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High-resolution supernova neutrino spectra represented by a simple fit

To study the capabilities of supernova neutrino detectors, the instantaneous spectra are often represented by a quasi-thermal distribution of the form f(E) = E^alpha e^{-(alpha+1)E/E_{av}} where E_{av} is the average energy and alpha a numerical parameter. Based on a spherically symmetric supernova model with full Boltzmann neutrino transport we have, at a few representative post-bounce times, re-converged the models with vastly increased energy resolution to test the fit quality. For our examples, the spectra are well represented by such a fit in the sense that the counting rates for a broad range of target nuclei, sensitive to different parts of the spectrum, are reproduced very well. Therefore, the mean energy and root-mean-square energy of numerical spectra hold enough information to provide the correct alpha and to forecast the response of multi-channel supernova neutrino detection.