Afterglow Light Curves from Impulsive Relativistic Jets with an Unconventional Structure

The jet structure in gamma-ray burst (GRB) sources is still largely an open question. The leading models invoke either (i) a roughly uniform jet with sharp edges, or (ii) a jet with a narrow core and wide wings where the energy per solid angle drops as a power law with the angle $\theta$ from the jet symmetry axis. Recently, a two component jet model has also been considered, with a narrow uniform jet of initial Lorentz factor $\Gamma_0 \gtrsim 100$ surrounded by a wider uniform jet with $\Gamma_0 \sim 10-30$. Some models predict more exotic jet profiles, such as a thin uniform ring (i.e. the outflow is bounded by two concentric cones of half opening angle $\theta_c$ and $\theta_c + \Delta\theta$, with $\Delta\theta \ll \theta_c$) or a fan (a thin outflow with $\Delta\theta \ll 1$ along the rotational equator, $\theta_c = \pi/2 + \Delta\theta/2$). In this paper we calculate the expected afterglow light curves from such jet structures, using a simple formalism that is developed here for this purpose, and could also have other applications. These light curves are qualitatively compared to observations of GRB afterglows. It is shown that the two component jet model cannot produce sharp features in the afterglow model due to the deceleration of the wide jet or the narrow jet becoming visible at lines of sight outside of it. We find that a "ring" shaped jet or a "fan" shaped jet produce a jet break in the afterglow light curve that is too shallow compared to observations, where the change in the temporal decay index across the jet break is about half of that for a uniform conical jet. For a "ring" jet, the jet break is divided into two distinct and smaller breaks, the first occurring when $\gamma\Delta\theta \sim 1-2$ and the second when $\gamma\theta_c \sim 1/2$.