Tidal evolution of the Uranian satellites II. An explanation of the anomalously high orbital inclination of Miranda

Miranda and Umbriel have passed through the 3:1 mean-motion commensurability if the specific dissipation function ( Q) of Uranus is less than about 39,000. There are three second-order inclination resonances associated with this commensurability. Temporary capture into either of the resonances involving the orbital inclination of Miranda can account for the anomalously high (∼4°) current inclination of Miranda. As the satellites approach the commensurability at low orbital inclinations, the coupling between the resonances is very weak, and capture into either of the resonances involving the orbital inclination of Miranda is likely. The evolution of the system after capture into to one of these resonances is initially described well by the standard theory of evolution through isolated mean-motion resonances. However, as the orbital inclination of Miranda increases, and the coupling between the resonances becomes stronger, the separatrices associated with the resonances broaden into chaotic zones and eventually merge, creating a sizable chaotic region. Escape from resonance occurs via a qualitatively new dynamical mechanism. The trajectory encounters low-order commensurabilities between the libration frequency of the resonant argument and other fundamental frequencies in the system. If the trajectory is captured into any of these secondary resonances, it is dragged into the chaotic region, whereupon the system can escape the mean-motion commensurability. Miranda retains a high orbital inclination comparable to the current value. Since the anomalously large inclination of Miranda is a natural outcome of passage through the 3:1 commensurability, the requirement that the satellites have encountered this resonance constrains the Q of Uranus to be less than 39,000.