Shock revival in core-collapse supernovae (CCSNe) may be due to the neutrino mechanism. While it is known that in a neutrino-powered CCSN, explosion begins when the neutrino luminosity of the proto-neutron star exceeds a critical value, the physics of this condition in time-dependent, multidimensional simulations are not fully understood. citet{Pejcha2012} found that an `antesonic condition exists for time-steady spherically symmetric models, potentially giving a physical explanation for the critical curve observed in simulations. In this paper, we extend that analysis to time-dependent, spherically symmetric polytropic models. We verify the critical antesonic condition in our simulations, showing that models exceeding it drive transonic winds whereas models below it exhibit steady accretion. In addition, we find that (1) high spatial resolution is needed for accurate determination of the antesonic ratio and shock radius at the critical curve, and that low resolution simulations systematically underpredict these quantities, making explosion more difficult at lower resolution; (2) there is an important physical connection between the critical mass accretion rate at explosion and the mass loss rate of the post-explosion wind: the two are directly proportional at criticality, implying that, at criticality, the wind kinetic power is tied directly to the accretion power; (3) the value of the post-shock adiabatic index $Gamma$ has a large effect on the length and time scales of the post-bounce evolution of the explosion larger values of $Gamma$ result in a longer transition from the accretion to wind phases.
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