Inverse Lidov-Kozai resonance for an outer test particle due to an eccentric perturber

Abstract

Aims. We analyze the behavior of the argument of pericenter ω 2 of an outer particle in the elliptical restricted three-body problem, focusing on the ω 2 resonance or inverse Lidov-Kozai resonance. Methods. First, we calculate the contribution of the terms of quadrupole, octupole, and hexadecapolar order of the secular approximation of the potential to the outer particle's ω 2 precession rate (dω 2 /dτ). Then, we derive analytical criteria that determine the vanishing of the ω 2 quadrupole precession rate (dω 2 /dτ) quad for different values of the inner perturber's eccentricity e 1 . Finally, we use such analytical considerations and describe the behavior of ω 2 of outer particles extracted from N-body simulations developed in a previous work. Results. Our analytical study indicates that the values of the inclination i 2 and the ascending node longitude Ω 2 associated with the outer particle that vanish (dω 2 /dτ) quad strongly depend on the eccentricity e 1 of the inner perturber. In fact, if e 1 < 0.25 (> 0.40825), (dω 2 /dτ) quad is only vanished for particles whose Ω 2 circulates (librates). For e 1 between 0.25 and 0.40825, (dω 2 /dτ) quad can be vanished for any particle for a suitable selection of pairs (Ω 2 , i 2 ). Our analysis of the N-body simulations shows that the inverse Lidov-Kozai resonance is possible for small, moderate and high values of e 1 . Moreover, such a resonance produces distinctive features in the evolution of a particle in the (Ω 2 , i 2 ) plane. In fact, if ω 2 librates and Ω 2 circulates, the extremes of i 2 at Ω 2 = 90 • and 270 • do not reach the same value, while if ω 2 and Ω 2 librate, the evolutionary trajectory of the particle in the (Ω 2 , i 2 ) plane evidences an asymmetry respect to i 2 = 90 • . The evolution of ω 2 associated with the outer particles of the N-body simulations can be very well explained by the analytical criteria derived in our investigation.

The Lidov-Kozai Oscillation and Hugo von Zeipel

“…The eccentric Lidov-Kozai oscillation in the outer problem is also formulated, and applied to solar system dynamics (e.g. Naoz et al, 2017;Zanardi et al, 2017;Vinson and Chiang, 2018;de Elía et al, 2019). Here again, we insist that this line of orbital phenomenon should be called, consistent with our discussion, the eccentric von Zeipel-Lidov-Kozai oscillation.…”

Formation of polar terrestrial circumbinary planets

“…All planets around eccentric binaries experience inclination oscillations that take place over tens of thousands of years as a result of the asymmetric potential of the binary (Verrier & Evans 2009;Farago & Laskar 2010). Previous test particle studies reveal complex dynamics due to a set of resonances with an interior perturber that induce large oscillations in particle eccentricity and inclination as well as librating orbits (orbits whose argument of periapsis oscillates about a fixed point) and orbit flipping (orbits that flip from prograde to retrograde) of the test particle (Naoz et al 2017;Vinson & Chiang 2018;de Elía et al 2019). Such resonances can lead to the chaotic and unstable evolution of a body and explain the semi-major axis dependence on eccentricity variation we observe in our simulations.…”

Radial-velocity variation of a tertiary star orbiting a binary black hole in coplanar and non-coplanar triples: short- and long-term anomalous behavior

“…It is also possible that the orbit of the outer star is affected by the inverse-KL and other eccentricityinclination resonances(e.g. de Elía et al 2019;Naoz et al 2017Naoz et al , 2020Vinson & Chiang 2018), which may enhance the outer eccentricity effectively, depending on the initial conditions. For example, Vinson & Chiang (2018) showed that the eccentricity of an outer test particle can be enhanced up to ∼ 0.2 and ∼ 0.3 by the inverse KL resonance and octupole resonance on out + in −2Ω out , respectively.…”