An energy-conserving dynamical model of GRB afterglows from magnetized forward and reverse shocks

Abstract

In the dynamical models of gamma-ray burst (GRB) afterglows, the uniform assumption of the shocked region is known as provoking total energy conservation problem. In this work we consider shocks originating from magnetized ejecta, extend the energy-conserving hydrodynamical model of Yan et al. (2007) to the MHD limit by applying the magnetized jump conditions from Zhang & Kobayashi (2005). Compared with the non-conservative models, our Lorentz factor of the whole shocked region is larger by a factor $\lesssim\sqrt{2}$. The total pressure of the forward shocked region is higher than the reversed shocked region, in the relativistic regime with a factor of about 3 in our interstellar medium (ISM) cases while ejecta magnetization degree $\sigma<1$, and a factor of about 2.4 in the wind cases. For $\sigma\le 1$, the non-conservative model loses $32-42$% of its total energy for ISM cases, and for wind cases $25-38$%, which happens specifically in the forward shocked region, making the shock synchrotron emission from the forward shock less luminous than expected. Once the energy conservation problem is fixed, the late time light curves from the forward shock become nearly independent of the ejecta magnetization. The reverse shocked region doesn't suffer from the energy conservation problem since the changes of the Lorentz factor are recompensed by the changes of the shocked particle number density. The early light curves from the reverse shock are sensitive to the magnetization of the ejecta, thus are an important probe of the magnetization degree.