A New Approach to Two-Planet Problems. Orthogonal Vector Method Applied to the HD 206893 System

An extended version of the two-planet problem of secular evolution of orbits with small eccentricities and mutual inclination is studied in the linear approximation. Our method allows us to study the orbits of exoplanets with an arbitrary orientation relative to the picture plane. To describe the orbits, instead of the osculating Lagrange elements, a system of three orthogonal vectors is used for the first time—the Laplace vector, the Hamilton vector, and the orbital momentum vector of the planet. Considering these vectors as new variables, in a linear approximation, a system of differential equations of motion was obtained and analytically solved. The mutual gravitational energy of Gaussian rings in the quadratic approximation is used as the perturbation function. The method is applied to the exoplanetary system HD206893, for which the time dependences of the orientation vectors, as well as the osculating Lagrange elements, are calculated.