### 11.2 Electron-positron pair annihilation

This process was proposed by Chiu & Morrison [98]. It requires the presence of positrons and is,
therefore, important at high temperatures and low densities. Such conditions prevail in the outer layer of
a newly-born neutron star. The matter there is opaque to photons. Electrons, positrons and
photons are in thermodynamic equilibrium, with number densities , , , and the
corresponding chemical potentials , , , respectively. As the number of photons is
not fixed, their chemical potential . Therefore, equilibrium with respect to reactions
implies . Electrons and positrons can be treated as ideal relativistic Fermi gases. An electron
or a positron of momentum has energy
The electron and positron number densities are given by (writing )
and
Charge neutrality implies
where is the density of nuclei. The calculation of the neutrino emissivity from reactions
is described in detail, e.g., in Yakovlev et al. [428]. Here we limit ourselves to a qualitative discussion of two
limiting cases. Let us first consider the case of nondegenerate electrons and positrons, ; such
conditions prevail in the supernova shock and in the shocked envelope of a newly born proto-neutron star.
Then, . The mean energies of electrons, positrons, photons and neutrinos are then
“thermal”, . The cross section for process (260) is quadratic in the
center-of-mass energy. Therefore, the temperature dependence of can be evaluated as
Let us now consider the opposite limit of degenerate ultra-relativistic electrons, and
. The positron density is then exponentially small. This is because is
large and negative, so that . Therefore, the pair annihilation
process is strongly suppressed for degenerate electrons, with decreasing temperature or increasing
density. Detailed formulae for , valid in different density-temperature domains, are given
in [428].
The pair annihilation process can be affected by a strong magnetic field . General expressions for
for arbitrary were derived by Kaminker et al. [230, 228]. In these papers one can also find
practical expressions for a hot, nondegenerate plasma in arbitrary , as well as interpolating expressions
for in a plasma of any degeneracy and in any . In a hot, nondegenerate plasma with
, must be huge to affect . However, at , even
may quantize the motion of positrons and increase substantially their number density.
Consequently, strongly increases al low densities. This is visualized in Figure
62.