From Størmer to Schwarzschild: Analytical Dynamics of Charged Particles in a Dipole Magnetosphere

Abstract

We investigate the conservative dynamics of relativistic charged particles in a Schwarzschild spacetime threaded by an externally supported dipolar magnetic field. The gravitational sector is treated exactly in isotropic coordinates, while the electromagnetic field is described by the leading asymptotic term of the relativistic exterior dipole solution. This formulation provides an analytically tractable relativistic extension of the classical Størmer problem and allows the circular-orbit and stability structure to be studied explicitly. For the outward-Lorentz branch, characterized by a positive signed magnetic coupling for positive azimuthal motion, we find a non-trivial competition between radial and vertical stability. Weak magnetic coupling shifts the radially marginal orbit inward, whereas at stronger coupling vertical instability becomes the limiting mechanism and the inner boundary of stable circular motion moves outward. The associated vertical epicyclic mode progressively softens and vanishes at the vertical stability boundary. The marginally bound sequence exhibits a distinct strong-coupling behaviour, illustrating that energetic and stability boundaries need not evolve together. The radial and vertical epicyclic frequencies also generate characteristic commensurabilities that may provide natural sites for nonlinear orbital coupling. Because the magnetic field is treated as a test field, these charged-particle effects leave the vacuum Schwarzschild null-geodesic structure unchanged. Magnetic signatures in compact-object observations must therefore arise through the dynamics and radiation of charged matter, plasma propagation effects, or physics beyond the test-field approximation. The framework developed here provides an analytical basis for studying nonlinear dynamics, radiation reaction, and more realistic compact-object magnetospheres.