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WHEN DO SUPERNOVA NEUTRINOS OF DIFFERENT FLAVORS HAVE SIMILAR LUMINOSITIES BUT DIFFERENT SPECTRA ?

Muon and tau neutrinos ($ν_x$) interact with protoneutron star matter only via neutral currents and exchange energy with the stellar gas predominantly by neutrino-electron scattering and neutrino-pair processes. In contrast, electron neutrinos and antineutrinos ($ν_e$ and $\barν_e$) are frequently absorbed and produced in charged-current mediated reactions with nucleons. Therefore the emergent $ν_e$ and $\barν_e$ originate from layers with lower temperatures further out in the star and are emitted with much lower characteristic spectral temperatures. In addition, a major contribution to the $ν_e$ and $\barν_e$ opacities is due to absorptions, while the opacity of $ν_x$ is strongly dominated by scattering reactions with nucleons and nuclei in which the $ν_x$ energy is (essentially) conserved. Therefore the $ν_x$ distribution is nearly isotropic when $ν_x$ decouple energetically and their outward diffusion is slowed down. In a generalized form to include this effect, the Stefan-Boltzmann Radiation Law can account for both the facts that $ν_e$ ($\barν_e$) and $ν_x$ emerge from the star with similar luminosities but with very different characteristic spectral temperatures. Simple analytical expressions to estimate the effect are given. If, as recently argued, even at densities significantly below nuclear matter density neutral-current scatterings were associated with considerable energy transfer between neutrino and target particle, one might expect spectral temperatures of $ν_x$ much closer to those of $ν_e$ and $\barν_e$. This is of relevance for the detection of neutrino signals from supernovae.