Neutrino energy loss from the plasma process at all temperatures and densities.

We present a unified approach which is accurate at all temperatures and densities for calculating the energy loss from a stellar plasma due to the plasma process, the decay of photons and plasmons into neutrino pairs. To allow efficient numerical calculations, an analytic approximation to the dispersion equations for photons and plasmons is developed. It is correct to order α in the classical, degenerate, and relativistic limits for all momenta k and is correct at small k for all temperatures and electron densities. Within the same approximations, concise expressions are derived for the transverse, longitudinal, and axial vector components of the neutrino emissivity. The emission of neutrinos can be an important energy loss mechanism for very hot or dense stars. The collective effects of the stellar plasma can significantly alter the production rate of neutrinos. The most dramatic example is the “plasma process”, the decay of photons and plasmons into neutrino pairs, a process that owes its very existence to plasma effects. It was pointed out by Adams, Ruderman, and Woo [1] in 1963 that the plasma process could be the dominant energy loss mechanism for very hot and dense stars. It was recently shown that the formulae for the plasma process that have been used in all previous work are inaccurate at relativistic temperatures and electron densities [2], underestimating the emissivity by a factor as large as 3.185. The purpose of this paper is to provide a unified treatment of the plasma process that is accurate at all temperatures and densities and also allows efficient numerical calculations. We introduce a simple analytic approximation to the dispersion equations for photons and plasmons which becomes exact in the classical limit, the degenerate limit, and the relativistic limit, and interpolates smoothly between these limits. Within the same approximation, we obtain simple expressions for the transverse, longitudinal, and axial vector components of the neutrino emissivity from the plasma process. The derivation of the analytic dispersion equations for photons and plasmons is presented in Appendix A. The effective neutrino-photon interaction that is responsible for the plasma process is discussed in Appendix B and the decay rate of a photon or plasmon into a neutrino pair is calculated in Appendix C. Because a plasma contains mobile charged particles, an electromagnetic wave propagating through the plasma consists of coherent vibrations of both the electromagnetic field and the density of charged particles. These coherent vibrations behave qualitatively differently from electromagnetic waves in the vacuum in that there are longitudinal waves as well as transverse waves, and they propagate at less than the speed of light. The quantization of the electromagnetic waves in a plasma gives rise to a spin-1 particle with 1 longitudinal and 2 transverse spin polarizations. It is common in the literature to refer to all 3 polarization states as “plasmons”, to emphasize that their dispersion relations depend on the properties of the plasma. The longitudinal and transverse modes are then awkwardly labelled “longitudinal plasmons” and “transverse plasmons”. While the longitudinal mode owes its very existence to the plasma, the transverse mode simply has its dispersion relation at low frequencies modified by the plasma. For this reason and for the sake of concise terminology, the