Constraints on Core Collapse from the Black Hole Mass Function

We model the observed black hole mass function under the assumption that black hole formation is controlled by the compactness of the stellar core at the time of collapse. Low compactness stars are more likely to explode as supernovae and produce neutron stars, while high compactness stars are more likely to be failed supernovae that produce black holes with the mass of the helium core of the star. Using three sequences of stellar models and marginalizing over a model for the completeness of the black hole mass function, we find that the compactness xi(2.5) above which 50% of core collapses produce black holes is xi(2.5)=0.24 (0.15 < xi(2.5) < 0.37) at 90% confidence). While models with a sharp transition between successful and failed explosions are always the most likely, the width of the transition between the minimum compactness for black hole formation and the compactness above which all core collapses produce black holes is not well constrained. The models also predict that f=0.18 (0.09 < f < 0.39) of core collapses fail assuming a minimum mass for core collapse of 8Msun. We tested four other criteria for black hole formation based on xi(2.0) and xi(3.0), the compactnesses at enclosed masses of 2.0 or 3.0 rather than 2.5Msun, the mass of the iron core, and the mass inside the oxygen burning shell. We found that xi(2.0) works as well as xi(2.5), while the compactness xi(3.0) works significantly worse, as does using the iron core mass or the mass enclosed by the oxygen burning shell. As expected from the high compactness of 20-25Msun stars, black hole formation in this mass range provides a natural explanation of the red supergiant problem.