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.