  The energy requirements of gamma ray bursts have in past been poorly
constrained because of three major uncertainties: The distances to bursts, the
degree of burst beaming, and the efficiency of gamma ray production. The first
of these has been resolved, with both indirect evidence (the distribution of
bursts in flux and position) and direct evidence (redshifted absorption
features in the afterglow spectrum of GRB 970508) pointing to cosmological
distances. We now wish to address the second uncertainty. Afterglows allow a
statistical test of beaming, described in an earlier paper. In this paper, we
modify a standard fireball afterglow model to explore the effects of beaming on
burst remnant dynamics and afterglow emission. If the burst ejecta are beamed
into angle zeta, the burst remnant's evolution changes qualitatively once its
bulk Lorentz factor Gamma < 1/zeta: Before this, Gamma declines as a power law
of radius, while afterwards, it declines exponentially. This change results in
a broken power law light curve whose late-time decay is faster than expected
for a purely spherical geometry. These predictions disagree with afterglow
observations of GRB 970508. We explored several variations on our model, but
none seems able to change this result. We therefore suggest that this burst is
unlikely to have been highly beamed, and that its energy requirements were near
those of isotropic models. More recent afterglows may offer the first practical
applications for our beamed models.
