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 $\theta_{13}$. The Earth effects show significant dependences on the neutrino path length inside the Earth and the value of $\Delta m^{2}_{12}$. We investigate rather optimistically a possibility that we can probe the value of $\Delta m^{2}_{12}$ by the Earth effects. We assume that $\theta_{12}$ 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 implication for the value of $\Delta m^{2}_{12}$ by comparing the observed energy spectrum to the predicted one. When SK detects neutrinos from supernova at 10kpc which traveled through the Earth (nadir angle $<$ 80 degree), $\Delta m^{2}_{12}$ can be determined with an accuracy of $\sim 10$. In much of the neutrino-detection-time-$\Delta m^{2}_{12}$ plane, $\Delta m^{2}_{12}$ might be determined with an accuracy equal to or better than $\pm 0.5 \times 10^{-5} {\rm eV}^{2}$.