Orbital stability in static axisymmetric fields

TL;DR

This paper studies the stability of test-particle orbits in axisymmetric gravitational and electromagnetic fields, highlighting the role of toroidal magnetic fields and their limitations in providing gyroscopic stabilization.

Contribution

It extends previous analyses to include toroidal magnetic fields and demonstrates their inability to stabilize orbits in source-free regions, with specific conditions where stabilization is possible.

Findings

Toroidal magnetic fields do not alter circular orbit locations.

Gyroscopic stabilization by toroidal fields is impossible in source-free regions.

Certain non-vanishing toroidal fields can stabilize positively charged particles.

Abstract

We investigate the stability of test-particle equilibrium orbits in axisymmetric, but otherwise arbitrary, gravitational and electromagnetic fields. We extend previous studies of this problem to include a toroidal magnetic field. We find that, even though the toroidal magnetic field does not alter the location of the circular orbits, it enters the problem as a gyroscopic force with the potential to provide gyroscopic stability. This is in essence similar to the situation encountered in the reduced three-body problem where rotation enables stability around the local maxima of the effective potential. Nevertheless, we show that gyroscopic stabilization by a toroidal magnetic field is impossible for axisymmetric force fields in source-free regions because in this case the effective potential does not possess any local maxima. As an example of an axisymmetric force field with sources, we consider the classical problem of a rotating, aligned magnetosphere. By analyzing the dynamics of halo and equatorial particle orbits we conclude that axisymmetric toroidal fields that are antisymmetric about the equator are unable to provide gyroscopic stabilization. On the other hand, a toroidal magnetic field that does not vanish at the equator can provide gyroscopic stabilization for positively charged particles in prograde equatorial orbits.