Dynamics of spin effects of compact binaries

Abstract

For a 10-dimensional post-Newtonian canonical conservative Hamiltonian system of spinning compact binaries, where the orbital part is accurate to the third-order post-Newtonian expansion and the spin-orbit contributions are up to the next-to-next-to-leading order, its dynamics is integrable and not at all chaotic due to the presence of five independent isolating integrals, including the total energy, three components of the total angular momentum vector and the length of the orbital angular momentum. As the spin-spin effects of the two spinning bodies are further included, only the length of the orbital angular momentum is no longer conserved, so that the dynamics becomes typically non-integrable. Numerical simulations support the onset of chaos. Above all, many chaotic orbits whose initial radii are larger than 10M and whose Lyapunov times are less than the corresponding inspiral decay times are found. In addition, a threshold value of the maximum ratio of the spin-spin Hamiltonian to the whole Hamiltonian for distinguishing between the ordered and chaotic cases is also given.