Supernova Neutrino Opacity from Nucleon-Nucleon Bremsstrahlung and Related Processes

Elastic scattering on nucleons, \nu N -> N \nu, is the dominant supernova (SN) opacity source for \mu and \tau neutrinos. The dominant energy- and number-changing processes were thought to be \nu e^- -> e^- \nu and \nu\bar \nu <-> e^+ e^- until Suzuki (1993) showed that the bremsstrahlung process \nu\bar \nu NN <-> NN was actually more important. We find that for energy exchange, the related ``inelastic scattering process'' \nu NN <-> NN \nu is even more effective by about a factor of 10. A simple estimate implies that the \nu_\mu and \nu_\tau spectra emitted during the Kelvin-Helmholtz cooling phase are much closer to that of \nu\bar_e than had been thought previously. To facilitate a numerical study of the spectra formation we derive a scattering kernel which governs both bremsstrahlung and inelastic scattering and give an analytic approximation formula. We consider only neutron-neutron interactions, we use a one-pion exchange potential in Born approximation, nonrelativistic neutrons, and the long-wavelength limit, simplifications which appear justified for the surface layers of a SN core. We include the pion mass in the potential and we allow for an arbitrary degree of neutron degeneracy. Our treatment does not include the neutron-proton process and does not include nucleon-nucleon correlations. Our perturbative approach applies only to the SN surface layers, i.e. to densities below about 10^{14} g cm^{-3}.