# Resonant orbits for a spinning particle in Kerr spacetime

## TL;DR

This paper investigates orbital resonances of spinning particles in Kerr spacetime, analyzing conditions for resonance in nearly circular and equatorial orbits, and explores the influence of black hole spin on these phenomena.

## Contribution

It provides a detailed derivation of prograde resonant orbits for spinning particles in Kerr spacetime, considering the effects of black hole spin and specific orbital configurations.

## Key findings

- Resonance occurs when fundamental frequencies are commensurate in nearly circular orbits.
- Resonance between radial and azimuthal frequencies occurs in equatorial orbits.
- Black hole spin influences the location and nature of resonant orbits.

## 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.