Analytical study of gyrophase, pitch angle, and normal speed of particles leaving an ideal magnetohydrodynamic discontinuity surface

Particle populations that have velocity distributions with only a small spread of gyrophase angles are commonly observed in the vicinity of magnetohydrodynamic (MHD) discontinuity surfaces such as collisionless shocks. Previous theoretical particle trajectory studies have concentrated on ion behavior at an ideal planar Earth's bow shock and have either assumed that a gyrotropic incident initial velocity distribution is reflected at the surface or instead focused on unique fixed initial gyrophase and pitch angle values specified by the generation mechanism assumed for the particle. In this analytical study of trajectories of particles departing an ideal planar MHD surface we demonstrate that a particle's initial gyrophase and pitch angle determine completely whether it will escape the surface or return to it, regardless of its initial energy. We identify the region in initial gyrophase-pitch angle space which leads to trajectories that return to the surface of the discontinuity. The speed normal to the surface of a returning particle, which can affect its ability to traverse the discontinuity, is shown to increase or decrease compared to its initial value according only to the orientation of its guiding-center motion in the frame of reference in which the discontinuity is at rest and the incoming plasma flow is aligned with the constant magnetic field. The dependence of our results on the direction of the upstream magnetic field is illustrated. Our general analytical results are discussed in the context of observations at the Earth's bow shock.