Earth effects on supernova neutrinos and their implications for neutrino parameters

We perform a detailed study of the Earth matter effects on supernova neutrinos with neutrino oscillation parameter LMA and small θ₁₃. The Earth effects show significant dependences on the neutrino path length inside the Earth and the value of Δm₁₂². We investigate rather optimistically the possibility that we can probe the value of Δm₁₂² by the Earth effects. We assume that θ₁₂ and the direction of the supernova are known with enough accuracy and that the resonance that occurs at higher density in supernova envelope is completely nonadiabatic. Further the neutrino spectra before neutrinos go through the Earth are assumed to be known. Then we show that, making use of these dependences, we can obtain an implication for the value of Δm₁₂² by comparing the observed energy spectrum to the predicted one. When SK detects neutrinos from supernova at 10 kpc which traveled through Earth (nadir angle <80^∘), Δm₁₂² can be determined with an accuracy of ~10%. In much of the neutrino-detection-time-Δm₁₂² plane, Δm₁₂² might be determined with an accuracy equal to or better than ±0.5×10^-5eV².