Librating Kozai-Lidov Cycles with a Precessing Quadrupole Potential Are Analytically Approximately Solved

The very long term evolution of the hierarchical restricted three-body problem with a slightly aligned precessing quadrupole potential is investigated analytically for librating Kozai-Lidov cycles (KLCs). Klein & Katz presented an analytic solution for the approximate dynamics on a very long timescale developed in the neighborhood of the KLCs' fixed point where the eccentricity vector is close to unity and aligned (or anti-aligned) with the quadrupole axis and for a precession rate equal to the angular frequency of the secular Kozai-Lidov equations around this fixed point. In this paper, we generalize the analytic solution to encompass a wider range of precession rates. We show that the analytic solution approximately describes the quantitative dynamics for systems with librating KLCs for a wide range of initial conditions, including values that are far from the fixed point, which is somewhat unexpected. In particular, using the analytic solution, we map the strikingly rich structures that arise for precession rates similar to the Kozai-Lidov timescale (ratio of a few).