Mu- and tau-neutrino spectra formation in supernovae

The mu- and tau -neutrinos emitted from a proto-neutron star are produced b y nucleonic bremsstrahlung NN --> NNv (v) over bar and pair annihilation e( +)e(-) --> v (v) over bar reactions that freeze out at the "energy sphere." Before escaping from there to infinity, the neutrinos diffuse through the "scattering atmosphere," a layer in which their main interaction is elastic scattering on nucleons vN --> Nv. If these collisions are taken to be isoe nergetic, as in all numerical supernova simulations, the neutrino flux spec trum escaping to infinity depends only on the medium temperature T-ES and t he thermally averaged optical depth <(<tau>)over bar>(ES) at the energy sph ere. For <(<tau>)over bar>(ES) = 10-50, one finds for the spectral flux tem perature of the escaping neutrinos T-flux = 0.5-0.6T(ES). Including energy exchange (nucleon recoil) in vN --> Nv can shift T-flux both up and down. D eltaT(flux) depends on <(<tau>)over bar>(ES),on the scattering atmosphere's temperature profile, and on T-ES. Based on a numerical study, we find that for typical conditions, DeltaT(flux)/T(flux)is between -10% and -20% and e ven for extreme parameter choices does not exceed -30%. The exact value of DeltaT(flux)/T-flux is surprisingly insensitive to the assumed value of the nucleon mass; i.e., the exact efficiency of energy transfer between neutri nos and nucleons is not important as long as it can occur at all. Therefore , calculating the v(mu) and v(tau) spectra does not seem to require a preci se knowledge of the nuclear medium's dynamical structure functions.