Inner third-body perturbations including the inclination and eccentricity of the perturbing body

Abstract

In the field of the orbital perturbations dealt with an approach based on the mean orbital elements theory, the outer third-body gravitational attraction has been widely investigated. On the contrary, since it represents a less common case in the Solar System, the inner third-body perturbation has only recently been considered. The aim of this paper is to provide a more rigorous formulation of the inner third-body perturbation using a double-averaged analytical model. The disturbing potential function of the inner third-body is expanded in Legendre polynomials up to the second order. Afterwards, it is averaged first with respect to the orbital period of the disturbing body and then with respect to the orbital period of the probe. This procedure eliminates the short periodic motion terms. By applying to the double-averaged disturbing potential the Lagrange Planetary Equations, the equations which describe the long-term and the secular variations of the classical orbital elements have been obtained: they present an analogy with those related to the planetary oblateness. Lastly, several cases of inner third-body perturbation in the Solar System are discussed, with the conclusion that this is a disturbance of relevance for binary asteroidal systems.