Phase-plane analysis for a spinning particle orbiting a Schwarzschild black hole with second-order spin corrections

Abstract

We study the motion of a particle driven by second-order spin corrections according to the Tulczyjew spin condition as it orbits a Schwarzschild black hole. We convert the Mathisson-Papapetrou-Dixon equations to first-order form and subject them to a phase-plane analysis. We characterize how the number of fixed points depend on the orbital angular momentum and spin of the particle. We find that the spin contributes to the total angular momentum, so that even if the orbital angular momentum is small a bound orbit can still exist because of the spin.