Fall-Back Disks in Long and Short GRBs

Abstract

We present numerical time-dependent calculations for fall-back disks relevant to GRBs in which the disk of material surrounding the black hole (BH) powering the GRB jet modulates the mass flow, and hence the strength of the jet. Given the initial existence of a small mass <10^{-4} msun near the progenitor with a circularization radius ~10^{10}-10^{11} cm, an unavoidable consequence will be the formation of an "external disk" whose outer edge continually moves to larger radii due to angular momentum transport and lack of a confining torque. For long GRBs, if the mass distribution in the initial fall-back disk traces the progenitor envelope, then a radius ~10^{11} cm gives a time scale ~10^4 s for the X-ray plateau. For late times t>10^7 s a steepening due to a cooling front in the disk may have observational support in GRB 060729. For short GRBs, one expects most of the mass initially to lie at small radii <10^8 cm; however the presence of even a trace amount ~10^{-9} msun of high angular momentum material can give a brief plateau in the light curve. By studying the plateaus in the X-ray decay of GRBs, which can last up to ~10^4 s after the prompt emission, Dainotti et al. find an apparent inverse relation between the X-ray luminosity at the end of the plateau and the duration of the plateau. We show that this relation may simply represent the fact that one is biased against detecting faint plateaus, and therefore preferentially sampling the more energetic GRBs. If, however, there were a standard reservoir in fall-back mass, our model can reproduce the inverse X-ray luminosity-duration relation. We emphasize that we do not address the very steep, initial decays immediately following the prompt emission, which have been modeled by Lindner et al. as fall-back of the progenitor core, and may entail the accretion of > 1 msun.