Context :

Aims : 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 \omega _ { 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 \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 developed in a previous work .

Results : 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 } .
The evolution of \omega _ { 2 } associated with the outer particles of the N-body simulations can be very well explained by the analytical criteria derived in our investigation .

Conclusions :