Fitting the trajectories of particles in the equatorial plane of a magnetic dipole with epicycloids

In this paper we discuss epicycloid approximation of the trajectories of charged particles in axisymmetric magnetic fields. Epicycloid trajectories are natural in the Guiding Center approximation and we study in detail the errors arising in this approach. We also discuss how using exact results for particle motion in the equatorial plane of a magnetic dipole the accuracy of this approximation can be significantly extended. We also show that the epicycloids approximate the trajectory of a charged particle more accurately than the position of the particle along the trajectory.