Epicycloid fits to trajectories of particles confined to the equatorial plane of a magnetic dipole

Abstract

Trajectories of charged particles in two-dimensional axisymmetric magnetic fields bear obvious resemblance to epicyclods. The main focus of this study is the systematic assessment of the accuracy of the epicycloid approximation for particle trajectories under such conditions. We primarily discuss particle motion constrained to the equatorial plane of a magnetic dipole. It is shown that with well chosen epicycloid parameters, this approximation can remain accurate for conditions well outside the area of expected validity of the Guiding Center approach. Calculating the epicycloid parameters using exact results for particle motion in the equatorial plane of a magnetic dipole allows to retain the accuracy of this approximation to surprisingly high particle energies. We also show that this approach allows us to approximate the trajectory of a charged particle more accurately than its velocity.