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We consider the neutrino energy-loss rate due to the pair neutrino process, the photoneutrino process, and the plasma neutrino process, in a hot plasma, including the ntnbution from positrons present in the black-body radiation. We derive some analytic expressions for the pair neutrino rate and include the plasma effects on the photoneutrino rate. A discussion of the approximations involved in the derivation of the plasma neutrino rate is also given. An analytic formula is derived to reproduce the total energy rate of all three processes; it has a 5-15 per cent precision in the temperature range of K and is also fairly good from 10 to 1011 K for all relevant densities Finally, the total rate of the three processes combined is compared with some other processes with the recombination process and the Urca process being singled out.
Beaudet et al. (Fri,) studied this question.