  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. We show here that, within the
context of leading models of GRB jet evolution, measurement of the GRB beaming
angle using optical orphan searches is extremely difficult, perhaps impossible
in practice. This is because in the leading model of GRB jets, the effective
afterglow beaming angle scales with the jet angle 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 (e.g. 20 deg) as for arbitrarily small jet angles (<< 0.1
deg). For nearly isotropic GRB geometry, or for radio afterglow searches in
which the jet has become non-relativistic, 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 one, 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.
