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 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 - 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.