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We investigate the criteria for successful core-collapse supernova explosionsby the neutrino mechanism. We find that acritical-luminosity/mass-accretion-rate condition distinguishes non-explodingfrom exploding models in hydrodynamic one-dimensional (1D) and two-dimensional(2D) simulations. We present 95 such simulations that parametrically explorethe dependence on neutrino luminosity, mass accretion rate, resolution, anddimensionality. While radial oscillations mediate the transition between 1Daccretion (non-exploding) and exploding simulations, the non-radial standingaccretion shock instability characterizes 2D simulations. We find that it isuseful to compare the average dwell time of matter in the gain region with thecorresponding heating timescale, but that tracking the residence timedistribution function of tracer particles better describes the complex flows inmulti-dimensional simulations. Integral quantities such as the net heatingrate, heating efficiency, and mass in the gain region decrease with time innon-exploding models, but for 2D exploding models, increase before, during, andafter explosion. At the onset of explosion in 2D, the heating efficiency is$\sim$2% to $\sim$5% and the mass in the gain region is $\sim$0.005 M$_{\sun}$to $\sim$0.01 M$_{\sun}$. Importantly, we find that the critical luminosity forexplosions in 2D is $\sim$70% of the critical luminosity required in 1D. Thisresult is not sensitive to resolution or whether the 2D computational domain isa quadrant or the full 180$^{\circ}$. We suggest that the relaxation of theexplosion condition in going from 1D to 2D (and to, perhaps, 3D) is of ageneral character and is not limited by the parametric nature of this study.

Criteria for Core-Collapse Supernova Explosions by the Neutrino Mechanism