Force function of a slightly elliptical Gaussian ring and its generalization to a nearly coplanar system of rings

Abstract

An analytical expression of the force function of a slightly elliptical Gaussian ring is proposed. The obtained power expansion of ellipse eccentricity contains the terms up to the third power inclusive. The coefficients of this expansion depend on the coordinates of the sampling point and semi-major axis of the ellipse. In the advanced application to the system of the major moons of Uranus for the investigation into secular mutual perturbations, the generalization of the force function is performed for the case of a nearly coplanar system of rings. The force function obtained in this work is more accurate than the earlier known ones and includes the terms up to the third power inclusive with respect to the eccentricities and equatorial inclinations of the moons’ orbits.