The eccentric Kozai-Lidov effect as a resonance phenomenon.

*(English)*Zbl 1382.70008Summary: Exploring weakly perturbed Keplerian motion within the restricted three-body problem, M. L. Lidov [“The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies”, Planet. Space Sci. 9, No. 10, 719–759 (1962)] and, independently, Y. Kozai [“Secular perturbations of asteroids with high inclination and eccentricity”, Astron. J. 67, 591–598 (1962; doi:10.1086/108790)] discovered coupled oscillations of eccentricity and inclination (the KL cycles). Their classical studies were based on an integrable model of the secular evolution, obtained by double averaging of the disturbing function approximated with its first non-trivial term. This was the quadrupole term in the series expansion with respect to the ratio of the semimajor axis of the disturbed body to that of the disturbing body. If the next (octupole) term is kept in the expression for the disturbing function, long-term modulation of the KL cycles can be established [E. B. Ford, B. Kozinsky and F. A. Rasio, “Secular evolution of hierarchical triple star systems”, Astrophys. J. 535, No. 1, 385–401 (2000); S. Naoz et al., “Hot jupiters from secular planet-planet interactions”, Nature 473, 187–189 (2011; doi:10.1038/nature10076); B. Katz, S. Dong and R. Malhotra, “Long-term cycling of Kozai-Lidov cycles: extreme eccentricities and inclinations excited by a distant eccentric perturber”, Phys. Rev. Lett. 107, No. 18, Article ID 181101, 5 p. (2011; doi:10.1103/PhysRevLett.107.181101)]. Specifically, flips between the prograde and retrograde orbits become possible. Since such flips are observed only when the
perturber has a nonzero eccentricity, the term “eccentric Kozai-Lidov effect” (or EKL effect) was proposed by Y. Lithwick and S. Naoz [“The eccentric Kozai mechanism for a test particle”, Astrophys. J. 742, No. 2, Article No. 94, 8 p. (2011; doi:10.1088/0004-637X/742/2/94)] to specify such behavior. We demonstrate that the EKL effect can be interpreted as a resonance phenomenon. To this end, we write down the equations of motion in terms of “action-angle” variables emerging in the integrable Kozai-Lidov model. It turns out that for some initial values the resonance is degenerate and the usual “pendulum” approximation is insufficient to describe the evolution of the resonance phase. Analysis of the related bifurcations allows us to estimate the typical time between the successive flips for different parts of the phase space.

Reviewer: Reviewer (Berlin)

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\textit{V. V. Sidorenko}, Celest. Mech. Dyn. Astron. 130, No. 1, Paper No. 4, 23 p. (2018; Zbl 1382.70008)

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