Linear Approximation in the Two-Planet Problem with Arbitrary Orbital Inclinations. Evolution of the TOI-1130 Exosystem

A new formulation of the two-planet problem is proposed and investigated, where orbits with small eccentricities and mutual inclinations can have an arbitrary orientation relative to the main (picture) plane. The model enables the study of a broad class of exoplanetary systems with orbital inclination angles different from $${\pi\mathord{\left/{\vphantom{\pi 2}}\right.\kern-1.2pt}2}$$ . To derive the equations for the secular evolution of the orbits, the mutual gravitational energy of Gaussian rings, expressed as a series up to second-order small terms, is used instead of the classical perturbation function. A theoretical method has been developed in which, for each orbit, a vector perpendicular to the orbital plane and two Poincaré variables are introduced instead of the osculating Lagrange elements, without loss of information. A closed system of 10 differential equations is obtained and solved analytically. The method is applied to investigate the secular evolution of the TOI-1130 exoplanetary system.