1 Introduction

We analyse the annihilation of an electron and a positron producing emission of one photon in a strong magnetic field. It is likely that this annihilation process plays an important role in understanding the radiation mechanism in pulsars and in $\gamma$-ray burst events. With the observational discovery of magnetars (Vasisht & Gotthelf 1997; Kouveliotou et al. 1999), superstrong magnetic fields, in the range $\sim10^{13}-10^{15}$ Gauss, are likely to exist, and calculations valid for such strong fields are of great interest.

This process has been considered by previous authors. One photon pair annihilation was first studied by Klepikov (1954), who evaluated the lifetime of highly relativistic $\rm e^+-e^-$ beams traversing a magnetic field. Daugherty & Bussard (1980) obtained the cross-section $\sigma_{1\gamma}$ and $\sigma_{2\gamma}$for the case that electron and positron are in the Landau ground state. They analyzed the cross section for Gaussian distributions of parallel momentum. Harding (1986) studied the one-photon annihilation rate allowing electron and positron to be in excited Landau states. Wunner et al. (1986) considered the effect of electron and positron polarization, and found that the dominant annihilation rate is for the case that the electron spin is opposite to and the positron spin parallel to the field. Kaminker et al. (1987) studied the two-photon pair annihilation process for the case that the electron and positron are at rest and in the Landau ground state. They found that the magnetic field broadens the spectra and leads to asymmetry in the line around mc² and that the angular distribution depends on magnetic field, B, and photon energy, $\omega$, with the peak emission perpendicular to the field. They argue that the annihilation radiation is linearly polarized, the degree and orientation depending on B, emission angle to the field, $\theta$, and $\omega$. The polarization may reach tens of percent, even for comparatively small fields $B=0.1\times B_{\rm cr}$. The critical value of the magnetic field is defined as $B_{\rm cr} = \frac {m^{2}c^{3}}{e\hbar} =4.414 \ 10^{13}$ Gauss. The polarization is large enough to be detected. However, detectors capable of detecting polarization have not yet been flown in space (although some are planned).

Other emission processes in strong magnetic fields include one and two photon emission (bremsstrahlung) by electrons. The first order (one photon) emission process is discussed by Harding & Preece (1987) Latal (1986), Semionova (1983), Herold et al. (1982), White (1978) and White (1974). Bussard et al. (1986) calculate first and second order Compton scattering, in which there is one photon in both the initial and final states, in the weak field limit ( $B << B_{\rm cr}$). Semionova & Leahy (1999) calculate the second order emission process, for the different photon polarizations and electron spins, and for any value of magnetic field.

The purpose of this paper is to study the polarization of the annihilation radiation by generalizing previous calculations. The calculation is valid for non-zero values of the parallel momentum of the electron or positron and also valid for any value of magnetic field. To make this possible we use the wave functions of Sokolov & Ternov (1983) (for details see the discussion in Graziani 1983 and Semionova & Leahy 1999). We consider the different cases of electron and positron spin and of Landau level, and also calculate cross-sections for the different photon polarizations.