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

Abstract

We analyze the behavior of the argument of pericenter ω2 of an outer particle in the elliptical restricted three-body problem, focusing on the ω2 resonance or 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 ω2 precession rate (dω2 /dτ). Then, we derive analytical criteria that determine the vanishing of the ω2 quadrupole precession rate (dω2/dτ)quad for different values of the inner perturber´s eccentricity e1. Finally, we use such analytical considerations and describe the behavior of ω2 of outer particles extracted from N-body simulations developed in a previous work. Our analytical study indicates that the values of the inclination i2 and the ascending node longitude Ω2 associated with the outer particle that vanish (dω2 /dτ)quad strongly depend on the eccentricity e1 of the inner perturber. In fact, if e1 < 0.25 (>0.40825), (dω2/dτ)quad is only vanished for particles whose Ω2 circulates (librates). For e1 between 0.25 and 0.40825, (dω2 /dτ)quad can be vanished for any particle for a suitable selection of pairs (Ω2 , i2). Our analysis of the N-body simulations shows that the inverse Lidov-Kozai resonance is possible for small, moderate and high values of e1. Moreover, such a resonance produces distinctive features in the evolution of a particle in the (Ω2 , i2 ) plane. In fact, if ω2 librates and Ω2 circulates, the extremes of i2 at Ω2 = 90◦ and270◦ do not reach the same value, while if ω2 and Ω2 librate, the evolutionary trajectory of the particle in the (Ω2, i2) plane evidences an asymmetry respect to i2 = 90◦ . The evolution of ω2 associated with the outer particles of the N-body simulations can be very well explained by the analytical criteria derived in our investigation.