Interesting dynamics at high mutual inclination in the framework of the Kozai problem with an eccentric perturber

Abstract

We study the dynamics of the 3D three‐body problem of a small body moving under the attractions of a star and a giant planet which orbits the star on a much wider and elliptic orbit. In particular, we focus on the influence of an eccentric orbit of the outer perturber on the dynamics of a small highly inclined inner body. Our analytical study of the secular perturbations relies on the classical octupole Hamiltonian expansion (third‐order theory in the ratio of the semimajor axes), as third‐order terms are needed to consider the secular variations of the outer perturber and potential secular resonances between the arguments of the pericentre and/or longitudes of the node of both bodies. Short‐period averaging and node reduction (by adoption of the Laplace plane reference frame) reduce the problem to two degrees of freedom. The 4D dynamics is analysed through representative planes which identify the main equilibria of the problem. As in the circular problem (i.e. perturber on a circular orbit), the ‘Kozai‐bifurcated’ equilibria play a major role in the dynamics of an inner body on a quasi‐circular orbit: its eccentricity variations are very limited for mutual inclination between the orbital planes smaller than ∼40°, while they become large and chaotic for higher mutual inclination. Particular attention is also given to a region around 35° of mutual inclination, detected numerically by Funk et al. and consisting of long‐time stable and particularly low‐eccentricity orbits of the small body. Using a 12th‐order Hamiltonian expansion in eccentricities and inclinations, in particular its action‐angle formulation obtained by Lie transforms from Libert & Henrard, we show that this region presents an equality of two fundamental frequencies and can be regarded as a secular resonance. Our results also apply to binary star systems where a planet is revolving around one of the two stars.

Secular resonance of inner test particles in hierarchical planetary systems

“…The curves of Tω and TΩ given in Fig. 1 are in good agreement with the ones shown in Libert & Delsate (2012) (see Fig. 8 in their work).…”

“…by means of Lie-series transformation method to determine the fundamental frequencies, similar to the method used in Libert & Henrard (2008). Then, Libert & Delsate (2012) showed that this region presents an equality of two fundamental frequencies (please refer to Fig. 8 in their work), and thus they concluded that the long-term stable dynamics at inclination of ∼35 • is due to the existence of the secular resonance associated with σ = ω − Ω.…”

“…They found an interesting dynamical region around the mutual inclination of ∼35 • where the low-eccentricity orbits are long-time stable, indicating that the inclined (∼35 • ) quasi-circular Earthmass companion is possible to survive in the habitable zone of extrasolar systems (Funk et al 2011). Under the double-averaged model at the octupole-level approximation, Libert & Delsate (2012) computed the maximum variation of eccentricity as a function of the mutual inclination (see Fig. 6 in their work) and reproduced the long-time stable region at inclination of ∼35 • .…”

“…The planetary system consists of a central body with mass m Sun , a giant planet with mass m Jupiter and a test particle with infinitesimal mass. The same parameters as the ones in Libert & Delsate (2012) are used in the simulations: the eccentricity of the giant planet is assumed as ep = 0.3, the semimajor axis ratio as α = 0.05 and the initial eccentricity of the test particle as e = 1 × 10 −6 .…”

“…8 in their work), and thus they concluded that the long-term stable dynamics at inclination of ∼35 • is due to the existence of the secular resonance associated with σ = ω − Ω. Based on our double-averaged Hamiltonian in power series of the semimajor axis ratio (to be presented in Section 2), we formulated the secular Hamiltonian by using the same transformation method adopted by Libert & Delsate (2012) and reproduced the fundamental frequencies ( ω and Ω) as well as the corresponding periods (Tω and TΩ) for test particles moving in low-eccentricity regions. The fundamental frequencies as well as their respective periods are reported in Fig.…”