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We present numerical time-dependent calculations for fall-back disks relevantto GRBs in which the disk of material surrounding the black hole (BH) poweringthe GRB jet modulates the mass flow, and hence the strength of the jet. Giventhe initial existence of a small mass <10^{-4} msun near the progenitor with acircularization radius ~10^{10}-10^{11} cm, an unavoidable consequence will bethe formation of an "external disk" whose outer edge continually moves tolarger 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 tracesthe progenitor envelope, then a radius ~10^{11} cm gives a time scale ~10^4 sfor the X-ray plateau. For late times t>10^7 s a steepening due to a coolingfront 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; howeverthe presence of even a trace amount ~10^{-9} msun of high angular momentummaterial can give a brief plateau in the light curve. By studying the plateausin the X-ray decay of GRBs, which can last up to ~10^4 s after the promptemission, Dainotti et al. find an apparent inverse relation between the X-rayluminosity at the end of the plateau and the duration of the plateau. We showthat this relation may simply represent the fact that one is biased againstdetecting faint plateaus, and therefore preferentially sampling the moreenergetic GRBs. If, however, there were a standard reservoir in fall-back mass,our model can reproduce the inverse X-ray luminosity-duration relation. Weemphasize that we do not address the very steep, initial decays immediatelyfollowing the prompt emission, which have been modeled by Lindner et al. asfall-back of the progenitor core, and may entail the accretion of > 1 msun.

Fall-Back Disks in Long and Short GRBs