On Beaming Effects in Afterglow Light Curves

The most luminous gamma-ray bursts can be explained in terms of models involving stellar-mass central engines only if the ejecta are beamed. As was pointed out by Rhoads, the dynamics of the blast wave, formed by the beamed ejecta sweeping the external gas, can be significantly modified by the sideways expansion. This is because, in this case, the surface of the blast-wave increases faster than it does just as a result of the radial divergence, so the blast wave deceleration rate increases faster. According to analytical estimates, the effect becomes important shortly after the bulk Lorentz factor of the blast wave drops below the inverse of the initial opening angle of the beamed ejecta and is accompanied by a sharp break in the afterglow light curve. However, our numerical studies, which follow both the dynamical evolution of the blast wave and the evolution of the electron energy distribution and take into account the light-travel effects related to the lateral size of the source, show that the break of the light curve is weaker and much smoother than the one analytically predicted. A prominent break emerges only for a model without the sideways expansion.