Equation of state in the gravitational collapse of stars

The equation of state in stellar collapse is derived from simple considerations, the crucial ingredient being that the entropy per nucleon remains small, of the order of unity (in units of k), during the entire collapse. In the early regime, ρ∼10¹⁰-10¹³ g/cm³, nuclei partially dissolve into α-particles and neutrons; the α-particles go back into the nuclei at higher densities. At the higher densities, nuclei are preserved right up to nuclear matter densities, at which point the nucleons are squeezed out of the nuclei. The low entropy per nucleon prevents the appearance of drip nucleons, which would add greatly to the net entropy. We find that electrons are captured by nuclei, the capture on free protons being negligible in comparison. Carrying the difference of neutron and proton chemical potentials μ_n-μ_p in our capture equation forces the energy of the resulting neutrinos to be low. Nonethelesd, neutrino trapping occurs at a density of about ρ = 10¹² g/cm³. The fact that the ensuing development to higher densities is adiabatic makes our treatment in terms of entropy highly relevant. The resulting equation of state has an adiabatic index of roughly 4 3 coming from the degenerate leptons, but lowered slightly by electrons changing into neutrinos and by the nuclei dissolving into α-particles (although this latter process is reversed at the higher densities), right up to nuclear matter densities. At this point the equation of state suddenly stiffens, with Γ going up to Γ ≈ 2.5 and bounce at about three times nuclear matter density. In the later stages of the collapse, only neutrinos of energy {less-than or approximate}10 MeV are able to get out into the photosphere, and these appear to be insufficient to blow off the mantle and envelope of the star. We do not carry our description into the region following the bounce, where a shock wave is presumably formed, and, therefore, we cannot answer the question as to whether the shock wave, in conjunction with neutrino transport, can dismantle the star, but a one-dimensional treatment shows the shock wave to be very promising in this respect.