2. Legendre expansion of emission-absorption TP kernels

Following Bruenn (1985) the contribution of thermal pair production and absorption of neutrino-antineutrino pairs to the right hand side of the Boltzmann equation (TP collision term) is

where () is the neutrino (antineutrino) invariant distribution function, () is the neutrino (antineutrino) energy in the frame comoving with the matter, () the cosine of the angle between the neutrino (antineutrino) momentum and the polar axis, is the azimutal angle, and is the angle between neutrino and antineutrino directions. In what follows, the explicit dependence on t and r of the distribution functions will be omitted and we will assume axial symmetry with respect to the polar axis. The superscripts a and p refer to absorption and production, respectively.

Assuming electrons and positrons in equilibrium at a temperature T, the following relation between the absorption and production kernels is satisfied

where the temperature is measured in units of energy. Expanding as follows

where are the Legendre polynomials and are the Legendre moments of the production kernel, the TP collision term can be written as

In order to obtain the expressions for , we need to evaluate the following integral over electron energy E

where and are summarized in Table 1 (click here) for the different neutrino types and is the Fermi-Dirac distribution function

being the electron degeneracy parameter, and

In the previous expression stands for the step function.

Although , and are complicated functions of , and , the integrals J_l can be done analytically, and the results for l=0,1 were first obtained by Bruenn (1985). We have calculated the integrals for l=2,3 and their dependence on E is simply a polynomial law of degree 2l+5 with coefficients being functions of and .

 

Table 1: Coefficients for different neutrino species  

Then, taking into account the following relation

the functions can be expressed in a simple way in terms of the dimensionless variables and .

where

The explicit expressions for a_ln, c_ln and d_ln coefficients are in Appendix A and the method to evaluate the G_n(a,b) integrals is detailed in Appendix B.