On the Laplace relation among the satellites of Uranus

Three of Uranus' satellites are involved in an interaction similar to the Laplace relation among the Galilean satellites of Jupiter. However, there are no pair-wise commensurabilities in the Uranus system as there are among the Galilean satellites. Thus analysis by Fourier expansion of disturbing functions becomes too unwieldy for practical calculation. Instead, the unexpanded perturbations are numerically averaged over a time scale short compared to the important variations in the Laplace relation. Perturbations of Miranda's orbit are estimated to be of observable magnitude for reasonable masses of Ariel and Umbriel.