e r e of a pulsar. The photons e m i t t e d by such p a r t i c l e s create pairs, which, in their turn, radiate s y n c h r o t r o n photons. T h e e n e r g y is s u b d i v i d e d until the m a g n e t o s p h e r e b e c o m e s transparent for the s y n c h r o t r o n photons. The energy of such photons is of order 102 MeV, and the energy of the l e a s t e n e r g e t i c but most n u m e r o u s s e c o n d a r y p a r t i c l e s p r o d u c e d in the last s t a g e of the subd i v i s i o n is of the same order. These particles, it is assumed, are the ones that radiate in the radio range, w h i c h makes it p o s s i b l e to o b s e r v e pulsars. However, such a scheme "works" only for fields B < 5-i012 G. If the m a g n e t i c field on the surface of a n e u t r o n star is g r e a t e r than 5-1012 G, the s u b d i v i s i o n of the e n e r g y terminates at m u c h h i g h e r energies, since the pairs will be p r o d u c e d in low L a n d a u levels and cannot emit s y n c h r o t r o n photons. It is well k n o w n that the m a g n e t i c fields on the surfaces of n e u t r o n stars are e s t i m a t e d from the d e c e l e r a t i o n of the r o t a t i o n under the a s s u m p t i o n that the d e c e l e r a tion is due to m a g n e t i c - d i p o l e radiation. It is f o u n d that of the 300 k n o w n p u l s a r s the m a g n e t i c field is g r e a t e r than 5-1012 G for oniy a few. We note h o w e v e r that the d e t e r m i n a t i o n of the m a g n e t i c field contains an u n d e t e r m i n e d p a r a m e t e r such as the moment of inertia of the n e u t r o n star. In addition, losses of angular v e l o c i t y may be due to o u t f l o w of matter, so that it is p o s s i b l e that the real fields are indeed always less than 5.1012 G. W h a t e v e r the case, it can be c o n c l u d e d that for n e u t r o n stars with B >i 5"1012: G cascade p r o c e s s e s will be suppressed, and if the radio e m i s s i o n m e c h a n i s m is indeed a s s o c i a t e d w i t h such cascade processes, then n e u t r o n stars w i t h B > 5-1012 G will not be o b s e r v e d as radio pulsars.

LITERATURE CITED 1. M. A. R u d e r m a n a n d P . G. S u t h e r l a n d , Astrophys. J . , 1 9 6 , 51 ( 1 9 7 5 ) . 2. E. T a d e m a r u , A s t r o p h y s . J . , 1 8 3 , 625 ( 1 9 7 3 ) . 3. V. B. B e r e s t e t s k i i , E. M. L i f s h i t z , an d L. Po P i t a e v s k i i , Quantum E l e c t r o d y n a m i c s [in Russian], N a u k a , Moscow ( 1 9 8 0 ) . 4. A. A. S o k o l o v a n d I . M. T e r n o v , T h e R e l a t i v i s t i c Electron [in Russian], Nauka~ Moscow (1974). 5. R. N. M a n c h e s t e r a n d J . H. T a y l o r , P u l s a r s , W. H. F r e e m a n ( 1 9 7 7 ) o 6. J . A r o n s an d E. T. S c h a r l e m a n , A s t r o p h y s . J . , 2 3 1 , 854 ( 1 9 7 8 ) .

272