Analytic Understanding of the Resonant Nature of Kozai–Lidov Cycles with a Precessing Quadrupole Potential

Abstract

The very long-term evolution of the hierarchical restricted three-body problem with a slightly aligned precessing quadrupole potential is studied analytically. This problem describes the evolution of a star and a planet that are perturbed either by a (circular and not too inclined) binary star system or by one other star and a second more distant star, as well as a perturbation by one distant star and the host galaxy or a compact-object binary system orbiting a massive black hole in nonspherical nuclear star clusters. Previous numerical experiments have shown that when the precession frequency is comparable to the Kozai–Lidov timescale, long-term evolution emerges that involves extremely high eccentricities with potential applications for a broad scope of astrophysical phenomena, including systems with merging black holes, neutron stars, or white dwarfs. By averaging the secular equations of motion over the Kozai–Lidov cycles (KLCs), we solve the problem analytically in the neighborhood of the KLC fixed point where the eccentricity vector is close to unity and aligned with the quadrupole axis and for a precession rate similar to the Kozai–Lidov timescale. In this regime the dynamics is dominated by a resonance between the perturbation frequency and the precession frequency of the eccentricity vector. While the quantitative evolution of the system is not reproduced by the solution far away from this fixed point, it sheds light on the qualitative behavior.

Keywords

• Newtonian gravitation
• Keplerian orbit
• Eccentricity
• Three-body problem