Hierarchical three-body problem at high eccentricities = simple pendulum II: octupole including Brown's Hamiltonian

The very long-term evolution of the hierarchical restricted three-body problem with a massive perturber is analysed analytically in the high eccentricity regime. Perturbations on the time-scale of the outer orbit can accumulate over long time-scales and be comparable to the effect of the octupole term. These perturbations are described by Brown's Hamiltonian - having different forms in the literature. We show that at the high eccentricity regime - the effect of Brown's Hamiltonian is an azimuthal precession of the eccentricity vector and can be solved analytically. In fact, the dynamics are equivalent to a simple pendulum model allowing an explicit flip criterion.