Overall Evolution of Jetted Gamma‐Ray Burst Ejecta

Whether gamma-ray bursts are highly beamed or not is a very difficult butimportant problem that we are confronted with. Some theorists suggest thatbeaming effect usually leads to a sharp break in the afterglow light curveduring the ultra-relativistic phase, with the breaking point determined by$\gamma = 1 / \theta_0$, where $\gamma$ is the Lorentz factor of the blastwaveand $\theta_0$ is the initial half opening angle of the ejecta, but numericalstudies tend to reject the suggestion. We note that previous studies areuniformly based on dynamics that is not proper for non-relativistic blastwaves.Here we investigate the problem in more detail, paying special attention to thetransition from the ultra-relativistic phase to the non-relativistic phase. Dueto some crucial refinements in the dynamics, we can follow the overallevolution of a realistic jet till its velocity is as small as $\beta c \sim10^{-3} c$. We find no obvious break in the optical light curve during therelativistic phase itself. However, an obvious break does appear at thetransition from the relativistic phase to the Newtonian phase if the physicalparameters involved are properly assumed. Generally speaking, the Newtonianphase is characterized by a sharp decay of optical afterglows, with the powerlaw timing index $\alpha \sim 1.8$ - 2.1. This is due to the quick lateralexpansion at this stage. The quick decay of optical afterglows from GRB 970228,980326, and 980519, and the breaks in the optical light curves of GRB 990123and 990510 may indicate the presence of highly collimated $\gamma$-ray burstejecta.Comment: 24 pages, submitted to ApJ, Eq.(26) (28) revised according to the referee's report, Fig. 8 redrawn, references updated and some new reference items adde