A theoretical explanation of the density of backstreaming ion beams at oblique collisionless shocks

Ion beams with energies several times the incoming flow energy propagate upstream away from oblique collisionless shocks, that is, when the angle between the direction normal to the shock and upstream magnetic field is broadly around 45°. The properties of these beams are well-known from measurements at Earth's bow shock, and numerical and simulation studies. It appears that the ions in the beams are initially solar wind ions that are reflected once or several times at the shock. This paper proposes a theoretical explanation for the origin of the backstreaming ion beams using a simple model of the shock as a thin flat discontinuity surface that reflects specularly any incident ions with insufficient energy to overcome the shock's electrostatic potential jump. Our analysis identifies a portion of the upstream velocity distribution that is occupied by ions that will eventually populate the backstreaming beam. Integration of the incident distribution over this restricted region in velocity space yields the beam density and its dependence on upstream Mach number, plasma beta, and magnetic field orientation. The model is applied to Maxwellian and Kappa incident distributions and the results are found to be consistent with those documented in the literature. The significant depletion of alpha particles in the beams, relative to abundances in the incoming plasma, is reproduced by the model and explained by reference to the different plasma betas of the incident proton and alpha populations and the diverse effect of the shock potential on species of different mass.