Title: Spinning Black Hole Pairs
Description: Black hole binaries will be an important source of gravitational radiation for both ground-based and future space-based gravitational wave detectors. The study of such systems will offer a unique opportunity to test the dynamical predictions of general relativity when gravity is very strong. To date, most investigations of black hole binary dynamics have focused attention on restricted scenarios in which the black holes do not spin (and thus are confined to move in a plane) and/or in which they stay on quasi-circular orbits. However, spinning black hole pairs in eccentric orbits are now understood to be astrophysically equally important. These spinning binaries exhibit a range of complicated dynamical behaviors, even in the absence of radiation reaction. Their conservative dynamics is complicated by extreme perihelion precession compounded by spin-induced precession. Although the motion seems to defy simple decoding, we are able to quantitatively define and describe the fully three-dimensional motion of arbitrary mass-ratio binaries with at least one black hole spinning and expose an underlying simplicity.

To do so, we untangle the dynamics by constructing an instantaneous orbital plane and showing that the motion captured in that plane obeys elegant topological rules. In this thesis, we apply the above prescription to two formal systems used to model black hole binaries. The first is defined by the conservative 3PN Hamiltonian plus spin-orbit coupling and is particularly suitable to comparable-mass binaries. The second is defined by geodesics of the Kerr metric and is used exclusively for extreme mass-ratio binaries. In both systems, we define a complete taxonomy for fully three-dimensional orbits. More than just a naming system, the taxonomy provides unambiguous and quantitative descriptions of the orbits, including a determination of the zoom-whirliness of any given orbit. Through a correspondence with the rational numbers, we are able to show that all of the qualitative features of the well-studied equatorial geodesic motion around Schwarzschild and Kerr black holes are also present in more general black hole binary systems.

This includes so-called zoom-whirl behavior, which turns out to be unexpectedly prevalent in comparable-mass binaries in the strong-field regime just as it is for extreme mass-ratio binaries. In each case we begin by thoroughly cataloging the constant radius orbits which generally lie on the surface of a sphere and have acquired the name "spherical orbits". The spherical orbits are significant as they energetically frame the distribution of all orbits. In addition, each unstable spherical orbit is asymptotically approached by an orbit that whirls an infinite number of times, known as a homoclinic orbit. We further catalog the homoclinic trajectories, each of which is the infinite whirl limit of some part of the zoom-whirl spectrum and has a further significance as the separatrix between inspiral and plunge for eccentric orbits. We then show that there exists a discrete set of orbits that are geometrically closed n-leaf clovers in a precessing orbital plane. When viewed in the full three dimensions, these orbits do not close, but they are nonetheless periodic when projected into the orbital plane.

Each n-leaf clover is associated with a rational number, q, that measures the degree of perihelion precession in the precessing orbital plane. The rational number q 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 orbits in both systems, in or out of the equatorial plane. A first significant conclusion that can be drawn from this analysis is that all generic orbits in the final stages of inspiral under gravitational radiation losses are characterized by precessing clovers with few leaves, and that no orbit will behave like the tightly precessing ellipse of Mercury. We close with a practical application of our taxonomy beyond the illumination of conservative dynamics. The numerical calculation of the first-order (adiabatic) approximation to radiatively evolving inspiral motion in extreme mass-ratio binaries is currently hindered by prohibitive computational cost.

Motivated by this limitation, we explain how a judicious use of periodic orbits can dramatically expedite both that calculation and the generation of snapshot gravitational waves from geodesic sources.