The effect on the GRB efficiency estimation from the jet expanding sideways

The gamma-ray burst (GRB) efficiency of converting kinetic energy into γ-rays is a very important quantity, which can potentially confirm or rule out the commonly accepted internal-shocks model. Several authors have been estimate the efficiency from the afterglow and GRB emission. Recently, some authors imply the efficiency derived from the latest observations from swift with this method is very high, which can challenge the standard internal-shocks model. We re-consider the efficiency estimating formula, and find the usually used formula is right only for the case that the jet opening angle is the same during the prompt and afterglow phases. However, in fact the jet opening angle will enlarge increasingly from GRB phase to afterglow one due to the expanding sideways of the ejecta. To obtain a more accurate efficiency, the beaming-corrected radiation and kinetic energy in the prompt and afterglow phases should be used. Adopting the typical parameters of GRB, we numerically calculate the jet opening angle evolution, derive the beaming-corrected energy, and then work out the efficiency. We find the efficiency is a factor of about 0.52 and 0.66 less than the typical value derived by Fan & Piran (2006) and Lloyd-Ronning & Zhang (2004), respectively. That means the efficiency of the pre-swift and swift GRBs is generally estimated high. After the factor is considered, the revised GRBs efficieny may still be understood in the internal model.