Heating of directly transmitted ions at low Mach number perpendicular shocks: new insights from a statistical physics formulation

The heating of directly transmitted ions at low Mach number, quasi-perpendicular collisionless shocks is rapid, greater than adiabatic, and exhibits a distinct T ⊥ > T ∥ anisotropy. In this paper we present a theoretical study of the evolution of the ion velocity distribution across a stationary one-dimensional perpendicular model shock profile. A Lagrangian/Hamiltonian formulation of the ion equations of motion is introduced. We argue that the classical statistical physics solution of Liouville's equation in terms of the energy (Hamiltonian) is not applicable to the case of a laminar perpendicular shock. Assuming a Maxwellian incident ion velocity distribution, it is possible to obtain the analytical form for the distribution through the shock in terms of functions of upstream parameters that are independent of the incident temperature. Unlike the classical Hamiltonian solution, we show that contours of equal phase space probability do not correspond to contours of equal energy. It is this property of the velocity distribution that makes anisotropic heating possible. We recover the observed results that the distribution is stretched across the magnetic field direction as it passes through the shock and that it rotates as a whole around the field in the downstream region. We are able to show that in the low-temperature limit, the shape of the distribution remains Gaussian but that this is not the case for higher temperatures. For this Gaussian approximation, lower and upper bounds for the variance of the downstream velocity and therefore the heating are obtained. An efficient method for the numerical computation of the distribution through the shock is proposed and evaluated for typical shock parameter combinations. The downstream behaviour of the distribution is also elucidated.