Abstract
If gamma-ray burst (GRB) emission is strongly collimated, then GRBs occur throughout the universe at a rate much higher than is detected. Since the emission from the optical afterglow is thought to be more isotropic than the gamma-ray emission, it has been hypothesized that a search for orphan afterglows (those without the triggering GRB) would allow strong constraints to be placed on the degree of GRB collimation. In this paper we establish that, within the context of leading models of GRB jet evolution, measurement of the GRB beaming angle θjet using optical orphan searches is extremely difficult, perhaps impossible in practice. We show that this is because, in the leading model of GRB jets, the effective afterglow beaming angle scales with the jet angle, Ωopt ∝ Ωjet for small angles, and so the ratio of detected orphan afterglows to GRBs is independent of the jet opening angle. Thus, the number of expected afterglow detections is the same for moderate jet angles (~20°) as for arbitrarily small jet angles (≪01). For nearly isotropic GRB geometry, or for radio afterglow searches in which the jet has become nonrelativistic, the ratio of afterglows to GRBs may give information on collimation. However, using a simple model we estimate the expected number of orphan detections in current supernova surveys and find this number to be less than 1 for all jet opening angles. Even for future supernova surveys, the small detection rate and lack of dependence on collimation angle appear to ruin the prospects of determining GRB beaming by this method. Radio searches may provide the best hope to find the missing orphans.
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