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

Abstract

We analyze the behavior of the argument of pericenter ${\omega }_2$ of an outer particle in the elliptical restricted three-body problem, focusing on the inverse Lidov-Kozai resonance. 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 ${\omega }_2$ precession rate $(d{\omega }_2/d\tau )$. Then, we derive analytical criteria that determine the vanishing of the ${\omega }_2$ quadrupole precession rate $(d{\omega }_2/d\tau {)}_{\text{quad}}$ for different values of the inner perturber's eccentricity $e_1$. Finally, we use such analytical considerations and describe the behavior of ${\omega }_2$ of outer particles extracted from N-body simulations. Our analytical study indicates that the values of the inclination $i_2$ and the ascending node longitude ${\Omega }_2$ associated with the outer particle that vanish $(d{\omega }_2/d\tau {)}_{\text{quad}}$ strongly depend on the eccentricity $e_1$ of the inner perturber. In fact, if $e_1<$ 0.25 ($>$ 0.40825), $(d{\omega }_2/d\tau {)}_{\text{quad}}$ is only vanished for particles whose ${\Omega }_2$ circulates (librates). For $e_1$ between 0.25 and 0.40825, $(d{\omega }_2/d\tau {)}_{\text{quad}}$ can be vanished for any particle for a suitable selection of pairs (${\Omega }_2$, $i_2$). 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 (${\Omega }_2$, $i_2$) plane. In fact, if ${\omega }_2$ librates and ${\Omega }_2$ circulates, the extremes of $i_2$ at ${\Omega }_2=$ 90${}^{\circ }$ and 270${}^{\circ }$ do not reach the same value, while if ${\omega }_2$ and ${\Omega }_2$ librate, the evolutionary trajectory of the particle in the (${\Omega }_2$, $i_2$) plane evidences an asymmetry respect to $i_2=$ 90${}^{\circ }$.

Accepted for publication in Astronomy and Astrophysics, 16 pages, 11 figures, 1 table

References in corpus (1)

• Secular Dynamics of an Exterior Test Particle: The Inverse Kozai and Other Eccentricity-Inclination Resonances

Cited by in corpus (4)

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• Properties of von Zeipel-Lidov-Kozai oscillations in triple systems at the quadrupole order: relaxing the test particle approximation
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