Expanding relativistic shells and gamma-ray burst temporal structure

Many models of gamma-ray bursts (GRBs) involve a shell expanding at extreme relativistic speeds. The shell of material expands in a photon-quiet phase for a period t0 and then becomes gamma-ray active, perhaps due to inhomogeneities in the interstellar medium or the generation of shocks. Based on kinematics, we relate the envelope of the emission of the event to the characteristics of the photon-quiet and photon-active phases. We initially assume local spherical symmetry wherein, on average, the same conditions prevail over the shell's surface within angles the order of Γ–1, where Γ is the Lorentz factor for the bulk motion. The contribution of the curvature to the temporal structure is comparable to the contribution from the overall expansion. As a result, GRB time histories from a shell should have an envelope similar to "FRED" (fast rise, exponential decay) events in which the rise time is related to the duration of the photon-active phase and the fall time is related to the duration of the photon-quiet phase. This result depends only on local spherical symmetry and, since most GRBs do not have such envelopes, we introduce the "shell symmetry" problem: the observed time history envelopes of most GRBs do not agree with that expected for a relativistic expanding shell.Although FREDs have the signature of a relativistic shell, they may not be due to a single shell, as required by some cosmological models. Some FREDs have precursors in which the peaks are separated by more than the expansion time required to explain FRED shape. Such a burst is most likely explained by a central engine; that is, the separation of the multiple peaks occurs because the central site produced multiple releases of energy on timescales comparable to the duration of the event. Alternatively, there still could be local spherical symmetry of the bulk material, but with a low "filling factor"; that is, only a few percent of the viewable surface (which is already very small, 4πΓ–2) ever becomes gamma-ray active.Long complex bursts present a myriad of problems for the models. The duration of the event at the detector is ~t0/(2Γ2). The long duration cannot be due to large t0, since it requires too much energy to sweep up the interstellar medium. Nor can it be due to small Σ if the time variation is due to ambient objects, since the density of such objects is unreasonable (~1018Γ–4 pc–3 for typical parameters). Long events must explain why they almost always violate local spherical symmetry or why they have low filling factors.Both precursor and long complex events are likely to be "central engines" that produce multiple releases of energy over ~100 s. One promising alternative scenario is one in which the shell becomes thicker than the radius of the curvature within Γ–1. Then it acts as a parallel slab, eliminating the problems associated with local spherical symmetry.