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The 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 $ι$, 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 $r-θ$. Each $n$-leaf clover is associated with a rational number, $1+q_{rθ}=ω_θ/ω_r$, that measures the degree of perihelion precession in the precessing orbital plane. The rational number $q_{rθ}$ 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.