Tidal evolution of the Uranian satellites III. Evolution through the Miranda-Umbriel 3:1, Miranda-Ariel 5:3, and Ariel-Umbriel 2:1 mean-motion commensurabilities

Tidal evolution has significantly affected the history of the Uranian satellite system. The presence of large chaotic zones at past mean-motion commensurabilities among the Uranian satellites has resulted in significant changes in the orbital elements of some of the satellites, while allowing them to escape from the commensurabilities and evolve to their present nonresonant configuration. Miranda and Umbriel have probably passed through the 3:1 commensurability, resulting in Miranda's current anomalously high inclination, and constraining the Q of Uranus to be less than 39,000 (W. C. Tittemore and J. Wisdom 1989, Icarus 78, 63-89) . The orbits of both satellites become chaotic during evolution through the eccentricity resonances associated with the 3:1 commensurability. During this phase of evolution, the orbital eccentricity of Miranda can be driven up to a value of about 0.05. Miranda can then escape from the 3:1 commensurability with a relatively large orbital eccentricity, which can damp to the current value in the time since resonance passage. Tidal friction may have heated the interior of Miranda to a temperature near the eutectic melting point of NH ₃ · H ₂O, but most likely did not result in the melting of significant quantities of water ice. Miranda and Ariel passed through the 5:3 mean-motion commensurability if the Q of Uranus is less than about 12,000. During evolution through this commensurability, the semimajor axis ratio ( a_M/ a_A) decreased. As the orbits enter a large chaotic zone associated with this commensurability, both the eccentricity and inclination of Miranda's orbit jump to values up to six times higher than the values approaching the resonance. Upon escaping from the resonance, the orbit of Miranda may have retained moderately high eccentricity and inclination, or e_M and i_M may have decreased back to values comparable to those approaching the resonance. If the Q of Uranus is smaller than about 11,000, Ariel and Umbriel would have encountered the 2:1 mean-motion commensurability. However, capture into this resonance is very likely if the eccentricities of the orbits approaching the resonance were comparable to the current values: the probability of escape would not have been significant unless the initial eccentricities were of order 0.03 or larger. It is therefore unlikely that these satellites encountered this resonance, constraining the Q of Uranus to be greater than 11,000. The specific dissipation function of Uranus is therefore well constrained: 11,000 < Q < 39,000.