Hierarchical three-body problem at high eccentricities = simple pendulum – III. Precessing quadrupole

ABSTRACT The very long-term evolution of the hierarchical restricted three-body problem with a slightly aligned precessing quadrupole potential is investigated analytically and solved for both rotating and librating Kozai–Lidov cycles (KLCs) with high eccentricities. We describe the finding of a striking similarity between librating and rotating KLCs for some range of precession rates. We show that the main effect occurs in both categories when the KLC frequency is equal to the precession rate of the perturbing potential. We solve the resonant dynamics analytically and show that it is equivalent to a simple pendulum model allowing us to map the strikingly rich structures that arise for precession rates similar to the Kozai–Lidov time-scale (ratio of a few) and explain the similarity and when it vanishes. Additionally, we show that the regular KLCs at high eccentricities can also be described using a simple pendulum.