Dynamical Structures under Nonrestricted Hierarchical Planetary Systems with Different Mass Ratios

Secular dynamics have been extensively studied in both the inner and outer restricted hierarchical three-body systems. In the inner restricted problem, the quadrupole-order resonance (i.e., the well-known Kozai resonance) causes large coupled oscillations of eccentricity and inclination when the maximum inclination is higher than 39.2 degrees, and the octupole-order resonance leads to the behavior of orbital flips. In the outer restricted problem, the behavior of orbital flips is due to the quadrupole-order resonance. Secular dynamics under the inner and outer restricted systems are distinctly different. The mass ratio of inner and outer bodies could change the ratio of circular orbital angular momenta beta, which significantly influences dynamical structures of the system. But this influence is still unclear. In this paper, we focus on nonrestricted hierarchical planetary systems where beta > 1 and investigate the secular dynamics by changing mass ratios. Dynamical structures are systematically explored from four aspects: periodic orbits, secular resonances, orbital flips, and chaos detection. We find that (a) it tends to lead to more bifurcations in the host family of prograde periodic orbits associated with Kozai resonance with smaller beta; (b) with the decrease of beta, fewer orbits inside the octupole-order resonance can realize flip; (c) for given initial conditions, the forbidden region appears in the retrograde region and becomes larger as beta decreases, meaning that the mutual inclination cannot reach a very high value if beta is small; and (d) chaotic orbits are distributed in the low-eccentricity, high-inclination region when beta > 1.