Stars as Laboratories for Fundamental Physics - MPP Theory Group

132 Chapter 4
where ρ/n B ≈ m N was used. The dimensionless opacity is
ˆκ −1 ≡ 15 ∫ ∞
dx
x4 e 2x 2x
8π 4 0 (e x − 1) 3 s(x) . (4.29)
For the nondegenerate case s(x) was given in Eq. (4.7), for the degenerate
one in Eq. (4.9). Then one finds numerically ˆκ ND = 0.46 and
ˆκ D = 1.53. The respective Γ σ ’s were given in Eqs. (4.7) and (4.9).
4.5 Neutrino Opacity
4.5.1 Elastic Scattering
The trapping of neutrinos in a hot SN core allows them to escape only
by diffusion. This implies that they transport energy in a fashion similar
to radiative transfer. Apart from proper spectral weights as in the
Rosseland mean opacity, the main figure of merit that determines the
transport efficiency is the neutrino mean free path (mfp). While the
charged-current absorption of ν e ’s is very important for practical SN
core-cooling calculations it has no direct bearing on the main issues of
interest here.
However, all neutrino flavors interact by neutral-current interactions
which allow for the scattering on nucleons. The inverse mean free path
is then simply the cross section times the nucleon density. If there is
only one species of nondegenerate nucleons one easily finds
λ −1 = (C 2 V + 3C 2 A) G 2 Fn B ω 2 1/π, (4.30)
where ω 1 is the energy of the incident neutrino, assumed to be much
smaller than m N so that recoil effects can be neglected. For a mixture
of protons and neutrons one has to take a proper average with the
coupling constants C V and C A for protons and neutrons (Appendix B).
For ν e ’s which have a large chemical potential, a Pauli blocking factor
must be included, and the same for nucleons if they are degenerate.
4.5.2 Pair Absorption
Because the neutrinos are assumed to be trapped there is an ambient
bath of ν’s and ν’s which allows for neutrino absorption by the inverse
bremsstrahlung process νν NN → NN, i.e. Fig. 4.5 read from right to
left. The rate for this process is closely related to that for pair emission: