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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 n − e, n + e, n γ, and the corresponding chemical potentials μe−, μe+, μ γ, respectively. As the number of photons is not fixed, their chemical potential μ γ = 0. Therefore, equilibrium with respect to reactions
+ − e + e ↔ 2γ (255 )
implies μe− = − μe+. Electrons and positrons can be treated as ideal relativistic Fermi gases. An electron or a positron of momentum ppp has energy
∘ -------------- ε(ppp ) = ε(p) = (mec2 )2 + p2c2. (256 )
The electron and positron number densities are given by (writing μe− = μe)
∫ ∞ 2 ne− = --1-- dp -----------p------------, (257 ) π2 ℏ3 0 1 + exp [(ε(p) − μe)∕kBT ]
and
1 ∫ ∞ p2 ne+ = ----- dp -------------------------. (258 ) π2 ℏ3 0 1 + exp[(ε(p) + μe)∕kBT ]
Charge neutrality implies
ne+ + ne− = nNZ , (259 )
where nN is the density of nuclei. The calculation of the neutrino emissivity Qpair from reactions
+ − e + e −→ νx + ¯νℓ, ℓ = e,μ,τ , (260 )
is described in detail, e.g., in Yakovlev et al. [428Jump To The Next Citation Point]. Here we limit ourselves to a qualitative discussion of two limiting cases. Let us first consider the case of nondegenerate electrons and positrons, kBT > μe; such conditions prevail in the supernova shock and in the shocked envelope of a newly born proto-neutron star. Then, ne+ ≈ ne− ≫ nNZ. The mean energies of electrons, positrons, photons and neutrinos are then “thermal”, ⟨ε⟩ ∼ kBT. The cross section σe+e−→ν¯ν for process (260View Equation) is quadratic in the center-of-mass energy. Therefore, the temperature dependence of Qpair ν can be evaluated as
Qpair∝ ⟨εν⟩ne − ne+σe+e−→ ν¯ν ∝ T × T 3 × T3 × T 2 ∝ T 9. (261 ) ν
Let us now consider the opposite limit of degenerate ultra-relativistic electrons, kBT ≪ μe and μe ≫ mec2. The positron density is then exponentially small. This is because μe+ = − μe is large and negative, so that ne+ ∝ exp(− μe∕kBT ) ≪ ne−. Therefore, the pair annihilation process is strongly suppressed for degenerate electrons, with decreasing temperature or increasing density. Detailed formulae for pair Q ν, valid in different density-temperature domains, are given in [428Jump To The Next Citation Point].

The pair annihilation process can be affected by a strong magnetic field BBB. General expressions for Qpair ν for arbitrary BBB were derived by Kaminker et al. [230228]. In these papers one can also find practical expressions for a hot, nondegenerate plasma in arbitrary B BB, as well as interpolating expressions for pair Qν in a plasma of any degeneracy and in any BBB. In a hot, nondegenerate plasma with T ≳ 1010 K, B ≫ 1015 G must be huge to affect Qpaνir. However, at T ≲ 109 K, even B ∼ 1014 G may quantize the motion of positrons and increase substantially their number density. Consequently, B ∼ 1014 G strongly increases Qpair ν al low densities. This is visualized in Figure 62View Image.


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