We present results of a systematic study of failing core-collapse supernovae and the formation of stellar-mass black holes (BHs). Using our open-source general-relativistic 1.5D code GR1D equipped with a three-species neutrino leakage/heating scheme and over 100 presupernova models, we study the effects of the choice of nuclear equation of state (EOS), zero-age main sequence (ZAMS) mass and metallicity, rotation, and mass-loss prescription on BH formation. We find that the outcome, for a given EOS, can be estimated, to first order, by a single parameter, the compactness of the stellar core at bounce. By comparing protoneutron star (PNS) structure at the onset of gravitational instability with solutions of the Tolman-Oppenheimer-Volkof equations, we find that thermal pressure support in the outer PNS core is responsible for raising the maximum PNS mass by up to 25% above the cold NS value. By artificially increasing neutrino heating, we find the critical neutrino heating efficiency required for exploding a given progenitor structure and connect these findings with ZAMS conditions, establishing, albeit approximately, for the first time based on actual collapse simulations, the mapping between ZAMS parameters and the outcome of core collapse. We also study the effect of progenitor rotation and find that the dimensionless spin of nascent BHs may be robustly limited below a* = Jc/GM 2 = 1 by the appearance of nonaxisymmetric rotational instabilities.
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