Probing the neutrino mass hierarchy and the 13-mixing with supernovae

We consider in details the effects of the 13-mixing (sin² theta₁₃) and of the type of mass hierarchy/ordering (sign Delta m²₁₃) on neutrino signals from the gravitational collapses of stars. The observables (characteristics of the energy spectra of nuₑ and antinuₑ events) sensitive to sin² theta₁₃ and signDelta m²₁₃ have been calculated. They include the ratio of average energies of the spectra, rE = /, the ratio of widths of the energy distributions, rGamma, the ratios of total numbers of nuₑ and antinuₑ events at low energies, S, and in the high energy tails, Rₓ₀₈₋. We construct and analyze scatter plots which show the predictions for the observables for different intervals of sin² theta₁₃ and signs of Delta m²₁₃, taking into account uncertainties in the original neutrino spectra, the star density profile, etc. . Regions in the space of observables rE, rGamma, S, Rₓ₀₈₋ exist in which certain mass hierarchy and intervals of sin² theta₁₃ can be identified or discriminated. We elaborate on the method of the high energy tails in the spectra of events. The conditions are formulated for which sin² theta₁₃ can be (i) measured, (ii) restricted from below, (iii) restricted from above. We comment on the possibility to determine sin² theta₁₃ using the time dependence of the signals due to the propagation of the shock wave through the resonance layers of the star. We show that the appearance of the delayed Earth matter effect in one of the channels (nuₑ or antinuₑ) in combination with the undelayed effect in the other channel will allow to identify the shock wave appeareance and determine the mass hierarchy.