Periodic solutions for resonance 4/3: Application to the construction of an intermediary orbit for Hyperion's motion

The motion of Hyperion is an almost perfect application of second kind and second genius orbit, according to Poincare's classification. In order to co nstruct such an orbit, we suppose that Titan's motion is an elliptical one and that the observed frequencies are such that 4n(H) - 3n(T) + 3n(omega) = 0, where n(H), n(T) are the mean motions of Hyperion and Titan, n(omega) i s the rate of rotation of Hyperion's pericenter. We admit that the observed motion of Hyperion is a <k(T)over bar> T periodic motion ((T) over bar = 2 pi/(n) over bar) such as n(H) = N-H . (n) over bar/k; 3n(T) = N-T . (n) ov er bar/k; n(omega) = - (n) over bar/k. Then, (4N(H) - N-T - 3)(n) over bar/ k = 0. N-H, N-T, k is an element of N+. With that hypothesis we show that H yperion's orbit tends to a particular periodic solution among the periodic solutions of the Keplerian problem, when Titan's mass tends to zero. The co ndition of periodicity allows us to construct this orbit which represents t he real motion with a very good approximation.