Specularly reflected ions, shock foot thicknesses, and shock velocity determination in space

The magnetic foot of a quasi-perpendicular, supercritical collisionless shock is spatially coincident with an caused by gyrating ions nearly specularly reflected from the shock. The reflected ions are turned around by the upstream magnetic field and returned to the shock after completing a partial gyration. An expression is derived for the turnaround distance of specularly reflected ions for arbitrary orientations of the incident velocity vector and the upstream magnetic field. This expression is then used to derive a formula for calculating the shock speed in the spacecraft frame from a single point measurement of the time required for the magnetic foot to transit a spacecraft. The derived formulas for turnaround distance and shock speed differ from previously published equations for these parameters and in some geometries give quantitatively very different results.