Can radio occultations constrain Uranus or Neptune's internal rotation periods?
The shapes of fluid planets bear the signatures of rotational flattening and atmospheric flows. Precise knowledge of their shapes and wind profiles may therefore reveal their interior rotation rates. We re-examine this idea for the ice giants, where missions like the Uranus Orbiter and Probe could use radio occultations to measure atmospheric heights near 1 bar at multiple latitudes, complementing Voyager 2's near-equatorial constraint for Uranus. Applying geodetic calculations and considering zonal wind uncertainties, we find that only a narrow range for Uranus's 1-bar polar radius, $R_{\rm pol}=24,968.6\pm4.7$ km, is consistent with Uranus's winds, occultations, and gravity field, even treating Uranus's interior spin as a free parameter. This is because the isobaric shape depends on the total rotation of the isobaric surface, which is already well constrained by observations, irrespective of what portion is attributed to bulk rotation versus winds. Occultations will, however, be valuable for testing our underlying assumption that the winds manifest the full differential rotation that sets the shape. The apparent north-south asymmetry in Uranus's winds, if permanent, produces a 5-km difference between the northern and southern polar radii, measurable with suitable radio occultations. Neptune's much more uncertain winds yield ~100 km variations in polar and equatorial radii. We confirm that Uranus and Neptune's magnetic rotation periods yield nonzero mean dynamical heights for their atmospheres. Accurate results for Uranus and Neptune require that the full latitude-dependent rotation be incorporated when fitting radii from occultations. Only significantly faster interior rotation -- periods close to 15 h in both Uranus and Neptune -- would minimize their dynamical heights.