From supernovae to neutron stars: crust formation time

A neutron star is born as a hot, lepton-rich protoneutron star (PNS) and cools via neutrino emission, eventually allowing heavy ions in the outer layers to crystallize into a solid crust. We develop a simple analytic estimate for the onset time of this crust formation during the late, post-convective PNS cooling phase. Using a diffusion-based neutrino luminosity and the resulting entropy evolution together with an approximately isentropic interior structure, we obtain the time-dependent density and temperature at the neutrinosphere. We then impose the Coulomb crystallization condition for heavy nuclei, expressed through the Coulomb coupling parameter, and determine when the neutrinosphere temperature first falls below the crystallization threshold evaluated at the neutrinosphere density. This procedure yields closed expressions for the entropy at crystallization and the corresponding crust-formation time, with explicit dependence on the PNS mass and radius, an effective diffusion/cooling normalization, and composition parameters such as the ionic charge $Z$ and heavy-nuclei mass fraction. For canonical microphysics, we find that the first solid phase typically appears at $t_{\mathrm{crust}}\sim 100$-$500\,\mathrm{s}$. These closed-form scalings provide a useful late-time analytic benchmark for the onset of crust formation and clarify its dependence on PNS and composition parameters.