The Uranus system from occultation observations (1977-2006): Rings, pole direction, gravity field, and masses of Cressida, Cordelia, and Ophelia

From an analysis of 31 Earth -based stellar occultations and three Voyager 2 occultations spanning 1977-2006 (French et al. 2023a), we determine the keplerian orbital elements of the centerlines of the nine main Uranian rings to high accuracy, with typical RMS residuals of 0.2-0.4 km and 1-o- formal errors in a, ae, and a sin i of order 0.1 km, registered on an absolute radius scale accurate to 0.2 km at the 2-o- level. The A ring shows more substantial scatter, with few secure detections. We identify a host of free and forced normal modes in several of the ring centerlines and inner and outer edges. In addition to the previously -known free modes m = 0 in the y ring and m = 2 in the o ring, we find two additional outer Lindblad resonance (OLR) modes (m = -1 and -2) and a possible m = 3 inner Lindblad resonance (ILR) mode in the y ring. No normal modes are detected for rings 6, 5, 4, ?, or fi. Five separate normal modes are forced by small moonlets: the 3:2 inner ILR of Cressida with the rj ring, the 6:5 ILR of Ophelia with the y ring, the 23:22 ILR of Cordelia with the o ring, the 14:13 ILR of Ophelia with the outer edge of the ? ring, and the counterpart 25:24 OLR of Cordelia with the ring's inner edge. The phases of the modes and their pattern speeds are consistent with the mean longitudes and mean motions of the satellites, confirming their dynamical roles in the ring system. We find no evidence of normal modes excited by internal planetary oscillations. We determine the width-radius relations for nearly all of the detected modes, with positive width-radius slopes for ILR modes (including the m = 1 elliptical orbits) and negative slopes for most of the detected OLR modes, supporting the standard self -gravity model for ring apse alignment. We find no convincing evidence for librations of any of the rings. The Uranus J2000 pole direction at epoch TDB 1986 Jan 19 12:00 is ?P = 77.311327 +/- 0.000141 degrees and oP = 15.172795 +/- 0.000618 degrees. The slight pole precession predicted by Jacobson (2023) is not detectable in our orbit fits, and the absolute radius scale is not strongly correlated with the pole direction. From Monte Carlo fits to the measured apsidal precession and nodal regression rates of the eccentric and inclined rings, we determine the zonal gravitational coefficients J2 = (3509.291 +/- 0.412)x 10-6, J4 = (-35.522 +/- 0.466)x 10-6, and J6 fixed at 0.5x10-6, with a correlation coefficient P(J2, J4) = 0.9861, for a reference radius R = 25559 km. This result differs significantly from both earlier and more recent results (Jacobson 2014, 2023), owing to our inclusion of previously neglected systematic effects, such as the offset of semimajor axes of the geometric ring centerlines from their estimated dynamical centers of mass and the significant contributions of Cordelia and Ophelia to the precession rate of the ? ring. Although we cannot set useful independent limits on J6, we obtain strong joint constraints on combinations of J2, J4, and J6 that are consistent with our measurements. These can be used to limit the range of realistic models of the planet's internal density distribution and wind profile with depth. The observed anomalous apsidal and nodal precession rates of the ? and fi rings are consistent with the presence of unseen moonlets with masses and orbital radii predicted by Chancia and Hedman (2016). The y ring's putative m = 3 mode does not appear to be forced by a satellite, whose predicted size would be too large to have avoided prior detection. If this mode is excluded from the orbit fit, the solution for the y ring has a very large anomalous apsidal precession rate of unknown origin. From the amplitudes and resonance radii of normal modes forced by moonlets, we determine the masses of Cressida, Cordelia, and Ophelia. Their estimated densities decrease systematically with increasing orbital radius and generally follow the radial trend of the Roche critical density for a shape parameter ? = 1.6.