Not yet reviewed by Pith; the record is open.

Thus far, judging the fate of a massive star (either a neutron star (NS) or a black hole) solely by its structure prior to core collapse has been ambiguous. Our work and previous attempts find a non-monotonic variation of successful and failed supernovae with zero-age main-sequence mass, for which no single structural parameter can serve as a good predictive measure. However, we identify two parameters computed from the pre-collapse structure of the progenitor, which in combination allow for a clear separation of exploding and non-exploding cases with only few exceptions (~1-2.5%) in our set of 621 investigated stellar models. One parameter is M4, defining the normalized enclosed mass for a dimensionless entropy per nucleon of s=4, and the other is mu4 = d(m/M_sun)/d(r/1000 km) at s=4, being the normalized mass-derivative at this location. The two parameters mu4 and M4*mu4 can be directly linked to the mass-infall rate, Mdot, of the collapsing star and the electron-type neutrino luminosity of the accreting proto-NS, L_nue ~ M_ns*Mdot, which play a crucial role in the "critical luminosity" concept for the theoretical description of neutrino-driven explosions as runaway phenomenon of the stalled accretion shock. All models were evolved employing the approach of Ugliano et al. for simulating neutrino-driven explosions in spherical symmetry. The neutrino emission of the accretion layer is approximated by a gray transport solver, while the uncertain neutrino emission of the 1.1 M_sun proto-NS core is parametrized by an analytic model. The free parameters connected to the core-boundary prescription are calibrated to reproduce the observables of Supernova 1987A for five different progenitor models.