Secular Resonances in Nonrestricted Hierarchical Triple Systems

In this paper, the averaged Hamiltonian of a nonrestricted hierarchical triple system truncated at the third order is investigated. First, each secular resonant term is studied. For the well-studied secular quadrupole theory, it is analytically reformulated in a different manner in our work. The resonance width is numerically determined and displayed on the root 1 - e(1)(2) - root 1 - e(2)(2) plane (also denoted as the (e) over tilde (1) - (e) over tilde (2) plane). In terms of the octupole terms, we show that for a near-planar configuration of the system, considerable variations of both the eccentricities of the inner and outer orbits can be generated by a single resonant term. The resonance width for every secular resonant angle from the octupole terms is also numerically determined and displayed on the (e) over tilde (1) - (e) over tilde (2) plane. The results show that an orbit flip with a near-perpendicular initial mutual inclination is possible for each secular resonance. By displaying the resonance widths of different resonant terms on the same (e) over tilde (1) - (e) over tilde (2) plane, we intuitively show the overlap of different secular resonances. Then, the full averaged Hamiltonian with both quadrupole and octupole terms is investigated using the Poincare surface of section, with a special focus on the orbit flip. For the cases we exploited, we find that the near-planar flip of the inner orbit can be either regular or chaotic while the outer orbit flip is generally chaotic.