Critical Conditions for Core-Collapse Supernovae

The explosion of a core-collapse supernova can be approximated by the breakdown of steady-state solutions for accretion onto a proto-neutron star (PNS). We analytically show that as the neutrino luminosity exceeds a critical value L_c, the neutrinosphere pressure exceeds the hydrostatic limit even for an optimal shock radius R. This yields L_c \propto M^2 T^2 (with logarithmic corrections) and R \propto M/T, in agreement with numerical results, where M, T are the PNS mass, neutrino temperature. The near-critical flow can be approximated as a ballistic shell on top of an isothermal layer.