The role of neutrino heating and convection in the explosions of Type-II supernovae is reviewed. The neutrino-driven mechanism of supernova explosions is based upon the fact that high-energetic neutrinos streaming up from the hotter interior must transfer energy to the cooler layers adjacent to the nascent neutron star. While this energy deposition is unavoidable, there is still controversy about the point whether it is able to drive and power a Type-II supernova event or not. To investigate this question one-dimensional hydrodynamical simulations have been performed for the longtime evolution of the collapsed stellar core after the bounce at nuclear matter density and after the associated formation of the supernova shock. In these studies the parameters describing the neutrino emission have been varied and the influence of the temporal contraction of the central part of the nascent neutron star has been tested. A detailed analysis of these models reveals that the possibility to obtain explosions is very sensitive to the physical conditions at and in the protoneutron star and to the properties of the matter accreted into the shock. It critically depends on the neutrino fluxes during the shock revival phase. Yet, above a certain threshold for the neutrino luminosity, stable explosions can be obtained in spherical symmetry. Recently, convection has become a topic which has attracted a lot of attention, partly inspired by the existence of convectively unstable regions in the the post-collapse core, partly by the fact that effects from large-scale mixing processes have been observed in SN 1987A. We repeated our calculations in two dimensions to answer the question whether and which convective processes occur during the early phase of the supernova explosion and to which extent they help the star to explode as a consequence of the neutrino energy deposition. Our results show strong convective processes in the collapsed stellar core in two spatially separate regions. One region of convection lies inside the neutrino sphere and another one is located in the neutrino-heated layer below the shock front. The convective mixing around the neutrino sphere is mainly driven by the negative lepton gradient, which is maintained due to rapid loss of leptons from the semitransparent layers at the neutrino sphere. This is found to considerably speed up the deleptonization of the outer layers of the collapsed stellar core. Even 0.5 seconds after bounce the convective motion is still present in the protoneutron star. Three-dimensional simulations revealed that rising and sinking convective elements have angular sizes between 10 ° and 15 °, which causes corresponding anisotropies of the neutrino emission from the core. The large-scale convective overturn that takes place between the shock and the position of maximum neutrino heating is able to efficiently transport energy from the heated layer into the postshock region. This helps to stabilize the shock revival during the critical phase. The overturn pattern shows downflows of matter in narrow flux tubes and rising bubbles with typical angular extensions of 30 ° to 60 ° (in two dimensions). The material falling towards the neutron star loses lepton number, but readily absorbs energy in the neutrino-heated region, before it rises again. This matter is not accreted onto the protoneutron star. After about 0.4 seconds the turbulent phase outside of the protoneutron star is followed by the spherically symmetrical neutrino-wind phase. The supernova shock is strongly deformed and velocity, density, temperature, and entropy in the postshock region exhibit large-scale variations with a contrast of up to a factor of 3. This must have consequences for the formation and spatial distribution of nucleosynthesis products and might help to explain the dumpiness, anisotropies, envelope and mantle mixing, and large velocities of iron group nuclei which were observed in the ejecta of SN 1987A and other supernovae. Although in this respect of essential importance, convection in the postshock region does not seem to be indispensable to get successful Type-II explosions via the neutrino-heating mechanism, nor do our simulations allow for the conclusion that its presence inevitably makes the mechanism successful and the explosions very energetic. As in spherical symmetry, the explosion energy is very sensitive to the neutrino energy deposition and the size of the neutrino fluxes during a period of several hundred milliseconds after core bounce.
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