A Mechanical Model for Magnetized Relativistic Blastwaves

The evolution of a relativistic blastwave is usually delineated under the assumption of pressure balance between forward- and reverse-shocked regions. However, such a treatment usually violates the energy conservation law, and is inconsistent with existing magnetohydrodynamic numerical simulation results. A mechanical model of non-magnetized blastwaves was proposed in previous work to solve the problem. In this paper, we generalize the mechanical model to the case of a blastwave driven by an ejecta with an arbitrary magnetization parameter $\sigma_{\rm ej}$. We test our modified mechanical model by considering a long-lasting magnetized ejecta and found that it is much better than the pressure-balance treatment in terms of energy conservation. For a constant central engine wind luminosity $L_{ \ rm ej} = 10^{47} { \ rm erg ~ s^{-1}}$ and $ \ sigma_{ \ rm ej} \u3c 10$, the deviation from energy conservation is negligibly small at small radii but only reaches less than $25{{\ \rm per\ cent}}$ even at 1019 cm from the central engine. For a finite lifetime of the central engine, the reverse shock crosses the magnetized ejecta earlier for the ejecta with a higher $\sigma_{\rm ej}$, which is consistent with previous analytical and numerical results. In general, the mechanical model is more precise than the traditional analytical models with results closer to those of numerical simulations