Dynamics of black hole pairs. I. Periodic tables

Abstract

Although the orbits of comparable-mass, spinning black holes seem to defy simple decoding, we find a means to decipher all such orbits--in the absence of radiation reaction. The conservative dynamics is complicated by extreme perihelion precession compounded by spin-induced precession. We are able to quantitatively define and describe the fully three-dimensional motion of comparable-mass binaries with one black hole spinning and expose an underlying simplicity. To do so, we untangle the dynamics by capturing the motion in the orbital plane and explicitly separate out the precession of the plane itself. Our system is defined by the conservative third-order post-Newtonian Hamiltonian plus spin-orbit coupling for one spinning black hole with a nonspinning companion. Our results are twofold: (1) We derive highly simplified equations of motion in a nonorthogonal orbital basis, and (2) 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 rationals, we are able to show that zoom-whirl behavior is prevalent in comparable-mass binaries in the strong-field regime, as it is for extreme-mass-ratio binaries in the strong field. Amore » 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. The gravitational waveform produced by these low-leaf clovers will reflect the natural harmonics of the orbital basis - harmonics that, importantly, depend only on radius. The significance for gravitational wave astronomy will depend on the number of windings the pair executes in the strong-field regime. The third-order post-Newtonian system studied provides an example of a general method that can be applied to any effective description of black hole pairs.« less