Rapid fading of optical afterglows as evidence for beaming in gamma-ray bursts

Based on the refined dynamical model proposed by us earlier for beamed $\gamma$-ray burst ejecta, we carry out detailed numerical procedure to study those $\gamma$-ray bursts with rapidly fading afterglows (i.e., $\sim t^{-2}$). It is found that optical afterglows from GRB 970228, 980326, 980519, 990123, 990510 and 991208 can be satisfactorily fitted if the $\gamma$-ray burst ejecta are highly collimated, with a universal initial half opening angle $\theta_0 \sim 0.1$. The obvious light curve break observed in GRB 990123 is due to the relativistic-Newtonian transition of the beamed ejecta, and the rapidly fading optical afterglows come from synchrotron emissions during the mildly relativistic and non-relativistic phases. We strongly suggest that the rapid fading of afterglows currently observed in some $\gamma$-ray bursts is evidence for beaming in these cases.