Conditions for Shock Revival by Neutrino Heating in Core-Collapse Supernovae

Energy deposition by neutrinos can rejuvenate the stalled bounce shock and can provide the energy for the supernova explosion of a massive star. This neutrino-heating mechanism, however, is not finally accepted or proven as the trigger of the explosion. Part of the problem is that the complexity of the hydrodynamic models often hampers a clear and simple interpretation of the results. This demands a deeper theoretical understanding of the requirements of a successful shock revival. A toy model is presented here for discussing the neutrino heating phase analytically by a time-dependent treatment, which allows one to calculate the radius and velocity of the supernova shock from global properties of the gain layer as solutions of an initial value problem. A criterion is derived for the requirements of shock revival. It confirms the existence of a minimum neutrino luminosity needed for shock expansion, but also demonstrates the importance of a sufficiently large mass infall rate to the shock. The possibility of very energetic neutrino-driven explosions seems excluded because the total specific energy transferred to nucleons is limited by about 1e52 ergs per solar mass (about 5 MeV per nucleon) and the total mass in the gain layer is typically only around 0.1 solar masses. Energy transport by convection from the region of maximum heating to radii closer behind the shock is found to support the explosion by reducing the energy loss associated with the inward advection of neutrino-heated matter through the gain radius. (abridged)