Orbital Flips Caused by the Eccentric Von Zeipel–Lidov–Kozai Effect in Nonrestricted Hierarchical Planetary Systems

Abstract

Abstract The eccentric von Zeipel–Lidov–Kozai effect is widely applied to diverse astrophysical settings. In the restricted hierarchical three-body system, the topic of orbital flips has been extensively studied. However, it is far from being understood for nonrestricted circumstances. In this work, the dynamics of orbital flips are investigated under the Hamiltonian model at the octupole-level approximation for nonrestricted hierarchical planetary systems, where the outer planet is more massive than the inner one. Numerical distribution of flipping orbits shows that there are three major flipping regions, distributed in low-, intermediate-, and high-eccentricity spaces. Poincaré sections indicate that those islands of libration centered near i tot = 90° can lead to orbital flips. Thus, we refer to the behavior of orbital flips as a resonance phenomenon. From this viewpoint, dynamical models of orbital flips can be described by a separable Hamiltonian, which can be treated by a means of perturbation theory. The resonant model for orbital flips is formulated based on the adiabatic invariant approximation and then phase portraits are generated by plotting level curves of adiabatic invariants with the given Hamiltonian. By analyzing phase portraits, analytical boundaries of libration and circulation zones causing orbital flips are obtained. As expected, the numerical and analytical conditions that allow orbits to flip agree well with each other. The phenomenon of orbital flips in nonrestricted hierarchical problems can be well understood with the help of dynamical structures of secular resonance.