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

Aims. The eccentric von Zeipel-Lidov-Kozai (ZLK) effect is widely used to explain dynamical phenomena in a variety of astrophysical systems. The purpose of this work is to clarify the dynamical essence of the eccentric ZLK effect by constructing an inherent connection between this effect and the dynamics of secular resonance in restricted hierarchical planetary systems. Methods. Dynamical structures of apsidal resonance were studied analytically by means of perturbative treatments. The resonant model was formulated by averaging the Hamiltonian (up to octupole order) over rotating ZLK cycles, producing an additional motion integral. The phase portraits under the resonant model can be used to analyse dynamical structures, including resonant centres, dynamical separatrices, and islands of libration. Results. By analysing phase portraits, five branches of libration centres and eight libration zones are found in eccentricity-inclination space. The analytical results of the libration zone and the numerical distributions of the resonant orbit agree very well, indicating that the resonant model for apsidal resonances is valid and applicable. Additionally, we found that in the test-particle limit, the distributions of flipping orbits are dominated by the apsidal resonances that are centred at an inclination of i = 90 degrees. Conclusions. The eccentric ZLK effect is dynamically equivalent to the effect of apsidal resonance in restricted hierarchical planetary systems. The dynamical response of the eccentric ZLK effect (or of the effect of apsidal resonance) is to significantly excite the eccentricities and/or inclinations of test particles in the very long-term evolution.