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 calculated 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 (d ω 2 ∕d τ ). Then, we derived analytical criteria that determine the vanishing of the ω 2 quadrupole precession rate (d ω 2 /d τ ) quad for different values of the inner perturber’s eccentricity e 1 . Finally, we used such analytical considerations and described 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 (d ω 2 /d τ ) quad strongly depend on the eccentricity e 1 of the inner perturber. In fact, if e 1 < 0.25 (>0.40825), (d ω 2 /d τ ) quad is only vanished for particles whose Ω 2 circulates (librates). For e 1 between 0.25 and 0.40825, (d ω 2 /d τ ) 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 shows evidence of an asymmetry with 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.
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The stationary points of the hierarchical three-body problem
ABSTRACT We study the stationary points of the hierarchical three body problem in the planetary limit (m1, m2 ≪ m0) at both the quadrupole and octupole orders. We demonstrate that the extension to octupole order preserves the principal stationary points of the quadrupole solution in the limit of small outer eccentricity e2 but that new families of stable fixed points occur in both prograde and retrograde cases. The most important new equilibria are those that branch off from the quadrupolar solutions and extend to large e2. The apsidal alignment of these families is a function of mass and inner planet eccentricity, and is determined by the relative directions of precession of ω1 and ω2 at the quadrupole level. These new equilibria are also the most resilient to the destabilizing effects of relativistic precession. We find additional equilibria that enable libration of the inner planet argument of pericentre in the limit of radial orbits and recover the non-linear analogue of the Laplace–Lagrange solutions in the coplanar limit. Finally, we show that the chaotic diffusion and orbital flips identified with the eccentric Kozai–Lidov mechanism and its variants can be understood in terms of the stationary points discussed here.
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- Award ID(s):
- 1739160
- NSF-PAR ID:
- 10211967
- Date Published:
- Journal Name:
- Monthly Notices of the Royal Astronomical Society
- Volume:
- 499
- Issue:
- 2
- ISSN:
- 0035-8711
- Page Range / eLocation ID:
- 1682 to 1700
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/mnras/staa2602
