Mu‐ and Tau‐Neutrino Spectra Formation in Supernovae

The mu- and tau-neutrinos emitted from a proto-neutron star are produced bynucleonic bremsstrahlung N + N -> N + N + nu + nu-bar and pair annihilation e^++ e^- -> nu + nu-bar, reactions which freeze out at the "energy sphere." Beforeescaping from there to infinity the neutrinos diffuse through the "scatteringatmosphere," a layer where their main interaction is elastic scattering onnucleons nu + N -> N + nu. If these collisions are taken to be iso-energetic asin all numerical supernova simulations, the neutrino flux spectrum escaping toinfinity depends only on the medium temperature T_ES and the thermally averagedoptical depth tau_ES at the energy sphere. For tau_ES = 10-50 one finds for thespectral 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_fluxboth up or down. Delta T_flux depends on tau_ES, on the scattering atmosphere'stemperature profile, and on T_ES. Based on a numerical study we find that fortypical conditions Delta T_flux/T_flux is between -10% and -20%, and even forextreme parameter choices does not exceed -30%. The exact value of DeltaT_flux/T_flux is surprisingly insensitive to the assumed value of the nucleonmass, i.e. the exact efficiency of energy transfer between neutrinos andnucleons 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 preciseknowledge of the nuclear medium's dynamical structure functions.Comment: 30 pages, 24 figures, Revised version with several clarifications in the text, no changes of substanc