Secular dynamics in hierarchical three-body systems

The secular approximation for the evolution of hierarchical triple configurations has proven to be very useful in many astrophysical contexts, from planetary to triple-star systems. In this approximation the orbits may change shape and orientation but the semimajor axes are constant. For example, for highly inclined systems, the Kozai-Lidov mechanism can produce large-amplitude oscillations of the eccentricities. Here we re-derive the secular evolution equations including both quadrupole and octupole orders using Hamiltonian perturbation theory. Our new derivation corrects an error in previous treatments of the problem. Already to quadrupole order, our results disagree with the previous "standard" treatment; they agree only in the test-particle limit where one of the bodies in the inner binary has negligible mass compared to that of the outer perturber. Assuming, as done in previous treatments, that the z-component of the inner orbit's angular momentum (perpendicular to the invariable plane) is conserved can produce erroneous results for various astrophysical systems of interest.