Resonant orbits for a spinning particle in Kerr spacetime

Abstract

In the present article, we study the orbital resonance corresponds to an extended object approximated up to the dipole order term in Kerr spacetime. We start with the Mathisson-Papapetrou equations under the linear spin approximation and primarily concentrate on two particular events. First, when the orbits are nearly circular and executing a small oscillation about the equatorial plane and second, a generic trajectory confined on the equatorial plane. While in the first case, all the three fundamental frequencies, namely, radial $\Omega_r$, angular $\Omega_{\theta}$, azimuthal $\Omega_{\phi}$ can be commensurate with each others and give rise to the resonance phenomenon, the later is only accompanied with the resonance between $\Omega_r$ and $\Omega_{\phi}$ as we set $\theta=\pi/2$. We provide a detail derivation in locating the prograde resonant orbits in either of these cases and also study the role played by the spin of the black hole. The implications related to spin-spin interactions between the object and black hole are also demonstrated.