Quadrupole and octupole order resonances in non-restricted hierarchical planetary systems

TL;DR

This paper investigates quadrupole and octupole order resonances in non-restricted hierarchical planetary systems using perturbative methods, revealing detailed dynamical structures and resonance locations.

Contribution

It provides an analytical and perturbative framework to understand quadrupole and octupole resonances, including phase portraits and resonance zones in hierarchical planetary systems.

Findings

Identified the distribution of libration and circulation regions.

Located the nominal position of octupole-order resonance.

Discovered five flipping regions related to orbit flips.

Abstract

Nonrestricted hierarchical three-body configurations are common in various scales of astrophysical systems. Dynamical structures of the quadrupole-order resonance (the von Zeipel-Lidov-Kozai resonance) and the octupole-order resonance (the apsidal resonance) under the nonrestricted hierarchical planetary systems are investigated in this work by taking advantage of perturbative treatments. Under the quadrupole-order Hamiltonian model, the distribution of libration and circulation regions as well as the distribution of flipping region are analytically explored in the parameter space spanned by the conserved quantities. The fundamental frequencies of system are produced and then the nominal location of octupole-order resonance is identified. From the viewpoint of perturbative theory, the quadrupole-order Hamiltonian determines the unperturbed dynamical model and the octupole-order Hamiltonian plays an role of perturbation to the quadrupole-order dynamics. The resonant Hamiltonian for octupole-order resonances is formulated by means of averaging theory, giving rise to a new constant of motion. Phase portraits are produced to analyse dynamical structures of octupole-order resonance, including resonant centres, saddle points, dynamical separatrices and islands of libration. By analysing phase portraits, it is found that there are four branches of libration centre and eight libration zones in the considered space. Applications to orbit flips show that there are five flipping regions.