Supernova neutrinos: Earth matter effects and neutrino mass spectrum

We perform a detailed study of the Earth matter effects on supernova neutrinos. The dependences of these effects on the properties of the original neutrino fluxes, on the trajectory of the neutrinos inside the Earth and on the oscillation parameters are described. We show that, for a large fraction (60 %) of the possible arrival times of the signal, the neutrino flux crosses a substantial amount of the matter of the Earth at least for one of the existing detectors. For oscillation parameters from the LMA solution of the solar neutrino problem the Earth matter effect consists in an oscillatory modulation of the $\barnue$ and/or $\nu_e$ energy spectra. The relative deviation with respect to the undistorted spectra can be as large as 20-30 % for E> 20 MeV and 70-100 % for E> 40 MeV. For parameters from the SMA and LOW solutions the effect is localized at low energies (E<10 MeV) and is not larger than 10 %. The Earth matter effects can be revealed (i) by the observation of oscillatory distortions of the energy spectra in a single experiment and (ii) by the comparison between the spectra at different detectors. For a supernova at distance D=10 Kpc, comparing the results of SuperKamiokande (SK), SNO and LVD experiments one can establish the effect at (2-3) \sigma level, whereas larger statistical significance ((4-5) \sigma) is obtained if two experiments of SK-size or larger are available. Studies of the Earth matter effect will select or confirm the solution of the solar neutrino problem, probe the mixing U_{e 3} and identify the hierarchy of the neutrino mass spectrum.