Stars as Laboratories for Fundamental Physics - MPP Theory Group

96 Chapter 3
Fig. 3.3. Dimensionless total cross sections ˆσ + and ˆσ − for the photoneutrino
process γe − → e − νν according to Eq. (3.12) with ω the initial photon energy
in the CM frame.
3.2.5 Energy-Loss Rates
The Compton-type processes are typically important when the electrons
are nondegenerate (otherwise bremsstrahlung dominates) and
nonrelativistic (otherwise e + e − annihilation dominates). In these limits
one may use the cross sections without Pauli blocking corrections.
Because the recoil of the target electron is neglected, the energy ω of
a photon impinging on an electron is identical with the energy carried
away by the new boson or neutrino pair. Therefore, the energy-loss
rate per unit volume is a simple integral over the initial-state photon
phase space, weighted with their Bose-Einstein occupation numbers,
Q = n e
∫
2 d 3 k
(2π) 3
σ ω
e ω/T − 1 , (3.15)
where n e is the number density of electrons, T the temperature, and
the factor 2 is for two photon polarization states.
In this expression the photon “plasma mass” ω P has been neglected.
If ω P ∼ > 3T , corresponding to a typical thermal photon energy, the photon
dispersion relation would have to be included properly in both the
phase-space integration and in the cross section calculation. However,
these are insignificant fine points for the cases to be studied below.
In the nonrelativistic limit it is easy to estimate a suppression factor
F deg by electron degeneracy. If recoil effects can be neglected, the