The role of secondary resonances in the orbital history of Miranda

Abstract

We have investigated in some detail the role of secondary resonances in the tidal evolution of Miranda and Umbriel through the 1:3 mean motion commensurability. The constraints placed by the present values of the orbital elements narrow the investigations to the I_M², e_Me_U, and the e_M² primary resonances. Tidal evolution in any of these primary resonances is described well by the analytical single resonance theory up to the point when a low-order secondary resonance is encountered. (A secondary resonance arises when the libration frequency of a primary resonance is commensurate with the circulation frequency of a nearby primary resonance). We present a simple “perturbed-pendulum” model to understand the origin and dynamics of these secondary resonances. Capture into a secondary resonance leads to chaotic motion and the eventual disruption of the primary mean motion resonance. In order to study the long-time evolution, we have used two approaches to speed up the computations: frequency scaling and algebraic mappings. We find that although several important features of the dynamics of the primary resonances are invariant under frequency scaling, certain other phenomena associated with secondary resonances are not. Since these phenomena are crucial to the dynamics, the numerical integration of the full equations of motion with frequency scaling is inadequate for our purpose. Therefore, we have relied primarily on the use of algebraic mappings in our numerical studies. We show that the present value (4.34°) of Miranda's orbital inclination is probably the result of capture into the I_M² primary resonance and subsequent capture into the $\text{3}\text{1}$ secondary resonance. We also find that there is a 10–20% probability that the eccentricity-type primary resonances could have increased Miranda's orbital eccentricity to values as high as 0.035, which would persist after the disruption of the mean motion commensurability. Damping of an eccentricity as high as 0.035 due to tidal dissipation in the satellite could help explain the bizarre surface features on Miranda.