Secular frequencies of 3-D exoplanetary systems

Using a 12th order expansion of the perturbative potential in powers of the eccentricities and the inclinations, we study the secular effects of two non-coplanar planets which are not in mean-motion resonance. By means of Lie transformations (which introduce an action-angle formulation of the Hamiltonian), we find the four fundamental frequencies of the 3-D secular three-body problem and compute the long-term time evolutions of the Keplerian elements. To find the relations between these elements, the main combinations of the fundamental frequencies common to these evolutions are identified by frequency analysis. This study is performed for two different reference frames: a general one and the Laplace plane. We underline the known limitations of the linear Laplace-Lagrange theory and point out the great sensitivity of the 3-D secular three-body problem to its initial values. This analytical approach is applied to the exoplanetary system μ Andromedae in order to search whether the eccentricities evolutions and the apsidal configuration δ ω̄ observed in the coplanar case are maintained for increasing initial values of the mutual inclination of the two orbital planes.