Charged particles in dipole magnetosphere of neutron stars: epicyclic oscillations in and off-equatorial plane

We study the motion of charged test particles around a magnetized neutron star described by the Schwarzschild gravitational field of mass M and a dipole magnetic field characterized by parameter b, representing the ratio of electromagnetic to gravitational forces. The neutron star's radius is assumed at R=3M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=3M$$\end{document}. Circular orbits "in" and "off" the equatorial plane, determined by the dipole field's symmetry plane, are analyzed based on background and particle parameters; their stability against radial and latitudinal perturbations is established. The existence of chaotic charged particle motion in belts around off-equatorial orbits, influenced by sufficiently strong repulsive magnetic force, is demonstrated. The belts' distance from the neutron star varies with parameter b for different charged particles. Frequencies of epicyclic oscillatory motion related to "in" and "off" equatorial circular orbits, alongside the orbital frequency, are determined. The magnetically modified geodesic model is applied to fit observational data from twin-peak, high-frequency quasi-periodic oscillations in binary systems containing neutron stars. Rough fitting to the data for circular orbits for particles under magnetic attraction (repulsion) is shown. The possible fittings imply strong limits on the magnetic parameter, b similar to 0.01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \sim 0.01$$\end{document} (b similar to-10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \sim -10$$\end{document}) for magnetic attraction (repulsion), suggesting minor influence of electromagnetic forces and test particles with small specific charges like dust or plasmoids in strong magnetic fields around neutron stars.