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We investigated the structure of the spherically symmetric accretion flowsthrough the standing shock wave onto the proto-neutron star in the post-bouncephase of the collapse-driven supernova. We assume that the accretion flow is ina steady state controlled by the neutrino luminosity and mass accretion ratethat are kept constant. We obtain solutions of the steady Euler equations for awide range of neutrino luminosity and mass accretion rate. We employ arealistic EOS and neutrino-heating rates. More importantly, we take intoaccount the effect of convection phenomenologically. For each mass accretionrate, we find the critical neutrino luminosity, above which there exists nosteady solution. These critical points are supposed to mark the onset of theshock revival. As the neutrino luminosity increases for a given mass accretionrate, there appears a convectively unstable region at some point before thecritical value is reached. We introduce a phenomenological energy flux byconvection so that the negative entropy gradient should be canceled out. Wefind that the convection lowers the critical neutrino luminosity substantially.We also consider the effect of the self-gravity. It is found that theself-gravity is important only when the neutrino luminosity is high. Thecritical luminosity, however, is little affected if the energy transport byconvection is taken into account.Comment: accepted by ApJ, 20 pages, 8 figure

Standing Accretion Shocks in the Supernova Core: Effects of Convection and Realistic Equations of State