The dynamics of precessing binary black holes using the post-Newtonian approximation

We investigate the (conservative) dynamics of binary black holes using the Hamiltonian formulation of the post-Newtonian (PN) equations of motion. The Hamiltonian we use includes spin-orbit coupling, spinspin coupling, and mass monopole/spin-induced quadrupole interaction terms. We investigate the qualitative effects of these terms on the orbits; in the case of both quasicircular and eccentric orbits, we search for the presence of chaos (using the method of Lyapunov exponents) for a large variety of initial conditions. For quasicircular orbits, we find no chaotic behavior for black holes with total mass 10‐40M� when initially at a separation corresponding to a Newtonian gravitational-wave (GW) frequency less than � 150 Hz. Only for rather small initial radial distances (corresponding to a GW frequency larger than � 150 Hz), for which spin-spin induced oscillations in the radial separation are rather important, do we find chaotic solutions, and even then they are rare. Moreover, these chaotic quasicircular orbits are of questionable astrophysical significance, since they originate from direct parametrization of the equations of motion rather than from widely separated binaries evolving to small separations under gravitational radiation reaction. In the case of highly eccentric orbits, which for ground-based interferometers are not astrophysically favored, we again find chaotic solutions, but only at pericenters so small that higher order PN corrections, especially higher spin PN corrections, should also be taken into account. Taken together, our surveys of quasicircular and eccentric orbits find chaos only for orbits that are either of dubious astrophysical interest for ground-based interferometers or which violate the approximations required for the equations of motion to be physically valid at the post-Newtonian order considered.