Neutrino and anti-neutrino transport in accretion disks

We numerically solve the one dimensional Boltzmann equation of the neutrino and anti-neutrino transport in accretion disks and obtain the fully energy dependent and direction dependent neutrino and anti-neutrino emitting spectra, under condition that the distribution of the mass density,temperature and chemical components are given. Then, we apply the resulting neutrino and anti-neutrino emitting spectra to calculate the corresponding annihilation rate of neutrino pairs above the neutrino dominated accretion disk and find that the released energy resulting from the annihilation of neutrino pairs can not provide sufficient energy for the most energetic short gamma ray bursts whose isotropic luminosity can be as high as $10^{52}$ ergs/s unless the high temperature zone where the temperature is beyond 10 MeV can stretch over 200 km in the disk. We also compare the resulting luminosity of neutrinos and anti-neutrinos with the results from the two commonly used approximate treatment of the neutrino and anti-neutrino luminosity: the Fermi-Dirac black body limit and a simplified model of neutrino transport, i.e., the gray body model, and find that both of them overestimate the neutrino/anti-neutrino luminosity and their annihilation rate greatly. Additionally, as did in Sawyer (2003), we also check the validity of the two stream approximation, and find that it is a good approximation to high accuracy.