Harmonic structure of generic Kerr orbits

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 $\ensuremath{\iota}$, there exists a discrete set of orbits that are geometrically $n$-leaf clovers in a precessing orbital plane. When viewed in the full three dimensions, these orbits are periodic in $r\ensuremath{-}\ensuremath{\theta}$. Each $n$-leaf clover is associated with a rational number, $1+{q}_{r\ensuremath{\theta}}={\ensuremath{\omega}}_{\ensuremath{\theta}}/{\ensuremath{\omega}}_{r}$, that measures the degree of perihelion precession in the precessing orbital plane. The rational number ${q}_{r\ensuremath{\theta}}$ 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.

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