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

By a new method we study the actual variant of the two-planet problem on the secular evolution of planetary orbits with small eccentricities and mutual inclination, having arbitrary orientation with respect to the main (picture) plane. A model has been developed that describes a wide class of exoplanetary systems with an orbital inclination angle different from π / 2. The orbits of the planets are modeled by 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 R of normal to the plane of the ring and two Poincaré variables (p,q); 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 components of the orientation vector R of the same name for each of the orbits, as well as the values (e,i,Ω), occur strictly in the opposite phase. The eccentricities of both orbits oscillate with a period of Tk = 182.3 years, and the inclinations of the orbits and the longitudes of the ascending nodes change in the libration mode with the same period Tg = 174.5 years. The lines of the orbital angles rotate unevenly counterclockwise with periods of secular rotation Tg2 = 178.3 years (for a light planet), and Tg1 = 8140 years (for a more massive planet).