Exact solutions, trajectories and radiation patterns in the classical relativistic St\"{o}rmer problem

TL;DR

This paper extends the classical Störmer problem to relativistic regimes, deriving exact solutions and analyzing radiation patterns of charged particles in magnetic dipole fields.

Contribution

It provides a covariant derivation of relativistic equations of motion, exact planar solutions, and detailed radiation analysis, advancing understanding of relativistic particle dynamics in magnetic fields.

Findings

Exact solutions for planar relativistic motions are obtained.

Relativistic effects significantly alter particle trajectories.

Distinct peaks in radiation spectra indicate preferred emission frequencies.

Abstract

We investigate the relativistic generalization of the classical St\"{o}rmer problem, which describes the motion of charged particles in a purely magnetic dipole field. By incorporating special relativistic effects, the particle dynamics is governed by a strongly nonlinear system of second-order differential equations derived from the Lorentz force law. We present a rigorous and fully covariant derivation of the relativistic equations of motion, together with the associated conservation laws. An exact solution for planar motions is obtained in parametric form, providing analytical insight into the structure of the trajectories. In addition, we perform a detailed numerical analysis of the particle dynamics across both nonrelativistic and relativistic regimes, exploring a range of initial conditions and highlighting the impact of relativistic corrections. The electromagnetic radiation emitted by the accelerated charges is also examined. We compute the time dependence of the total radiated power and determine the corresponding frequency spectrum. Our results provide a comprehensive characterization of magnetic dipole--type radiation associated with St\"{o}rmer-like motion. In particular, the power spectral density consistently exhibits distinct peaks, indicating the presence of preferred frequency bands in the emitted radiation.