A Synthetic Theory of Motion for Titan-Hyperion

Abstract

We present an iterative method allowing to synthetize a semi-numerical solution for the equations of motion of the resonant Saturn's satellites Titan-Hyperion (limited now to the planar problem). The current theory of Hyperion by Taylor, Sinclair & Message (1987) gives the greatest terms of the long-period part of the solution (depending on two angles: the libration angle τ, and the angular distance of the pericenters ζ). Using it as a first approximation, this solution is substituted numerically in the exact Lagrange equations of motion for Titan and Hyperion, computed for many values of the three angles: τ, ζ and ϕ (the mean synodic longitude). Then, a multivariable Fourier transform allows to reconstruct the equations in these three angles, that is in same form as the initial one with, in addition, the short-period terms. Then, a solution may be obtained and used as a better approximation in an iterative process. Besides a complete determination of the short-period perturbations of Hyperion obtained here completely for the first time, some long-period perturbations of Titan by Hyperion are also found which would be non negligible at the 10 km level.