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

The variability observed in many complex gamma-ray bursts (GRBs) is inconsi stent with causally connected variations in a single, symmetric, relativist ic shell interacting with the ambient material ("external shocks"). Rather, either the central site must produce similar to 10(50) ergs s(-1) for hund reds of seconds ("internal shocks"), or the local spherical symmetry of the shell must be broken on an angular scale much smaller than Gamma(-1), wher e Gamma is the bulk Lorentz factor for the shell. The observed variability in the external shock models arises from the number of causally connected r egions that (randomly) become active. We define the surface filling factor to be the ratio of the area of causally connected regions that become activ e to the observable area of the shell. From the observed variability in 52 BATSE bursts, we estimate the surface filling factor to be typically simila r to 5 x 10(-3), although some values are near unity. We find that the surf ace filling factor, f, is similar to 0.1 Delta T/T in both the constant Gam ma phase (which probably produces the GRB) and the decelerating phase (whic h 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 f illing factor must be small(10(-3)) in the X-ray afterglow phase as well. C ompared with the energy required for an isotropic shell, E-iso, explanation s 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 argu ment that GRBs must be internal shocks.