Periapsis shift in magnetized stationary and axisymmetric spacetimes

In this work, we conduct a detailed study of the precession of charged particles in stationary and asymmetric spacetimes with external magnetic fields. Specifically, we develop the post-Newtonian method and the quasi-circular approximation to derive the periapsis shift respectively for two common types of magnetic fields, the dipolar one and the asymptotically uniform one. It is found using the PN method for magnetic fields decaying as fast or faster than a magnetic dipole that the magnetic effect in the periapsis shift appears from the same order of the traditional frame-dragging term due to the spacetime spin. The magnetic field is found to enhance (or decrease) the periapsis shift when the Lorentz force is attractive (or repulsive). When the repulsive Lorentz force is strong enough, the periapsis shift can become negative. For magnetic fields with a slower decay rate, the periapsis shift of quasi-circular orbits exhibits a more complex dependence on the Lorentz force. The periapsis shift increases as the attractive Lorentz force increases from zero but will decrease eventually. However, when the Lorentz force is repulsive, the orbit develops local helical loops and the periapsis shift approaches $-2\pi$. The results for the dipolar magnetic field are applied to the periapsis shift of Mercury around the Sun and the S2 around the Sgr A* to constrain the dipole moment of the center and the charge of the orbiting object.