Legendre expansion of the $\vec{{\nu}\bar{\nu}} \rightleftharpoons {\bf e}^+ {\bf e}^-$ kernel: Influence of high order terms

We calculate the Legendre expansion of the rate of the process +$ e + +e up to 3 rd order ex- tending previous results of other authors which only con- sider the 0 th and 1 st order terms. Using dierent closure relations for the moment equations of the radiative trans- fer equation we discuss the physical implications of taking into account quadratic and cubic terms on the energy de- position outside the neutrinosphere in a simplied model. The main conclusion is that 2 nd order is necessary in the semi-transparent region and gives good results if an ap- propriate closure relation is used.