ON SYMMETRICAL PLANETARY COROTATIONS

Exact corotations are equilibrium points in the phase space of the asteroidal elliptic restricted problem of three bodies averaged over the synodic period, at a mean-motions resonance. If the resonant critical angle is theta = (p + q)lambda(jup) - p lambda - q pi, exact corotafions are double resonant motions defined by the conditions d theta/dt = 0 and d(pi - pi(jup))/dt = 0. The first condition is characteristic of the periods resonance (p + q) : p and the second one is a secular resonance equivalent to that usually known as the v(5)-resonance. This paper presents the symmetric solutions theta = 0 (mod pi), pi = pi(jup) (mod pi). Corotations have a coherence property which is unique in non-collisional Celestial Mechanics: An elementary calculation shows that, in the neighbourhood of these solutions, the motions cluster around p independent longitude values and are, in each cluster, as close together as theta and pi are close to the equilibrium values.