| Abstract
| - 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, though investigated by numerical simulations and analytic studies, is not finally accepted or proven as the trigger of the explosion. Part of the problem is that different groups have obtained seemingly discrepant results, and 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 developed here for discussing the neutrino heating phase analytically. The neutron star atmosphere between the neutrinosphere and the supernova shock can well be considered to be in hydrostatic equilibrium, with a layer of net neutrino cooling below the gain radius and a layer of net neutrino heating above. Since the mass infall rate to the shock is in general different from the rate at which gas is advected into the neutron star, the mass in the gain layer varies with time. Moreover, the gain layer receives additional energy input by neutrinos emitted from the neutrinosphere and the cooling layer. Therefore the determination of the shock evolution requires a time-dependent treatment. To this end the hydrodynamical equations of continuity and energy are integrated over the volume of the gain layer to obtain conservation laws for the total mass and energy in this layer. The radius and velocity of the supernova shock can then be calculated from global properties of the gain layer as solutions of an initial value problem, which expresses the fact that the behavior of the shock is controlled by the cumulative effects of neutrino heating and mass accumulation in the gain layer. The described toy model produces steady-state accretion and mass outflow from the nascent neutron star as special cases. The approach is useful to illuminate the conditions that can lead to delayed explosions and in this sense supplements detailed numerical simulations. On grounds of the model developed here, a criterion is derived for the requirements of shock revival. It confirms the existence of a minimum neutrino luminosity that is needed for shock expansion, but also demonstrates the importance of a sufficiently large mass infall rate to the shock. If the neutrinospheric luminosity or accretion rate by the shock are too low, the shock is weakened because the gain layer loses more mass than is resupplied by inflow. On the other hand, very high infall rates damp the shock expansion and above some threshold, the development of positive total energy in the neutrino-heating layer is prevented. Time-dependent solutions for the evolution of the gain layer show that the total specific energy transferred to nucleons by neutrinos is limited by about 10 52 erg $ $$M_{odot}^{-1}$ (~5 MeV per nucleon). This excludes the possibility of very energetic explosions by the neutrino-heating mechanism, because the typical mass in the gain layer is about 0.1 $M_{odot}$ and does not exceed a few tenths of a solar mass. The toy model also allows for a crude discussion of the global effects of convective energy transport in the neutrino-heating layer. Transfer of energy from the region of maximum heating to radii closer behind the shock mainly reduces the loss of energy by the inward flow of neutrino-heated matter through the gain radius.
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