Collapsar Gamma-Ray Bursts: how the luminosity function dictates the duration distribution

Jets in long-duration $\gamma$-ray bursts (GRBs) have to drill through the collapsing star in order to break out of it and produce the $\gamma$-ray signal while the central engine is still active. If the breakout time is shorter for more powerful engines, then the jet-collapsar interaction acts as a filter of less luminous jets. We show that the observed broken power-law GRB luminosity function is a natural outcome of this process. For a theoretically motivated breakout time that scales with jet luminosity as $L^{-\chi}$ with $\chi\sim 1/3-1/2$, we show that the shape of the $\gamma$-ray duration distribution can be uniquely determined by the GRB luminosity function and matches the observed one. This analysis has also interesting implications about the supernova-central engine connection. We show that not only successful jets can deposit sufficient energy in the stellar envelope to power the GRB-associated supernovae, but also failed jets may operate in all Type Ib/c supernovae.