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

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$^{\circ}$.