On the Optical Light Curves of Afterglows from Jetted Gamma-ray Burst Ejecta: Effects of Parameters

Due to some refinements in the dynamics, we can follow the overall evolution of a realistic jet numerically till its bulk velocity being as small as $\beta c \sim 10^{-3} c$. We find no obvious break in the optical light curve during the relativistic phase itself. However, an obvious break does exist at the transition from the relativistic phase to the non-relativistic phase, which typically occurs at time $t \sim 10^6$ --- $10^{6.5}$ s (i.e., 10 --- 30 d). The break is affected by many parameters, such as the electron energy fraction $\xi_{\rm e}$, the magnetic energy fraction $\xi_{\rm B}^2$, the initial half opening angle $\theta_0$, and the medium number density $n$. Increase of any of them to a large enough value will make the break disappear. Although the break itself is parameter-dependent, afterglows from jetted GRB remnants are uniformly characterized by a quick decay during the non-relativistic phase, with power law timing index $\alpha \geq 2.1$. This is quite different from that of isotropic fireballs, and may be of fundamental importance for determining the degree of beaming in $\gamma$-ray bursts observationally.