2000ApJ...543...90H

Whether gamma-ray bursts are highly beamed or not is a very difficult but important problem that we are confronted with. Some theorists suggest that beaming effect usually leads to a sharp break in the afterglow light curve during the ultrarelativistic phase, with the breaking point determined by γ=1/θ₀, where γ is the Lorentz factor of the blast wave and θ₀ is the initial half-opening angle of the ejecta, but numerical studies tend to reject the suggestion. We note that previous studies are uniformly based on dynamics that is not proper for nonrelativistic blast waves. Here we investigate the problem in more detail, paying special attention to the transition from the ultrarelativistic phase to the nonrelativistic phase. Due to some crucial refinements in the dynamics, we can follow the overall evolution of a realistic jet until its velocity is as small as βc∼10^–3c. We find no obvious break in the optical light curve during the relativistic phase itself. However, an obvious break does appear at the transition from the relativistic phase to the Newtonian phase if the physical parameters involved are properly assumed. Generally speaking, the Newtonian phase is characterized by a sharp decay of optical afterglows, with the power-law timing index α∼1.8-2.1. This is due to the quick lateral expansion 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 990123 and 990510 may indicate the presence of highly collimated gamma-ray burst ejecta.