A Systematic Study about Orbit Flips of Test Particles Caused by Eccentric Von Zeipel–Lidov–Kozai Effects

Abstract

The problem of orbit flips caused by eccentric von Zeipel–Lidov–Kozai effects is systematically investigated by means of three approaches, including Poincaré sections, dynamical system theory (periodic orbits and invariant manifolds), and perturbation treatments. Poincaré sections show that orbit flips are due to the existence of islands of libration centered at inclination of 90°, dynamical system theory shows that orbit flips are due to the existence of polar periodic orbits and invariant manifolds, and perturbative treatments indicate that orbit flips are due to the libration of a certain critical argument. Using these approaches, the boundaries of flipping regions in the entire parameter space are produced, and they are in excellent agreement with one another. Through analysis, the essence of flipping orbits is reached: (a) flipping orbits are a kind of quasiperiodic trajectory around polar periodic orbits and invariant manifolds at the same level of Hamiltonian provide boundaries of flipping regions, and (b) flipping orbits are a kind of resonant trajectory, and resonant width measures the size of flipping regions.

Dynamical essence of the eccentric von Zeipel-Lidov-Kozai effect in restricted hierarchical planetary systems

“…In this section, we produce phase portraits and then analyse the phase portraits to estimate resonant width (Lei 2021a;Lei & Li 2021;Lei 2022). At last, we construct a connection between the libration zones of apsidal resonance and numerical distributions of resonant orbit (or flipping orbit).…”

“…In this work, secular resonances are investigated for an inner test particle moving around a central star under the gravitational perturbation from a distant planet 1 . Such a dynamical model is called restricted hierarchical planetary system, which is widely adopted as the basic dynamical model to study secular dynamics in varieties of astrophysical systems Li et al 2014a,b;Sidorenko 2018;Luo et al 2016;Lei et al 2018;Lei 2019;Katz et al 2011;Antognini 2015;Lei 2021bLei ,a, 2022.…”

“…For simplicity, we denote the critical argument of apsidal resonance as σ = h * + sign(H * )g * in the test-particle limit 8 . Based on the set of Delaunay variables (g, h, G, H), Sidorenko (2018) defined the critical argument as σ = h + sign(H)g, which is also adopted by Lei (2022) in his study. Additionally, Katz et al (2011) introduced the longitude Ω e = Ω + arctan (tan ω cos i) to describe the very long-term behaviours caused by eccentric ZLK effect.…”

“…Additionally, Katz et al (2011) introduced the longitude Ω e = Ω + arctan (tan ω cos i) to describe the very long-term behaviours caused by eccentric ZLK effect. Discussions about the relation between Ω e and σ = h + sign(H)g are made in Lei (2022).…”

“…It is noted that a similar transformation to Eq. ( 11) is introduced by Sidorenko (2018) and Lei (2022) but based on the set of canonical variables (g, h, G, H).…”