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Tilted near-circular charged-particle orbits around a weakly magnetized Schwarzschild black hole

A. M. Al Zahrani*

  • *Contact author: amz@kfupm.edu.sa
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Phys. Rev. D 114, 084087 – Published 28 October, 2026

DOI: https://doi.org/10.1103/6hkn-95rc

Abstract

We study charged-particle orbits that are tilted with respect to the magnetic equatorial plane around a Schwarzschild black hole immersed in a weak, asymptotically uniform magnetic field. The initial data are specified by the orbit’s guiding radius, tilt angle, and signed magnetic coupling. We choose a simultaneous radial and polar turning point and impose instantaneous radial force balance. These conditions determine the canonical azimuthal angular momentum and the conserved energy. We show that exact nonequatorial constant-radius timelike orbits do not exist when the magnetic coupling is nonzero. The constructed orbits must therefore breathe radially. We introduce a polar quasi-invariant and derive its exact evolution equation. In the weak-coupling regime, we obtain Jacobi-elliptic and driven-oscillator approximations for the polar and radial motions, as well as a second-order correction to the leading Larmor precession rate. We test these results using long numerical integrations, both magnetic orientations, a parameter scan, and convergence tests. At stronger coupling, Poincaré sections and variational equations reveal interleaved regular and chaotic windows. The sampled quasi-invariant distinguishes coherent resonant motion from broadband behavior, although its variation alone is not a criterion for chaos. The analytical and numerical frequencies agree at the percent level, and the magnetic interaction shifts the geodesic three-to-two polar-radial crossing. We finally discuss possible applications to tilted accretion flows and quasiperiodic oscillations, together with the limitations of the test-particle model.

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Phys. Rev. D 114, 084087– Published 28 October, 2026
  • Received 29 July 2026
  • Accepted 23 September 2026
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