Volume 99B, number 3 PHYSICS LETTERS 19 February 1981

COSMIC GAMMA-RAYS ASSOCIATED WITH ANNIHILATION OF RELATIVISTIC e + - e - PAIRS

F.A. AHARONIAN Yerevan Physics lnstitute, 375036 Yerevan, USSR

and

A.M. ATOYAN Physical Faculty, Yerevan University, 375049 Yerevan, USSR

Received 27 August 1980

Results of electron-positron annihilation spectrum investigations are reported. The radiation spectra of an e÷-e - rela- tivistic maxwellian plasma are presented. We consider the particular case of positron annihilation on electrons at rest. It is shown that for positrons with a suprathermal power law distribution the radiation due to annihilation exceeds bremsstrah- lung up to at least co ~ 100 mc 2.

The annihilation o f electron-posi tron pairs is one of the essential mechanisms for cosmic gamma-ray pro- duction. This process has recently drawn increasing attention, stimulated by the detection of characteristic annihilation lines from different astrophysical objects; during a solar flare on 4 August 1972 [1], in the di- rection of the galactic center [2,3], presumably from the pulsar NP0532 in the Crab Nebula [4], in a gamma burst on 5 March 1979 [5] etc.

The annihilation spectrum for a nonrelativistic ( T < 10 7 K) plasma has been studied in detail in refs. [6,7]. The annihilation of relativistic positrons on electrons at rest has been considered by Stecker [8] ; however, the spectrum has not been obtained analyti- cally.

The radiation spectrum of relativistic e+-e - pairs has not yet been discussed in the literature ,1. This process, however, may be rather important in a num- ber of cases since the high energy electrons and posi- trons can be produced copiously in the vicinity o f various relativistic objects (accreting black holes and neutron stars, pulsar magnetospheres etc.). The anni-

tl Recently we have learned from R.A. Sunyaev about Monte- Carlo calculations of the annihilation spectrum in a relati- vistic e ÷- e- plasma performed by A.A. Zdziarski.

hilation problem is, of course, very important in cos- mology, first of all in baryon-symmetric big-bang models [9].

In this letter we present the main results of a high- energy particle (T e > 1 keV) annihilation spectrum in- vestigation. The restriction on the kinetic energy T e of a colliding pair allows one to neglect the Coulomb correction term as well as the annihilation through the positronium state previously formed [6].

It is convenient to express the differential annihila- tion cross section of a positron p(+4) = (p+, ie+) and an electron p(__4) = (p_ , ie_) [10] leading to the produc- tion of two photons K (4) = (K, ico) andK~ 4) = (K1, ico 1),

"~,~ (4)K(4), t~ through the relativistic invariants ~ 1 = -,-t~+ ~ = (4)r(4) , (4) (4) 2 - 1/2 2 " - - 2 p _ K , v e + e _ = [(p+ p_ ) - I ] , ~ d~2(in

units with m = c = I~ = 1):

r 2 w 2 d ~ 2 [ 4 ( ~ 1 K-~) do - v ' e+e_(K 1 + K2) +

- 4 (~1 + ~2) 2 + (~--~12 + ~ ) 1 - (1)

Here r 0 is the classical radius of the electron, d~2 is the solid angle o f the K (4) photon; v' is the relative velocity o f the colliding particles. I f the momenta p+ and'p_ are fixed, the energy of the photon in the direc-

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Volume 99B, number 3 PHYSICS LETTERS 19 February 1981

tion of observation is definite and can be found from the conservation law

p(_4) + p(+4) K(4) = K~4). (2)

However, if the initial conditions permit some free- dom to choose the momenta p+, p _ , then we should detect a spectrum of photons that can be obtained by averaging the cross section (1) in accordance with the given electron-positron distribution.

(1) In order to get the annihilation radiation spec- trum of isotropically distributed mono-energetic par- ticles with fixed energies e+ and e , one should take the average o f the product v ' . do over the solid angles d~2+ and d~2_. The detailed analytical calculations and discussions are given elsewhere [11 ]. The £mal result is rather cumbersome to be written out in this letter. Here we discuss only the main features of the spectrum.

In fig. 1 the averaged differential cross sections for various energies of annihilating pairs are shown. For the case e+ ~ e_ the cross section reveals two maxima, symmetrical with respect to the point o f the minimum of the spectrum coo = (e+ + e )/2. This fea- ture has an obvious physical explanation. Indeed, from the energy conservation law it follows that if one of the photons is produced in the range (co, co + dco), then the second one must be produced in the range (2co 0-co, 2co 0 - co + dco). Thus the total number of photons in these intervals is equal.

The maxima of the spectrum are at photon ener- gies

1 co(max) = $(e+ + e_ g IP+ -- P_ I), (3)

and the energy of the photon varies between

_< ~<! e e +p+ -~(e++e_ - P + - - P - ) ' - ' - c o " - ' - 2( ++ - + P - )

(p_+ -ip_+ I). (3a) It should be noted that for the case e+ = e_ the

cross section increases logarithmically to infinity. However, that does not lead to a divergence of the total number of photons.

(2) The annihilation spectrum when one of the in- teracting particles is at rest, represents a matter of separate interest. It can be obtained directly from expressions (1)and (2). Using eq. (2) and assuming p(_4) __. (0, i) we find

8

6 o

,.03

L~

o 1 ~ 5 c.o (mc z)

| 5

~E 2

0 2 c, 6 8 10 12 co (mc 2)

Fig. 1. The differential free-annihilation cross sections, aver- aged over the directions of electrons and positrons with total energies: (a) (1) e+ = 3 me 2, e_ = 2 me2; (2) e+ = e = 3 rnc 2. (b) (I)e+ = 11 mc 2, e_ = 1.1 rnc2; (2) %= 11 mc2,e_ = 2 mc2; (3) e+ = l l m c 2,e_ = 3 m c 2.

co(e+ - p+cos O) = e+ + 1 - w ,

d cos 0 = (e+ + 1)(p+co2) -1dec . (4)

Now we should take the average of (do- v') over the direction of the annihilating positrons (41r)-ld~2+ -~

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Volume 99B, number 3 PHYSICS LETTERS 19 February 1981

½d cos 0, where 0 is the angle betweenp+ andK. Using (1) and (4) one arrives at

dw e+ e+ + 1 -

1 )2(+ e + + l - co e + + ] - - 6o

(s)

In obtaining this expression we also integrated over th6 direction of observation too (fd~.. . ~ 41r...).

(3) These results can be used in various astrophysi- cal problems, of which we here discuss the following two.

(a) Hot maxwellian plasma. In fig. 2 we plot the annihilation spectra for various temperatures in a max- wellian plasma. The spectrum has a maximum with a halfwidth proportional to the temperature T(A ~ 1.25 k T ) for k T > 0.5 m c 2.

For nonrelativistic temperatures A ~- X/~ this re- suit is in agreement with previous investigations [6] performed mainly on the basis of the Doppler broad- ening of the 511 keV annihilation line.

- l f i t0

10 -16

E

10 -18

-19

10 O.i

, y , 2:=2mo2

1 oa(mc 2) 10 I00

Fig. 2. The radiation spectrum of an e+-e - plasma due to an- nihilation (solid lines) for plasma temperatures k T = 1 mc 2 (curve 1), k T = 2mc 2 (curve 2) and kT= 5 mc 2 (curve 3). The bremsstrahlung spectra for the corresponding temperatures are given by the dashed lines (for details see ref. [ 11 ] ).

However, it should be stressed that accurate calcu- lations, based on the cross section of the process, reveal not only the broadening of the 511 keV line, but also a shift o f its maximum to higher energies:

~(max) = (1 + a k T / m c 2 ) m c 2 , (6)

where a is about 1 and slightly depends on the tem- perature T. In the relativistic limit a tends to 1.2 and for nonrelativistic plasmas a = 0.75. For example, for k T = 25 keV we get 6o(Tmax) = 530 keV. It is interest- ing to note that a similar line has been observed in the direction of the Galactic center [2].

(b) Radiation due to annihilation of high-energy positrons on electrons at rest. In astrophysics the most wide spread distributions of suprathermal particles can be well fitted by power laws. To be definite we considered a positron distribution f ( T + ) = A T ~ ~ for T+ > T O and / ' (7 '+) = constant for T+ < TO, where T+ = e+ - m e 2 is the kinetic energy of the positron. The coefficient A is chosen so as to satisfy the condi- tion W = f f ( T + ) e + d e + = 1 erg/cm 3.

In fig. 3 are given the high-energy tails of the brems- strahlung and annihilation spectra for various ct. The comparison of these curves shows that bremsstrahlung

-11

I0

- 13

I0

- 15

iO

-17 Io

10 o 101 co (mc 2) iO 2 ~0 a

Fig. 3. The high energy tail of the bremsstrahlung and annihi- lation radiation spectra (electron are at rest).

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Volume 99B, number 3 PHYSICS LETTERS 19 February 1981

becomes equal to the radiation of annihilation at pho- ton energies co ~ I00 - 1000 MeV.

The high-energy tail of the annihilation spectrum can be found for arbitrary distribution function f(e+) of suprathermal positrons. Indeed, on annihilation of a relativistic positron (% >> 1) and an electron at rest, one of the two produced photons has primarily an energy of about 0 .5-1 mc 2, while the energy of the second one is co ~ %. Therefore the high energy pho- tons (co >> mc 2) should give the spectrum

q(co) ~ at(co = e+)f(co), (7)

where ot(e+) is the total cross section of the process [10l:

ot(e+) = Orr2/e+)[ln(2e+)- 1] (mc 2 = 1). (8)

In the particular case o f a power-law positron dis- tribution f(e+) ~ eT a, it follows that

q(co) ~ co-(a+1) [ln(2co) - 1 ] . (9)

Finally it should be noted that if mono-energetic beams of electrons and positrons are formed by some means, then two peaks in the gamma-ray spectrum will be produced. The energies (3) of these peaks depend on the energies of the annihilating particles and are thus not fixed. Taking this feature into account, it seems attractive to look for peculiar (unexpected) gamma-ray lines like the unidentified lines in gamma- ray transients [ 12].

The authors would like to thank R.A. Sunyaev for discussions which have stimulated the present investi- gation. We are also grateful to prof. P.H. Bezirganian and prof. G.A. Vartapetian for their support of this work.

References

[1] E.L. Chupp et al., Nature 241 (1973) 333. [2] R.C. Haymes et al., Astrophys. J. 201 (1975) 593. [3] M. Leventhal, C.J. McCaUum and P.D. Stang, Astrophys.

J. Lett. 225 (1978) Lll. [4] M. Leventhal, C.J. MeCaUum and A.C. Watts, Astrophys.

J. 216 (1977) 491. [5 ] E.P. Mazets, S.V. Golenetskii, V.N. II'inskii, R.L. Apteka~

and Yu.A. Guryan, Nature 288 (1979) 587. [6] C.J. Crannel, G. Joyce, R. Ramaty and C. Werntz, Astro-

phys. J. 210 (1976) 582. [7] R.W. Bussard, R. Ramaty and R.J. Drachman, Astro-

phys. J. 228 (1979) 928. [8] F.W. Stecker, Cosmic gamma-rays, NASA-SP-249,

Washington (1971). [9] H. Alfven, Rev. Mod. Phys. 37 (1965) 652.

[10] A.I. Akhieser, V.B. Berestetskii, Quantum electrody- namics Onterscience, New York, 1965).

[11] F.A. Aharonian, A.M. Atoyan and R.A. Sunyaev, pre- print Yerevan Physics Institute (1980).

[ 12] A.S. Jacobson, J.C. Ling, W.A. Mahoney and J.B. Willet, in: Gamma-ray spectroscopy in astrophysics, eds. T.L. Cline and R. Ramaty (NASA TM79619, 1978) p. 228.

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