Gamma-ray bursts and relativistic shells: The surface filling factor

The variability observed in many complex gamma-ray bursts (GRBs) is inconsistent with causally connected variations in a single, symmetric, relativistic shell interacting with the ambient material ("external shocks"). Rather, either the central site must produce similar to 10(50) ergs s(-1) for hundreds of seconds ("internal shocks"), or the local spherical symmetry of the shell must be broken on an angular scale much smaller than Gamma(-1), where Gamma is the bulk Lorentz factor for the shell. The observed variability in the external shock models arises from the number of causally connected regions that (randomly) become active. We define the surface filling factor to be the ratio of the area of causally connected regions that become active to the observable area of the shell. From the observed variability in 52 BATSE bursts, we estimate the surface filling factor to be typically similar to 5 x 10(-3), although some values are near unity. We find that the surface filling factor, f, is similar to 0.1 Delta T/T in both the constant Gamma phase (which probably produces the GRB) and the decelerating phase (which probably produces the X-ray afterglows). Here, AT is a typical timescale of variability, and T is the time since the initial signal. We analyze the 2 hr flare seen by ASCA 36 hr after the GRB and conclude that the surface filling factor must be small(10(-3)) in the X-ray afterglow phase as well. Compared with the energy required for an isotropic shell, E-iso, explanations for a low surface filling factor can either require more energy (f(-1) E-iso similar to 10(56) ergs) or less energy [(Delta T/4T)E-2(iso) similar to 10(49) ergs]. Thus, the low filling factor cannot be used as a strong argument that GRBs must be internal shocks.