equal in the first approximation to the "exterior" induction, i.e., in our case (Sz/3)M.

In this approximation there are no Meissner currents, which corresponds to the normal

state, so that the distribution of the induction inside and outside the sphere is given

by (i) and (2).

The corrections to (i) and (2) associated with the incomplete vanishing of the

Meissner currents have (for a >> ~) smallness of order Hc!/4~M.

Thus, in the case (6) the induction of the sphere is practically not screened.

Thus, if the "drag currents" of a neutron star or the "conduction currents" on

a superconducting sphere are sufficiently large for the magnetization produced by them

to satisfy the condition (6), then the magnetic field outside is a dipole field with

moment proportional to the volume of the sphere, while inside, because of the dense mesh of superconducting vortices, the field is uniform. In particular; for neutron

stars the magnetic field within them may be of order 6-1012-8-1013 G, and the dipole moment of order 1030-1032 G-cm -3.

LITERATURE CITED

1. D. M. S e d r a k y a n , A s t r o f i z i k a , 18, 417 ( 1 9 8 1 ) . 2. P. G. De G e n n e s , S u p e r c o n d u c t i v i t y o f M e t a l s and A l l o y s , B e n j a m i n ~ New York ( 1 9 6 6 ) .

POSITRONS IN COSMIC RAYS AND THE GALACTIC GA~9~A

RADIATION ASSOCIATED WITH THEM

F. A. Agaronyan, V. G. Kirillov-Ugryumov,

and Yu. D. Kotov

The possibility of recovering the energy spectrum and determining the charge composition of the electron--positron component of the cosmic rays by in- vestigating the diffuse galactic gamma radiation in the energy range Ey 50 MeV is discussed.

i. Introduction

Recent years have brought a number of observational indications of very effective generation of positrons in various astronomical objects. In the center of the Galaxy

[I, 2], in flare processes on the Sun [3, 4], in the neighborhood of pulsars [5-7], and

in compact radio sources [S]. Many models of cosmic-ray sources also predict the pro-

duction and acceleration of positrons [9-17]. In the majority of these models~ electrons

and positrons with energy ~i GeV are considered. There is no doubt that information

about the spectrum and charge composition of the electron--positron component of the

cosmic rays in this range of energies could make an important contribution to the solu- tion of the problem of the origin of cosmic rays. Analysis of the charge composition

of the electron--positron component shows that the main fraction of the high-energy

(>i GeV) electrons observed near the Earth is accelerated directly in the sources, whereas the high-energy positrons are evidently the product of interaction of the

proton--nuclear component of the cosmic rays (see, for example, [18]). Unfortunately, data on the electron--positron component at energies below several hundred MeV are very sparse and ambiguous due to the effects of solar modulation. Moreover, it is possible that the low-energy electrons observed near the Earth have a local origin (for example, are accelerated in Jupiter's magnetosphere [19]). Because of the

opacity of the interstellar medium for radio waves with ~ < I0 }.~z, corresponding to

synchrotron radiation of electrons with E e < 1 GeV, radio observations are also not in a position to enable us to draw conclusions about the low-energy part of the

Erevan Phys{cs Institute; Moscow Engineering-Physics Institute. Translated from Astrofizika, Vol. 19, No. I, pp. 139-152, January-March, 1983. Original article submitted May 31, 1982; accepted for publication November 6, 1982.

82 0571-7132/83/1901-0082507.50 �9 1983 Plenum Publishing Corporation

electron--positron component of the cosmic rays in the Galaxy.

The main hopes of obtaining information about the electron--positron component in

this energy range are associated with gamma astronomy [20]. in particular, in [21]

Fichtel et el. emphasized the importance of gamma astronomy at medium energies (10-30

MeV) to obtain information about galactic electrons. An important argument here is

the circumstance that bremsstrahlung of electrons and positrons makes the main con-

tribution to the region of energies 10-30 MeV of photons generated in interstellar

space by interactions of cosmic rays. Moreover, in [22] it was pointed out that

bremsstrahlung can dominate up to Ey ~ i00 MeV, for which there is serious support

confirmed by observations [23].

In the discussion of the mechanisms of generation of gamma radiation in inter-

stellar space no allowance is made, as a rule, for the contribution of photons pro-

duced by the annihilation of positrons of the electron--positron component in flight.

However, as is shown below, in the range of gamma-ray energies 0.5 ~ Ey ~ 50 MeV

the contribution of annihilation radiation is very appreciable, and under certain

conditions may exceed the contributions of the other possible generation mechanisms,

including bremsstrahlung.

Thus, allowance for annihilation as well as bremsstrahlung of electrons makes

it possible not only to recover more accurately the spectrum of the electron--positron

component but also to obtain important information about the charge composition of

the electron--positron component in the galactic disk. From this point of view, there

is undoubted interest in gamma-astronomy investigations in the range Ey ~ 50 MeV from

definite parts of the Galaxy in which there can be a high concentration of relativistic

electrons. In the first place, this applies to the remnants of young supernovae, the

central region of the Galaxy, and the discrete sources of x-ray and gamma radiation.

2. Annihilation of Positrons in Flight

In interactions with electrons of the ambient medium, positrons are annihilated

with the production of gamma rays, the annihilation occurring both before and after

thermalization of the positrons. As a result of annihilation of the thermalized

positrons the 0.511 MeV line is formed and, possibly, a three-photon continuum with

~ 0.511MeV, whereas the annihilation of superthermal positrons in flight leads to a continuous spectrum [24]:

d[ ~r:mc I (race mce ) (mce mce )2 d~ -- E P 2 \ ~ + E---i, mc = o~ -- \ ~-~ + E + mc ~ -- w +

( c,~ @ E + m c e - - o ~ ) ] E + rnc 2 - - ~ ~o .

(1)

1/2 ( E + mc 2 - Pc ) ~o7 ~ 1/2 ( E + mc ~ + Pc) , (1 ' )

w h e r e m, P , a n d E a r e t h e r e s t m a s s , m o m e n t u m , a n d t o t a l e n e r g y o f t h e p o s i t r o n t h a t i s a n n i h i l a t e d , a n d ~ i s t h e p h o t o n e n e r g y .

T h e s p e c t r u m ( 1 ) h a s t w o m a x i m a s i t u a t e d s y m m e t r i c a l l y w i t h r e s p e c t t o t h e p o i n t o f m i n i m u m ~0 = 1 / 2 ( E + m c 2 ) . T h e m a x i m a a r e a t t a i n e d a t t h e t w o l i m i t i n g p o i n t s o f t h e

k i n e m a t i c r e g i o n ( 1 ' ) a n d , t h e r e f o r e , h a v e a s y m m e t r i c p r o f i l e s . I f r e l a t i v i s t i c p o s i t r o n s (E >> mc 2 ) a r e a n n i h i l a t e d , t h e n , a s f o l l o w s f r o m ( 1 ) , t h e m a x i m a a r e s i t u a t e d a t t h e p o i n t s w ! = m c 2 / 2 a n d ~2 = E + m c 2 / 2 .

T h e s p e c t r u m o f p h o t o n s p r o d u c e d b y i n t e r a c t i o n s o f p o s i t r o n s w i t h t h e s u r r o u n d i n g m a t t e r i s m a d e up o f a n n i h i l a t i o n r a d i a t i o n a n d b r e m s s t r a h l u n g . B e c a u s e o f t h e a b s e n c e i n t h e a n n i h i l a t i o n c r o s s s e c t i o n o f t h e f i n e s t r u c t u r e c o n s t a n t ~ = e 2 / h c = 1 / 1 3 7 , a n d also the different behavior of the cross sections of these two processes as a function

of the positron energy, there must exist energy regions in which one or other process is dominant.

To determine these regions, we compare the so-called radiation loss cross sections ~r, which are defined by

83

Ej For bremsstrahlung when E >> mc 2 [26]

~b (m ~-~ 4 [In (2E/mc - l /3 ) ] ~.Z(Z + ~ ) r,,.

where Z is the atomic number of matter of the target.

For annihilation radiation, it is possible to find ~r (A) by substituting formula

(i) (divided by the positron velocity) in (2). However, for relativistic positrons

we can with good accuracy set

' ( E ) z : ?oZ c

w h e r e z ( E ) = ~ r ~ ( m c ~ / E ) [ l a ( 2 E / m c i ) ' ~ l ] i s t h e t o t a l a n n i h i l a t i o n c r o s s s e c t i o n ( s e e , f o r

e x a m p l e , [ 2 5 ] ) . H e r e , we h a v e u s e d t h e c i r c u m s t a n c e t h a t w h e n a r e l a t i v i s t i c p o s i t r o n i s

a n n i h i l a t e d o n e o f t h e e m i t t e d p r o t o n s i s f o u n d p r e f e r e n t i a l l y i n t h e r e g i o n Qf e n e r g i e s

~ ( 0 . 5 - 1 ) m c 2 w h i l e t h e o t h e r h a s ~ ~ E ( s e e a b o v e )

C o m p a r i n g (3 ) a n d ( 4 ) , we f i n d t h a t a t e n e r g i e s

(2 )

(3)

(4 )

g ~ s ~- mci/~ ( Z + 1 ) (5 )

t h e a n n i h i l a t i o n r a d i a t i o n l o s s e s e x c e e d t h e b r e m s s t r a h l u n w l o s s e s . I n p a r t i c u l a r , i n

a m e d i u m c o n s i s t i n g p r e d o m i n a n t l y o s h y d r o g e n , E c r ~- 70 m c " . I t s h o u ! d h o w e v e r b e n o t e d t h a t i n t h e h a r d p a r t o f t h e s p e c t r u m , i n p a r t i c u l a r a t ~ ~ E , t h e a n n i h i l a t i o n r a d i a t i o n

may a p p r e c i a b l y e x c e e d t h e b r e m s s t r a h l u n g e v e n w h e n E > E c r . T h i s f e l l o w s d i r e c t l y f r o m

c o m p a r i s o n o f t h e b e h a v i o r s o f t h e d i f f e r e n t i a l c r o s s s e c t i o n s o f t h e t w o p r o c e s s e s . T h e

s p e c t r u m o f t h e b r e m s s t r a h l u n g p h o t o n s a t f i x e d E d e c r e a s e s i n a c c o r d a n c e w i t h t h e l a w ~ - 1 t o c~ ~ 1 / 2 E a n d t h e n s t e e p e n s s t r o n g l y , b e c o m i n g z e r o a t t h e t h r e s h o l d p o i n t ,~ =

E - - m c 2 , w h e r e a s t h e s p e c t r u m o f t h e a n n i h i l a t i o n r a d i a t i o n , b e g i n n i n g a t ,~ ~ E / 2 , i n -

c r e a s e s m o n o t o n i c a l l y a n d r e a c h e s a maximum a t ~ ~ E.

We now a s s u m e t h a t t h e p o s i t r o n s h a v e a c e r t a i n e n e r g y d i s t r i b u t i o n N ( E ) . T h e n t h e n u m b e r o f e m i t t e d a n n i h i l a t i o n p h o t o n s i n u n i t e n e r g y i n t e r v a l n e a r w i n 1 cm - S

d u r i n g o n e s e c o n d i s

S 5 - = , ~ N ( E ) dE, (6) d~o

where n e is the concentration of electrons of the medium.

To be specific, we consider the positron power-law spectra most characteristic of

astrophysical objects:

jKE -~ E ~ < E < E i , N ( E ) = I O E < E1; E > E 2 ,

where E, as before, is the total energy of a positron.

Calculations were made for three types of spectra (7):

I) ~ = 1.5, 2) ~ = 2.5 without cutoff of the spectrum at low energies (ioeo~ E 1 =

1 mc2),2and 3) ~ = 2.5 (cutoff energy E 1 = 20 mc2). In all three cases~ we took E 2 =

400 mc .

Figure 1 shows the generation functions dQ/d~ for annnihilation radiation (con-

tinuous curves) and bremsstrahlung (broken curves) in a hydrogen median normalized to

= 1 cm - 3 a n d t o p o s i t r o n e n e r g y d e n s i t y we ~ i N ( E ) ( E - - m c ~ ) d E : i e r g / c m 3 o n e

The e x p r e s s i o n s f r o m [26] w e r e u s e d f o r t h e b r e m s s t r a h l u n g c r o s s s e c t i o n on n e u t r a l

h y d r o g e n .

As can be seen from Fig. I, the annihilation radiation spectrum corresponding to

the power law distribution (7) of the positrons without cutoff below (curves IA and 2A)

has a broad maximum around w - 1 mc 2, and, for the employed normalization to w e = 1

erg/cm 3, the value of the maximum depends strongly on the exponent ~ of the spectrum.

(7)

84

| I I I i . ' ' ~ - ' ' 7

i/ J • !,,/ / ~ ~ .3A ' % . . ~ \ \ /

i:i _,[-I'/, H! .. " - ~ \ , % X , , ~ !

l ' -J L _ _ I 1 J ~ . r ~ ! ! l [ l l l l , * ~ ~

1 t 5 2 1 10 ~00 {.j (mc2~

Fig. i. Generation functions of annihilation

radiation (A) and bremmstrahlung (B) of posi-

trons with the energy spectrum (7) in hydrogen.

In this region of energies, the contribution of the annihilation radiation appreciably

exceeds (by 1-2 orders of magnitude) the contribution of the bremsstrahlung. The in-

tensities of generation of annihilation and bremsstrahlung photons become equal at

energies ~* = 20 mc 2 (a = 1.5) and ~* = 80 mc 2 (a = 2.5). Note that the value of ~*

for the hard positron spectra (~ ~ 2) also depend on the upper cutoff energy E 2 of

the spectrum. Indeed, as follows from Eq. (I), for fixed value of ~ the annihilation

cross section decreases with increasing E, whereas the bremsstrahlung cross section

increases logarithmically. Therefore, the high-energy positrons present in large

numbers in the case of the hard spectrum generate photons predominantly by bremsstrahlung.

Very different is the spectrum of the generated radiation in the case of a positron

spectrum with "cutoff" at low energies. The spectrum of bremsstrahlung photons in the

region w ~ E 1 becomes gentle and instead of the previous dependence -~-a has the form

~w -I. In the annihilation radiation spectrum the maximum around ~ = mc 2 disappears

and instead of it there are two maxima at w ~ mc2/2 and ~ ~ E 1 . However, the first

maximum ~ ~ mc 2 is below the bremsstrahlung spectrum (for E 1 ~ 2 mc2), while the second,

for ~ ~ E 1 ~ 137 mc 2, can be identified above the bremsstrahlung spectrum. For ~ < E 1

the bremsstrahlung spectrum is predominant, and in the region ~ > E 1 the ratio of the

two contributions is obviously the same as for the positron spectrum without cutoff

at E 1 .

Hitherto, we have discussed the spectrum of gamma rays generated in interactions

of superthermal positrons with surrounding gas. If not only positrons but also electrons

are present in the generation region, this obviously leads to an increased role of

bremsstrahlung. In particular, if the electrons and positrons have equal energy dis-

tributions, and the fraction of positive electrons is R = N+/(N_ + N+), the values of

the generation function of the annihilation photons in Fig. 1 must be multiplied by

R < I.

T h u s , a l l o w a n c e f o r t h e a n n i h i l a t i o n p h o t o n s i n a d d i t i o n t o t h e b r e m s s t r a h l u n g m a k e s i t p o s s i b l e i n p r i n c i p l e o n t h e b a s i s o f a n a n a l y s i s o f m e a s u r e m e n t s o f t h e s p e c t r a o f c o s m i c gamma r a y s i n t h e r a n g e g l O 0 mc 2 t o o b t a i n i n f o r m a t i o n a b o u t t h e

e n e r g y d i s t r i b u t i o n a n d t h e c h a r g e c o m p o s i t i o n o f t h e l o w - e n e r g y p a r t o f t h e e l e c t r o n - - p o s i t r o n c o m p o n e n t o f t h e c o s m i c r a y s i n v a r i o u s a s t r o p h y s i c a l o b j e c t s . I t s h o u l d

b e n o t e d t h a t t h i s a s s e r t i o n i s j u s t i f i e d o n l y u n d e r t h e c o n d i t i o n R > 0 . 1 a n d p r o v i d e d t h e r e a r e n o o t h e r c o m p e t i n g m e c h a n i s m s o f g e n e r a t i o n o f gamma r a y s i n t h e g i v e n e n e r g y range.

3. Diffuse Gamma Radiation of the Galactic Disk

3.1. Electron--Positron Component of the Cosmic Rays in Interstellar Space. It has

by now been firmly established that the electrons observed near the Earth consist of

85

primary a n d secondary particles. The main processes of generation of secondary electrons

in interactions of cosmic rays with the interstellar gas are as follows: a) production

and decay of ~• mesons, b) B decay of secondary neutrons and radioactive nuclei~ c) the production of 6 electrons. Decay of charged mesons is predominant in the region of

electron energies E e ~ i00 MeV, and the processes b) and e) play an important part at

lower energies. It follows from the calculations of [27] that around E e ~ i00 MeV

about five times as many positrons as electrons are produced by the decay of ~• mesons.

Since the observations of the charge composition R = N+/(N+ + N_.) of the electron--

positron component of the cosmic rays reveal a quite different dependence on E e (see,

for example, [28]), to explain R(E) satisfactorily one can assume the existence of

two independent components of the electron--positron component. The first, which is

predominant in the region E e > 1 GeV, consists mainly of electrons and has primary

origin. The abundance of positrons in the second, low-energy component (E e ~ I GeV), is evidently very large (at least R - 0.5). This component may be due to decay of + ~- mesons produced by cosmic rays in the interstellar medium~ It is also impossible

to rule out a probability of primary origin of the low-energy component of the electron-- positron component of the cosmic rays.

There are various arguments in favor of a secondary origin of the low-energy com-

ponent. The most important of them is the good agreement between the calculated spectrum

of ~-electron generation and the observed intensity of electrons with E e ~ 1-20 MeV.

But then (ignoring the small probability of existence of antimatter in certain regions

of the Galaxy*) one would expect a small abundance of positrons in the energy range

E e ~ 2 0 MeV.

A t t h e s a m e t i m e , C l i n e a n d H o n e s [28 ] h a v e r e p o r t e d d e t e c t i o n o f a n a n o m a l o u s l y h i g h f l u x o f p o s i t r o n s i n t h e e n e r g y r a n g e E e 0 . 5 - 2 MeV ( ~ 0 . 2 p a r t i c l e s , c m - 2 . s e c - 1 ) , t h i s b e i n g c o m p a r a b l e w i t h t h e e l e c t r o n f l u x . I n t e r p r e t a t i o n o f t h i s r e s u l t b y g e n - e r a t i o n o f p o s i t r o n s t h r o u g h b o m b a r d m e n t o f t h e i n t e r s t e l l a r g a s b y c o s m i c r a y s l e a d s t o i n s u p e r a b l e d i f f i c u l t i e s .

S i n c e we a r e c o n c e r n e d w i t h p o s i t r o n s w i t h E e < 10 MeV, t h e m o s t l i k e l y m e c h a n i s m of their generation associated with cosmic rays is inelastic collisions leading to the production of B+-active nuclei, in the first place IIc, 13N, 140, 150o Since dis-

integration reactions of nuclei of the group (CNO) have a maximum at energies of the

bombarding particles of ~10-30 MeV/nucleon, the main contribution to the generation of

these nuclei is made by subcosmic rays. The equilibrium spectrum of positrons in inter-

stellar space produced by the interaction of cosmic rays was calculated by Ramaty et al.

[30]. It follows from their results that to explain the observed flux of positrons around 1 MeV one requires a density of subcosmic rays that is too high, w ~ 50 eV/cm 3,

and this is not confirmed by other observational data. Indeed, such a high density of

subcosmic rays leads to ionization of interstellar hydrogen and the production of light nuclei, in particular 7Li in the reaction 4He(~, p)7Li at the rate

~H ~ F i C R ( E ) ~ I ( E ) d E ~ I O - l U s e e _ 1 d Li = , - - ~ n H t z ~ ( E ) : ~ ( E ) d E ~ 2 . 1 0 -~4 s e e - 1 4t

w h i c h e x c e e d s b y a t l e a s t t w o o r d e r s o f m a g n i t u d e t h e u p p e r l i m i t s o f t h e c o r r e s p o n d i n g q u a n t i t i e s o b t a i n e d f r o m o b s e r v a t i o n s : ~H 5 2 " 1 0 - 1 5 s e e - 1 a n d d / d t ( L i / H ) ~ 3 ' 1 0 - 2 7 s e c - I

[ 31 ] .

I n a d d i t i o n , a n a l y s i s o f t h e c o n d i t i o n s o f f o r m a t i o n o f x r a y s a s a r e s u l t o f K c a p t u r e o f e l e c t r o n s o f t h e m e d i u m b y f a s t i r o n n u c l e i [ 3 2 ] a n d o f b r e m s s t r a h t u n g o f

s u b c o s m i c r a y s [ 3 3 ] a n d c o m p a r i s o n o f t h e c a l c u l a t e d f l u x e s w i t h t h e o b s e r v e d r a d i a t i o n i n t e n s i t y i n t h e i n t e r v a l 2 - 1 0 keV i n t h e d i r e c t i o n o f t h e c e n t e r o f t h e G a I a x y m a k e s i t p o s s i b l e t o e s t a b l i s h a s t r i n g e n t u p p e r l i m i t o n t h e d e n s i t y o f s u b c o s m i c r a y s (w Z 5 e V / c m 3) i n d i s t a n t r e g i o n s o f t h e G a l a x y a s w e l l , w h e r e t h e p a r a m e t e r s %H a n d d / d t ( L i / H ) a r e u n k n o w n i n t h e m a j o r i t y o f c a s e s .

T h e a b o v e b o u n d o n t h e d e n s i t y o f s u b c o s m i c r a y s m a k e s i t p o s s i b l e t o a s s e r t t h a t t h a t t h e p o s i t r o n s g e n e r a t e d i n t h e i n t e r s t e l l a r m e d i u m a r e l e s s t h a n 10% o f t h e o b s e r v e d f l u x . T h e r e f o r e , i f new i n d e p e n d e n t m e a s u r e m e n t s c o n f i r m t h e r e l i a b i l i t y o f t h e r e s u l t s

* S e e , h o w e v e r t h e c o n c l u s i o n s o f [ 2 9 ] b a s e d o n t h e a n o m a l o u s l y h i g h f l u x o f l o w - e n e r g y

a n t i p r o t o n s i n c o s m i c r a y s .

86

of [28] (unfortunately, there are as yet no such data), this would be an indication of

the existence of primary positrons in the cosmic rays in space near the Sun. However,

because of modulation effects associated with the motion of cosmic rays in the helio-

sphere, it is possible that the low-energy electron--positron component of the cosmic

rays detected near the Earth had an energy of several hundred MeV outside the hello-

sphere (for more details, see, for example, [34]). It is therefore possible that we

are quite unable to obtain information about low-energy positrons from observations

near the Earth. In addition, both the charge composition as well as the intensity

of the electron--positron component of cosmic rays at energies E e < 1 GeV in distant

regions of the Galaxy may be quite different from what they are near the Sun [20].

Indeed, it follows from the large-scale distribution of galactic gamma rays with

Ey ~ i00 MeV [35] that there is a strong inhomogeneity in the distribution of cosmic

rays (Epncr ~ 0.5-30 GeV/nucleon if the gamma radiation is generated by the decay of o

mesons and E e ~ 0.i-I GeV if it is due to bremsstrahlung) in the disk. Moreover, gamma-

astronomy data obtained recently by the satellite COS B give serious grounds for believing

that the cosmic rays observed near the Earth have a local origin and are apparently

accelerated by stars of the classes 0 and B in the Gould belt [36].

The production and acceleration of positrons is predicted by many modern models

of cosmic rays. Favorable conditions for the generation of positrons can arise in

the magnetospheres of pulsars [9-13], in the neighborhood of the conjectured Kerr

black hole at the center of the Galaxy [15, 37], during nucleosynthesis in supernovae

[17, 37, 38], etc., the positrons being accelerated as effectively as the electrons in

these models. Observation of the annihilation line in the radiation spectra from a

number of astrophysical objects (the Sun [3, 4], the compact source at the center of

the Galaxy [2], the unidentified sources of gamma bursts [5] and transients [6, 7])

undoubtedly indicate effective generation of positrons in these sources. However;

the line 0.511 MeV is formed by the annihilation of thermal positrons and, therefore,

does not give information about accelerated positrons. Therefore, the question of

the existence of primary positrons in the cosmic rays is still open.

Since observations of nonthermal radio emission give reliable information about

galactic electrons only in the region of energies E e > 1GeV (because of the absorption

of radio waves with frequency v < i0 ~z in the interstellar medium), the main hopes

for obtaining information about the electron--positron component of the cosmic rays

with E e < 1 GeV are associated with gamma astronomy, namely, the possibility of detect-

ing bremsstrahlung gamma rays generated in interstellar space [21]. However, as follows

from Fig. !, if the fraction of positrons in the electron--positron component is suf-

ficiently great, then besides bremsstrahlung the annihilation of positrons in flight

will make an appreciable contribution to the production of photons with energy Ey 50 MeV.

3.2. Gamma-Radiation Generation Functions in the Galaxy. In order to compare

the contributions of the different mechanisms of production of photons in the inter-

stellar medium, we consider the generation functions of the corresponding quantities.

In calculations of the generation functions of the processes of annihilation and

bremsstrahlung we considered two types of electron spectra -- the two most widely

discussed in the literature:

a common power-law spectrum with exponent ~ = 2.8 [22],

J~ (E~) ~ 0.01 E9 zs cm-2-sec-l.sr-l-GeV-l;

a spectrum with inflection at E e = 2 GeV [21],

16.810 -3E71s cm-2"sec-l'sr-l'GeV-l, Ee~ 2 GeV,

j~(Ee) = {1.4.10_2E~2.8 cm_2.sec_l.sr_l.GeV_l, K~ > 2 GeV.

Since the behavior of the spectrum at low energies is quite undetermined [18, 22],

we considered different variants of its behavior for E e ~ i0 MeV:

[J=J~(E) E > 2 ~eV; Spectrum. (1) iJ = J~ (2 MeV) = const E ~ ) MeV.

87

} 7-., " \

- 2 B / " - . \

o ~ ~ ", . . . . . . ~ , , , ~ , ,

X / \ '%",.\ \ 1

0,1 I iO 100 ~000

Ey (l~,'Ie V )

Fig. 2. Generation functions of gamma radia- tion in interstellar space: A) annihilation,

B) bremsstrahlung of electron--positron com- ponent of the cosmic rays [33], IC) inverse

Coompton scattering on microwaveo background, 7r ) decay of secondary ~ mesons [39], LRN) line radiation of excited nuclei [40].

Spectrum[l') ==J~(E) A'~10 MeV, --= 0 E~10 MeV.

Spectrum (2) {] = J~ (~:).

Spectrum (2'){~ i J2 (E)0 EEl> !0 MeV ; . < 10 :~,{eV.

In the calculations we assume that R = N+/(N_ + N+) = 0.5 in the complete energy

range; the density of the interstellar gas is n = 1 cm -3. The generation functions

for bremsstrahlung and annihilation radiation corresponding to these electron spectra

are shown in Fig. 2, in which we have also plotted the generation functions of inverse

Compton scattering of electrons by the background 2.~K microwave radiation, bremsstrahlung

of the proton--nucleus component of the cosmic rays on electrons of the medium [3],

gamma rays from the decay of z mesons [39], and line gamma radiation of excited nuclei

[40]. As one would expect, the behavior of the electron spectr[~ at low energies has a

strong influence on the generation function of the annihilation radiation in the MeV

energy range. For spectra with a high content of low-energy positrons (the spectra

(i) and (2)) the contribution of annihilation radiation exceeds the contribution of

the other considered process, including inverse Compton scattering on optical and infra-

red radiation, as calculated in [41]. It follows from Fig. 2 that the investigation of

diffuse galactic radiation at energies Ey ~ 50 MeV makes it possible not only to recover

the spectrum of the electron--positron component of the cosmic rays in interstellar

space but also to find the abundance of positrons with E e ~ 200 MeV (because of the

falling spectra of the electron--positron component, it is these poSitrons that make

the main contribution to the considered region of gamma rays). Obviously, this asser-

tion is valid-for R > 0.i.

88

,-~ Apotto 16

I0 -s \ ~ \ \

~ 1 0 -

I0 I00 ~000

Ey (~,~eV)

F i g . 3. E x p e c t e d f l u x e s o f gamma r a y s i n t h e d i r e c t i o n o f t h e c e n t e r o f t h e G a l a x y . I and I I c o r r e s p o n d t o t h e s p e c t r a ( 1 ' ) and (2) o f t h e e l e c t r o n - - p o s i t r o n c o m p o n e n t ; t h e b r o k e n and c o n t i n u o u s c u r v e s a r e , r e s p e c t i v e l y , t h e b r e m s s t r a h l u n g and t h e t o t a l ( b r e m s s t r a h l u n g + a n n i h i l a t i o n ) r a d i a t i o n o f t h e e l e c t r o n - - p o s i t r o n c o m p o n e n t o f t h e c o s m i c r a y s .

In Fig. 3, we give the expected fluxes of gamma rays in the direction of the galactic

center. They were obtained by multiplying the generation functions by (nH1) , the amount of hydrogen (atomic and molecular) in a column with unit section in the direction toward the center of the Galaxy. We determined (nHl) from the condition that the observed gamma

o radiation in the region E~ ~ 300 MeV be entirely due to the decay of ~ mesons [40]. We note that the employed-value also agrees with the estimates of <nH2) obtained by means

of radio observations at wavelengths 21 cm and 2.6 mm and analysis of the absorption of

soft x rays [42].

The spectrum (i) of the electron--positron component, which agrees well with the

data in both the gamma range (Ey ~ 50 MeV) [23] and in the x-ray range (2-6 keV) [43],

does not contradict the results of measurements of the gamma radiation in the interval

1-5.5 ~eV from the galactic disk made with Apollo 16 [44] only under the condition that R ~ 0.I. But if R = 0.5, the spectrum of the electron--positron component must

be less steep than (i) in the region of energies E e ~ i0 ~eV. In particular, the

spectra (I') and (2) of the electron--positron component do not contradict the results of [44]. At the same time, as follows from Fig. 3, the total (annihilation + bremsstrahlung)

spectrum of the radiation of the electron--positron component (continuous curves) differs

strongly from the purely bremsstrahlung spectrum in the interval 1-50 MeV. Unfortunately,

this region of energies has hitherto been little investigated, and data are available

only on the integrated fluxes. However, the currently planned large-aperture gamma

telescopes with good energy resolution will make it possible to identify the features in the spectrum due to the contribution of the annihilation radiation.

The annihilation line 0.511 MeV has frequently been observed in the direction of

the galactic center during the last ten years. Because of the poor angular resolution

of the detectorS, the origin of this radiation remained unclear for a long time. Among many explanations in the literature there was wide discussion of the possibility of

formation of this line in the interstellar medium by annihilation of thermal positrons ejected from sources (for example, pulsars) with ultrarelativistic velocities. However,

in [45], on the basis of analysis of the observed intensities of photons in the line 0.5!i MeV and in the continuum with E~ > i0 MeV, it was shown that the positrons

responsible for the observed annihilation line enter the region of formation of the

89

radiation with low energies (E e ~ 20 MeV). This rules out the possibility of explaining this line by positrons in the electron--positron component of the cosmic rays thermalized in the interstellar gas. This conclusion was recently confirmed by the discovery by the satellite HEAO-3 of variability in the intensity of the 0.511 MeV line [2] during a time of %0.5 year. At the same time, it should be noted that some of the positrons of the electron--positron component of the cosmic rays will, on being decelerated in the interstellar medium, be annihilated with formation of the 0.:511MeV line. However, the fraction of such positrons depends on various factors describing the propagation, thermalization, and annihilation of the electron--positron component in interstellar space (the ratio between the thermalization and positron loss times, the temperature

and degree of ionization in the annihilation region, etc.), and this question requires additional investigation.

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23. J. A Paul, K. Bennet, G. F. Bignami, et al., Astron. Astrophys., 633, L3! (1978). 24. F. A Aharonian and A. M. Atoyan, Phys. Lett., 99B, 301 (1981). 25. A. I Akhiezer and B. V. Berestetskii, Quantum Electrodynamics, New York (1965). 26. G. R Blumenthal and R. J. Gould, Rev. Mod. Phys., 42, 237 (1970). 27. R. Ramaty and R. E. Lingenfelter, J. Geophys. Res., 71, 3687 (1966). 28. T. L. Cline and E. W. Hones, ll-th ICRC, Budapest (1970)~ p. 159. 29. R. J. Protheroe, Astrophys. J., 251, 378 (1981). 30. R. Ramaty, F. W. Stecker, and D. Misra, J. Geophys. Res., 75, 1141 (1970). 31. M. Meneguzzi and H. Reeves, Astron. Astrophys., 40, 91 (1975). 32. R. W. Bussard, R. Ramaty, and K. Omidvar, Astrophys. J., 220, 353 (1978) o 33. F. A. Aharonian, S. R. Kelner, V. G. Kirillov-Ugryumov, and Yu. D. Kotov 16-th ICRC,

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90

37. R. Ramaty and R. E. Lingenfelter, NASA Tech. Memor. 82066 (1980). 38. J. J. Burgen, S. A. Stephens, and B. N. Swanenburg, Astrophys. Space Sci., 8, 20

(1970). 39. S. A. Stephens and G. D. Badhwar, Astrophys. Space Sci., 76, 213 (1981). 40. R. Ramaty, B. Kozlovsky, and R. E. Lingenfelter, Astrophys. J. Suppl. Ser., 400, 487

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PROPAGATION OF OBLIQUE WAVES IN A RELATIVISTIC

ELECTRON--POSITRON PLASMA

M. E. Gedalin and G. Z. Machabeli

The propagation of oblique waves in a magnetoactive relativistic electron--

positron plasma is investigated. It is shown that there exists a purely

transverse electromagnetic wave with linear polarization and a "vacuum"

nature of the dispersion. On the basis of the obtained results and in

the framework of the maser mechanism of the radio emission of pulsars, the linear polarization of the radio pulses of the pulsar NP 0532 is

given a qualitative explanation.

i. Introduction

The study of the excitation and propagation of waves in a magnetized relativistic

electron--positron plasma is very important for interpreting the radiation of pulsars.

Sturrock [I] showed that in the polar regions cut by open lines of force (lines of

force that go beyond the light cylinder) electron--positron pairs are produced. In

the case of the pulsar NP 0532, which is in the Crab Nebula, these pairs form a two- component electron--positron plasma, which flows from the polar regions of the pulsar with mean velocity corresponding to the Lorentz factor 7p ~ 102~and-density np ~ 1020

cm -3. The plasma is penetrated by an electron--positron beam with density n b ~ 1017

cm -3 and Lorentz factor 7b ~ 106 . The plasma--beam system, which is in the strong

magnetic field of the pulsar (B 0 ~ 1012 G), can be described by a one-dimensional

momentum distribution function, since the transverse component of the momentum in the field B 0 is rapidly de-excited by synchroton radiation.

The linear and quasilinear stage in the development of the instability of waves

propagating along magnetic field, k = (0, O, kz) , in such a plasma has been analyzed in detail in various studies [2-4]. Numerous studies have been made of the linear

theory of the pulsar plasma (see, for example, [5-9]). This interest is due to the

hope of constructing a sensible model of the radio emission of pulsars.

The magnetic field is assumed to be a dipole field, i.e., one that expands with increasing distance from the surface of the pulsar. Therefore, if a wave initially excited along the magnetic field (k I = 0) propagates along the line of sight, a

nonzero transverse component of the wave vector k will appear with the passage of

time, so that it is necessary to investigate oblique waves.

In an ordinary electron--ion plasma it is not possible to separate potential and nonpotential oscillations in the case of propagation at an angle to the magnetic field.

Abastumani Astrophysical Observatory. Translated from Astofizika, Vol. 19, No. i, pp. 153-159, January-March, 1983. Original article submitted February 18, 1982; accepted for publication November 6, 1982.

0571-7132/83/1901-0091507.50 �9 1983 Plenum Publishing Corporation 91