Jet Sideways Expansion Effect on Estimating the Gamma-Ray Burst   Efficiency
  The high efficiency of converting kinetic energy into gamma-rays estimated with late-time afterglows in Gamma-Ray Burst (GRB) phenomenon challenges the commonly accepted internal-shock model. However, the efficiency is still highly uncertain because it is sensitive to many effects. In this Letter we study the sideways expansion effect of jets on estimating the efficiency. We find that this effect is considerable, reducing the efficiency by a factor of $\sim0.5$ for typical parameters, when the afterglow data $\sim 10$ hr after the GRB trigger are used to derive the kinetic energy. For a more dense circumburst medium, this effect is more significant. As samples, taking this effect into account, we specifically calculate the efficiency of two bursts whose parameters were well constrained. Almost the same results are derived. This suggests that the sideways expansion effect should be considered when the GRB efficiency is estimated with the late afterglow data. 
Xiaohong Zhao, J. M. Bai