The \mu - and \tau -neutrinos emitted from a proto-neutron star are produced by nucleonic bremsstrahlung NN \to NN \nu \bar { \nu } and pair annihilation e ^ { + } e ^ { - } \to \nu \bar { \nu } , 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 \to 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 \bar { \tau } _ { ES } at the energy sphere .
For \bar { \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 \to N \nu can shift T _ { flux } both up or down . \Delta T _ { flux } depends on \bar { \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 .