The harmonic structure of generic Kerr orbits

TL;DR

This paper classifies complex three-dimensional Kerr orbits using a set of periodic orbits called n-leaf clovers, revealing their structure through a rational number related to perihelion precession, and showing how all bound orbits relate to these fundamental patterns.

Contribution

It introduces a classification scheme for Kerr orbits based on periodic n-leaf clovers, providing a new geometric and analytical framework for understanding their structure.

Findings

Existence of discrete n-leaf clover orbits characterized by a rational precession number

Monotonic relationship between the precession number and orbital energy and eccentricity

Bound orbits can be approximated by these periodic n-leaf clovers, forming a structural skeleton

Abstract

Generic Kerr orbits exhibit intricate three-dimensional motion. We offer a classification scheme for these intricate orbits in terms of periodic orbits. The crucial insight is that for a given effective angular momentum $L$ and angle of inclination $\iota$, there exists a discrete set of orbits that are geometrically $n$-leaf clovers in a precessing {\it orbital plane}. When viewed in the full three dimensions, these orbits are periodic in $r-\theta$. Each $n$-leaf clover is associated with a rational number, $1+q_{r\theta}=\omega_\theta/\omega_r$, that measures the degree of perihelion precession in the precessing orbital plane. The rational number $q_{r\theta}$ varies monotonically with the orbital energy and with the orbital eccentricity. Since any bound orbit can be approximated as near one of these periodic $n$-leaf clovers, this special set offers a skeleton that illuminates the structure of all bound Kerr orbits, in or out of the equatorial plane.