The thermal evolution of a hot, dense medium cooling through neutrino and photon emission is analyzed. It is assumed that the medium is a plane-parallel, semi-infinite atmosphere in hydrostatic equilibrium and that it is initially isothermal. It is noted that the thermal history of the neutrino photosphere is governed by a nonlinear diffusion equation and that at late times the solution of the equation acquires a self-similar form. Since the photon opacity everywhere exceeds the neutrino opacity, the photon flux is markedly below the neutrino flux and can be treated as a small perturbation. The temperature profile, together with the neutrino and photon fluxes, is obtained as a function of depth and time by a combination of numerical and analytic calculations. The main results are summarized by a convenient set of compact, closed-form similarity formulas. These formulas are then used in examining the cooling of a young neutron star during approximately the first 15 sec of its lifetime, when the star is hot, optically thick to photons and neutrinos, and quasi-static out to the neutrino photosphere.
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Salpeter et al. (1981) studied this question.