Mu- and Tau-Neutrino Spectra Formation in Supernovae

The mu- and tau-neutrinos emitted from a proto-neutron star are produced by nucleonic bremsstrahlung N + N -> N + N + nu + nu-bar and pair annihilation e^+ + e^- -> nu + nu-bar, reactions which freeze out at the "energy sphere." Before escaping from there to infinity the neutrinos diffuse through the "scattering atmosphere," a layer where their main interaction is elastic scattering on nucleons nu + N -> N + nu. If these collisions are taken to be iso-energetic as in all numerical supernova simulations, the neutrino flux spectrum escaping to infinity depends only on the medium temperature T_ES and the thermally averaged optical depth tau_ES at the energy sphere. For tau_ES = 10-50 one finds for the spectral flux temperature of the escaping neutrinos T_flux = 0.5-0.6 T_ES. Including energy exchange (nucleon recoil) in nu + N -> N + nu can shift T_flux both up or down. Delta T_flux depends on tau_ES, on the scattering atmosphere's temperature profile, and on T_ES. Based on a numerical study we find that for typical conditions Delta T_flux/T_flux is between -10% and -20%, and even for extreme parameter choices does not exceed -30%. The exact value of Delta T_flux/T_flux is surprisingly insensitive to the assumed value of the nucleon mass, i.e. the exact efficiency of energy transfer between neutrinos and nucleons is not important as long as it can occur at all. Therefore, calculating the nu_mu and nu_tau spectra does not seem to require a precise knowledge of the nuclear medium's dynamical structure functions.