In the Rome meeting I presented a derivation of the dynamical behavior of a beamed gamma ray burst (GRB) remnant and its consequences for the afterglow light curve. (Cf. Rhoads 1999 [Paper I]). Here, I summarize these results and apply them to test the range of beaming angles permitted by the optical light curve of GRB 970508.

Suppose that ejecta from a GRB are emitted with initial Lorentz factor
into a cone of opening half-angle and expand into an
ambient medium of uniform mass density with negligible
radiative energy losses. Let the initial kinetic energy and rest mass
of the ejecta be *E _{0}* and

The swept-up mass is determined by the working surface area: , where and
are the sound speed and time since the burst in the frame of
the blast wave + accreted material. Once , and the dynamical evolution with radius *r* changes
from to (Rhoads 1998, & Paper I). The relation between observer frame time
and radius *r* also changes, from to . Thus, at early times , while at late times . The
characteristic length scale is , and the characteristic observed transition time
between the two regimes is , where *z* is the burst's
redshift.

We assume that swept-up electrons are injected with a power law energy
distribution for , with *p* > 2, and contain a
fraction of . This power law extends up to the cooling
break, , at which energy the cooling time is
comparable to the dynamical expansion time of the remnant. Above
, the balance between electron injection (with
) and cooling gives .

We also assume a tangled magnetic field containing a fraction of . The comoving volume and burster-frame volume *V*
are related by , so that
and .

The resulting spectrum has peak flux density at an observed frequency . Additional spectral features occur at the frequencies of optically thick synchrotron self absorption (which we shall neglect) and the cooling frequency (which is important for optical observations of GRB 970508). The cooling break frequency follows from the relations (Sari et al. 1998; Wijers & Galama 1998) and . In the power law regime, , , and ; while in the exponential regime, ,, and . The spectrum is approximated by a broken power law, ,with for , for , and for .

The afterglow light curve follows from the spectral shape and the time behavior of the break frequencies. Asymptotic slopes are given in Table 1. For the regime, we study the evolution of break frequencies numerically. The results for and are given in Paper I. For , a good approximation is

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