Relativistic shocks in conductive media

Abstract

Relativistic shocks are present in all high-energy astrophysical processes involving relativistic plasma outflows interacting with their ambient medium. While a well understood process in the context of relativistic hydrodynamics and ideal magnetohydrodynamics, there is little to no understanding of their propagation in media with a finite electrical conductivity. This work presents a method for the derivation and solution of the jump conditions for relativistic shocks propagating in MHD media with a finite electrical conductivity. The covariant expressions of the jump conditions are derived and the algebraic equations expressing the Rankine-Hugoniot conditions are solved numerically. This method is employed for the solution of the Riemann problem for the case of a forward and reverse shock which form during the interaction of a gamma-ray burst ejecta with the circumburst medium in order to determine the kinematics of the resulting blastwave and the dynamical conditions in its interior. Our solutions clearly depict the impact of the plasma's electrical conductivity in the properties of the post-shock medium. Two characteristic regimes are identified with respect to the value of a dimensionless parameter which has a linear dependence on the conductivity. For small values of this parameter the shock affects only the hydrodynamic properties of the propagation medium and leaves its electromagnetic field unaffected. No current layer forms in the shock front; thus this is called the current-free regime. For large values of this parameter the ideal MHD regime is retrieved. We also show that the assumption of a finite electrical conductivity can lead to higher efficiencies in the conversion of the ejecta energy into thermal energy of the blastwave through the reverse shock. The theory developed in this work can be applied to the construction of Riemann solvers for resistive relativistic MHD.