Structures of secular resonances for inner test particles in hierarchical planetary systems

Abstract

In this study, dynamics of secular resonances for inner test particles are investigated under the octupole-level approximation by taking non-perturbative approaches. In practice, webs of the major secular resonances are produced by identifying families of stable periodic orbits and the associated stable libration zones are obtained by analysing Poincaré surfaces of section. By taking different values of the factor ( measures the contribution of octupole terms), the influences of the octupole-order terms upon the dynamical structures are evaluated. Under the condition of = 0 (without octupole-order contribution), the dynamical model is totally integrable and there is only Kozai resonance arising in the phase space. When the factor is different from zero, the dynamical structure in the phase space becomes complicated due to varieties of secular resonances. Numerical results further indicate that (a) distributions of libration centres and stable libration zones remain qualitatively similar with different values of , (b) Kozai resonance disappears in the low-eccentricity region due to the chaotic motion and (c) the chaotic area arising in the low-eccentricity region increases with the factor . Secular resonances are the source of many important dynamical phenomena, such as chaos, orbit alignment and orbit flipping, and thus, the results presented in this work could be useful to understand the secular dynamics for those high-eccentricity and/or high-inclination objects in hierarchical planetary systems. KeywordsSecular resonances • Poincaré surfaces of section • Families of periodic orbits B Hanlun Lei

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

“…For such a kind of non-integrable system, Poincaré surface of section provides a powerful tool to numerically explore the global structures in the phase space. The technique of Poincaré sections has been used in different contexts Li et al 2014a;Naoz et al 2017;Malhotra & Zhang 2020;Lei & Li 2021;Lei 2021b). In this section, our purpose is to analyze the surfaces of section at different levels of Hamiltonian to find libration islands causing orbital flips.…”

“…. On the Poincaré sections, the smooth curves represent regular orbits, including librating trajectories inside the islands and circulating trajectories outside the islands, and the scattering points filling with a certain region on the sections stand for chaotic motions Li et al 2014a;Lei 2021b).…”

“…In all panels of Figure 2, we use ΔH to stand for the size of libration islands causing orbit flips. 7 By analyzing the Poincaré sections (Malhotra & Zhang 2020;Lei & Li 2021;Lei 2021b), it is possible to generate the size of flipping regions as a function of the Hamiltonian ( ) H D  . According to the definition of H, we can produce the boundaries of flipping regions in the space (e 0 , i 0 ), as shown in Figure 3.…”