Mass fall-back and accretion in the central engine of gamma-ray bursts

We calculate the rate of in-fall of stellar matter on an accretion disc during the collapse of a rapidly rotating massive star and estimate the luminosity of the relativistic jet that results from accretion on to the central black hole. We find that the jet luminosity remains high for about 102 s, at a level comparable to the typical luminosity observed in gamma-ray bursts (GRBs). The luminosity then decreases rapidly with time for about ∼103 s, roughly as ∼t−3; the duration depends on the size and rotation speed of the stellar core. The rapid decrease of the jet power explains the steeply declining X-ray flux observed at the end of most long-duration GRBs. Observations with the Swift satellite show that, following the steep decline, many GRBs exhibit a plateau in the X-ray light curve (XLC) that lasts for about 104 s. We suggest that this puzzling feature is due to continued accretion in the central engine. A plateau in the jet luminosity can arise when the viscosity parameter α is small, ∼10−2 or less. A plateau is also produced by continued fall-back of matter – either from an extended stellar envelope or from material that failed to escape with the supernova ejecta. In a few GRBs, the XLC is observed to drop suddenly at the end of the plateau phase, while in others the XLC declines more slowly as ∼t−1−t−2. These features arise naturally in the accretion model depending on the radius and mean specific angular momentum of the stellar envelope. The total energy in the disc-wind accompanying accretion is found to be about 1052 erg. This is comparable to the energy observed in supernovae associated with GRBs, suggesting that the wind might be the primary agent responsible for the explosion. The accretion model thus provides a coherent explanation for the diverse and puzzling features observed in the early XLC of GRBs. It might be possible to use this model to invert gamma-ray and X-ray observations of GRBs and thereby infer basic properties of the core and envelope of the GRB progenitor star.