Libration effects for retrograde satellites in the restricted three-body problem

Abstract

Numerical explorations of the restricted problem have shown that for stable large nonperiodic retrograde satellite orbits, the motion can be decomposed into a fast ‘reference motion’ and a slow libration aroundB ₂ We study here this libration in the circular plane Hill's case, for which the ‘reference motion’ is elliptic. We establish the equations of motion for the coordinates of the centre of this ellipse. We find two integrals of motion: the first is the semi-major axis of the ellipse; the second is essentially Jacobi's integral, translated into the new coordinates. We give a formula for the period of the libration and we find its limiting value for small libration amplitudes. A numerical verification gives very good agreement for all these results.