The Uranus System from Occultation Observations (1977-2006): Rings, Pole Direction, Gravity Field, and Masses of Cressida, Cordelia, and Ophelia
Authors: French Richard , Hedman Matthew , Nicholson Philip , McGhee-French Colleen 
Published: February 2023

&lt;p&gt;From an analysis of 31 Earth-based stellar occultations and three &lt;em&gt;Voyager&lt;/em&gt; 2 occultations spanning 1977&amp;#8211;2006 (R. French &lt;em&gt;et al.&lt;/em&gt;&amp;#160;2023), &amp;#160;we determine the keplerian orbital elements of the centerlines (COR) of the nine main Uranian rings to high accuracy, with typical RMS residuals of 0.2 &amp;#8211; 0.4 km and 1-&amp;#963; formal errors in &lt;em&gt;a&lt;/em&gt;, &lt;em&gt;ae&lt;/em&gt;, and &lt;em&gt;a&lt;/em&gt; sin&amp;#160;&lt;em&gt;i&lt;/em&gt; of order 0.1 km, registered on an absolute radius scale accurate to 0.2 km at the 2-&amp;#963; level.&amp;#160; The Uranus pole direction at epoch TDB 1987 Jan 1 12:00 is &amp;#945;&lt;em&gt;&lt;sub&gt;P&lt;/sub&gt;&lt;/em&gt;=77.311346&amp;#177;0.000128&amp;#176; and &amp;#948;&lt;sub&gt;&lt;em&gt;P&lt;/em&gt;&lt;/sub&gt;=15.172664&amp;#177;0.000487&amp;#176;. The pole precession predicted by Jacobson(2014) is not detectable. We identify a host of free and forced normal modes in several of the ring centerlines and inner and outer edges (IER/OER). In addition to the free modes &lt;em&gt;m&lt;/em&gt;=0&amp;#160; in the &amp;#947; ring and &lt;em&gt;m&lt;/em&gt;=2 in the &amp;#948; ring, we find two additional OLR modes (&lt;em&gt;m&lt;/em&gt;=&amp;#8211;1 and &amp;#8211;2) and a possible &lt;em&gt;m&lt;/em&gt;=3 mode in the &amp;#947; ring. No normal modes were detected for rings 6, 5, 4, &amp;#945;, or &amp;#946;. Five separate normal modes are forced by small moonlets: the 3:2 inner Lindblad resonance (ILR) of Cressida with the &amp;#951; ring, the 6:5 ILR of Ophelia with the &amp;#947; ring, the 23:22 ILR of Cordelia with the &amp;#948; ring, the 14:13 ILR of Ophelia with the outer edge (OER) of the &amp;#949; ring, and the counterpart 25:24 OLR of Cordelia with the ring&#039;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. From the amplitudes and resonance radii of forced modes, we determine the masses of Cressida, Cordelia, and Ophelia. When combined with their sizes, we estimate their densities, which vary systematically with orbital radius and closely match the Roche critical density for a shape parameter &amp;#947;=0.6. The measured anomalous apsidal and nodal precession rates of the &amp;#945; and &amp;#946; rings are consistent with the presence of unseen moonlets with masses and orbital radii predicted by Chancia &lt;em&gt;et al.&lt;/em&gt; (2016). The resonance radii, amplitudes, and phases of the &amp;#949; ring edge waves driven by Cordelia and Ophelia provide quantitative support for the shepherding model for narrow ring confinement. Separate orbit fits to the ring edge measurements yield width-radius relations for nearly all of the detected modes, with positive width-radius slopes for ILR modes (including the &lt;em&gt;m&lt;/em&gt;=1 elliptical orbits) and negative slopes for most of the detected OLR modes, as expected on dynamical grounds. 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 &lt;em&gt;J&lt;/em&gt;&lt;sub&gt;2&lt;/sub&gt;=(3510.482&amp;#177;0.384)&amp;#215;10&lt;sup&gt;&amp;#8211;6&lt;/sup&gt; and &lt;em&gt;J&lt;sub&gt;4&lt;/sub&gt;&lt;/em&gt;=(&amp;#8211;34.540&amp;#177;0.426)&amp;#215;10&lt;sup&gt;&amp;#8211;6&lt;/sup&gt; with a correlation coefficient &amp;#961;(&lt;em&gt;J&lt;/em&gt;&lt;sub&gt;2&lt;/sub&gt;,&lt;em&gt;J&lt;sub&gt;4&lt;/sub&gt;&lt;/em&gt;)=0.986, for a reference radius &lt;em&gt;R&lt;/em&gt;=25559 km. Although we cannot set useful independent limits on &lt;em&gt;J&lt;/em&gt;&lt;sub&gt;6&lt;/sub&gt;, we obtain strong joint constraints on combinations of &lt;em&gt;J&lt;/em&gt;&lt;sub&gt;2&lt;/sub&gt;, &lt;em&gt;J&lt;sub&gt;4&lt;/sub&gt;&lt;/em&gt;, and &lt;em&gt;J&lt;sub&gt;6&lt;/sub&gt;&lt;/em&gt; that are consistent with our measurements, which will be useful in limiting the range of internal density and wind models for Uranus.&lt;/p&gt;