Moment equations of neutrinos in supernova

We derive a series of moment equations describing the motion and flavor transformation of neutrinos in supernova. We find a particular series of moments of neutrino density matrix in supernova. The emission angle distribution of neutrinos is described by this series of moments. We expand the equation of neutrinos using these moments and obtain moment equations. We find that these moments have very good property of convergence and the infinite series of equations can be truncated to equations with a small set of moments. Using a small set of moment equations the required computational power is reduced by about two orders of magnitude compared to that in multi-angle simulation. The study on non-linear flavor transformation of neutrinos is substantially simplified using these equations. Two flavor system of neutrinos is also considered and new equations describing the flavor polarization vectors of neutrinos are found.