Roche Limits and Tidal Heating Budgets for Ice-Giant Moons and Rings

This paper examines the tidal environment of ice-giant rings and moons, set by the Roche limit and by tidal dissipation. Using only published input values and transparent formulae, the calculation addresses where the Roche limit lies for icy material around each planet compared with the observed ring systems, and how much tidal heating representative moons receive at their measured eccentricities. The headline results are: roche_limit_km_from_centre_uranus = 67033; roche_limit_km_from_centre_neptune = 70819; roche_limit_radii_uranus = 2.64. The method, inputs and limitations are stated in full so that any reader can reproduce or rescale every number from the deposited calculation script. The central finding is that at their present near-circular orbits the computed heating rates are minute, so the dramatic resurfacing recorded on Miranda and Triton must reflect past epochs of higher eccentricity or resonance, not the present tidal budget. Agreement and disagreement with independent published estimates are reported at their actual size, and the conclusions are limited to what the budget arithmetic supports.