Legendre expansion of the neutrino-antineutrino annihilation kernel: Influence of high order terms

We calculate the Legendre expansion of the rate of the process $\nu + \bar{\nu} \leftrightarrow e^+ + e^-$ up to 3rd order extending previous results of other authors which only consider the 0th and 1st order terms. Using different closure relations for the moment equations of the radiative transfer equation we discuss the physical implications of taking into account quadratic and cubic terms on the energy deposition outside the neutrinosphere in a simplified model. The main conclusion is that 2nd order is necessary in the semi-transparent region and gives good results if an appropriate closure relation is used.Comment: 14 pages, 4 figures. To be published in A&A Supplement Serie