Mutual Gravitational Energy of Gaussian Rings and the Problem of Perturbations in Celestial Mechanics

A new approach to the study of long-period and secular perturbations in celestial mechanics is developed. In contrast to the traditional use of the apparatus of the perturbing Lagrange function, we rely on the mutual potential energy of elliptic Gaussian rings. This approach is important due to the fact that instead of averaging the expression for the perturbing Lagrange function obtained in a very complicated way, it is methodologically simpler to immediately calculate the mutual energy of Gaussian rings. In this paper, we consider the problem for two Gaussian rings with one common focus, with small eccentricities, a small angle of mutual inclination, and an arbitrary angle between the lines of apsides. An expression for the mutual energy of such a system of rings is obtained in the form of a series up to the terms of the 4th order of smallness inclusively. This expression is used to derive and solve a system of differential equations describing the evolution of rings in an ecliptic reference frame. The method is used for a detailed study of the two-planetary Sun–Jupiter–Saturn problem. The results complement and refine the results of other authors. The new expression of the perturbing function can be applied not only to the planetary problem, in which all inclinations should be small, but also to the problem with nonplanetary rings found around small celestial bodies.