The role of secondary resonances in the orbital history of Miranda

A simple 'perturbed pendulum' model is developed in order to ascertain the origin and dynamics of the secondary resonances in the tidal evolution of Miranda and Umbriel through the 1:3 mean motion commensurability. Long-term evolution is addressed on the basis of the computation-accelerating method of algebraic mappings. It is found that the present 4.34-deg value of Miranda's orbital inclination is probably a result of capture into the primary resonance, as well as subsequent capture into the 3/1 secondary resonance. The damping of an eccentricity as high as 0.035 by tidal dissipation in the satellite may account for the surface features of Miranda.