neutrino model
------------------------

We use the SN neutrino model adopted in [@yk06a; @yk06b; @ys08]. The neutrinos are assumed to carry the energy of $E_{\nu,total} = 3 \times 10^{53}$ erg [@wh90]. The neutrino luminosity is equally partitioned among three flavors of neutrinos and antineutrinos. It is assumed to decay exponentially with the decay time of 3 s [@yk06a; @yk06b; @ys08; @wh90; @yk05]. The energy spectra of neutrinos emitted from a proto-neutron star are assumed to obey Fermi-Dirac distributions with zero chemical potential. The temperatures of $\nu_e$, $\bar{\nu}_e$, and $\nu_x$, where $\nu_x$ correspond to $\nu_{\mu,\tau}$ and $\bar{\nu}_{\mu,\tau}$, are set to be $(T_{\nu_e}, T_{\bar{\nu}_e}, T_{\nu_x})$ = (3.2, 5, 6) MeV [@yk06a; @yk06b; @ys08].

Neutrinos change their flavors along the propagation in the stellar interior. We consider three flavor neutrino oscillations taking account of the MSW effect and the RSF conversion. The equation of the neutrino flavor change is written as (e.g. [@as03a; @af03]) $$i \frac{d}{dr}
\left( \begin{array}{c}
\nu \\ \bar{\nu}
\end{array} \right)
=\left\{ \left( \begin{array}{cc}
H & MB_\bot \\ -MB_\bot & H \end{array} \right)
+ \left( \begin{array}{cc}
V & 0 \\ 0 & \bar{V} \end{array} \right) \right\}
\left( \begin{array}{c} \nu \\ \bar{\nu} \end{array} \right)$$ where $$\nu = \left(\begin{array}{c}
\nu_e \\ \nu_\mu \\ \nu_\tau 