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 _cM2T2 (with logarithmic corrections) and RM/T, in agreement with numerical results, where M and T are the PNS mass and neutrino temperature, respectively. The near-critical flow can be approximated as a ballistic shell on top of an isothermal layer.