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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 tex2html_wrap_inline1636 (tex2html_wrap_inline1638) is the neutrino (antineutrino) invariant distribution function, tex2html_wrap_inline1640 (tex2html_wrap_inline1642) is the neutrino (antineutrino) energy in the frame comoving with the matter, tex2html_wrap_inline1644 (tex2html_wrap_inline1646) the cosine of the angle between the neutrino (antineutrino) momentum and the polar axis, tex2html_wrap_inline1648 is the azimutal angle, and tex2html_wrap_inline1650 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 tex2html_wrap_inline1662 as follows
where tex2html_wrap_inline1664 are the Legendre polynomials and tex2html_wrap_inline1666 are the Legendre moments of the production kernel, the TP collision term can be written as

In order to obtain the expressions for tex2html_wrap_inline1668, we need to evaluate the following integral over electron energy E
where tex2html_wrap_inline1672 and tex2html_wrap_inline1674 are summarized in Table 1 (click here) for the different neutrino types and tex2html_wrap_inline1676 is the Fermi-Dirac distribution function

tex2html_wrap_inline1678 being the electron degeneracy parameter, and

In the previous expression tex2html_wrap_inline1680 stands for the step function.

Although tex2html_wrap_inline1682, tex2html_wrap_inline1684 and tex2html_wrap_inline1686 are complicated functions of tex2html_wrap_inline1644, tex2html_wrap_inline1640 and tex2html_wrap_inline1642, the integrals Jl 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 tex2html_wrap_inline1640 and tex2html_wrap_inline1642.


tex2html_wrap_inline1712 tex2html_wrap_inline1714


tex2html_wrap_inline1718 tex2html_wrap_inline1720
tex2html_wrap_inline1674 tex2html_wrap_inline1724 tex2html_wrap_inline1724
Table 1: Coefficients tex2html_wrap_inline1708 for different neutrino species  

Then, taking into account the following relation
the tex2html_wrap_inline1668 functions can be expressed in a simple way in terms of the dimensionless variables tex2html_wrap_inline1730 and tex2html_wrap_inline1732.



The explicit expressions for aln, cln and dln coefficients are in Appendix A and the method to evaluate the Gn(a,b) integrals is detailed in Appendix B.

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