A Two-Planet Problem with an Arbitrary Inclination of a Pair of Orbits. Secular Evolution of the Kepler-117 Exosystem

A new method is used to study a current version of the two-planet problem on the secular evolution of planetary orbits with small eccentricities and mutual inclinations, having an arbitrary orientation relative to the main (picture) plane. A model has been developed that describes a wide class of exoplanetary systems with an inclination angle of orbits different from \(\pi {\text{/}}2.\) The orbits of the planets are modeled by the Gaussian rings, the perturbing function is represented by the mutual gravitational energy of these rings in the form of a series up to terms of second order of smallness. To describe the evolution of orbits, instead of osculating Keplerian elements, a new set of variables is introduced: the unit vector \({\mathbf{R}}\) of normal to the plane of the ring and two Poincaré variables \(\left( {p,q} \right);\) for eight independent variables, a system of differential equations is obtained and analytically solved. The method is applied to study the secular evolution of the two-planet system Kepler-117 (KOI-209) with non-resonant orbits of exoplanets. It has been established that in this system the oscillations of the same components of the orientation vector \({\mathbf{R}}\) for each of the orbits, as well as the values \(\left( {e,i,{{\Omega }}} \right),\) occur strictly in antiphase. The eccentricities of both orbits oscillate with the period \({{T}_{\kappa }} \approx 182.3\;{\text{years}},\) and the inclinations of the orbits and the longitudes of the ascending nodes change in the libration mode with the same period \({{T}_{g}} \approx {\text{174}}.5\;{\text{years}}.\) The lines of the orbital apsides rotate unevenly counterclockwise with the periods of secular rotation \({{T}_{{{{g}_{2}}}}} \approx 178.3\;{\text{years}}\) (for a light planet), and \({{T}_{{{{g}_{1}}}}} \approx 8140\;{\text{years}}\) (for a more massive planet).