Hydrodynamic and MHD equations across the bow shock and along the surfaces of planetary obstacles

Examinations of the magnetohydrodynamic (MHD) equations across a bow shock are presented. These equations are written in the familiar Rankine-Hugoniot set, and an exact solution to this set is given which involves the upstream magnetosonic Mach number, plasma beta, polytropic index, and theta(B-v), as a function of position along the shock surface. The asymptotic Mach cone angle of the shock surface is also given as a function of the upstream parameters, as a set of transcendental equations. The standoff position of a detached bow shock from an obstacle is also reviewed. In addition, a detailed examination of the hydrodynamic equations along the boundary of the obstacle is performed. Lastly, the MHD relations along the obstacle surface are examined, for specific orientations of the upstream interplanetary magnetic field (IMF) in relation to the upstream flow velocity vector.