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