Numerical Models of Protoneutron Stars and Type-II Supernovae --- Recent   Developments
  The results of recent multi-dimensional simulations of type-II supernovae are reviewed. They show that convective instabilities in the collapsed stellar core might play an important role already during the first second after the formation of the supernova shock. Convectively unstable situations occur below and near the neutrinosphere as well as in the neutrino-heated region between the nascent neutron star and the supernova shock after the latter has stalled at a radius of typically 100-200 km.   While convective overturn in the layer of neutrino energy deposition clearly helps the explosion to develop and potentially provides an explanation of strong mantle and envelope mixing, asphericities, and non-uniform 56Ni distribution observed in supernova SN 1987A, its presence and importance depends on the strength of the neutrino heating and thus on the size of the neutrino fluxes from the neutron star. Convection in the hot-bubble region can only develop if the growth timescale of the instabilities and the heating timescale are both shorter than the accretion timescale of the matter advected through the stagnant shock. For too small neutrino luminosities this requirement is not fulfilled and convective activity cannot develop, leading to very weak explosions or even fizzling models, just as in the one-dimensional situation.   Convectively enhanced neutrino luminosities from the protoneutron star can therefore provide an essential condition for the explosion of the star. Very recent two-dimensional, self-consistent, general relativistic simulations of the cooling of a newly-formed neutron star demonstrate and confirm the possibility that Ledoux convection, driven by negative lepton number and 
H.-Thomas Janka (Max-Planck-Institut f"ur Astrophysik, Garching,   Germany)