Integral equations and Rankine–Hugoniot relations for collisionless anisotropic plasma with heat fluxes

A system of integral equations is suggested to address discontinuous motions of the collisionless anisotropic plasma with heat fluxes. For continuous motions, this system reduces to differential equations of the 8-moment approximation. The closed system of Rankine–Hugoniot (RH) jump relations is derived. In special cases, the derived jump relations are identical to those that were previously used for the parallel shocks and also match the relations suggested by the conservative form of equations for the perpendicular shock propagation. An additional integral equation in this approximation describes the magnetic moment, generalizing the first adiabatic invariant from the Chew–Goldberger–Low equations. The two new RH relations for heat fluxes are derived from the proposed integral equations for the two additional quantities: the parallel energy flux associated with the parallel degree of freedom, and the magnetic moment flux parallel component.