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

Richard French, Matthew Hedman, Philip Nicholson, Pierre-Yves Longaretti, Colleen Mcghee-French

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## To cite this version:

Richard French, Matthew Hedman, Philip Nicholson, Pierre-Yves Longaretti, Colleen Mcghee-French. The Uranus system from occultation observations (1977–2006): Rings, pole direction, gravity field, and masses of Cressida, Cordelia, and Ophelia. Icarus, 2024, 411, pp.115957. 10.1016/j.icarus.2024.115957. hal-04778362

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# The Uranus System from Occultation Observations (1977-2006): Rings, Pole Direction, Gravity Field, and Masses of Cressida, Cordelia, and Ophelia

RICHARD G. FRENCH,¹ MATTHEW M. HEDMAN,² PHILIP D. NICHOLSON,³ PIERRE-YVES LONGARETTI,⁴ AND
COLLEEN A. MCGHEE-FRENCH¹

¹Department of Astronomy, Wellesley College, Wellesley MA 02481 USA

²Department of Physics, University of Idaho, Moscow, ID 83844 USA

³Astronomy Department, Cornell University, Ithaca NY 14853 USA

⁴Institut de Planétologie et d'Astrophysique de Grenoble, Grenoble, FR

## ABSTRACT

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 (COR) of the nine main Uranian rings to high accuracy, with typical RMS residuals of 0.2 – 0.4 km and 1-σ 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-σ level. The λ 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 γ ring and *m* = 2 in the δ 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 γ ring. No normal modes are detected for rings 6, 5, 4, α, or β. Five separate normal modes are forced by small moonlets: the 3:2 inner ILR of Cressida with the η ring, the 6:5 ILR of Ophelia with the γ ring, the 23:22 ILR of Cordelia with the δ ring, the 14:13 ILR of Ophelia with the outer edge of the ε 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 pole direction at epoch TDB 1986 Jan 19 12:00 is $\alpha_P = 77.311327 \pm 0.000141^\circ$ and $\delta_P = 15.172795 \pm 0.000618^\circ$. 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 $J_2 = (3509.291 \pm 0.412) \times 10^{-6}$, $J_4 = (-35.522 \pm 0.466) \times 10^{-6}$, and $J_6$ fixed at $0.5 \times 10^{-6}$, with a correlation coefficient $\rho(J_2, J_4) = 0.9861$, 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 ε ring. Although we cannot set useful independent limits on $J_6$, we obtain strong joint constraints on combinations of $J_2, J_4$, and $J_6$ 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 α and β rings are consistent with the presence of unseen moonlets with masses and orbital radii predicted by Chancia and Hedman (2016). The γ 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 $\gamma$ 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 $\gamma = 1.6$.

Keywords: occultations, planets: rings

# 1. INTRODUCTION

Prior to the discovery of the Uranian rings in 1977 (Elliot et al. 1977; Millis et al. 1977), Saturn was the only planet known to have rings, which in that pre-Voyager era were thought to be broad and diffuse, owing to viscous spreading associated with interparticle collisions. This simple paradigm was overturned by the detection of the narrow, eccentric, inclined Uranian rings that somehow manage to avoid circularization due to differential precession and to be sharp-edged and radially confined in the absence of direct evidence for shepherd satellites for most of the rings. The discovery of Jupiter's dusty rings by *Voyager 1* in 1979 (Smith et al. 1979) expanded the scope and variety of rings in the solar system, and the subsequent *Voyager* Saturn encounters revealed a bewildering variety of complex ring structure, much of it produced by the gravitational influence of nearby satellites. The *Cassini* mission's 13 year reconnaissance of the Saturn system greatly expanded our knowledge of the structure and dynamical environment of the rings (Colwell et al. 2009; Schmidt et al. 2009; Cuzzi et al. 2009; Charnoz et al. 2009; Cuzzi et al. 2018). From the *Voyager 2* flyby of Uranus in 1986, a closer view of its rings revealed the faint narrow $\lambda$ ring, sheets of dusty ring material between the narrow rings, and a gallery of nearby tiny moons (Smith et al. 1986). Subsequent observations revealed additional details of this dusty and satellite-rich system (de Pater et al. 2002, 2006; Showalter and Lissauer 2006; de Pater et al. 2013). The census of giant plant ring systems in the solar system was extended by the detection of the Neptune's ring arcs from stellar occultation observations (Hubbard 1986a; Hubbard et al. 1986b; Manfroid et al. 1986) and *Voyager 2* images (Smith et al. 1989). Several tiny trans-Neptunian objects have rings: 10199 Chariklo's rings were discovered during a stellar occultation in 2013 (Braga-Ribas et al. 2014) and observed from the James Webb Space Telescope in 2022 (Santos-Sanz 2022), Centaur 2060 Chiron has possible ring material (Ortiz et al. 2015), and a 70 km-wide ring was detected in orbit around dwarf planet Haumea (Ortiz et al. 2017). Further afield, the search is on for evidence of ringed extra-solar planets (see Sicardy et al. (2018) for a recent discussion).

This observed wide variety of ring phenomena leaves us with many unanswered questions about their physical properties, internal structure, dynamics, origins, and evolution (Esposito and De Stefano 2018). The observational basis to advance our understanding of the narrow Uranian rings in particular rests in large part on an extensive set of stellar occultation observations obtained from 1977 to 2006, described in Paper 1 (French et al. 2023a). Here, we make use of these observations to determine accurate orbits for the Uranian rings and ring edges, the planet's pole direction and gravity field, and the masses of the moonlets Cressida, Cordelia, and Ophelia.

The paper is organized as follows: In Section 2, we summarize the Earth-based and *Voyager 2* Uranus ring occultation observations used for this work. Section 3 presents our geometric model of the Uranus ring system and our kinematical model for the rings and ring edges. In Section 4 we present the orbits of the ten narrow rings and ring edges and identify a host of free and forced normal modes. Then, in Section 5, we determine the direction of the planet's pole and the absolute ring radius scale, and in Section 6 we examine the widths, shapes, and masses of the rings. Next, in Section 7, we solve for the zonal gravity coefficients $J_2$ and $J_4$, taking into account several important systematic effects that have been ignored in previous analyses. In Section 8 we summarize the evidence for anomalous precession in several rings and the possibility that it is caused by nearby unseen small satellites, and in Section 9 we estimate the masses and densities of Cressida, Cordelia, and Ophelia from their associated forced normal modes in the rings. In Section 10, we discuss the broader context of the dynamics of narrow ringlets, compare the Uranus and Saturn narrow rings, and identify persistent unsolved problems. Our main conclusions are summarized in Section 11.
## 2. OBSERVATIONS

The observations underlying the present work include 31 Earth-based stellar occultations by the narrow Uranian rings, sometimes viewed with multiple telescopes and at multiple wavelengths, and three *Voyager* occultations: one by the Radio Science Subsystem (RSS; Tyler et al. (1986); Gresh et al. (1989)) and stellar occultations of $\sigma$ Sgr and $\beta$ Per observed by the Photopolarimeter (PPS; Lane et al. (1986)) and the Ultraviolet Spectrometer (UVS; Holberg et al. (1987)). Paper 1 provides a detailed description of the Earth-based observations and a summary of the *Voyager* data. Here, we limit ourselves to a brief overview of the observations, emphasizing characteristics of the data and event geometry that affect the accuracy of the ring measurements used for this study and the scientific interpretation of our results.

### 2.1. Earth-based stellar occultations

For most of the 1977–2006 interval of occultation observations, the Uranus system presented a nearly pole-on view as seen from Earth, a geometry that favored the determination of the absolute radius scale of the rings while limiting the initial accuracy of their inclinations and of the planet’s pole direction. The majority of the Earth-based occultation events were observed between 1980–1990, when Uranus was traversing the Milky Way and high-SNR occultation opportunities were frequent. After the initial discovery observations, which were made at visual wavelengths, most occultations were observed in the infrared K band ($\lambda \sim 2.2 \mu$m), where a strong methane absorption band minimizes the planet’s signal within the photometric aperture. Most of the IR observations were made using InSb aperture photometers, centered on the occultation star but also containing the background signal from the rings, planet, and sky. Many events were recorded continuously in what is known as DC mode, most suitable for photometric conditions. In other cases, especially for faint stars or events with a strong background signal, observations were conducted in chopping mode, where the secondary mirror of the telescope rapidly nodded between the event star and the nearby sky, with filtering electronics recording the difference between the brightness within the aperture at these two alternate positions.

As an example of some of the best data used in this work, **Fig. 1** shows the observations of the May 24, 1985 occultation of U25 from Palomar Observatory’s 5 m Hale telescope. The bright star was observed in chopping mode under photometric conditions with excellent seeing and accurate tracking, resulting in high-SNR lightcurves with stable baselines. The figure shows the normalized ingress and egress signals as a function of equatorial plane radius. The radial misalignment of several of the ingress/egress ring event pairs is due primarily to their orbital eccentricities, most notably for the outermost $\epsilon$ ring. Most ring events are sharp and unresolved at this resolution, with the exception of low optical depth companions exterior to the $\eta$ ring and interior to the $\delta$ ring (barely visible here in the ingress profile). Only the eccentric $\epsilon$ ring is radially resolved, narrower near periapse (upper panel) than near apoapse. The orbital radius of the elusive $\lambda$ ring is labeled, but it was not detected during either ingress or egress during this occultation. (The sharp dip in the signal just exterior to the egress $\epsilon$ ring is a calibration check of the normalized intensity of the occulted star.)

A gallery of the individual ring profiles for the U25 Palomar occultation is shown in **Fig. 2**, plotted in units of normalized flux as a function of ring plane radius (lower axis) and event time (upper axis) and arranged in parallel rows for ingress and egress, increasing radially from the innermost ring 6 to the outermost $\epsilon$ ring. The gap reflects the non-detection of the $\lambda$ ring, noted above. Each profile is labeled by the ring name and event direction (I for ingress, E for egress), and a Quality Index (QI) in parentheses. As described in more detail in Paper 1, every individual Earth-based ring profile used in this study has been assigned a subjective quality index (QI), ranging from 1 (high-SNR profile with sharp ring edges well-matched by the model fit) to 4 (unreliable detection). For the observations shown, QI=1 for every ring event, but this is not the usual case – see Paper 1 for a complete survey of all occultations.

Most of the Uranian rings are at most a few km in width and are unresolved radially in the lightcurves, owing to the combined smoothing effects of Fresnel diffraction and the finite angular diameter of the occultation star. In some cases, the resolution is further limited by intrinsic instrumental time constants or by the sampling interval of the recorded data. For this event, the Fresnel diffraction scale $F \sim \sqrt{\lambda D/2} = 1.73$ km, where $\lambda$ is the observed wavelength and D
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Observations of the May 24, 1985 occultation of U25 from Palomar Observatory's 5 m Hale telescope at $\lambda = 2.2$ µm (after Fig. 4.2 Nicholson et al. (2018a)). The orbital eccentricities of several of the narrow rings are evident from their radial misalignments between ingress (top) and egress (bottom). Only the egress $\epsilon$ ring profile shows internal structure at this scale. Both the $\eta$ and $\delta$ rings have adjacent low-optical depth shoulders, not considered in this work. There is no hint of the elusive $\lambda$ ring in this very high SNR observation, indicative of its azimuthal variability in optical depth and/or its dusty nature, rendering it less evident in IR observations.</figcaption></figure>

is the distance between the observer (or spacecraft) and the ring plane, and the projected stellar diameter at Uranus was assumed to be $d_* = 5.254$ km. In this case, the stellar smoothing dominated that by diffraction, but there are visible diffraction fringes in several of the observed profiles and both effects were included in the models. (Smoothing due to the $\lambda = 2 - 2.4$ µm response of the K-band filter was also taken into account.) Each profile has been fitted to a square-well model in which the ring is assumed to be sharp-edged and radially uniform in opacity. The underlying square-well model is shown as a box function in red, and the corresponding smoothed model is overdrawn on the data. In this case, the models match the observations extremely well, as indicated in the assignment of $QI=1$ for all profiles.

In the examples shown above, diffraction effects are suppressed somewhat by the significant smoothing due to the relatively large projected diameter of the occulted star at Uranus. For fainter and more distant occultation stars, however, stellar smoothing is less significant and diffraction by the ring edges is more prominent. **Figure 3** shows the premier example of this: the egress $\gamma$ ring profile observed from the Anglo-Australian Telescope (AAT) during the remarkable 4-day occultation of U36 in March/April 1987, when Uranus reversed direction at the retrograde point in its geocentric orbit and the consequent skyplane velocity of the star relative the projected ring edge was only $v_\perp = 0.90$ km s$^{-1}$. The projected diameter of U36 was 0.548 km, significantly smaller than the Fresnel scale $F = 1.76$ km. In this example, multiple diffraction fringes are closely matched by a model of the $\gamma$ ring as a 0.73 km-wide square-well with a fractional transmission $f = 0.077$.

For our present purposes, the key observables from the full set of individual Earth-based occultation ring profiles are the times of the midline and edges from the fitted square-well models, which we treat as our best estimates of
<figure><img src="image_4.png" /></figure>

**Figure 2.** Gallery of individual ring profiles from the May 24, 1985 observations of the U25 event from Palomar Observatory's 5 m Hale telescope, plotted as a function of ring plane radius (lower axis) and observed time in UTC (upper axis). The profiles are arranged in parallel rows (ingress, then egress), from the innermost ring 6 to the outermost $\epsilon$ ring. Square-well model fits are overplotted on each profile, showing the smoothing effects of diffraction and the finite angular size of the occulted star, and the geometric square-well model is shown as a box function in each panel. Rings are labeled by name, event direction (I for ingress, E for egress), and Quality Index (1 in all cases shown here – see text for details).

the midline and edges of the actual ring.<fnref n="1" /> As described below, we determine the geometry of the Uranus ring system from least-squares fits to these ring midpoint and edge location times. In our final ring orbit models, the post-fit RMS radius errors of the midpoint measurements for the nine main rings are 0.212–0.401 km and somewhat larger for the ring edges, but in all cases well below the characteristic Fresnel scale for the observations.

Under these circumstances, it is appropriate to ask whether the square-well model accurately determines the widths and midlines of the narrow ring profiles that are at the heart of our investigation. We address this question in Appendix A, where we compare square-well model fits to the diffraction-limited *Voyager* RSS ring observations with the measured widths of the corresponding spatially resolved diffraction-corrected profiles. Our summary conclusion

<footnote n="1">¹ The equivalent width and mean opacity are also determined from the square-well fits, but we do not make use of this information in this study.</footnote>
<figure><img src="image_5.png" /><figcaption>**Figure 3.** The egress γ ring profile observed from the AAT during the stellar occultation of U36 on April 2, 1987. The best-fitting square-well diffraction model is overplotted on the data, and the underlying box model of the ring is shown in red.</figcaption></figure>

is that the square-well model demonstrably provides estimates of ring widths that are accurate at the level of a few hundred m for rings with intrinsically sharp edges, and are likely to be somewhat less accurate for rings with more gradual edges. To the extent that the fitted ring width can be viewed as equivalent to independent measurements of the inner and outer ring edges whose average defines the midline of the ring, the corresponding accuracy of the ring midline measurements is reduced by a factor of $1/\sqrt{2}$ to about 0.2 km, a bit smaller than but of the same order as the RMS error in the ring orbit fits presented below. In cases where one edge is sharp and the other is more gradual, the two test cases examined above show no systematic sense of error, with one ring width being underestimated and the other example being overestimated. These limitations should be kept in mind when assessing the accuracy of the ring orbit model fits to the observed measurements of the ring midlines and edges.

## 2.2. *Imaging observations*

Direct images of the Uranian rings provide additional information about their photometric properties that is complementary to the results from stellar occultations. Although we make no direct use of imaging data in this paper, preliminary (Showalter 2011) and more detailed recent (Hedman et al. 2023) analysis of *Voyager 2* wide- and narrow-angle camera (WAC/NAC) images have revealed periodic azimuthal brightness variations in several of the rings that are indicative of normal modes, some of which are also seen in our analysis. *Voyager* and *Hubble Space Telescope* (HST) observations of small Uranian moons have provided updated estimates of the ephemeris of Ophelia that we will compare to our results in Section 9 below. Ongoing and upcoming imaging observations from *JWST* will provide additional valuable information about the ring structure as well.

# 3. GEOMETRIC MODEL OF THE URANUS RING SYSTEM

## 3.1. *General description*

We determine the orbital elements of the rings and the pole direction and gravity field of Uranus from model fits to the measured midpoints of the occultation ring profiles from the complete set of observations described in Section 2, obtained from square-well model fits for the Earth-based stellar occultations and *Voyager* RSS data, and from direct measurement of midpoints of the spatially resolved *Voyager* PPS stellar occultation profiles. We make use of our well-tested orbit fitting code (RINGFIT) that implements the solar system barycenter (SSBC) occultation
geometry described in detail by French et al. (1993), with minor modifications described by French et al. (2010).<fnref n="2" /> The code makes extensive use of the ICY interface to NASA’s NAIF SPICE toolkit (Acton 1996), which provides access to planetary ephemerides and spacecraft trajectory files. We use the J2000 heliocentric reference frame, the IAU 1976 model for the Earth shape (Abalakin 1981), and the ITRF93 Earth rotation model (Boucher et al. 1994). We account for general relativistic deflection by Uranus (including the effect of $J_2$) for Earth-based stellar occultations only, solving for the deflection at the time the occultation ray is closest to Uranus in the sky plane, rather than the time the occultation ray penetrates the ring plane, as implied by Eqs. (A20) – (A23) of French et al. (1993). (The difference amounts to only a few meters in the derived ring plane radius.)

### 3.1.1. Occultation stars

Until recently, the typical astrometric uncertainty in the parallax- and proper motion-corrected occultation star positions was substantially larger than the uncertainty in the Uranus ephemeris, and orbit fits to the rings included fitted corrections to the predicted star positions at the epoch of each occultation. With the release of the *Gaia* EDR3 and DR3 catalogs (Gaia Collaboration et al. 2021, 2022), the situation has reversed, with small but measurable systematic errors in the Uranus ephemeris in *ura111.bsp* and *de440.bsp* exceeding the star position uncertainties. To reduce these systematic errors, the *ura178* series of ephemerides (described in Section 3.3 below) was developed using the astrometric constraints provided by the full set of ring occultation events in Paper 1 for all but the $\gamma$ and $\lambda$ rings (R. Jacobson, pers. comm). We cross-referenced the predicted star positions for all Earth-based Uranus ring occultations, using VizieR (Ochsenbein et al. 2000), to obtain the J2000 *Gaia* DR3 catalog positions given in Table 2 of Paper 1. The *Voyager 2* stellar occultation stars $\sigma$ Sgr and $\beta$ Per are too bright to allow accurate *Gaia* DR3 positions, and for these stars we make use of the *Hipparcos* catalog positions (Perryman et al. 1997). As part of our orbit solution, we fit for corrections to selected star positions and proper motions under the assumption that the revised Uranus ephemeris is free of systematic errors.

### 3.1.2. Telescope coordinates

The geocentric coordinates of many of the telescopes used for this study are listed in multiple sources that are not always in agreement. For this work, we used modern GPS coordinates, when available. In all cases, these closely matched our estimates from Google Earth, and we therefore used Google Earth to estimate the locations of telescopes not otherwise known. **Table 1** lists our adopted coordinates for each telescope, including the observatory name, ID, telescope diameter, E longitude, latitude, altitude relative to the IAU 1976 Earth model, and geocentric coordinates *x*, *y*, *z* and radius $\rho$. These were converted to a custom SPK kernel file of observatory locations (*ObsCodes_Uranus_20220212.spk*) for use in our SPICE-based geometry calculations in our ring orbit fitting code.

<footnote n="2">² Detailed comparisons of our calculations of the event geometry of an inclined and eccentric ring occultation event with those by R. Jacobson, using an independent code, agree at the 0.002 km level in the derived ring plane radius for a given Earth-received event time.</footnote>
<table><caption>Table 1. Observatory and telescope coordinates</caption><thead><tr><th>Observatory</th><th>ID</th><th>Diam (cm)</th><th>E longitude</th><th>Latitude</th><th>alt (m)</th><th>x (km)</th><th>y (km)</th><th>z (km)</th><th>ρ (km)</th></tr></thead><tbody><tr><td>Centro Astron. Hispano-Aleman (CAHA)</td><td>CAL</td><td>123</td><td>+357</td><td>27</td><td>11.675</td><td>+37</td><td>13</td><td>15.38</td><td>2161</td><td>5081.895</td><td>-226.036</td><td>3838.254</td><td>6372.515</td></tr><tr><td rowspan="2">Cerro Tololo Interamerican Obs. (CTIO)</td><td>807</td><td>400</td><td>+289</td><td>11</td><td>36.917</td><td>-30</td><td>10</td><td>10.78</td><td>2380</td><td>1815.052</td><td>-5213.997</td><td>-3187.844</td><td>6375.149</td></tr><tr><td>C60</td><td>150</td><td>+289</td><td>11</td><td>35.700</td><td>-30</td><td>10</td><td>09.30</td><td>2380</td><td>1815.029</td><td>-5214.030</td><td>-3187.804</td><td>6375.149</td></tr><tr><td rowspan="3">European Souther Observatory</td><td>ESO</td><td>360</td><td>+289</td><td>16</td><td>05.900</td><td>-29</td><td>15</td><td>39.50</td><td>2400</td><td>1838.338</td><td>-5258.792</td><td>-3100.341</td><td>6375.460</td></tr><tr><td>ES2</td><td>200</td><td>+289</td><td>15</td><td>48.000</td><td>-29</td><td>15</td><td>28.20</td><td>2317</td><td>1837.913</td><td>-5259.044</td><td>-3099.997</td><td>6375.378</td></tr><tr><td>ES1</td><td>104</td><td>+289</td><td>15</td><td>41.900</td><td>-29</td><td>15</td><td>24.00</td><td>2321</td><td>1837.780</td><td>-5259.161</td><td>-3099.886</td><td>6375.382</td></tr><tr><td>NASA Infrared Telescope Facility (IRTF)</td><td>IRT</td><td>320</td><td>+204</td><td>31</td><td>40.932</td><td>+19</td><td>49</td><td>34.46</td><td>4212</td><td>-5464.326</td><td>-2493.467</td><td>2151.039</td><td>6379.907</td></tr><tr><td rowspan="2">Las Campanas Observatory</td><td>LAS</td><td>250</td><td>+289</td><td>17</td><td>45.784</td><td>-29</td><td>00</td><td>26.67</td><td>2270</td><td>1845.370</td><td>-5270.713</td><td>-3075.720</td><td>6375.410</td></tr><tr><td>LAV</td><td>100</td><td>+289</td><td>17</td><td>59.053</td><td>-29</td><td>00</td><td>43.22</td><td>2270</td><td>1845.627</td><td>-5270.361</td><td>-3076.166</td><td>6375.409</td></tr><tr><td>Lowell Observatory</td><td>688</td><td>180</td><td>+248</td><td>27</td><td>47.436</td><td>+35</td><td>05</td><td>49.42</td><td>2204</td><td>-1918.477</td><td>-4861.173</td><td>3647.949</td><td>6373.311</td></tr><tr><td>McDonald Observatory</td><td>711</td><td>270</td><td>+255</td><td>58</td><td>40.667</td><td>+30</td><td>40</td><td>18.27</td><td>2103</td><td>-1330.794</td><td>-5328.793</td><td>3235.718</td><td>6374.709</td></tr><tr><td>Mount Stromlo</td><td>414</td><td>190</td><td>+149</td><td>00</td><td>31.968</td><td>-35</td><td>19</td><td>08.72</td><td>770</td><td>-4466.831</td><td>2683.001</td><td>-3667.254</td><td>6371.799</td></tr><tr><td>Observatorio del Teide</td><td>TEE</td><td>155</td><td>+343</td><td>29</td><td>20.740</td><td>+28</td><td>18</td><td>01.82</td><td>2395</td><td>5390.292</td><td>-1597.793</td><td>3007.004</td><td>6375.756</td></tr><tr><td>Palomar Observatory</td><td>675</td><td>508</td><td>+243</td><td>08</td><td>06.612</td><td>+33</td><td>21</td><td>22.57</td><td>1706</td><td>4678.798</td><td>11.472</td><td>4324.379</td><td>6371.149</td></tr><tr><td>Pic du Midi (OPMT)</td><td>586</td><td>200</td><td>+000</td><td>08</td><td>25.764</td><td>+42</td><td>56</td><td>14.52</td><td>2891</td><td>1815.052</td><td>-5213.997</td><td>-3187.844</td><td>6375.149</td></tr><tr><td></td><td>PI1</td><td>106</td><td>+000</td><td>08</td><td>31.900</td><td>+42</td><td>56</td><td>11.20</td><td>2862</td><td>4678.846</td><td>11.612</td><td>4324.285</td><td>6371.120</td></tr><tr><td>S. African Astronomical Obs. (SAAO)</td><td>SAA</td><td>188</td><td>+020</td><td>48</td><td>41.710</td><td>-32</td><td>22</td><td>44.16</td><td>1768</td><td>5041.283</td><td>1916.170</td><td>-3396.939</td><td>6373.809</td></tr><tr><td>Siding Spring Observatory (AAT)</td><td>413</td><td>390</td><td>+149</td><td>04</td><td>02.208</td><td>-31</td><td>16</td><td>31.14</td><td>1164</td><td>-4681.032</td><td>2805.172</td><td>-3292.625</td><td>6373.572</td></tr><tr><td>Siding Spring Observatory(ANU)</td><td>ANU</td><td>230</td><td>+149</td><td>03</td><td>44.597</td><td>-31</td><td>16</td><td>17.88</td><td>1149</td><td>-4680.963</td><td>2805.674</td><td>-3292.268</td><td>6373.559</td></tr><tr><td>UK Infrared Telescope (UKIRT)</td><td>UKI</td><td>380</td><td>+204</td><td>31</td><td>46.890</td><td>+19</td><td>49</td><td>20.70</td><td>4194</td><td>-5464.370</td><td>-2493.677</td><td>2150.635</td><td>6379.890</td></tr></tbody></table>

## 3.2. Kinematical model for ring orbits

The RINGFIT orbit fitting code solves for the geometric ring orbital elements that minimize the sum of squared residuals between the observed and model ring plane radii, using a standard kinematical model for all ring features. Following Nicholson et al. (2014b), our basic model is of a precessing, inclined keplerian ellipse, specified by

$$r(\lambda, t) = \frac{a(1 - e^2)}{1 + e \cos f}, \qquad (1)$$

where the true anomaly $f = \lambda - \varpi = \lambda - \varpi_0 - \dot{\varpi}(t - t_0)$. Here, $r$, $\lambda$, and $t$ are the radius, inertial longitude, and time of the observation at the ring (not at the observer), $a$ and $e$ are the ring's semimajor axis and eccentricity, $\varpi$ and $\dot{\varpi}$ are its longitude of periapse and apsidal precession rate, and $t_0$ is the epoch of the fit. For inclined rings, we include three additional parameters: $i$ (the inclination relative to the mean ring plane), $\Omega_0$ (the longitude of the ascending node) and $\dot{\Omega}$ (the nodal regression rate), and compute the intercept point of the occultation ray with the specified inclined ring plane. The zero-point for the inertial longitudes $\lambda, \varpi_0$, and $\Omega_0$ (as well as $\delta_m$ below) is the ascending node of Uranus's equator on Earth's equator of J2000, where the orientation of the Uranus pole is in the direction of positive angular momentum (i.e., 180° from the IAU definition of the Uranus north pole). The apsidal and nodal rates $\dot{\varpi}$ and $\dot{\Omega}$ can be treated as free parameters in the orbit fit, or alternatively, as in Nicholson et al. (2014b), the expected secular rates $\dot{\varpi}_{sec}$ and $\dot{\Omega}_{sec}$ can be calculated from the combined effects of Uranus's zonal gravity harmonics $J_2, J_4, J_6, ...$ based on the results of Borderies-Rappaport and Longaretti (1994), and the secular precession induced by the planet's satellites according to the expressions:

$$\dot{\varpi}_{\text{sec}} = \sqrt{\frac{GM}{a^3}} \left\{ \frac{3}{2} J_2 \left(\frac{R}{a}\right)^2 (1 + e^2 - 2 \sin^2 i) - \frac{15}{4} J_4 \left(\frac{R}{a}\right)^4 \right. \\ \left. + \left[ \frac{27}{64} J_2^3 - \frac{45}{32} J_2 J_4 + \frac{105}{16} J_6 \right] \left(\frac{R}{a}\right)^6 + \frac{1}{4} \sum_{j=1}^{M_o} \frac{m_j}{M} \alpha_{oj}^2 b_{3/2}^1(\alpha_{oj}) + \frac{1}{4} \sum_{j=1}^{M_i} \frac{m_j}{M} \alpha_{ij}^2 b_{3/2}^1(\alpha_{ij}) \right\} \quad (2)$$

$$\dot{\Omega}_{\text{sec}} = -\sqrt{\frac{GM}{a^3}} \left\{ \frac{3}{2} J_2 \left(\frac{R}{a}\right)^2 (1 + e^2 - \frac{1}{2} \sin^2 i) - \left[\frac{9}{4} J_2^2 + \frac{15}{4} J_4\right] \left(\frac{R}{a}\right)^4 \right. \\ \left. + \left[\frac{351}{64} J_2^3 + \frac{315}{32} J_2 J_4 + \frac{105}{16} J_6\right] \left(\frac{R}{a}\right)^6 + \frac{1}{4} \sum_{j=1}^{M_o} \frac{m_j}{M} \alpha_{oj}^2 b_{3/2}^1(\alpha_{oj}) + \frac{1}{4} \sum_{j=1}^{M_i} \frac{m_j}{M} \alpha_{ij}^2 b_{3/2}^1(\alpha_{ij}) \right\}, \quad (3)$$
where the summation is carried out over outer (subscript *o*) and inner (subscript *i*) satellites of mass $m_j$ and orbital radius $a_{oj}$ or $a_{ij}$, where for outer satellites $\alpha_{oj} = a/a_{oj}$ and for inner satellites $\alpha_{ij} = a_{ij}/a$, and $b_{3/2}^1(\alpha_{oj})$ and $b_{3/2}^1(\alpha_{ij})$ are Laplace coefficients as defined by Brouwer and Clemence (1961). (Note that the Laplace coefficient factor $\alpha_{oj}$ is squared, whereas $\alpha_{ij}$ is to the first power only.) **Figure 4** shows radial dependence of satellite-induced secular precession in the vicinity of the ten narrow Uranian rings, with the individual contributions shown for each satellite. As we will see, the uncertainty in the fitted apse rates is as small as $4.2 \times 10^{-6} \circ d^{-1}$ for the $\epsilon$ ring, comparable to the individual secular contributions of Oberon, Portia, and Puck to its precession rate.

The orbital semimajor axes *a* and masses used to compute the secular precession rate due to the major satellites are given in **Table 2**, from Jacobson (2023).<fnref n="3" />

<table><caption>**Table 2.** Major satellite orbits and masses from Jacobson (2023)</caption><thead><tr><th>Satellite</th><th>a (km)</th><th>GM<sub>sat</sub> (km<sup>3</sup> s<sup>-2</sup>)</th><th>M<sub>sat</sub>/M<sub>Ur</sub> × 10<sup>-6</sup></th><th>M<sub>sat</sub> (×10<sup>20</sup> kg)</th></tr></thead><tbody><tr><td>Ariel</td><td>190928.</td><td>82.30 ± 1.20</td><td>14.204 ± 0.207</td><td>12.331 ± 0.180</td></tr><tr><td>Umbriel</td><td>265981.</td><td>86.00 ± 1.50</td><td>14.843 ± 0.259</td><td>12.885 ± 0.225</td></tr><tr><td>Titania</td><td>436283.</td><td>230.60 ± 3.40</td><td>39.800 ± 0.587</td><td>34.550 ± 0.509</td></tr><tr><td>Oberon</td><td>583447.</td><td>207.60 ± 5.00</td><td>35.830 ± 0.863</td><td>31.104 ± 0.749</td></tr><tr><td>Miranda</td><td>129828.</td><td>4.20 ± 0.20</td><td>0.7249 ± 0.0345</td><td>0.6293 ± 0.0300</td></tr><tr><td>Puck</td><td>86004.</td><td>0.1275 ± 0.0425</td><td>0.0220 ± 0.0073</td><td>0.0191 ± 0.0064</td></tr></tbody></table>

The corresponding results for the minor satellites are given in **Table 3**, with semimajor axes *a* from Jacobson (1998) and satellite properties from Karkoschka (2001a,b). Each satellite is modeled as a prolate spheroid with semimajor and semiminor axes *A* and *B*, respectively, and volume *V* given by

$$V = \frac{4\pi}{3}AB^2 \quad (4)$$

using the dimensions and uncertainties in Table V of Karkoschka (2001b). We estimate the fractional mass uncertainty for an assumed satellite density from the fractional uncertainty in its volume $\sigma(V)/V$, computed from the propagated uncertainties $\sigma(\sqrt{AB})$ and $\sigma(B/A)$.

The corresponding mass estimates and uncertainties are computed for an assumed mean density of uncompressed solid water ice $\rho = 0.90 \text{ gm cm}^{-3}$, which can be scaled for alternate assumed densities. Also included are the directly measured masses and inferred densities of Cordelia, Ophelia, and Cressida determined from the amplitudes and inferred locations of forced resonances between these moons and ring edges (see Section 9.2 below).

In addition to the keplerian orbit, the orbit model allows for a combination of free or forced modes of radial distortion $\Delta r(m, \lambda, t)$ given by

$$\Delta r(m, \lambda, t) = -A_m \cos(m\theta), \quad (5)$$

where

$$\theta = \lambda - \Omega_P(t - t_0) - \delta_m. \quad (6)$$

Here, *m* is the number of radial minima and maxima in the pattern, $A_m$ and $\delta_m$ are the mode's radial amplitude and phase (the longitude at epoch of one of the *m* radial minima), respectively, and the pattern speed $\Omega_P$ is its angular rotation rate in inertial space. As explained in more detail by Nicholson et al. (2014b), $\Omega_P$ is expected to be close to that of a Lindblad resonance located at a ring particle's orbit, in which case

$$\Omega_P \approx [(m-1)n + \dot{\omega}_{\text{sec}}]/m, \quad (7)$$

where the mean motion *n* and apsidal precession rate $\dot{\omega}_{\text{sec}}$ are evaluated at the semimajor axis of the ring feature in question. A positive value of *m* corresponds to an inner Lindblad resonance (ILR)-type normal mode, expected at the

<footnote n="3">³ Since satellite masses are variously quoted in the literature in units of $GM_{\text{sat}}$, fractional planet mass $M_{\text{sat}}/M_{\text{Ur}}$, and kg, we list all three for convenient comparison.</footnote>
<table><caption>**Table 3.** Minor satellite orbits, dimensions, densities, and masses<sup>a</sup></caption><thead><tr><th>Satellite</th><th>a<br>km</th><th>√AB<br>km</th><th>B/A</th><th>A × B<br>km×km</th><th>V<br>×10<sup>5</sup> km<sup>3</sup></th><th>σ(V)<br>gm cm<sup>-3</sup></th><th>ρ<br>×10<sup>-3</sup> km<sup>3</sup> s<sup>-2</sup></th><th>GM<sub>sat</sub><br>×10<sup>-10</sup> kg</th><th>M<sub>sat</sub>/M<sub>Ur</sub><br>×10<sup>-16</sup> kg</th></tr></thead><tbody><tr><td>Cordelia</td><td>49752.000</td><td>21 ± 3</td><td>0.7 ± 0.2</td><td>25×18</td><td>0.339</td><td>0.349</td><td>[ 0.90]</td><td>2.04 ± 0.71</td><td>3.52 ± 1.23</td><td>3.05 ± 1.06</td></tr><tr><td>Cordelia<sup>b</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1.79<sup>+0.97</sup><sub>-0.49</sub></td><td>4.06 ± 0.38</td><td>7.00 ± 0.66</td><td>6.08 ± 0.57</td></tr><tr><td>Ophelia</td><td>53764.000</td><td>23 ± 4</td><td>0.7 ± 0.3</td><td>27×19</td><td>0.408</td><td>0.504</td><td>[ 0.90]</td><td>2.45 ± 1.24</td><td>4.23 ± 2.13</td><td>3.67 ± 1.85</td></tr><tr><td>Ophelia<sup>b</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.87<sup>+0.89</sup><sub>-0.30</sub></td><td>2.38 ± 0.22</td><td>4.11 ± 0.37</td><td>3.57 ± 0.32</td></tr><tr><td>Bianca</td><td>59165.000</td><td>27 ± 2</td><td>0.7 ± 0.2</td><td>32×23</td><td>0.709</td><td>0.299</td><td>[ 0.90]</td><td>4.26 ± 1.27</td><td>7.35 ± 2.20</td><td>6.38 ± 1.91</td></tr><tr><td>Cressida</td><td>61767.000</td><td>41 ± 2</td><td>0.8 ± 0.3</td><td>46×37</td><td>2.638</td><td>0.380</td><td>[ 0.90]</td><td>15.85 ± 6.02</td><td>27.35 ± 10.40</td><td>23.74 ± 9.03</td></tr><tr><td>Cressida<sup>b</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.70<sup>+0.44</sup><sub>-0.21</sub></td><td>12.27 ± 1.41</td><td>21.18 ± 2.44</td><td>18.39 ± 2.12</td></tr><tr><td>Desdemona</td><td>62659.000</td><td>35 ± 4</td><td>0.6 ± 0.2</td><td>45×27</td><td>1.374</td><td>0.375</td><td>[ 0.90]</td><td>8.25 ± 3.09</td><td>14.25 ± 5.34</td><td>12.37 ± 4.63</td></tr><tr><td>Juliet</td><td>64358.000</td><td>53 ± 4</td><td>0.5 ± 0.1</td><td>75×37</td><td>4.301</td><td>0.230</td><td>[ 0.90]</td><td>25.83 ± 5.95</td><td>44.59 ± 10.26</td><td>38.71 ± 8.91</td></tr><tr><td>Portia</td><td>66097.000</td><td>70 ± 4</td><td>0.8 ± 0.1</td><td>78×63</td><td>12.968</td><td>0.148</td><td>[ 0.90]</td><td>77.90 ± 11.55</td><td>134.44 ± 19.93</td><td>116.71 ± 17.30</td></tr><tr><td>Rosalind</td><td>69927.000</td><td>36 ± 6</td><td>1.0 ± 0.2</td><td>36×36</td><td>1.954</td><td>0.314</td><td>[ 0.90]</td><td>11.74 ± 3.69</td><td>20.26 ± 6.36</td><td>17.59 ± 5.52</td></tr><tr><td>Belinda</td><td>75255.000</td><td>45 ± 8</td><td>0.5 ± 0.1</td><td>64×32</td><td>2.745</td><td>0.327</td><td>[ 0.90]</td><td>16.49 ± 5.39</td><td>28.46 ± 9.30</td><td>24.71 ± 8.07</td></tr><tr><td>Puck</td><td>86004.000</td><td>81 ± 2</td><td>1.0 ± 0.1</td><td>81×81</td><td>22.261</td><td>0.078</td><td>[ 0.90]</td><td>133.72 ± 10.47</td><td>230.79 ± 18.06</td><td>200.35 ± 15.68</td></tr></tbody></table>

outer edge of a ringlet, while a negative value of $m$ corresponds to an outer Lindblad resonance (OLR)-type normal mode, expected at a ringlet's inner edge. In either case, $\Omega_P$ is positive. For orbits about an oblate planet, $n$ is given by Eq. (8) of Nicholson et al. (2014b). For each free mode, the additional fit parameters are $A_m$, $\Omega_P$ and $\delta_m$.

Equation (5) can also be used to describe radial perturbations forced by a Lindblad resonance with an external satellite, in which case the pattern speed, $m$-value, and phase are all determined by the satellite's orbital parameters; for a first-order Lindblad resonance, $\Omega_P$ is equal to the satellite's mean motion. The exact resonance location $a_{res}$ is specified implicitly by Eq. 7. Porco and Goldreich (1987) were the first to compute such satellite resonance locations in the vicinity of the known Uranian rings.

Our final expression for the modeled shape of each ringlet in its orbital plane is

$$r_{\text{mod}}(\lambda, t) = r(\lambda, t) + \sum_{i=1}^{M} \Delta r(m_i, \lambda, t), \qquad (8)$$

where $M$ is the number of modeled free or forced normal modes.

In addition to the kinematical orbit parameters for each ring, the least-squares fit incorporates several fitable physical and geometrical parameters, including the direction of the (possibly precessing) planetary pole, offsets to the catalog positions or proper motions of Earth-based occultation stars (or, alternatively, skyplane offsets to Uranus ephemeris), and time offsets for selected telescope observations. RINGFIT also allows for a variety of weighting schemes, including iteratively computing the weight of each ring's observations from its RMS residuals, or alternatively using the RMS residuals of each separate set of telescope observations to determine the weight of each data point.

<footnote><sup>a</sup> Except as noted, masses and uncertainties computed from satellite shapes from Table V of Karkoschka (2001b) and assumed density $\rho=0.90$ gm cm$^{-3}$.</footnote>

<footnote><sup>b</sup> Inferred mass from the amplitude(s) of normal mode(s) forced on ring edge(s) and corresponding density computed from the tabulated satellite volume $V$. Density uncertainty computed from the combined volume and mass uncertainties. See Section 9.2.</footnote>
<figure><img src="image_6.png" /></figure>

**Figure 4.** The magnitude of the secular apsidal and nodal precession rates due to Uranus's satellites, plotted as a function of radius in the vicinity of the ten narrow rings. The black line labeled 'total' includes all the satellites except Cordelia and Ophelia, whose individual contributions are plotted as the upper and lower solid red lines, respectively, bracketed by the uncertainties shown by dotted red lines. The blue lines labeled 'total (AUTOMP)' and '$\sigma$(AUTOMP)' are the sum of the satellite-induced precession rates from the six major satellites (Ariel, Umbriel, Titania, Oberon, Miranda, and Puck), and the quadratic sum of their uncertainties, calculated from the uncertainties in their masses (Jacobson 2014). The green lines show the radial dependence of the contributions to the precession rates of the rings by the gravitational coefficients $J_2$ and $J_4$, scaled to be within the vertical range of the plot, and by a representative value of $J_6 = 0.5 \times 10^{-6}$. The filled circles show the inferred anomalous precession rates for the $\alpha, \beta$, and $\gamma$ rings, with error bars indicating their $1-\sigma$ uncertainties (see Section 8). The filled square shows the inferred anomalous precession rate of the $\gamma$ ring for an alternate case in which the $m=3$ mode is omitted from the orbit model. The open circles at the radii of the eccentric rings 6, 5, 4, $\alpha, \beta$, and $\epsilon$ ring mark the formal uncertainties in their fitted apsidal precession rates. The dashed brown lines centered on the eccentric rings show the estimated magnitude of their contributions to predicted precession rates at nearby radii, for assumed masses of the rings. In all cases, these are quite small, falling below that due to Cressida.

### 3.3. Ephemerides

For this analysis, we used the recent JPL *ura178.bsp* Uranus satellite ephemerides, the *peph.ura178.bsp* planetary ephemerides, and the associated *Voyager 2* ephemeris *vgr2.ura178.bsp* at Uranus. These ephemerides were updated from the *ura111.bsp*, *ura116.bsp*, *vgr2.ura111.bsp*, and *de440.bsp* series (Jacobson 2014; Park et al. 2021), using constraints provided by the Uranus ring occultation data in Paper 1, excluding the $\gamma$ and $\lambda$ rings (Jacobson 2023). For the U0 observations, we used the SPICE kernel described in Paper 1 for the flight path of the Kuiper Airborne Observatory (KAO) during the occultation. The U137 and U138 Uranus ring occultations were observed
using the HST, for which we require accurate spacecraft ephemerides to compute the occultation geometry. There are two sources of HST ephemerides: JPL's NAIF website<fnref n="4" /> hosts a single kernel file (hst.bsp) that is updated regularly and contains NORAD-provided two-line elements (TLE) that have an estimated accuracy of a few km, and the Space Telescope Science Institute (STScI) hosts individual files provided by the Flight Dynamics Facility at the Goddard Space Flight Center (GSFC). The ephemeris accuracy depends primarily on the level of solar activity, which affects drag, and is estimated to be better than 300 m, under most ranges of solar activity. For our orbit fits, we used the more accurate STScI files, which we converted from their original *.orx format to FITS files, using the IRAF task hstephem, and then to standard NAIF SPK format using the NAIF tool mkspk. The two custom-produced SPK files are pg3f0000r.bsp for U137 and pg490000r.bsp for U138.

The complete list of kernel files used in our analysis is given in Table 4.

<table><caption>**Table 4.** Spice kernels</caption><thead><tr><th>File name<sup>a</sup></th><th>Description</th></tr></thead><tbody><tr><td>pleph.ura178.bsp</td><td>Planetary ephemerides, constrained by ring data in Paper 1</td></tr><tr><td>ura178.bsp</td><td>Major Uranus satellite ephemerides, constrained by ring data in Paper 1</td></tr><tr><td>vgr2.ura178.bsp</td><td>Voyager 2 trajectory at Uranus, constrained by ring data in Paper 1</td></tr><tr><td>ura115.bsp</td><td>Minor Uranus satellite ephemerides</td></tr><tr><td>earthstns_ittf93_040916.bsp</td><td>Geocentric coordinates of DSN groundstations</td></tr><tr><td>earth_720101_070426.bpc</td><td>Earth rotation model</td></tr><tr><td>ObsCodes_Uranus_20220212.spk</td><td>Custom geocentric observer coordinates</td></tr><tr><td>pg3f0000r.bsp</td><td>HST ephemeris for U137</td></tr><tr><td>pg490000r.bsp</td><td>HST ephemeris for U138</td></tr><tr><td>urkao_v1.bsp</td><td>Custom ephemeris for KAO (U0)</td></tr><tr><td>naif0012.tls</td><td>leap seconds</td></tr></tbody></table>

## 4. RING ORBIT FITS

The most recent comprehensive model for the orbital elements of the ten narrow Uranian rings was published by French et al. (1988), with updates provided in the review articles by French et al. (1991) and Nicholson et al. (2018a). Subsequently, more recent occultation data were incorporated into orbit models used to determine the geometry of the possible detection of the λ ring during the 1992 Jul 11 occultation of U103 (French et al. 1996), but no details were provided about the updated orbital elements. Jacobson (2014) utilized a subset of the both published and unpublished ring occultation data from 1977–1992 in an updated model for the keplerian orbits of eight of the ten narrow rings (excluding γ and δ, each of which was known to have significant normal modes), the gravity field of the planet, and the orientation of the pole. Finally, the tabulated residuals from an unpublished orbit fit were used by Chancia et al. (2017) to estimate the mass of the moon Cressida from its perturbation on the η ring.

Here, we extend these analyses by utilizing the full set of observations from 1977–2006 documented in Paper 1 to determine the orbital shapes of both the ring midlines and their edges. As a first step, we updated our search for normal modes, as described below.

### 4.1. Normal Modes

<footnote>a All kernels listed are available from ftp://ssd.jpl.nasa.gov/pub/eph or as part of the Uranus ring occultation support bundle on PDS. See Appendix A of French et al. (2023b).</footnote>

<footnote n="4">⁴ https://naif.jpl.nasa.gov/pub/naif/HST/kernels/spk/</footnote>
The first evidence for normal modes in narrow ringlets was provided by French et al. (1986), who identified an *m* = 2 ILR in the shape of the *δ* ring and an *m* = 0 OLR in the shape of the *γ* ring.<fnref n="5" /> Subsequently, tentative detections of edge waves in the *ε* ring associated with resonances with Cordelia and Ophelia were reported by French and Nicholson (1995), based on a subset of the observations presented here, and Chancia et al. (2017) identified the signature of an *m* = 3 mode in the *η* ring, forced by Cressida. From the wealth of Cassini occultation observations, a host of free and forced normal modes have also been identified in Saturn's ringlets and sharp ring edges, sometimes in astonishing numbers: the informally-designated “Strange” ringlet in the Cassini Division is both eccentric and inclined and has seven normal modes, and the inner edge of the Barnard Gap has 11 (French et al. 2016b)! (For a recent review of observations of narrow rings, gaps, and sharp edges at Saturn, Uranus, and Neptune, see Nicholson et al. (2018b) and references therein; for a detailed modern exposition of the theory of narrow rings and sharp edges, see Longaretti (2018).)

In our search for normal modes in the Uranian rings, we focus on the measured locations of the center of each ring (COR), although as noted above, dynamical arguments suggest that ILRs should be associated with outer edges of rings (OER) and OLRs with inner edges (IER). For ringlets only a few km wide, these distinctions may be unwarranted, and in any event the measurement of the COR from a square-well fit is necessarily affected by the locations of the IER and OER, each of which might be distorted by a local normal mode. Additional evidence for normal modes in the Uranian rings is provided by periodic azimuthal brightness variations in the rings in *Voyager* (Hedman et al. 2023) and *HST* images (Showalter 2011).

Our general procedure for identifying free and forced normal modes is as follows: For each ring feature (COR, IER, or OER), we begin with the best-fitting keplerian ellipse model and then scan the residuals over a range of pattern speeds Ω<sub>P</sub> and candidate wavenumbers from *m* = 1 to *m* = 30 for ILR-type perturbations, and from *m* = 0 to *m* = -30 for OLR-type perturbations, solving for the best-fitting amplitude *A*<sub>*m*</sub> and phase *δ*<sub>*m*</sub> at each pattern speed. The range of values scanned for Ω<sub>P</sub> for each mode is centered on the predicted value for the semimajor axis of the ring feature, based on Eq. 7, and is sufficiently broad to provide a sampling of the statistical significance of a putative detection. (In Section 4.5, we make use of similar information to set detection limits on the eccentricities and inclinations of some of the rings.) We then add the statistically significant detected modes to the kinematical model of the ring feature, fit for their amplitudes, pattern speeds and phases, and form a new set of residuals. We repeat the frequency scanning process to search for additional weaker modes. With the addition of successive normal modes, the RMS residual is reduced and the sensitivity to even weaker modes is increased.

We now describe the normal modes detected in this systematic search (none were identified at a statistically significant level for rings 6, 5, 4, α, β, or λ).

4.1.1. The *η* ring

The very narrow core of the *η* ring has no measurable eccentricity or inclination, but a normal mode scan reveals the presence of a forced *m* = 3 mode associated with Cressida, as first reported by Chancia et al. (2017). **Figure 5** shows the *η* ring COR *m* = 3 normal mode scan based on our complete set of observations. The pattern speed Ω*P* for the best fit is displaced from the expected value for a free normal mode, and instead matches the mean motion of Cressida, as expected for a first-order Lindblad resonance. (We present quantitative results for this and other mode searches below, once all identified modes have been incorporated into our final orbit models for ring midlines and edges.) No other *η* ring normal modes were detected at a statistically significant level.

4.1.2. The *γ* ring

<footnote n="5">⁵ French et al. (1986) interpreted the δ and γ ring modes as being forced by unseen satellites because the pattern speeds differed measurably from the expected values for free normal modes located at the semimajor axes of the rings, but in retrospect this was based on an a value of GM for Uranus that subsequently proved to be in error by five times its estimated formal error. With an improved determination of GM, the resonance radii of the modes closely matched the semimajor axes of the rings, as expected for free normal modes unassociated with satellites.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Normal mode scan of the η ring COR for *m* = 3. Upper panel: RMS residual as a function of Ω*P*, centered on the predicted value for an *m* = 3 ILR, marked by a thick vertical gray line. The best-fitting pattern speed Ω*P* (dashed vertical line) is faster than the predicted value, and matches the mean motion of Cressida. Lower panel: amplitude A*3* of the fitted *m* = 3 mode for each pattern speed. Our final orbit fit gives A*3* = 0.600 ± 0.069 km, very similar to the result from the normal mode scan.</figcaption></figure>

Unlike the δ and η rings, the γ ring is measurably eccentric, as demonstrated by the *m* = 1 COR normal mode scan of the residuals to a circular orbit model in **Fig. 6**. Here, no other modes are included, and the minimum RMS residual is quite large: 4.0 km. The best-fitting amplitude A₁ = 4.94 km from the scan, and the corresponding pattern speed marked by a vertical dashed line is very slightly faster than the predicted value for the ring's semimajor axis. We will return to this disagreement in Section 7.2, where we determine Uranus's gravity field from the observed apsidal and nodal precession rates of the rings.

The *m* = 0 normal mode amplitude is roughly equal to the *m* = 1 mode amplitude, as seen in the γ ring COR *m* = 0 normal mode scan (**Fig. 7**). Here, the best-fitting *m* = 1 mode from the final orbit fit has been included prior to the scan. (The upper limit to the RMS residual in the upper panel in this case is 4.2 km, a bit larger than in Fig. 6 owing to the slightly different best-fitting keplerian ellipses in the two cases.)

We extended our search for γ ring normal modes over the range *m* = -30 to 30, and we successively added each statistically significant mode to our RINGFIT model and repeated the search for additional modes until none were found. To illustrate the signatures of the additional detected γ ring normal modes, the best fitting final model for all other detected modes was subtracted from the measured radii prior to performing the normal mode scans shown in each figure below. We present these in decreasing order of fitted mode amplitude *A*<sub>*m*</sub>. Our search yielded the detection of the γ ring *m* = -1 OLR. Note that Eq. (7) implies that Ω<sub>*P*</sub> ~ 2*n* ~ 2290 ° d<sup>-1</sup>, much greater than the *m* = 1 apse rate ω̇ ≃ 1.751 ° d<sup>-1</sup>. The corresponding normal mode scan is shown in **Fig. 8**. The best-fitting pattern speed is slightly faster than that predicted for a free *m* = -1 mode at the COR, and the corresponding resonance radius is therefore somewhat interior to the midline of the ring. This is consistent with the anticipated association of OLRs with inner ring edges. The fitted amplitude A<sub>-1</sub> = 1.822 ± 0.067 km from our final orbit solution. In this and subsequent cases, the normal mode scans give slightly different values for the fitted amplitudes compared to our final orbit fit because the normal mode scans allow for an additional fitted parameter of a radial offset from the ring
<figure><img src="image_8.png" /></figure>

**Figure 6.** Normal mode scan of the residuals to a circular model for the $\gamma$ ring COR for $m = 1$. Upper panel: RMS residual as a function of $\Omega_P$, centered on the predicted value for an $m = 1$ ILR, marked by a thick vertical gray line. Lower panel: amplitude $A_1$ of the fitted $m = 1$ mode for each pattern speed. The best-fitting model based on this normal mode scan has $A_1 = 4.78$ km, compared to the final value $A_1 = 5.509 \pm 0.076$ km when all detected normal modes are included.

midline or edge, while in the final orbit fit all modes are constrained to be centered on the corresponding midline or edge.

We identified a second OLR in the $\gamma$ ring's midline: the $m = -2$ normal mode shown in **Fig. 9**. Once again, the best-fitting $\Omega_P$ is slightly faster than the predicted value for the COR, indicating that the resonance radius is interior to the COR as expected for an OLR. This mode is somewhat weaker than the $m = -1$ mode, with a fitted amplitude $A_{-2} = 0.690 \pm 0.076$ km from our final orbit solution.

We also detected the $m = 6$ ILR forced by Ophelia. **Figure 10** shows the $\gamma$ ring COR $m = 6$ normal mode scan, with a clear offset in the best-fitting $\Omega_P$ compared to the predicted value for the ring's semimajor axis, indicating that the resonance location is interior to the centerline of the ring. The best-fitting model has $A_6 = 0.637 \pm 0.063$ km from our final orbit solution. Similar scans (not shown) confirm that the $m = 6$ mode is also present on both ring edges.

Finally, there is evidence for a free $m = 3$ ILR with a resonance location near the outer edge of the $\gamma$ ring. The normal mode scan is shown in **Fig. 11**. The best-fitting pattern speed corresponds to a resonance location exterior to the outer edge of the ringlet. There is no known satellite with the appropriate orbit to force this mode, and to assess the statistical significance of the detection, we performed a scan over a very wide range of pattern speeds. The two left panels of the figure show that the best-fitting normal mode lies very close to the predicted pattern speed and that there are no comparably strong aliases over the large range of pattern speeds considered. The upper right panel shows a histogram of the RMS residuals for all of the individual pattern speeds in the normal mode scans at left, plotted as the number of results $N$ per bin of width 0.0005 km, with an arrow marking the value for the candidate $m = 3$ mode near the ring center. The lower right panel shows a similar histogram of the best-fitting amplitude for each pattern speed, showing the number of results $N$ per bin of width 0.0018 km. Again, the arrow marks the candidate mode, which is well above the noise threshold.
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Normal mode scan of the $\gamma$ ring COR for $m = 0$, after inclusion of best-fitting $m = 1$ mode from adopted orbit solution. Upper panel: RMS residual as a function of $\Omega_P$, centered on the predicted value for an $m = 0$ OLR, marked by a thick vertical gray line. Lower panel: amplitude $A_0$ of the fitted $m = 0$ mode for each pattern speed. The best-fitting model has $A_0 = 5.509 \pm 0.076$ km from our final orbit solution.</figcaption></figure>

The amplitude of the $m = 3$ mode is even stronger on the outer edge of the $\gamma$ ring, as shown from the normal mode scan shown in **Fig. 12**, where we again have scanned over a wide range of pattern speeds. In this case (as for other ring edges), the RMS residuals are larger than for the COR normal mode scans because the fitted ring widths from the square-well model are less accurate than the ring midlines, as discussed above. On the other hand, we found no evidence for an $m = 3$ mode near the $\gamma$ ring's inner edge – the corresponding normal mode scan is shown in **Fig. 13**. As we discuss below, including the $m = 3$ mode not only reduces the RMS residuals of the COR fit but also reduces the inferred anomalous precession rate of the $\gamma$ ring.

### 4.1.3. The $\delta$ ring

The $\delta$ ring has no detectable eccentricity or inclination, but the RMS residual of a circular orbit model is 2.35 km, nearly ten times that of the eccentric and inclined mode-free rings 6, 5, and 4, indicative of an unmodeled strong perturbation in the ring's shape. In previous orbit models, this was attributed to an $m = 2$ mode, and this signature is clearly present in the normal mode scan shown in **Fig. 14**. The upper panel shows a sharp decrease in the RMS error at the predicted pattern speed for the semimajor axis of the ring, marked by a solid vertical line. The best-fitting $\Omega_P$ is marked by a solid dashed line, which in this case coincides with the predicted value. The lower panel shows the fitted amplitude of the normal mode at each assumed pattern speed in the scan, with a best-fitting value $A_2 \simeq 3.2$ km.

After including the free parameters for the $m = 2$ mode of the $\delta$ ring, we formed a new set of residuals and repeated the normal mode scan for the same set of possible wavenumbers. The results of this search revealed a weak but statistically significant $m = 23$ mode in the COR data, shown in **Fig. 15**. The RMS residual is further reduced and the fitted amplitude $A_{23} \simeq 0.34$ km in the normal mode scan (similar to $A_{23} = 0.339 \pm 0.063$ km from our final orbit fit). The pattern speed is slower than that expected for a free $m = 23$ mode located at the semimajor axis of the COR, and the corresponding resonance radius is somewhat exterior to the ring midline. Instead, the pattern speed and phase of the observed $m = 23$ mode match the orbital characteristics of Cordelia, as originally proposed by Chancia et al.
<figure><img src="image_10.png" /></figure>

**Figure 8.** Normal mode scan of the $\gamma$ ring COR for $m = -1$, after inclusion of modes $m = -2, 0, 1, 3$, and 6 from adopted orbit solution. Upper panel: RMS residual as a function of $\Omega_P$, centered on the predicted value for an $m = -1$ OLR, marked by a thick vertical gray line. Lower panel: amplitude $A_{-1}$ of the fitted $m = -1$ mode for each pattern speed. The best-fitting model has $A_{-1} = 1.822 \pm 0.067$ km from our final orbit solution.

(2017), who calculated the expected normal mode amplitude $A_{23}$ for a range of assumed satellite masses and densities. Normal mode scans of $\delta$ ring IER and OER showed no detection of the $m = 23$ mode on either ring edge near the expected pattern speed, but the RMS noise level in these scans was substantially greater than the observed amplitude at the ring centerline, so the non-detections are not surprising. A search for the $m = 23$ mode in the width of the ring itself was also negative. No other $\delta$ ring normal modes were detected at a statistically significant level.

### 4.1.4. The $\epsilon$ ring

The inner and outer edges of the $\epsilon$ ring have long been associated with the nearby $m = -24$ OLR with Cordelia and the $m = 14$ ILR with Ophelia, respectively (Porco and Goldreich 1987). Chancia et al. (2017) showed that, for reasonable assumptions about the masses of the two satellites, the expected amplitudes of the corresponding edge waves would lie in the range $A_{-24} = 0.4-1.0$ km for the inner edge (IER) and a somewhat weaker $A_{14} = 0.25-0.7$ km at the OER. As noted above, French and Nicholson (1995) found tentative evidence for these edge waves. **Figure 16** shows the results of the $m = 14$ ILR normal mode scan of the $\epsilon$ ring OER. There is a convincing minimum in the RMS residuals at a pattern speed a bit faster than the value predicted for the ring edge, indicating that the resonance lies slightly interior to the outer edge. **Figure 17** shows the results of the $m = -24$ OLR normal mode scan of the $\epsilon$ ring IER. In this case, the best-fitting pattern speed is near Cordelia's mean motion $\Omega_P \simeq 1074.52$ $^\circ$ d$^{-1}$ and is slightly slower than that expected at the exact ring edge, indicating that the resonance is located within the ring itself, near the inner edge.

In Section 9, we confirm the association of these two normal modes with Cordelia and Ophelia on the basis of the fitted phases, pattern speeds, and amplitudes of the $\epsilon$ ring IER and OER edge waves compared to the expectations from satellite ephemerides, using the full set of observations reported here.

### 4.1.5. Signatures of the $\epsilon$ ring edge waves in the shape of the ring midline
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Normal mode scan of the $\gamma$ ring COR for $m = -2$, after inclusion of modes $m = -1, 0, 1, 3$, and 6 from adopted orbit solution. Upper panel: RMS residual as a function of $\Omega_P$, centered on the predicted value for an $m = -2$ OLR, marked by a thick vertical gray line. Lower panel: amplitude $A_{-2}$ of the fitted $m = -2$ mode for each pattern speed. The best-fitting model has $A_{-2} = 0.690 \pm 0.076$ km from our final orbit solution.</figcaption></figure>

While these normal modes are forced at the *edges* of the relatively wide $\epsilon$ ring and their dynamical influence might well be restricted to the immediate vicinity of the edges, the measured centerline of the ring should be offset by approximately one half of the sum of the local radial amplitudes of the two edge modes. We performed normal mode scans on the $\epsilon$ ring COR radius residuals for $m = 14$ and $m = -24$ and found weak but secure signatures of both modes, with the measured phases and pattern speeds being in excellent agreement with the values found at the corresponding ring edges, as discussed below in Section 9. **Figure 18** shows the results for $m = 14$ COR scan. Here, the vertical dashed line shows the best-fitting pattern speed near the value from our final orbit fit $\Omega_P = 956.418051 \pm 0.000269^\circ \text{ d}^{-1}$, close to the mean motion of Ophelia, with an amplitude $A_{14} = 0.383 \pm 0.071$ km, a bit more than half that at the OER, for which $A_{14} = 0.590 \pm 0.130$ km. **Figure 19** shows the results for the $m = -24$ COR scan. Here, the vertical dashed line is near the best-fitting $\Omega_P = 1074.522703 \pm 0.000142^\circ \text{ d}^{-1}$, very close to the mean motion of Cordelia (see Table 20), with an amplitude $A_{-24} = 0.443 \pm 0.061$ km from our final orbit fit, a bit less than half that at the IER, for which $A_{-24} = 1.011 \pm 0.107$ km.

### 4.1.6. Searches for other normal modes

We performed an orbit fit that included all of the normal modes identified above and used normal mode scans for wavenumbers between $m = -30$ and 30 to search for evidence of any additional statistically significant detections in the orbit fit residuals near the predicted pattern speeds, for all ring widths, edges and midlines. None were found. Chancia et al. (2017) identified two weak first-order resonances with predicted amplitudes $A_m \sim 0.1$ km for assumed moonlet densities $\rho = 0.9 \text{ g cm}^{-3}$: the 13:12 Cordelia resonance with the $\eta$ ring and the 2:1 Portia resonance with ring 6. Our $3-\sigma$ detection limits for these modes are 0.204 km and 0.168 km, respectively, based on Rayleigh distribution fits to the amplitudes in the corresponding normal mode scans, placing them just out of reach for secure identification in our observations.
<figure><img src="image_12.png" /></figure>

**Figure 10.** Normal mode scan of the γ ring COR for *m* = 6, forced by Ophelia, after inclusion of modes *m* = −2, −1, 0, 1, and 3 from adopted orbit solution. Upper panel: RMS residual as a function of Ω*, centered on the predicted value for an *m* = 6 ILR, marked by a thick vertical gray line. Lower panel: amplitude A* of the fitted *m* = 6 mode for each pattern speed. The best-fitting model has A* = 0.637 ± 0.063 km from our final orbit solution.

Internal planetary oscillations provide another possible source for normal modes in the rings. Saturn's C and B
rings provide a tapestry revealing dozens of such signatures in occultation profiles (Rosen et al. 1991; Colwell et al.
2009; Baillié et al. 2011; Hedman and Nicholson 2013, 2014; French et al. 2019; Hedman et al. 2019; French et al. 2021;
Hedman et al. 2022). Although the Uranus system of narrow rings provides incomplete radial coverage for possible
modes, Saturn's Maxwell ringlet – very similar in structure to the Uranus ϵ ring – is home to an *m* = 2 mode forced by
Saturn's internal oscillations, and provides an example of a narrow ringlet located precisely at the mode's resonance
radius (French et al. 2016a). As part of our normal mode survey, we searched for evidence of distortions in the shapes
of the ring midlines and edges near the predicted values for wavenumbers and pattern speeds excited by resonances
with fundamental Uranus normal modes (A'Hearn et al. 2022). We found no statistically significant matches with
radial amplitudes ΔA_m > 0.2 km, but the possibility remains that internal oscillations can reveal their presence in
azimuthal variations in ring brightness seen in images, an important task for the future.

4.2. *Adopted ring orbital elements and normal modes*

Having characterized a set of detected normal modes, we then solved for the ring orbital elements and Uranus system geometry iteratively and in several stages, taking into account the greater accuracy of the ring midline measurements than those of the ring edges. We first used the COR (ring midline) observations to fit for the planet's pole direction, time offsets for selected stations, and corrections to selected occultation star positions and proper motions, in addition to the orbital elements and normal modes of the nine principal rings. Next, we fitted separately for the IER/OER orbital elements and normal modes only, using the fitted widths and mid-times and assuming the system geometry, star positions, and station offset times established by the COR fit. Finally, we solved for the best circular orbit model for the λ ring and the skyplane offset positions of the secondary stars in the multiple-star U36 occultation and the binary star U102 occultation.
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Normal mode scan of the $\gamma$ ring COR for $m = 3$, performed over a wide range of pattern speeds to illustrate the statistical significance of the detection near the expected pattern speed. See text for details.</figcaption></figure>

The combined results of the separate unweighted COR and IER/OER orbit fits for the ten narrow rings are given in Tables 5 and 6. For these fits, the apsidal precession and nodal regression rates $\dot{\omega}$ and $\dot{\Omega}$ were included as separate free parameters for the rings with measurable eccentricity and inclination, respectively, rather than being constrained by the gravitational field of Uranus and the secular precession due to the satellites. We will return to these contributions below, when we evaluate the evidence for anomalous forced precession of individual rings and set constraints on $J_2, J_4$, and $J_6$ from the COR observations. Similarly, all normal mode pattern speeds and longitudes at epoch were included as separate free parameters, even for modes identified as being forced by Cressida, Cordelia, and Ophelia. Table 5 lists the fitted keplerian orbital elements to N ring event times for the IER/COR/OER of each ring, in increasing order of semimajor axis. (For the $\gamma$ ring COR, we include the results of the nominal fit that includes the $m = 3$ mode as well as the alternate fit in which the $m = 3$ mode is omitted.) All errors listed are $1-\sigma$ formal errors from the separate unweighted least-squares fits for the COR observations and for the IER/OER data (which include the fitted ring widths from the square-well models discussed in Appendix A). Eccentricities $ae$ and inclinations $a \sin i$ in square brackets were held fixed during orbit determination. The epoch of the fit $t_0$ is TDB 1986 Jan 19 12:00, near the time of the *Voyager 2* encounter with Uranus, chosen to be near the mid-point of the time interval of the entire set of observations so as to minimize the correlations between the fitted apse and node rates and the longitudes at epoch.<fnref n="6" />

The entries $\Delta\dot{\omega}$ and $\Delta\dot{\Omega}$ are the differences between the observed apse and node rates and the predicted values computed for the fitted geometric semimajor axis $a$ of the given feature. The predicted rates take into account the estimated secular apse or node precession due to both major and minor satellites.<fnref n="7" /> The entries $\Delta a_{\dot{\omega}}$ and $\Delta a_{\dot{\Omega}}$ are

<footnote n="6">⁶ This is the same epoch used by Jacobson (2014, 2023).</footnote>

<footnote n="7">⁷ The predicted rates tabulated here are based solely on the planet's gravitational field and the satellite-induced precession. As discussed in Section 7.1, in solving for the gravitational field, we take into account the estimated difference between the semimajor axes of the geometric centers of the rings and their estimated radially averaged centers of mass. For a normal mode forced by a satellite, the predicted mode pattern speed corresponds to the satellite's mean motion. We discuss these forced modes in Section 9.</footnote>
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Normal mode scan of the γ ring OER for *m* = 3, performed over a wide range of pattern speeds to illustrate the statistical significance of the detection near the expected pattern speed for the rings's outer edge.</figcaption></figure>

the corresponding differences for the fitted apse and node rates $\dot{\varpi}$ and $\dot{\Omega}$. In practice, we compute these iteratively to high precision, but they can be estimated to reasonable accuracy for $\Delta a_{\dot{\varpi}}$ and $\Delta a_{\dot{\Omega}}$ of order a few km from the linear approximations:

$$ \Delta a_{\dot{\varpi}} \approx \frac{\dot{\varpi}(\text{observed}) - \dot{\varpi}(\text{predicted})}{d\dot{\varpi}/da} = \frac{\Delta \dot{\varpi}}{d\dot{\varpi}/da} \qquad (9) $$

and

$$ \Delta a_{\dot{\Omega}} \approx \frac{\dot{\Omega}(\text{observed}) - \dot{\Omega}(\text{predicted})}{d\dot{\Omega}/da} = \frac{\Delta \dot{\Omega}}{d\dot{\Omega}/da}. \qquad (10) $$

Since $d\dot{\varpi}/da \propto -J_2a^{-9/2} < 0$ and $d\dot{\Omega}/da \propto +J_2a^{-9/2} > 0$, $\Delta\dot{\varpi}$ and $\Delta\dot{\Omega}$ have opposite signs, while $\Delta\dot{\Omega}$ and $\Delta a_{\dot{\varpi}}$ have the same signs.

Table 6 lists the fitted amplitudes $A_m$, phases $\delta_m$, and pattern speeds $\Omega_P$ for all detected normal modes. For the γ ring, we include the results of two separate fits, one that included the suspected *m* = 3 mode and one in which that mode is omitted. The quantity $\Delta\Omega_P$ is the difference between the fitted pattern speed and the predicted value computed for the fitted geometric semimajor axis *a* (Table 5) and our adopted gravity field of the planet and satellites; $\Delta a_{\dot{\Omega}}$ is the corresponding difference between the geometric semimajor axis and the predicted resonance radius for the fitted pattern speed, with the linear approximation:

$$ \Delta a_{\Omega_P} \approx \frac{\Omega_P(\text{observed}) - \Omega_P(\text{predicted})}{d\Omega_P/da} = \frac{\Delta\Omega_P}{d\Omega_P/da}. \qquad (11) $$

From Eq. (7), $d\Omega_P/da \propto -(m-1)a^{-5/2}$, so the sign of $\Delta a_{\Omega_P}$ depends on both the sign of $(m-1)$ and the sign of $\Delta\Omega_P$.
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Normal mode scan of the $\gamma$ ring IER for $m = 3$, showing no evidence for the mode in the vicinity of the expected pattern speed for this ring edge, shown as a thick vertical gray line.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 14.** Normal mode scan of the $\delta$ ring COR for $m = 2$. Upper panel: RMS residual as a function of $\Omega_p$, centered on the predicted value for an $m = 2$ ILR, marked by a thick vertical gray line. Lower panel: amplitude $A_2$ of the fitted $m = 2$ mode for each pattern speed. The best-fitting amplitude from the normal mode scan is $A_2 = 3.2$ km; from our final orbit fit, $A_2 = 3.169 \pm 0.059$ km.</figcaption></figure>
<figure><img src="image_17.png" /></figure>

**Figure 15.** Normal mode scan of the *δ* ring COR for *m* = 23. Upper panel: RMS residual as a function of *Ω*<sub>*P*</sub>, centered on the predicted value for an *m* = 23 ILR, marked by a thick vertical gray line. The best-fitting pattern speed *Ω*<sub>*P*</sub> (dashed vertical line) is slower than the predicted value but matches the mean motion of Cordelia *Ω*<sub>*P*</sub> = 1074.52° d<sup>-1</sup>. Lower panel: amplitude *A*<sub>23</sub> of the fitted *m* = 23 mode for each pattern speed. The best-fitting normal mode scan value is *A*<sub>23</sub> = 0.34 km, close to the value from the final orbit fit of *A*<sub>23</sub> ≃ 0.339 ± 0.063.

We will make use of these relations in Section 6, when we examine the width, shape, and differential precession of
the rings.
<figure><img src="image_18.png" /></figure>

**Figure 16.** Normal mode scan of $\epsilon$ ring OER for $m = 14$, showing a minimum in the RMS residuals with amplitude $A_{14} \simeq 0.60$ km at a pattern speed near that of Ophelia's mean motion $\Omega_P \simeq 956.42$ $^\circ$ d$^{-1}$ and slightly faster than that expected at the ring's outer edge (thick gray line).

<figure><img src="image_19.png" /></figure>

**Figure 17.** Normal mode scan of $\epsilon$ ring IER for $m = -24$, showing a minimum in the RMS residuals with amplitude $A_{-24} \simeq 1.0$ km at a pattern speed near that of Cordelia's mean motion $\Omega_P \simeq 1074.52$ $^\circ$ d$^{-1}$, and slightly slower than that expected at the ring's inner edge (thick gray line).
<figure><img src="image_20.png" /><figcaption>**Figure 18.** Normal mode scan of the ϵ ring COR for the *m* = 14 mode forced by Ophelia. The best-fitting pattern speed *Ω*<sub>P</sub> = 956.418051 ± 0.000269 ° d<sup>-1</sup>, close to the mean motion of Ophelia.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>rfrench@galileo Mon Dec 18 11:57:21 2023 normal_mode_scan_uranus_v20.pro 010-nm_v17-Ur017L-RF-V0307-all-modesoff_set1-m=-24_rgf_residuals_mid_wide_case0190_20231204a.ps

**Figure 19.** Normal mode scan of the ϵ ring COR for the *m* = −24 mode forced by Cordelia. The best-fitting *Ω*<sub>P</sub>=1074.522703±0.000142 ° d<sup>-1</sup>, close to the mean motion of Cordelia.</figcaption></figure>
<table><caption>Table 5. Uranian ring keplerian orbital elements</caption><thead><tr><th rowspan="2">Ring</th><th rowspan="2">Feature<sup>(a)</sup></th><th colspan="2">a (km)</th><th>ae (km)</th><th>ω̄<sub>0</sub> (°)<sup>(b)</sup></th><th>ω̇ (° d<sup>-1</sup>)</th><th>Δω̇ (° d<sup>-1</sup>)</th><th>Δa<sub>ω̇</sub> (km)</th></tr><tr><th>N</th><th>RMS (km)</th><th>a sin i (km)</th><th>Ω<sub>0</sub> (°)</th><th>Ω̇ (° d<sup>-1</sup>)</th><th>ΔΩ̇ (° d<sup>-1</sup>)</th><th>Δa<sub>Ω̇</sub> (km)</th></tr></thead><tbody><tr><td rowspan="6">6</td><td rowspan="2">IER</td><td>41835.920 ± 0.102</td><td></td><td>42.068 ± 0.157</td><td>181.733 ± 0.213</td><td>2.7620198 ± 0.0001046</td><td>-0.0003140</td><td>1.352 ± 0.450</td></tr><tr><td>45</td><td>0.830</td><td>44.747 ± 0.320</td><td>90.465 ± 0.498</td><td>-2.7569020 ± 0.0001882</td><td>-0.0000256</td><td>-0.110 ± 0.813</td></tr><tr><td rowspan="2">COR</td><td>41837.092 ± 0.096</td><td></td><td>42.499 ± 0.077</td><td>181.657 ± 0.112</td><td>2.7619579 ± 0.0000516</td><td>-0.0001040</td><td>0.448 ± 0.222</td></tr><tr><td>50</td><td>0.263</td><td>44.643 ± 0.104</td><td>89.727 ± 0.272</td><td>-2.7565287 ± 0.0000502</td><td>0.0000766</td><td>0.331 ± 0.217</td></tr><tr><td rowspan="2">OER</td><td>41838.237 ± 0.102</td><td></td><td>42.930 ± 0.158</td><td>181.526 ± 0.209</td><td>2.7619237 ± 0.0001028</td><td>0.0001277</td><td>-0.550 ± 0.442</td></tr><tr><td>45</td><td>0.603</td><td>44.446 ± 0.323</td><td>88.850 ± 0.507</td><td>-2.7560947 ± 0.0001898</td><td>0.0002455</td><td>1.061 ± 0.820</td></tr><tr><td rowspan="6">5</td><td rowspan="2">IER</td><td>42233.577 ± 0.089</td><td></td><td>80.052 ± 0.148</td><td>176.823 ± 0.089</td><td>2.6715990 ± 0.0000451</td><td>-0.0003241</td><td>1.456 ± 0.203</td></tr><tr><td>59</td><td>0.603</td><td>40.572 ± 0.270</td><td>297.500 ± 0.482</td><td>-2.6667586 ± 0.0001679</td><td>-0.0000154</td><td>-0.069 ± 0.757</td></tr><tr><td rowspan="2">COR</td><td>42234.893 ± 0.091</td><td></td><td>80.237 ± 0.076</td><td>176.819 ± 0.048</td><td>2.6715756 ± 0.0000227</td><td>-0.0000548</td><td>0.246 ± 0.102</td></tr><tr><td>67</td><td>0.212</td><td>40.951 ± 0.114</td><td>296.068 ± 0.259</td><td>-2.6663516 ± 0.0000445</td><td>0.0000998</td><td>0.450 ± 0.200</td></tr><tr><td rowspan="2">OER</td><td>42236.261 ± 0.089</td><td></td><td>80.463 ± 0.148</td><td>176.851 ± 0.089</td><td>2.6715450 ± 0.0000449</td><td>0.0002192</td><td>-0.985 ± 0.202</td></tr><tr><td>59</td><td>0.641</td><td>41.331 ± 0.270</td><td>294.460 ± 0.475</td><td>-2.6658983 ± 0.0001677</td><td>0.0002496</td><td>1.125 ± 0.756</td></tr><tr><td rowspan="6">4</td><td rowspan="2">IER</td><td>42569.511 ± 0.089</td><td></td><td>44.931 ± 0.146</td><td>256.398 ± 0.183</td><td>2.5978170 ± 0.0000927</td><td>-0.0006562</td><td>3.057 ± 0.431</td></tr><tr><td>57</td><td>0.744</td><td>23.183 ± 0.313</td><td>339.064 ± 1.489</td><td>-2.5940762 ± 0.0004674</td><td>-0.0005631</td><td>-2.630 ± 2.183</td></tr><tr><td rowspan="2">COR</td><td>42571.124 ± 0.091</td><td></td><td>45.347 ± 0.076</td><td>255.880 ± 0.093</td><td>2.5980293 ± 0.0000433</td><td>-0.0000977</td><td>0.455 ± 0.202</td></tr><tr><td>63</td><td>0.264</td><td>23.413 ± 0.076</td><td>336.828 ± 0.671</td><td>-2.5932795 ± 0.0001121</td><td>-0.0001117</td><td>-0.522 ± 0.524</td></tr><tr><td rowspan="2">OER</td><td>42572.743 ± 0.089</td><td></td><td>45.742 ± 0.143</td><td>255.294 ± 0.181</td><td>2.5981594 ± 0.0000908</td><td>0.0003798</td><td>-1.769 ± 0.423</td></tr><tr><td>57</td><td>0.727</td><td>24.329 ± 0.308</td><td>333.891 ± 1.361</td><td>-2.5925059 ± 0.0004191</td><td>0.0003157</td><td>1.475 ± 1.958</td></tr><tr><td rowspan="6">α</td><td rowspan="2">IER</td><td>44714.884 ± 0.083</td><td></td><td>32.512 ± 0.118</td><td>205.885 ± 0.213</td><td>2.1851508 ± 0.0000949</td><td>-0.0007266</td><td>4.229 ± 0.552</td></tr><tr><td>73</td><td>0.542</td><td>11.853 ± 0.276</td><td>206.062 ± 1.839</td><td>-2.1831510 ± 0.0005365</td><td>-0.0010535</td><td>-6.145 ± 3.130</td></tr><tr><td rowspan="2">COR</td><td>44718.473 ± 0.086</td><td></td><td>33.916 ± 0.062</td><td>206.074 ± 0.112</td><td>2.1855003 ± 0.0000510</td><td>0.0002392</td><td>-1.392 ± 0.297</td></tr><tr><td>81</td><td>0.249</td><td>12.005 ± 0.089</td><td>203.837 ± 0.894</td><td>-2.1813455 ± 0.0001311</td><td>0.0001373</td><td>0.801 ± 0.765</td></tr><tr><td rowspan="2">OER</td><td>44722.161 ± 0.084</td><td></td><td>35.335 ± 0.118</td><td>206.243 ± 0.195</td><td>2.1857991 ± 0.0000898</td><td>0.0011714</td><td>-6.819 ± 0.523</td></tr><tr><td>73</td><td>0.543</td><td>11.676 ± 0.289</td><td>202.596 ± 1.749</td><td>-2.1799630 ± 0.0005158</td><td>0.0008880</td><td>5.186 ± 3.012</td></tr><tr><td rowspan="2">β</td><td rowspan="2">IER</td><td>45656.770 ± 0.087</td><td></td><td>18.573 ± 0.128</td><td>319.589 ± 0.435</td><td>2.0309488 ± 0.0001626</td><td>-0.0004852</td><td>3.103 ± 1.040</td></tr><tr><td>71</td><td>0.574</td><td>3.799 ± 0.194</td><td>224.376 ± 8.293</td><td>-2.0272655 ± 0.0019207</td><td>0.0007997</td><td>5.128 ± 12.315</td></tr></tbody></table>

Table 5 continued on next page
<table><caption>Table 5 (continued)</caption><thead><tr><th rowspan="2">Ring</th><th rowspan="2">Feature<sup>(a)</sup></th><th colspan="2">a (km)</th><th>ae (km)</th><th>ω̄<sub>0</sub> (°)<sup>(b)</sup></th><th>ω̇ (° d<sup>-1</sup>)</th><th>Δω̇ (° d<sup>-1</sup>)</th><th>Δā (km)</th></tr><tr><th>N</th><th>RMS (km)</th><th>a sin i (km)</th><th>Ω<sub>0</sub> (°)</th><th>Ω̇ (° d<sup>-1</sup>)</th><th>ΔΩ̇ (° d<sup>-1</sup>)</th><th>Δā<sub>Ω̇</sub> (km)</th></tr></thead><tbody><tr><td rowspan="2"></td><td rowspan="2">COR</td><td>45661.056 ± 0.087</td><td>20.106 ± 0.071</td><td>317.925 ± 0.222</td><td>2.0308229 ± 0.0000857</td><td>0.0000590</td><td>-0.377 ± 0.548</td></tr><tr><td>78</td><td>0.266</td><td>4.004 ± 0.070</td><td>232.878 ± 4.169</td><td>-2.0279790 ± 0.0006048</td><td>-0.0005821</td><td>-3.733 ± 3.880</td></tr><tr><td rowspan="2"></td><td rowspan="2">OER</td><td>45665.286 ± 0.089</td><td>21.745 ± 0.131</td><td>316.622 ± 0.383</td><td>2.0308319 ± 0.0001456</td><td>0.0007290</td><td>-4.664 ± 0.932</td></tr><tr><td>71</td><td>0.468</td><td>3.851 ± 0.206</td><td>239.903 ± 7.558</td><td>-2.0287602 ± 0.0016596</td><td>-0.0020226</td><td>-12.970 ± 10.649</td></tr><tr><td rowspan="2">η</td><td rowspan="2">IER</td><td>47174.853 ± 0.093</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td>56</td><td>0.679</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td></tr><tr><td rowspan="2"></td><td rowspan="2">COR</td><td>47176.009 ± 0.088</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td>60</td><td>0.297</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td></tr><tr><td rowspan="2"></td><td rowspan="2">OER</td><td>47177.080 ± 0.093</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td>55</td><td>0.525</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td></tr><tr><td rowspan="2">γ</td><td rowspan="2">ier, including m = 3 normal mode</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>47624.606 ± 0.103</td><td>5.016 ± 0.122</td><td>47.274 ± 1.570</td><td>1.7516631 ± 0.0006611</td><td>0.0001409</td><td>-1.091 ± 5.119</td></tr><tr><td rowspan="2"></td><td rowspan="2">COR</td><td>47626.170 ± 0.089</td><td>5.306 ± 0.071</td><td>48.720 ± 0.858</td><td>1.7514650 ± 0.0004210</td><td>0.0001448</td><td>-1.121 ± 3.260</td></tr><tr><td>83</td><td>0.401</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td></tr><tr><td rowspan="2"></td><td rowspan="2">OER</td><td>47627.865 ± 0.100</td><td>5.470 ± 0.130</td><td>47.145 ± 1.458</td><td>1.7524301 ± 0.0005903</td><td>0.0013286</td><td>-10.286 ± 4.572</td></tr><tr><td>76</td><td>0.692</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td></tr><tr><td rowspan="2">γ</td><td rowspan="2">ier, excluding m = 3 normal mode</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>47624.606 ± 0.129</td><td>5.016 ± 0.153</td><td>47.274 ± 1.964</td><td>1.7516631 ± 0.0008271</td><td>0.0001409</td><td>-1.091 ± 6.405</td></tr><tr><td rowspan="2"></td><td rowspan="2">COR</td><td>47626.289 ± 0.094</td><td>5.314 ± 0.074</td><td>47.573 ± 0.881</td><td>1.7524471 ± 0.0003578</td><td>0.0011422</td><td>-8.842 ± 2.771</td></tr><tr><td>83</td><td>0.490</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td></tr><tr><td rowspan="2"></td><td rowspan="2">OER</td><td>47627.987 ± 0.122</td><td>5.524 ± 0.157</td><td>46.394 ± 1.775</td><td>1.7536847 ± 0.0006681</td><td>0.0025990</td><td>-20.112 ± 5.175</td></tr><tr><td>76</td><td>0.895</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td></tr><tr><td rowspan="2">δ</td><td rowspan="2">ier</td><td>48297.775 ± 0.082</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td>73</td><td>0.608</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td></tr><tr><td rowspan="2"></td><td rowspan="2">COR</td><td>48300.227 ± 0.082</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr></tbody></table>

Table 5 continued on next page
<table><caption>Table 5 (continued)</caption><thead><tr><th rowspan="2">Ring</th><th rowspan="2">Feature<sup>(a)</sup></th><th colspan="2">a (km)</th><th>ae (km)</th><th>ω̄<sub>0</sub> (°)<sup>(b)</sup></th><th>ω̇ (° d<sup>-1</sup>)</th><th>Δω̇ (° d<sup>-1</sup>)</th><th>Δa<sub>ω̇</sub> (km)</th></tr><tr><th>N</th><th>RMS (km)</th><th>a sin i (km)</th><th>Ω<sub>0</sub> (°)</th><th>Ω̇ (° d<sup>-1</sup>)</th><th>ΔΩ̇ (° d<sup>-1</sup>)</th><th>Δa<sub>Ω̇</sub> (km)</th></tr></thead><tbody><tr><td></td><td></td><td>80</td><td>0.273</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>OER</td><td>48302.752 ± 0.083</td><td></td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>72</td><td>0.683</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">λ</td><td rowspan="2">COR</td><td>50026.557 ± 1.314</td><td></td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td>8</td><td>3.477</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">ε</td><td rowspan="2">IER</td><td>51120.014 ± 0.077</td><td>386.530 ± 0.107</td><td>307.089 ± 0.016</td><td>1.3632690 ± 0.0000070</td><td>-0.0028286</td><td>30.128 ± 0.075</td><td></td></tr><tr><td>79</td><td>0.583</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">COR</td><td>51149.279 ± 0.081</td><td>405.894 ± 0.062</td><td>307.076 ± 0.009</td><td>1.3632575 ± 0.0000042</td><td>-0.0001081</td><td>1.154 ± 0.045</td><td></td></tr><tr><td>89</td><td>0.396</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>OER</td><td>51178.588 ± 0.079</td><td>425.242 ± 0.111</td><td>307.061 ± 0.015</td><td>1.3632618 ± 0.0000070</td><td>0.0026240</td><td>-28.026 ± 0.075</td><td></td></tr><tr><td></td><td></td><td>77</td><td>0.560</td><td>[0.00]<sup>(c)</sup></td><td></td><td></td><td></td><td></td></tr></tbody></table>

(a) COR: center of ring. RunIDs: ringfit_v1.9_Ur018M-RF-V0351-URA178-COR-v2 and ringfit_v1.9_Ur018M-RF-V0351-URA178-COR-lambda. IER: inner edge of ring, OER: outer edge of ring. RunID for all edges: ringfit_v1.9_Ur018M-RF-V0351-URA178-IER-OER-v4.

(b) The epoch is TDB 1986 Jan 19 12:00. The zero-point for inertial longitudes is the ascending node of Uranus's equator on Earth's equator of J2000, where the orientation of the Uranus pole is in the direction of positive angular momentum (i.e., 180° from the IAU definition of the Uranus north pole).

(c) Eccentricities *e* and inclinations *a sin i* in square brackets were held fixed during orbit determination.
<table><caption>Table 6. Uranian ring normal modes</caption><thead><tr><th>Ring</th><th>Feature<sup>(a)</sup></th><th>m</th><th>A<sub>m</sub> (km)</th><th>δ<sub>m</sub> (°)</th><th>Ω<sub>p</sub> (° d<sup>-1</sup>)</th><th>ΔΩ<sub>p</sub> (° d<sup>-1</sup>)</th><th>Δa<sub>p</sub> (km)</th></tr></thead><tbody><tr><td rowspan="3">η</td><td>IER</td><td>3</td><td>0.452 ± 0.122</td><td>77.208 ± 5.798</td><td>776.585539 ± 0.002490</td><td>0.086574</td><td>-3.499 ± 0.101</td></tr><tr><td>COR</td><td>3</td><td>0.600 ± 0.069</td><td>75.901 ± 2.359</td><td>776.584048 ± 0.001164</td><td>0.113697</td><td>-4.595 ± 0.047</td></tr><tr><td>OER</td><td>3</td><td>0.677 ± 0.124</td><td>75.946 ± 3.819</td><td>776.584059 ± 0.001818</td><td>0.140201</td><td>-5.666 ± 0.073</td></tr><tr><td colspan="8">γ ring, including m = 3 normal mode</td></tr><tr><td rowspan="6">γ</td><td rowspan="3">IER</td><td>0</td><td>6.137 ± 0.138</td><td>309.462 ± 1.178</td><td>1145.576306 ± 0.000538</td><td>-0.050871</td><td>1.411 ± 0.015</td></tr><tr><td>6</td><td>0.590 ± 0.110</td><td>33.405 ± 2.208</td><td>956.418079 ± 0.000877</td><td>-0.022759</td><td>0.754 ± 0.029</td></tr><tr><td>-2</td><td>1.099 ± 0.146</td><td>35.175 ± 2.940</td><td>1720.120475 ± 0.001212</td><td>-0.071814</td><td>1.325 ± 0.022</td></tr><tr><td rowspan="3">COR</td><td>0</td><td>5.509 ± 0.076</td><td>311.695 ± 0.750</td><td>1145.576845 ± 0.000352</td><td>0.006033</td><td>-0.167 ± 0.010</td></tr><tr><td>3</td><td>0.577 ± 0.073</td><td>54.989 ± 2.254</td><td>765.399832 ± 0.001152</td><td>-0.065364</td><td>2.706 ± 0.048</td></tr><tr><td>6</td><td>0.637 ± 0.063</td><td>32.518 ± 1.171</td><td>956.419622 ± 0.000474</td><td>0.025958</td><td>-0.861 ± 0.016</td></tr><tr><td rowspan="6">OER</td><td>-2</td><td>0.690 ± 0.079</td><td>26.843 ± 2.762</td><td>1720.117977 ± 0.001217</td><td>0.010437</td><td>-0.193 ± 0.022</td></tr><tr><td>-1</td><td>1.822 ± 0.067</td><td>25.852 ± 2.169</td><td>2292.904186 ± 0.001054</td><td>0.011240</td><td>-0.156 ± 0.015</td></tr><tr><td>0</td><td>4.702 ± 0.128</td><td>313.066 ± 1.574</td><td>1145.576675 ± 0.000698</td><td>0.066941</td><td>-1.857 ± 0.019</td></tr><tr><td>3</td><td>0.935 ± 0.129</td><td>56.867 ± 2.520</td><td>765.397669 ± 0.001028</td><td>-0.026590</td><td>1.101 ± 0.043</td></tr><tr><td>6</td><td>0.892 ± 0.117</td><td>28.027 ± 1.363</td><td>956.419529 ± 0.000560</td><td>0.076982</td><td>-2.552 ± 0.019</td></tr><tr><td>-1</td><td>1.955 ± 0.122</td><td>25.258 ± 3.578</td><td>2292.903174 ± 0.001739</td><td>0.132603</td><td>-1.836 ± 0.024</td></tr><tr><td colspan="8">γ ring, excluding m = 3 normal mode</td></tr><tr><td rowspan="6">γ</td><td rowspan="3">IER</td><td>0</td><td>6.137 ± 0.172</td><td>309.462 ± 1.474</td><td>1145.576306 ± 0.000673</td><td>-0.050871</td><td>1.411 ± 0.019</td></tr><tr><td>6</td><td>0.590 ± 0.138</td><td>33.405 ± 2.763</td><td>956.418079 ± 0.001097</td><td>-0.022759</td><td>0.754 ± 0.036</td></tr><tr><td>-2</td><td>1.099 ± 0.183</td><td>35.175 ± 3.678</td><td>1720.120475 ± 0.001517</td><td>-0.071814</td><td>1.325 ± 0.028</td></tr><tr><td rowspan="3">COR</td><td>0</td><td>5.450 ± 0.079</td><td>311.233 ± 0.762</td><td>1145.576581 ± 0.000376</td><td>0.010044</td><td>-0.279 ± 0.010</td></tr><tr><td>6</td><td>0.698 ± 0.066</td><td>30.504 ± 1.075</td><td>956.419653 ± 0.000442</td><td>0.029566</td><td>-0.980 ± 0.015</td></tr><tr><td>-2</td><td>0.635 ± 0.083</td><td>27.010 ± 3.104</td><td>1720.120139 ± 0.001183</td><td>0.019028</td><td>-0.351 ± 0.022</td></tr><tr><td rowspan="3">OER</td><td>-1</td><td>1.807 ± 0.070</td><td>26.105 ± 2.325</td><td>2292.903886 ± 0.001134</td><td>0.019506</td><td>-0.270 ± 0.016</td></tr><tr><td>0</td><td>4.724 ± 0.160</td><td>313.481 ± 1.841</td><td>1145.575881 ± 0.000844</td><td>0.070553</td><td>-1.958 ± 0.023</td></tr></tbody></table>

Table 6 continued on next page
<table><caption>Table 6 (continued)</caption><thead><tr><th>Ring</th><th>Feature<sup>(a)</sup></th><th>m</th><th>A<sub>m</sub> (km)</th><th>δ<sub>m</sub> (°)</th><th>Ω<sub>p</sub> (° d<sup>-1</sup>)</th><th>ΔΩ<sub>p</sub> (° d<sup>-1</sup>)</th><th>Δa<sub>p</sub> (km)</th></tr></thead><tbody><tr><td rowspan="2"></td><td rowspan="2"></td><td>6</td><td>1.057 ± 0.140</td><td>26.147 ± 1.358</td><td>956.419580 ± 0.000563</td><td>0.080719</td><td>-2.676 ± 0.019</td></tr><tr><td>-1</td><td>1.966 ± 0.151</td><td>22.643 ± 4.393</td><td>2292.903576 ± 0.002122</td><td>0.141832</td><td>-1.964 ± 0.029</td></tr><tr><td rowspan="3">δ</td><td>IER</td><td>2</td><td>2.349 ± 0.105</td><td>171.686 ± 1.527</td><td>562.516712 ± 0.000580</td><td>-0.042324</td><td>2.415 ± 0.033</td></tr><tr><td>COR</td><td>2</td><td>3.169 ± 0.059</td><td>170.383 ± 0.666</td><td>562.516205 ± 0.000258</td><td>0.000146</td><td>-0.008 ± 0.015</td></tr><tr><td></td><td>23</td><td>0.339 ± 0.063</td><td>6.844 ± 0.472</td><td>1074.523021 ± 0.000189</td><td>-0.072580</td><td>2.173 ± 0.006</td></tr><tr><td rowspan="3">ε</td><td>IER</td><td>2</td><td>4.035 ± 0.105</td><td>171.248 ± 0.885</td><td>562.516313 ± 0.000351</td><td>0.044481</td><td>-2.539 ± 0.020</td></tr><tr><td>COR</td><td>-24</td><td>1.011 ± 0.107</td><td>2.916 ± 0.259</td><td>1074.522889 ± 0.000099</td><td>-0.034140</td><td>1.082 ± 0.003</td></tr><tr><td></td><td>14</td><td>0.383 ± 0.071</td><td>14.105 ± 0.628</td><td>956.418015 ± 0.000269</td><td>-0.798192</td><td>28.425 ± 0.010</td></tr><tr><td rowspan="2"></td><td rowspan="2"></td><td>-24</td><td>0.443 ± 0.061</td><td>2.828 ± 0.346</td><td>1074.522703 ± 0.000142</td><td>0.888501</td><td>-28.177 ± 0.005</td></tr><tr><td>14</td><td>0.590 ± 0.130</td><td>14.904 ± 0.704</td><td>956.418119 ± 0.000298</td><td>0.024894</td><td>-0.887 ± 0.011</td></tr></tbody></table>

(a) COR: center of ring. RunIDs: ringfit_v1.9_Ur018M-RF-V0351-URA178-COR-v2 and ringfit_v1.9_Ur018M-RF-V0351-URA178-COR-lambda. IER: inner edge of ring, OER: outer edge of ring. RunID for all edges: ringfit_v1.9_Ur018M-RF-V0351-URA178-IER-OER-v4.

(b) The epoch is TDB 1986 Jan 19 12:00. The zero-point for inertial longitudes is the ascending node of Uranus's equator on Earth's equator of J2000, where the orientation of the Uranus pole is in the direction of positive angular momentum (i.e., 180° from the IAU definition of the Uranus north pole). All longitudes reduced to the minimum value of λ mod 360°/|m|.
### 4.3. Offset times for selected events

As discussed in Paper 1, the accuracy of the absolute timing of individual data sets is highly dependent on the availability of accurate time standards at each observatory, the methods used to incorporate time signals into the data streams, and the intrinsic and sometimes variable time delays introduced by filtering electronics and recording systems. For several occultations observed with multiple telescopes, there are significant systematic offsets between the predicted and observed event times. We incorporate these into our orbit fit by including these station offset times as free parameters, as given in Table 7. For each event, the absolute timing was anchored by observations from one or more telescopes that were assumed to have the most reliable time reference. Details of the timing for each of the Earth-based observations are included in Paper 1.

For the *Voyager* occultations, we fitted for along-track spacecraft time offsets relative to the nominal *Voyager* trajectory for the RSS occultation and the $\beta$ Per stellar occultation. The JPL solution for the *Voyager* trajectory in vgr2.ura178.bsp assumed an offset time $dt = 0$ for the $\sigma$ Sgr stellar occultation, and instead solved for an offset to the *Hipparcos* catalog position of this multiple star system. We have followed this prescription in our orbit fits as well.

<table><caption>**Table 7.** Fitted station offset times</caption><thead><tr><th>Event</th><th>Station</th><th>Offset (s)</th></tr></thead><tbody><tr><td rowspan="3">U12</td><td>ESO (2m)</td><td>-0.073 ± 0.016</td></tr><tr><td>Las Campanas (IR)</td><td>0.099 ± 0.022</td></tr><tr><td>Las Campanas (vis)</td><td>0.114 ± 0.019</td></tr><tr><td rowspan="6">U14</td><td>ESO (1m)</td><td>-0.091 ± 0.007</td></tr><tr><td>Las Campanas (IR)</td><td>0.064 ± 0.006</td></tr><tr><td>Pic du Midi</td><td>3.717 ± 0.009</td></tr><tr><td>Pic du Midi (1m)</td><td>0.760 ± 0.012</td></tr><tr><td>Teide (ingress)</td><td>0.040 ± 0.010</td></tr><tr><td>Teide (egress)</td><td>0.467 ± 0.010</td></tr><tr><td>U25</td><td>McDonald Obs.</td><td>-0.024 ± 0.006</td></tr><tr><td>U36A</td><td>IRTF</td><td>-8.769 ± 0.262</td></tr><tr><td>U103</td><td>ESO (2m)</td><td>0.055 ± 0.013</td></tr><tr><td>U103</td><td>CTIO</td><td>-0.035 ± 0.013</td></tr><tr><td>U134</td><td>SAAO (egress)</td><td>0.475 ± 0.170</td></tr><tr><td>U137</td><td>HST</td><td>0.635 ± 0.008</td></tr><tr><td rowspan="2">U144</td><td>CAHA (ingress)</td><td>0.428 ± 0.068</td></tr><tr><td>CAHA (egress)</td><td>0.724 ± 0.240</td></tr><tr><td>Vgr2 RSS</td><td>DSS-43</td><td>-0.013 ± 0.011</td></tr><tr><td>Vgr2 σ Sgr</td><td>PPS</td><td>[0.000]</td></tr><tr><td>Vgr2 β Per</td><td>PPS</td><td>-0.061 ± 0.016</td></tr></tbody></table>

### 4.4. Orbit fit residuals

The residuals of the separate COR and IER/OER fits summarized in Tables 5 and 6 are shown in Fig. 20. For the COR observations shown in the upper panel, the RMS residuals for the nine main rings range from 0.212 km for ring 5 to 0.401 km for the $\gamma$ ring including the $m = 3$ mode and 0.490 km when the $m = 3$ mode is absent. For the IER/OER observations, the RMS residuals range from 0.468 km for $\beta$ OER to 0.830 km for ring 6 IER, and 0.895 km for the $\gamma$ ring OER when the $m = 3$ mode is excluded from the fit. In all, 651 data points were included in the COR fit, for an RMS per degree of freedom of 0.348 km. In comparison, the IER/OER fit included 1174 data points and an RMS per degree of freedom of 0.655 km. As noted previously, the COR measurements are intrinsically more accurate, which prompted the separate fits for the ring midlines and the ring edges.
<figure><img src="image_22.png" /><figcaption>**Figure 20.** Ring plane residuals of the COR (top panel) and IER/OER (bottom panel) fits in Tables 5 and 6, including the $\gamma$ ring $m = 3$ normal mode and excluding the $\lambda$ ring. The dashed lines in each panel show the RMS residuals per degree of freedom for each fit.</figcaption></figure>

**Table 8** provides a more detailed view of the distribution of the COR fit RMS residuals by occultation event and observatory code. The total number of data points for each event is listed as $N_{tot}$, and for multi-station events, N gives the number of data points per station.

<table><caption>**Table 8.** RMS residuals by event and observatory</caption><thead><tr><th>Star</th><th>Date</th><th>Obs</th><th>N<sub>tot</sub></th><th>N</th><th>RMS (km)</th></tr></thead><tbody><tr><td>U0</td><td>1977-03-10</td><td>KAO</td><td>16</td><td></td><td>0.350</td></tr><tr><td>U2</td><td>1977-12-23</td><td>TEN</td><td>3</td><td></td><td>0.028</td></tr><tr><td>U5</td><td>1978-04-10</td><td>304</td><td>16</td><td></td><td>0.421</td></tr><tr><td>U9</td><td>1979-06-10</td><td>304</td><td>7</td><td></td><td>0.476</td></tr><tr><td>U11</td><td>1980-03-20</td><td>807</td><td>6</td><td></td><td>0.193</td></tr><tr><td>U12</td><td>1980-08-15</td><td></td><td>47</td><td></td><td>0.208</td></tr><tr><td></td><td></td><td>807</td><td></td><td>13</td><td>0.205</td></tr><tr><td></td><td></td><td>ESO</td><td></td><td>17</td><td>0.164</td></tr><tr><td></td><td></td><td>LAS</td><td></td><td>8</td><td>0.257</td></tr><tr><td></td><td></td><td>LAV</td><td></td><td>9</td><td>0.237</td></tr><tr><td>U13</td><td>1981-04-26</td><td>413</td><td>18</td><td></td><td>0.190</td></tr></tbody></table>

**Table 8 continued on next page**
<table><caption>Table 8 (continued)</caption><thead><tr><th>Star</th><th>Date</th><th>Obs</th><th>N<sub>tot</sub></th><th>N</th><th>RMS (km)</th></tr></thead><tbody><tr><td rowspan="8">U14</td><td rowspan="8">1982-04-22</td><td></td><td>87</td><td></td><td>0.324</td></tr><tr><td>586</td><td></td><td>8</td><td>0.480</td></tr><tr><td>807</td><td></td><td>18</td><td>0.274</td></tr><tr><td>ES1</td><td></td><td>14</td><td>0.162</td></tr><tr><td>LAS</td><td></td><td>16</td><td>0.305</td></tr><tr><td>LAV</td><td></td><td>10</td><td>0.451</td></tr><tr><td>PI1</td><td></td><td>5</td><td>0.363</td></tr><tr><td>TEE</td><td></td><td>8</td><td>0.338</td></tr><tr><td></td><td></td><td>TEN</td><td>8</td><td>0.237</td></tr><tr><td>U15</td><td>1982-05-01</td><td>414</td><td>17</td><td></td><td>0.371</td></tr><tr><td>U16</td><td>1982-06-04</td><td>675</td><td>18</td><td></td><td>0.278</td></tr><tr><td>U17B</td><td>1983-03-25</td><td>SAA</td><td>13</td><td></td><td>0.179</td></tr><tr><td rowspan="4">U23</td><td rowspan="4">1985-05-04</td><td></td><td>30</td><td></td><td>0.350</td></tr><tr><td>711</td><td></td><td>9</td><td>0.299</td></tr><tr><td>807</td><td></td><td>18</td><td>0.379</td></tr><tr><td>TEN</td><td></td><td>3</td><td>0.306</td></tr><tr><td rowspan="4">U25</td><td rowspan="4">1985-05-24</td><td></td><td>54</td><td></td><td>0.259</td></tr><tr><td>675</td><td></td><td>18</td><td>0.239</td></tr><tr><td>711</td><td></td><td>18</td><td>0.178</td></tr><tr><td>807</td><td></td><td>18</td><td>0.335</td></tr><tr><td>U28</td><td>1986-04-26</td><td>568</td><td>17</td><td></td><td>0.289</td></tr><tr><td>U34</td><td>1987-02-26</td><td>568</td><td>16</td><td></td><td>0.348</td></tr><tr><td rowspan="7">U36A</td><td rowspan="7">1987-04-02</td><td></td><td>25</td><td></td><td>0.264</td></tr><tr><td>413</td><td></td><td>3</td><td>0.217</td></tr><tr><td>568</td><td></td><td>5</td><td>0.332</td></tr><tr><td>807</td><td></td><td>8</td><td>0.314</td></tr><tr><td>ANU</td><td></td><td>1</td><td>0.058</td></tr><tr><td>IR2</td><td></td><td>2</td><td>0.149</td></tr><tr><td>UKI</td><td></td><td>6</td><td>0.190</td></tr><tr><td>U1052</td><td>1988-05-12</td><td>568</td><td>10</td><td></td><td>0.288</td></tr><tr><td>U65</td><td>1990-06-21</td><td>568</td><td>15</td><td></td><td>0.328</td></tr><tr><td>U83</td><td>1991-06-25</td><td>568</td><td>18</td><td></td><td>0.265</td></tr><tr><td>U84</td><td>1991-06-28</td><td>568</td><td>18</td><td></td><td>0.222</td></tr><tr><td>U102A</td><td>1992-07-08</td><td>568</td><td>6</td><td></td><td>0.385</td></tr><tr><td rowspan="3">U103</td><td rowspan="3">1992-07-11</td><td></td><td>27</td><td></td><td>0.281</td></tr><tr><td>675</td><td></td><td>12</td><td>0.230</td></tr><tr><td>ES2</td><td></td><td>15</td><td>0.316</td></tr><tr><td>U9539</td><td>1993-06-30</td><td>807</td><td>18</td><td></td><td>0.281</td></tr><tr><td rowspan="2">U134</td><td rowspan="2">1995-09-09</td><td></td><td>18</td><td></td><td>0.273</td></tr><tr><td>SA1</td><td></td><td>9</td><td>0.284</td></tr></tbody></table>

<caption>Table 8 continued on next page</caption>
<table><caption>**Table 8 (continued)**</caption><thead><tr><th>Star</th><th>Date</th><th>Obs</th><th>N<sub>tot</sub></th><th>N</th><th>RMS (km)</th></tr></thead><tbody><tr><td rowspan="3">U137</td><td rowspan="3">1996-03-16</td><td>SAA</td><td rowspan="3">22</td><td>9</td><td>0.261</td></tr><tr><td>568</td><td>0.336</td></tr><tr><td>HST</td><td>0.359</td></tr><tr><td rowspan="3">U138</td><td rowspan="3">1996-04-10</td><td></td><td rowspan="3">18</td><td>4</td><td>0.206</td></tr><tr><td>675</td><td>0.167</td></tr><tr><td>HST</td><td>0.165</td></tr><tr><td rowspan="3">U144</td><td rowspan="3">1997-09-30</td><td></td><td rowspan="3">15</td><td>9</td><td>0.168</td></tr><tr><td>CAE</td><td>0.445</td></tr><tr><td>CAI</td><td>0.450</td></tr><tr><td rowspan="3">U149</td><td rowspan="3">1998-11-06</td><td>SAA</td><td rowspan="3">12</td><td>5</td><td>0.382</td></tr><tr><td>568</td><td>0.369</td></tr><tr><td>688</td><td>0.395</td></tr><tr><td rowspan="2">U0201</td><td rowspan="2">2002-07-29</td><td>675</td><td rowspan="2">12</td><td>6</td><td>0.342</td></tr><tr><td>688</td><td>0.411</td></tr><tr><td>U0602</td><td>2006-09-20</td><td>568</td><td>14</td><td>0.367</td></tr></tbody></table>

### 4.5. Limits on eccentricity and inclination

In our adopted final orbit model, several of the narrow rings have no detectable eccentricity and/or inclination. Nevertheless, we can set upper limits on these quantities from a statistical analysis of the patterns of orbit fit radial residuals for a given ring. We illustrate this procedure with the $\eta$ ring. The top panel of Fig. 21 shows the results of a series of least-squares fits to the residuals in the measurements of the $\eta$ ring radius after subtraction of the best-fitting circular orbit and $m = 3$ normal mode, over a range of precession rates centered on the predicted apsidal rate appropriate for the ring’s fitted semimajor axis. The free parameters for each fit are the eccentricity and mean anomaly of the best-fitting ellipse with the assumed precession rate. The top panel shows the RMS residuals of each fit as a function of the assumed pattern speed, and the second panel shows the amplitude $ae$ of the best-fitting ellipse. The solid vertical lines mark the expected precession rate for an eccentric ring with the actual radius of the $\eta$ ring. If the ring were measurably eccentric, we would expect to see a sharp dip in the RMS residuals and a peak in the fitted amplitude, centered on this pattern speed. Instead, the best-fitting eccentric model for the $\eta$ ring is marked by the vertical dashed lines, far removed from the physically significant expected precession rate. The bottom panel of the figure shows a histogram of the distribution of fitted amplitudes ($\Delta a = ae$) from the middle panel. The overplotted solid line shows the best-fitting Rayleigh distribution to this histogram, appropriate for a random one-sided distribution:

$$N(\Delta a) = N_0 \frac{\Delta a}{\sigma^2} \exp(-\Delta a^2/2\sigma^2), \qquad (12)$$

where we fit for $N_0$ and $\sigma$. In this case, $\sigma(\Delta ae) = 0.063$ km, which is comparable to the 1-$\sigma$ formal uncertainties of the rings with measurable eccentricities (Table 5). Setting the eccentricity detection limit at 2-$\sigma$, the corresponding upper limit to the $\eta$ ring’s eccentricity is $ae \lesssim 0.126$ km.

Using the same approach to estimate a detection limit for inclination, Fig. 22 shows a scan over nodal regression rates for the $\eta$ ring, where we follow the prescription given by Eqs. (8)–(11) of French et al. (2016b) to relate the observed radial residual $\Delta r$ to the predicted radial displacement corresponding to the local vertical displacement of an
<figure><img src="image_23.png" /></figure>

**Figure 21.** Statistical limit on the eccentricity of the *η* ring. The upper two panels show the null results of an *m* = 1 normal mode scan, with no detectable dip in the RMS residuals near the predicted apsidal precession rate for the observed semimajor axis of the ring, shown as a thick vertical gray line. The bottom panel shows a histogram of the amplitudes given by the individual least-squares fits for each assumed pattern speed, along with the best-fitting Rayleigh distribution with *σ* = 0.063 km.

inclined ring. Once again, there is no indication of a measurable inclination at the expected pattern speed (marked by a thick vertical gray line), with $\sigma(\Delta a \sin i) = 0.145$ km. Adopting the same 2-$\sigma$ detection threshold, the upper limit to the inclination of the *η* ring is $a \sin i \lesssim 0.290$ km.

Proceeding in a similar fashion for the other rings, we obtain the results in **Table 9** for the 2-$\sigma$ upper limits on the eccentricities and inclinations of all rings with no detected *m* = 1 mode and/or inclination.

<table><caption>**Table 9.** 2-$\sigma$ eccentricity/inclinations limits</caption><thead><tr><th>Ring</th><th><i>ae</i> (km)</th><th><i>a sin i</i> (km)</th></tr></thead><tbody><tr><td><i>η</i></td><td>0.126</td><td>0.290</td></tr><tr><td><i>γ</i></td><td>—</td><td>0.286</td></tr><tr><td><i>δ</i></td><td>0.088</td><td>0.284</td></tr><tr><td><i>ε</i></td><td>—</td><td>0.158</td></tr></tbody></table>

### 4.6. *The λ ring*

The λ ring differs from the other nine narrow Uranian rings in the apparent wavelength dependence of its equivalent width (French et al. 1991) and in having an enhanced brightness in the forward scattering direction compared to
<figure><img src="image_24.png" /><figcaption>**Figure 22.** Statistical limit on the inclination of the *η* ring. The upper two panels show the null results of an *m* = 1 inclination scan, with no detectable dip in the RMS residuals near the predicted nodal regression rate for the observed semimajor axis of the ring, marked by a thick vertical gray line. The bottom panel shows a histogram of the amplitudes given by the individual least-squares fits for each assumed pattern speed, along with the best-fitting Rayleigh distribution with *σ* = 0.145 km.</figcaption></figure>

the other narrow rings (Smith et al. 1986; Ockert et al. 1987). From a comparison of *Voyager* UVS and PPS stellar occultations ($\lambda = 0.11$ µm and 0.27 µm, respectively) and Earthbased IR observations at $\lambda = 2.2$ µm, Kangas and Elliot (1987) modeled the wavelength dependence of the equivalent depth of the fitted square-well profiles by assuming a two-component population of particles large compared to the observed wavelength and of smaller particles that scattered in the Mie regime with $\tau \propto 1/\lambda$. They derived an upper limit of 6% for the contribution of large particles to the total optical depth at $\lambda = 0.11$ µm. Collectively, these results suggest that the *λ* ring is primarily composed of micron-sized dust. However, it is also azimuthally variable, having arcs and clumps (Ockert et al. 1987; Colwell et al. 1990; Showalter 1995), some of which might have larger particles detectable in the IR. A detection at infrared wavelengths was reported for the U103 occultation from CTIO and possibly from ESO as well (French et al. 1996). Kangas (1989) identified several additional candidate *λ* ring events from an analysis of Earth-based occultations at $\lambda = 2.2$ µm, and to these we add a likely detection from the *HST* observations of the U138 occultation at $\lambda = 0.362 - 0.705$ µm (see Paper 1 for observational details).

For the orbit fit presented here, we include the two secure *Voyager* PPS UV occultation detections (we omit the lower-SNR UVS observations for the same event, to avoid double-counting), five Earth-based IR candidate events, and the likely U138 *HST* detection, for a total of eight possible *λ* ring events, listed in **Table 10**. For each event, we include the observed radius $r_{obs}$, the residual *dr* relative to our adopted circular orbit fit, the ring profile equivalent width *EW*, and the wavelength of the observations. Radial profiles and square-well model fits to the six Earth-based candidate detections are shown in **Fig. 23**.
<table><caption>**Table 10.** Candidate λ ring occultation events.</caption><thead><tr><th>Ring</th><th>Obs</th><th>Dir</th><th>UTC</th><th>r<sub>obs</sub> (km)</th><th>dr (km)</th><th>EW (km)</th><th>λ (μm)</th></tr></thead><tbody><tr><td>U23</td><td>807</td><td>E</td><td>1985-05-04T06:05:50.0905</td><td>50022.817</td><td>-3.740</td><td>0.27</td><td>2.20</td></tr><tr><td>σ Sgr</td><td>PPS</td><td>I</td><td>1986-01-24T05:30:42.3000</td><td>50025.105</td><td>-1.452</td><td>0.19</td><td>0.27</td></tr><tr><td>σ Sgr</td><td>PPS</td><td>E</td><td>1986-01-24T08:06:57.1900</td><td>50022.528</td><td>-4.030</td><td>0.19</td><td>0.27</td></tr><tr><td>U28</td><td>568</td><td>I</td><td>1986-04-26T13:27:23.3009</td><td>50032.187</td><td>5.630</td><td>0.28</td><td>2.20</td></tr><tr><td>U28</td><td>568</td><td>E</td><td>1986-04-26T15:11:51.8047</td><td>50029.089</td><td>2.532</td><td>0.23</td><td>2.20</td></tr><tr><td>U103</td><td>807</td><td>I</td><td>1992-07-11T07:59:42.0300</td><td>50026.778</td><td>0.221</td><td>0.32</td><td>2.20</td></tr><tr><td>U103</td><td>ES2</td><td>I</td><td>1992-07-11T07:59:43.5982</td><td>50023.378</td><td>-3.179</td><td>0.48</td><td>2.20</td></tr><tr><td>U138</td><td>HST</td><td>E</td><td>1996-04-10T12:15:28.1639</td><td>50030.576</td><td>4.018</td><td>0.30</td><td>0.362-0.705</td></tr></tbody></table>

<figure><img src="image_25.png" /><figcaption>**Figure 23.** Candidate Earth-based occultation detections of the λ ring. (The substantial offset of the U103 ESO model curve relative to the square well shown in red is due to the large instrumental time constant for that dataset – see Paper 1.)</figcaption></figure>

No satisfactory keplerian elliptical model fitted these measurements, and our circular orbit fit gives $a = 50026.557 \pm 1.314$ km and an RMS residual of 3.477 km (Table 5), compared to the Jacobson (2014) value $a = 50024.16 \pm 0.96$, which was based on the combined *Voyager* PPS and UVS detections during the σ Sgr occultation and the vgr2.ura111.bsp ephemeris. A separate fit using the vgr.ura178.bsp ephemeris that included only the two *Voyager* points yielded $a = 50023.816 \pm 1.289$ km, close to the Jacobson (2014) result. With the addition of the HST detection, which we regard as convincing (in part because it was observed at wavelengths fairly close to the PPS sensitivity of $\lambda = 0.27 \mu m$), we obtained $a = 50026.069 \pm 2.37$ km with an RMS residual of 3.363 km.

Overall, the orbit fits to the λ ring show considerably more scatter than the other rings, even when restricted to the convincing *Voyager* detections. The *Voyager* images show evidence of an $m = 6$ pattern in the ring's radial position with an amplitude of $4.6 \pm 0.9$ km (Showalter 1995) that could be compatible with this excess scatter, but the occultation data are too sparse to confirm the existence of this pattern at this point. In the absence of additional secure observations, its semimajor axis should probably be regarded as uncertain at the level of a few km.
## 4.7. Multiple-star occultation geometry

Among the Uranus occultations reported in Paper 1, the ring profiles of two occultations revealed that they involved multiple-star systems. The occultation of U36 by Uranus and the rings was a remarkable multi-day event, lasting from 1987 Mar 30 through Apr 2 and occurring while Uranus was at the end of its retrograde loop as seen from Earth (Elliot et al. 1987). Besides the ring occultations of the primary star, additional secondary events involving three additional stars were observed, as described in detail in Paper 1. Similarly, star U102A was found to have a binary companion, U102B. We solved for the skyplane offsets $\Delta f$ (east) and $\Delta g$ (north) of secondary stars U36B and U36C relative to the primary star U36A (only one ring event was observed for U36D, preventing a unique determination of its offset position), and of star U102B relative to star U102A, from separate orbit fits to the secondary ring event times, with the results shown in **Table 11**.

<table><caption>**Table 11.** Skyplane positions of secondary stars</caption><thead><tr><th>Star</th><th>Δf (km)</th><th>Δg (km)</th></tr></thead><tbody><tr><td>U36B</td><td>+9.15 ± 0.32</td><td>+20.74 ± 0.77</td></tr><tr><td>U36C</td><td>-128.86 ± 0.32</td><td>+138.53 ± 0.77</td></tr><tr><td>U102B</td><td>+287.28 ± 0.42</td><td>-283.02 ± 1.80</td></tr></tbody></table>

## 4.8. Star positions, proper motions and planetary ephemeris offsets

Under the assumption that the Gaia DR3 star positions at the catalog epoch of TDB 2016 Jan 1 12:00 are more accurate than the proper motion-corrected positions, in our nominal orbit fit we solved for corrections to the catalog values of the proper motions of the Earth-based occultation stars. To provide alternative representations of the corrections to the relative positions of the star and planet for each occultation, we performed two additional fits: first, we assumed that the ura178 series of ephemerides were exact and used the unmodified star catalog proper motions, solving for corrections to the predicted star positions for each occultation; next, we used the unmodified catalog values to compute the proper motion and parallax-corrected star positions for each observation, and instead fitted for skyplane offsets $f_0$ and $g_0$ (east and north) of Uranus relative to the ephemeris position for each occultation. All three fits returned virtually identical pole directions and ring orbital elements, as expected. **Table 12** lists the results of these three separate fits, and includes for each star the proper motion correction in RA ($d\alpha \cos \delta/dt$) and Dec ($d\delta/dt$), corrections to the catalog positions $d\alpha$ and $d\delta$, and sky-plane ephemeris offsets $f_0$ and $g_0$, along with their correlation coefficient $\rho$ (which was identical for the three fits). Except for a few outliers, mostly associated with multiple star systems, the fitted star offsets $d\alpha$ and $d\delta$ for most of the remaining events are on the order of a few mas. Similarly, the sky-plane ephemeris offsets $f_0$ and $g_0$ are well under 100 km, confirming that the substantial systematic drift in time of the ura111 Uranus ephemeris that amounted to several hundred km in the sky plane by the time of the final ring occultation in 2006 has been effectively eliminated in the ura178 ephemeris.<fnref n="8" />

# 5. URANUS POLE DIRECTION AND RING PLANE RADIUS SCALE

We determined the uncertainty in the Uranus pole direction and ring plane radius scale from a series of test fits, described below. The results are included in **Table 13** and shown in **Figs. 24 and 25**.

## 5.1. Uranus pole direction

* Fit 1: Nominal pole direction and ring orbital elements. This fit included the COR measurements of Earth-based and *Voyager* observations of the nine principal rings, resulting in the adopted orbital elements and normal modes

<footnote n="8">⁸ The Gaia DR3 catalog ID for each star is included in Table 2 of Paper 1.</footnote>
<table><caption>**Table 12.** Fitted corrections to proper motions, star positions, and planet ephemeris</caption><thead><tr><th rowspan="2">Star</th><th colspan="2">Proper motion fit</th><th colspan="2">Star offset fit</th><th colspan="2">Ephemeris offset fit</th><th rowspan="2">ρ</th></tr><tr><th>dα cos δ/dt (mas/yr)</th><th>dδ/dt (mas/yr)</th><th>dα (mas)</th><th>dδ (mas)</th><th>f<sub>0</sub> (km)</th><th>g<sub>0</sub> (km)</th></tr></thead><tbody><tr><td>U0</td><td>0.04476 ± 0.00028</td><td>0.01458 ± 0.00049</td><td>-1.7370 ± 0.0108</td><td>-0.5660 ± 0.0189</td><td>22.583 ± 0.141</td><td>7.357 ± 0.246</td><td>-0.009</td></tr><tr><td>U2</td><td>-0.05768 ± 0.00127</td><td>0.08987 ± 0.03230</td><td>2.1930 ± 0.0483</td><td>-3.4165 ± 1.2273</td><td>-30.649 ± 0.678</td><td>47.658 ± 17.144</td><td>-0.952</td></tr><tr><td>U5</td><td>0.06834 ± 0.00034</td><td>-0.03155 ± 0.00038</td><td>-2.5782 ± 0.0130</td><td>1.1901 ± 0.0142</td><td>33.119 ± 0.167</td><td>-15.291 ± 0.182</td><td>-0.236</td></tr><tr><td>U9</td><td>-0.02314 ± 0.00062</td><td>0.04092 ± 0.00258</td><td>0.8459 ± 0.0227</td><td>-1.4966 ± 0.0943</td><td>-10.931 ± 0.293</td><td>19.324 ± 1.217</td><td>0.889</td></tr><tr><td>U11</td><td>0.02566 ± 0.00092</td><td>-0.01070 ± 0.00039</td><td>-0.9183 ± 0.0330</td><td>0.3828 ± 0.0140</td><td>12.094 ± 0.434</td><td>-5.043 ± 0.184</td><td>-0.163</td></tr><tr><td>U12</td><td>-0.02100 ± 0.00029</td><td>-0.03987 ± 0.00087</td><td>0.7429 ± 0.0104</td><td>1.4103 ± 0.0307</td><td>-10.111 ± 0.141</td><td>-19.197 ± 0.418</td><td>0.621</td></tr><tr><td>U13</td><td>0.00444 ± 0.00023</td><td>0.00634 ± 0.00066</td><td>-0.1540 ± 0.0080</td><td>-0.2202 ± 0.0228</td><td>1.997 ± 0.104</td><td>2.852 ± 0.295</td><td>0.280</td></tr><tr><td>U14</td><td>0.13015 ± 0.00019</td><td>0.08204 ± 0.00052</td><td>-4.3854 ± 0.0063</td><td>-2.7647 ± 0.0176</td><td>57.299 ± 0.082</td><td>36.120 ± 0.231</td><td>0.150</td></tr><tr><td>U16</td><td>0.00340 ± 0.00024</td><td>-0.00384 ± 0.00052</td><td>-0.1142 ± 0.0081</td><td>0.1289 ± 0.0176</td><td>1.481 ± 0.106</td><td>-1.673 ± 0.228</td><td>0.104</td></tr><tr><td>U15</td><td>0.03359 ± 0.00023</td><td>-0.00179 ± 0.00067</td><td>-1.1310 ± 0.0078</td><td>0.0602 ± 0.0226</td><td>14.717 ± 0.102</td><td>-0.785 ± 0.295</td><td>0.208</td></tr><tr><td>U17B</td><td>0.02862 ± 0.00031</td><td>-0.00861 ± 0.00051</td><td>-0.9378 ± 0.0103</td><td>0.2822 ± 0.0167</td><td>12.579 ± 0.138</td><td>-3.787 ± 0.224</td><td>0.337</td></tr><tr><td>U23</td><td>0.03614 ± 0.00024</td><td>0.04487 ± 0.00032</td><td>-1.1082 ± 0.0074</td><td>-1.3756 ± 0.0100</td><td>14.650 ± 0.098</td><td>18.184 ± 0.132</td><td>0.175</td></tr><tr><td>U25</td><td>0.30159 ± 0.00022</td><td>0.16083 ± 0.00029</td><td>-9.2306 ± 0.0068</td><td>-4.9224 ± 0.0088</td><td>121.117 ± 0.089</td><td>64.590 ± 0.115</td><td>0.043</td></tr><tr><td>U28</td><td>0.04119 ± 0.00027</td><td>-0.05310 ± 0.00051</td><td>-1.2227 ± 0.0081</td><td>1.5763 ± 0.0150</td><td>16.341 ± 0.108</td><td>-21.063 ± 0.201</td><td>0.183</td></tr><tr><td>U34</td><td>0.42907 ± 0.00042</td><td>-0.28186 ± 0.00523</td><td>-12.3768 ± 0.0121</td><td>8.1289 ± 0.1507</td><td>174.952 ± 0.171</td><td>-114.803 ± 2.128</td><td>0.845</td></tr><tr><td>U36A</td><td>0.31706 ± 0.00029</td><td>-0.36079 ± 0.00056</td><td>-9.1173 ± 0.0083</td><td>10.3772 ± 0.0160</td><td>125.254 ± 0.114</td><td>-142.893 ± 0.220</td><td>-0.078</td></tr><tr><td>U1052</td><td>0.00355 ± 0.00089</td><td>-0.04157 ± 0.00037</td><td>-0.0981 ± 0.0247</td><td>1.1490 ± 0.0102</td><td>1.314 ± 0.331</td><td>-15.392 ± 0.137</td><td>0.049</td></tr><tr><td>U65</td><td>0.04912 ± 0.00042</td><td>0.00392 ± 0.00053</td><td>-1.2539 ± 0.0108</td><td>-0.1001 ± 0.0136</td><td>16.736 ± 0.144</td><td>1.336 ± 0.181</td><td>0.192</td></tr><tr><td>U83</td><td>0.04739 ± 0.00044</td><td>0.07387 ± 0.00051</td><td>-1.1620 ± 0.0109</td><td>-1.8112 ± 0.0125</td><td>15.566 ± 0.146</td><td>24.263 ± 0.168</td><td>0.205</td></tr><tr><td>U84</td><td>0.05320 ± 0.00067</td><td>0.03366 ± 0.00036</td><td>-1.3038 ± 0.0164</td><td>-0.8252 ± 0.0089</td><td>17.460 ± 0.220</td><td>11.051 ± 0.119</td><td>0.193</td></tr><tr><td>U102A</td><td>-0.57815 ± 0.00051</td><td>0.49276 ± 0.00250</td><td>13.5766 ± 0.0119</td><td>-11.5716 ± 0.0587</td><td>-182.414 ± 0.160</td><td>155.471 ± 0.789</td><td>-0.409</td></tr><tr><td>U103</td><td>0.02529 ± 0.00097</td><td>0.05789 ± 0.00036</td><td>-0.5939 ± 0.0229</td><td>-1.3590 ± 0.0085</td><td>7.981 ± 0.307</td><td>18.263 ± 0.115</td><td>0.324</td></tr><tr><td>U9539</td><td>0.00114 ± 0.00031</td><td>-0.04155 ± 0.00172</td><td>-0.0257 ± 0.0070</td><td>0.9353 ± 0.0387</td><td>0.347 ± 0.094</td><td>-12.624 ± 0.523</td><td>-0.239</td></tr><tr><td>U134</td><td>-0.01846 ± 0.00286</td><td>0.01964 ± 0.01047</td><td>0.3749 ± 0.0580</td><td>-0.3988 ± 0.2127</td><td>-5.187 ± 0.800</td><td>5.504 ± 2.932</td><td>0.983</td></tr><tr><td>U137</td><td>-0.01975 ± 0.00051</td><td>-0.02044 ± 0.00370</td><td>0.3909 ± 0.0100</td><td>0.4045 ± 0.0733</td><td>-5.768 ± 0.148</td><td>-5.964 ± 1.081</td><td>-0.873</td></tr><tr><td>U138</td><td>-0.02093 ± 0.00072</td><td>-0.01165 ± 0.00272</td><td>0.4130 ± 0.0142</td><td>0.2300 ± 0.0537</td><td>-5.981 ± 0.206</td><td>-3.332 ± 0.779</td><td>-0.818</td></tr><tr><td>U144</td><td>-0.05555 ± 0.00208</td><td>0.03560 ± 0.00405</td><td>1.0136 ± 0.0380</td><td>-0.6498 ± 0.0740</td><td>-14.230 ± 0.532</td><td>9.126 ± 1.037</td><td>0.905</td></tr><tr><td>U149</td><td>-0.05626 ± 0.00056</td><td>-0.06285 ± 0.00189</td><td>0.9649 ± 0.0095</td><td>1.0781 ± 0.0324</td><td>-13.951 ± 0.138</td><td>-15.582 ± 0.469</td><td>-0.280</td></tr><tr><td>U0201</td><td>-0.69338 ± 0.00185</td><td>0.43748 ± 0.00102</td><td>9.3095 ± 0.0248</td><td>-5.8740 ± 0.0136</td><td>-128.665 ± 0.343</td><td>81.183 ± 0.189</td><td>0.566</td></tr><tr><td>U0602</td><td>-0.53994 ± 0.00258</td><td>-0.36195 ± 0.00403</td><td>5.0120 ± 0.0240</td><td>3.3595 ± 0.0374</td><td>-69.474 ± 0.332</td><td>-46.570 ± 0.519</td><td>-0.278</td></tr></tbody></table>

given in Tables 5 and 6. The uncertainties in the pole direction are the formal $1-\sigma$ errors from the unweighted least squares fit to the ring observations, plotted as the black error ellipse in the Fig. 24.

* Fit 2: Adopted pole direction, including Voyager trajectory uncertainties. Fit 1 assumed that the *Voyager vgr2.ura178.bsp* ephemeris was exact and did not take into account the formal uncertainties in the spacecraft position that were estimated as part of the ephemeris solution. We conducted a series of Monte Carlo simulations in which we randomly displaced the nominal spacecraft position by the corresponding estimated uncertainty during each of the three separate occultations (RSS, $\sigma$ Sgr, and $\beta$ Per) to derive an error ellipse representing the pole direction uncertainties arising solely from the estimated *Voyager* ephemeris errors. We convolved this probability density function with the pole direction error ellipse from Fit 1 to derive a modified $1-\sigma$ error ellipse and correlation coefficient $\rho(\alpha_P, \delta_P)$ that accounted for the combined uncertainties in our orbit fit and the systematic uncertainties in the spacecraft trajectory. Overall, the *Voyager* trajectory uncertainty has a relatively small effect on the final error budget: the fitted pole direction is the same as in Fit 1, but the error estimates have increased modestly from $\sigma(\alpha_P) = 0.000128^\circ$ to $0.000141^\circ$, and from $\sigma(\delta_P) = 0.000472^\circ$ to $0.000618^\circ$. This result is shown as a red error ellipse in Fig. 24 and represents our best estimate of the final pole direction and its $1-\sigma$ error.

* Fit 3: Precession of the Uranus pole direction. For Fit 1, we assumed that the Uranus pole direction is fixed in time, but it is predicted to experience slight precession caused by the weak periodic torques supplied by the sun and the planet's satellites. Jacobson (2023) developed a trigonometric series representation of these contributions to the pole direction over time, derived from numerical integrations of the *ura178* satellite ephemerides.

Over the course of the 29-year interval of the ring occultation observations considered here, the corresponding predicted direction of the pole varied periodically by $\pm 0.0002^\circ$ in $\alpha_P$ and $\pm 0.00015^\circ$ in $\delta_P$, following the dominant

<footnote>a Correlation coefficient.</footnote>
short-period term in the series with a period of 17.79 years (6494 days), comparable to the leading terms $E_1$ and $I_1$ in Miranda's eccentricity and inclination arguments (Laskar (1986), Table 3). For Fit 3, we adopted the trigonometric representation of the pole precession as a function of time, fitting for the pole direction at the TDB 1986 Jan 19.5 (i.e., Jan 19 12:00) reference epoch assumed for the ring orbital elements. The brick-red oscillatory path labeled ura178 is the pole direction given by this trigonometric series over the interval 1600–2600, beginning at the filled circle at the lower left of the path. The ura178 pole direction over the interval of the occultation observations 1977–2006 is shown as a thicker circular line near the center of the figure. The semimajor axes of the rings and the RMS residuals for this fit were virtually identical to those in Fit 1, indicating that the predicted pole precession has little effect on the derived ring system geometry, as was also found by Jacobson (2014, 2023). The error ellipse for Fit 3 is slightly displaced from the Fit 1 error ellipse. Based on this insensitivity of our results to the inclusion of pole precession, we adopt a fixed pole direction of Fits 1 and 2 for our preferred orbit solution, with the error bars from Fit 2.

*   **Fit 4:** Earth-based observations only. This fit included only the Earth-based ring occultation data and reveals the importance of the unique geometry of the *Voyager* occultations in helping to constrain the pole direction and radius scale. The corresponding elongated solid blue error ellipse in Fig. 24 reflects the strong correlation $ρ(α_P, δ_P) = -0.88$ resulting from the limited range of viewing geometry of the ring plane available from the ensemble of Earth-based observations. Although the *Voyager* data contributed only 38 ring measurements (compared to 613 Earth-based measurements), they significantly reduce the estimated error in the pole direction (and radius scale, as we show below). Note that the error ellipses for Fits 1 and 2 are nearly centered on the Fit 5 error ellipse, reflecting the fact that the *Voyager* ephemeris was constrained by both the Earth-based and spacecraft ring observations.

*   **Fit 5:** Jacobson (2023) solution. This pole solution was determined as part of the development of the ura178 series of ephemerides that made use of nearly all the Uranus ring occultation data in Paper 1, with the exception of the $γ$ and $λ$ rings. Extensive detailed comparisons of fits using only the COR ring data showed virtually perfect agreement between Jacobson's test cases (personal communication) and our Fit 1 results, a robust confirmation of the validity of our two independent orbit fitting codes. In contrast to Fits 1–4, the Fit 5 pole solution was based on a comprehensive fit to extensive historical astrometric and spacecraft navigational data, resulting in a somewhat larger error ellipse (displaced slightly from our adopted Fit 1 pole) that incorporates uncertainties in these additional data sources and provides a more conservative estimate of the accuracy of the pole determination.

*   **Fit 6:** Jacobson (2014) solution. The Jacobson (2014) pole direction at the J2000 epoch ($α = 77.310 ± 0.002°$ and $δ = 15.172 ± 0.002°$) was derived using a subset of the COR ring occultation observations included here, excluding the $γ$ and $δ$ rings and post-1991 data. The *Voyager* occultation geometry was calculated from the vgr2.ura111.bsp ephemeris, which was developed using extensive spacecraft navigation and satellite astrometry observations, but no ring occultation data. The Jacobson (2014) pole is shown as the dotted brown error ellipse with an assumed correlation $ρ(α_P, δ_P) = 0$. The systematic offset from the Fit 1 pole position is due primarily to differences between the updated vgr2.ura178.bsp ephemeris and the earlier spacecraft ephemeris, as well as to the use by Jacobson (2014) of keplerian ring orbital elements that modeled the $η$ ring as inclined and eccentric, excluded the $γ$ and $δ$ rings, and neglected the contributions of normal modes to the $η$ and $ε$ ring shapes.<fnref n="9" />

*   **Fit 7:** French et al. (1991) solution. The French et al. (1991) pole direction at the B1950 epoch ($α = 76.5969 ± 0.0034°$ and $δ = 15.1117 ± 0.0033°$) was derived using a subset of the ring occultation observations included here, excluding the post-1991 data, and based on an early *Voyager* ephemeris. The result (precessed to the J2000 frame) is shown as the dotted green error ellipse with an assumed correlation $ρ(α_P, δ_P) = 0$. The fitted pole direction is remarkably close to the Fit 1 result, although with a much larger error estimate. The similarity in pole directions for Fits 1 and 7 is due in part to the largely overlapping data sets and to a likely similarity in the two *Voyager* ephemerides.

<footnote n="9">⁹ In contrast, the Jacobson (2023) solution for the Uranus system geometry fitted for the same normal modes in the ring midlines as in our orbit fits, and closely matched our keplerian ring orbital elements.</footnote>
<table><caption>**Table 13.** Uranus pole direction and ring plane radius scale</caption><thead><tr><th>Fit</th><th>α<sub>P</sub> (deg)</th><th>δ<sub>P</sub> (deg)</th><th>ρ<sup>(a)</sup></th><th>Δα<sub>P</sub> (deg)</th><th>Δδ<sub>P</sub> (deg)</th><th>⟨Δa⟩ (km)</th><th>σ(Δa) (km)</th><th>Description</th></tr></thead><tbody><tr><td>1</td><td>77.311327 ± 0.000128</td><td>15.172795 ± 0.000472</td><td>-0.22</td><td>—</td><td>—</td><td>—</td><td>—</td><td>nominal pole direction</td></tr><tr><td>2</td><td>77.311327 ± 0.000141</td><td>15.172795 ± 0.000618</td><td>-0.42</td><td>—</td><td>—</td><td>—</td><td>—</td><td>w/ Vgr trajectory errors</td></tr><tr><td>3</td><td>77.311210 ± 0.000128</td><td>15.172762 ± 0.000471</td><td>-0.22</td><td>-0.000117</td><td>-0.000033</td><td>-0.001</td><td>0.003</td><td>ura178 pole</td></tr><tr><td>4</td><td>77.311246 ± 0.000274</td><td>15.173393 ± 0.002013</td><td>-0.88</td><td>-0.000081</td><td>0.000597</td><td>0.278</td><td>0.027</td><td>Earth-based only</td></tr><tr><td>5</td><td>77.311200 ± 0.000400</td><td>15.172400 ± 0.001700</td><td>-0.39</td><td>-0.000127</td><td>-0.000395</td><td>—</td><td>—</td><td>Jacobson (2023)</td></tr><tr><td>6</td><td>77.310000 ± 0.003000</td><td>15.172000 ± 0.002000</td><td>[1.0]</td><td>-0.001327</td><td>-0.000795</td><td>-0.015</td><td>0.091</td><td>Jacobson (2014)</td></tr><tr><td>7</td><td>77.310877 ± 0.003400</td><td>15.174564 ± 0.003300</td><td>[1.0]</td><td>-0.000450</td><td>0.001768</td><td>-0.055</td><td>0.092</td><td>French et al. (1991)</td></tr></tbody></table>

(a)Correlation coefficient $ρ(α_P, δ_P)$

<figure><img src="image_26.png" /><figcaption>**Figure 24.** Uranus pole direction in the J2000 reference frame from the fits listed in Table 13. Ellipses are shown as solid lines for Uranus ring and pole fits performed using the data in Paper 1, and as dotted lines for previously published results. See text for details of each fit.</figcaption></figure>

## 5.2. Ring plane radius scale

A secure dynamical association between satellite resonances and the narrow Uranian rings depends on the accuracy of the absolute radius scale. Figure 25 shows the differences $Δa$ in the semimajor axes of the nine main rings between the fits listed in Table 13 and our adopted geometric solution (Fits 1 and 2 have identical pole direction and ring orbital elements, and differ only in the uncertainty in the pole direction), plotted as a function of orbital radius and displaced horizontally for clarity. (Note that $Δa$ should not be confused with $Δa_ω$, $Δa_Ω$, and $Δa_{Ω_P}$ in Tables 5 and 6.) Table 13 includes $⟨Δa⟩$, the average offset in fitted semimajor axes from the Fit 1 values, and $σ(Δa)$, the standard deviation of the offsets. The error bars on each point in the figure correspond to the individual $1-σ$ uncertainties in the
<figure><img src="image_27.png" /><figcaption>**Figure 25.** Ring plane radius scale. The differences $\Delta a$ in the semimajor axes of the nine main rings between fits listed in Table 13 and our adopted geometric solution are shown as a function of orbital radius, displaced horizontally for clarity. The error bars on each point are the individual $1-\sigma$ uncertainties in the determination of the semimajor axis for each ring in each fit.</figcaption></figure>

determination of the semimajor axis for each ring in each fit. The Fit 1 error bars are those in our adopted orbit fit in Table 5. As described above, for Fit 2 we incorporated the uncertainties in the *Voyager* ephemeris into the overall error estimate for the pole direction. From this same series of Monte Carlo simulations, the RMS scatter in the fitted semimajor axes of the rings due to spacecraft trajectory uncertainties was quite small, varying from 0.033 km for ring 6 to 0.027 for the $\epsilon$ ring. The error bars shown (in red) for Fit 2 were computed by combining these small errors in quadrature with the formal errors in Fit 1. The results are so similar to those in Fit 1 that we have not modified the error bars for the orbital elements in Table 13 to include the very small contribution due to spacecraft ephemeris uncertainties. The fitted semimajor axes and orbital elements from Fit 3, which included pole precession, are virtually identical to those in Fits 1 and 2, with $\langle\Delta a\rangle = -0.001$ km and $\sigma(\Delta a) = 0.003$ km.

Fit 4 (shown as blue squares in Fig. 25), based solely on Earth-based data, deserves special attention. The fitted semimajor axes are systematically higher ($\langle\Delta a\rangle = 0.278$ km), and with substantially larger error bars than in Fits 1 and 2 that included the *Voyager* data. The systematic offset in radius scale is directly related to the corresponding offset in the fitted pole directions between the same two fits shown in Fig. 24, and reflects the strong correlation between the absolute radius scale and the assumed pole direction. This is understandable on a geometric basis insomuch as the separation of ingress and egress ring event times depends on the projection of the ring orbits into the skyplane resulting from the assumed orientation of the pole.

Finally, the radius scales of two previous determinations that included *Voyager* observations, but based on different spacecraft ephemerides, are very similar to our adopted solution: the Fit 6 (Jacobson 2014) semimajor axes differed from the Fits 1 and 2 values on average by only $\langle\Delta a\rangle = -0.015$ km, with an RMS scatter $\sigma(\Delta a) = 0.092$ km, and the corresponding Fit 7 (French et al. 1991) results were $\langle\Delta a\rangle = -0.053$ km and $\sigma(\Delta a) = 0.093$ km.
Based on the demonstrated importance of the *Voyager* observations to constraining the pole direction (Fig. 24), the resultant reduction in error estimates for the ring radii, and the similarity of the present and previously-published determinations of the radius scale based on independent *Voyager* ephemerides, we estimate that our adopted radius scale (Fits 1 and 2) is accurate to ~0.2 km at the 2-σ level.

## 6. RING WIDTHS, SHAPES, AND MASSES

We now turn to an examination of the widths, shapes, and masses of the rings, based on the centerline (COR) and edge (IER/OER) fits given in Tables 5 and 6. Since differential precession of an eccentric ringlet would result in streamline crossing on a dynamically short timescale compared to the presumed age of the ring, it is commonly assumed that narrow rings precess uniformly, enforced either by the collective effects of collisions and self-gravity or by external forcing from a satellite (see Longaretti (2018) for a detailed review of these and other proposed mechanisms). The self-gravity explanation was originally proposed by Goldreich and Tremaine (1979), motivated by the observed elliptical shape of the ϵ ring.

Borderies et al. (1983) developed a two-streamline self-gravity formalism that Nicholson et al. (2018b) applied to observations of narrow rings, and it provides a useful starting point for interpreting the observed variation of width with ring radius.<fnref n="10" /> In this model, differential precession is balanced by self-gravity for a free $m = 1$ mode (i.e., not forced by a satellite) when the following equilibrium condition (Eq. (10.109), Longaretti (2018) holds:

$$ \frac{\delta \epsilon}{\epsilon} = \frac{21\pi}{4} J_2 \left(\frac{R}{a}\right)^2 \frac{M_{\text{Ur}}}{M_r} \left(\frac{\delta a'}{a}\right)^3 \frac{1}{H(q_e^2)}, \quad (13) $$

where $ϵ$ and $δϵ$ are the mean eccentricity and the difference in eccentricities of the two streamlines, $a$ and $δa'$ are the mean value and difference in the streamline semimajor axes, $J_2$ is the second-order gravitational harmonic coefficient with reference radius $R$, $M_{Ur}$ is the planet mass, $M_r$ is the ring mass, $q_e = aδϵ/δa'$, and

$$ H(q_e^2) = \frac{1 - (1 - q_e^2)^{1/2}}{q_e^2(1 - q_e^2)^{1/2}}. \quad (14) $$

Relating the two-streamline quantities $ϵ$, $δϵ$ and $δa'$ to the observed eccentricities and semimajor axes of the ring edges, we have

$$ \epsilon = e = e_{\text{COR}}, \quad (15) $$

$$ \delta e = e_{\text{OER}} - e_{\text{IER}} \approx 2\delta\epsilon, \quad (16) $$

$$ \delta a = a_{\text{OER}} - a_{\text{IER}} \approx 2\delta a', \quad (17) $$

and

$$ a = a_{\text{COR}} \quad (18) $$

The eccentricity gradient $q_e$ is identical for both the two-streamline model and the actual ring observations:

$$ q_e = \frac{a\delta\epsilon}{\delta a'} = \frac{a\delta e/2}{\delta a/2} = \frac{a\delta e}{\delta a}. \quad (19) $$

The variation in the observed width with orbital radius around the elliptical ring is $dW/dr \approx a\delta e/ae = \delta e/e = 2a\delta\epsilon/ae = 2\delta\epsilon/e$. Explicitly including the factors of two in Eqs. (16) and (17) when translating from the two-streamline quantities $δϵ$ and $δa'$ to their observational counterparts $δe$ and $δa$, we have

$$ \frac{dW}{dr} = \frac{\delta e}{e} = \frac{2\delta \epsilon}{\epsilon}. \quad (20) $$

(The dimensionless quantity $dW/dr \approx \delta e/e$ should not be confused with $q_e = a\delta e/\delta a$.)

<footnote n="10">¹⁰ We express these in terms of the *observed* widths and shapes of the normal modes at the actual ring edges and centerlines, rather than in terms of the spacing and eccentricities of the two streamlines, as assumed by Borderies et al. (1983) and Longaretti (2018). These alternate approaches differ by a factor of two in the definitions of several important geometric quantities. We identity these differences where they occur.</footnote>
For $m \neq 1$, the corresponding result is

$$ \frac{dW_m}{dr} = \frac{\delta e_m}{e_m} = \frac{2\delta\epsilon_m}{\epsilon_m}, \qquad (21) $$

where $a\delta e_m$ is the difference in the mode amplitudes between the outer and inner ring edges and $ae_m$ is the COR mode amplitude $A_m$. (Note that the eccentricity gradient for $m \neq 1$ is $q_{em} = a\delta_{em}/\delta a$.) For modes with $m \le 0$, the self-gravity model predicts that the slope of the width-radius relation will be negative. In the derivation of Eqs. (20) and (21), an underlying assumption of the two-streamline formalism is that the fractional difference in mode amplitude $e_m$ is small, which we will see is not necessarily true for the $\eta, \gamma$, and $\delta$ rings.

An additional quantity of interest is the mean gradient in periapse longitude, given by

$$ q_{\varpi} = -ae\delta\varpi_0/\delta a, \qquad (22) $$

where $\delta\varpi_0 = \varpi_{OER} - \varpi_{IER}$ is the difference between the outer and inner ring edge pericenter longitudes. Eqs. (19) Eq. (22) together define a generalized $q$:

$$ \begin{aligned} q \cos \gamma &= q_e \\ q \sin \gamma &= q_{\varpi}, \end{aligned} \qquad (23) $$

where $\gamma$ is the phase lag of the minimum width of the ring relative to pericenter. Similar expressions apply for the case $m \neq 1$. The ring width varies as

$$ W(f) = \delta a[1 - q \cos(f - \gamma)], \qquad (24) $$

where $f = \lambda - \varpi$ is the true anomaly of the ring's centerline. As noted by Nicholson et al. (2018b), $\tan \gamma \approx -e\delta\varpi/\delta e$, and thus the phase lag can be much larger than the pericenter offset $\delta\varpi$ if $\delta e \ll e$. For a single mode, we require that $q < 1$ to avoid streamline crossing, although in the simultaneous presence of multiple modes this restriction may not apply to each mode individually.

In the idealized case of a uniformly precessing ring, the fitted apse rates of the IER, COR, and OER would be equal:

$$ \dot{\varpi}(a_{IER}) = \dot{\varpi}(a_{COR}) = \dot{\varpi}(a_{OER}). \qquad (25) $$

The inner edge of an eccentric ringlet must thus precess more slowly than expected for an isolated particle in orbit with semimajor axis $a_{IER}$, and the opposite is the case at the outer edge. Recalling the definitions from Section 4.2 given in Eqs. (9)–(11) (i.e., that the fit parameter $\Delta\dot{\varpi}$ is the difference between the observed precession rate and that predicted from the planet's gravity harmonics and satellite precession contributions at the edge in question), uniform precession predicts that at the IER, one would expect $\Delta\dot{\varpi} < 0$ and $\Delta a_{\dot{\varpi}} > 0$. Similarly, for the OER one would expect $\Delta\dot{\varpi} > 0$ and $\Delta a_{\dot{\varpi}} < 0$. For $m \neq 1$, the corresponding conditions at the IER are $\Delta\Omega_P < 0$ and $\Delta a_{\Omega_P} > 0$ and for inclined ringlets, $\Delta\dot{\Omega} > 0$ and $\Delta a_{\dot{\Omega}} > 0$.

Under equilibrium conditions, where the inner and outer ring edges precess at the same rate as the COR, the simple self-gravity model predicts that $\Delta a_{\dot{\varpi}}(\text{COR}) = 0$ and thus that

$$ \begin{aligned} \Delta a_{\dot{\varpi}}(\text{IER}) - \Delta a_{\dot{\varpi}}(\text{COR}) &= \Delta a_{\dot{\varpi}}(\text{IER}) = a_{\text{COR}} - a_{\text{IER}} > 0 \\ \Delta a_{\dot{\varpi}}(\text{OER}) - \Delta a_{\dot{\varpi}}(\text{COR}) &= \Delta a_{\dot{\varpi}}(\text{OER}) = a_{\text{COR}} - a_{\text{OER}} < 0. \end{aligned} \qquad (26) $$

With this dynamical framework in mind, we next describe the shapes, widths, width-radius relations, and comparative precession rates across each ring obtained from the fitted keplerian and normal mode elements from Tables 5 and 6. Table 14 lists the following quantities for each ring: mean semimajor axis $\bar{a}$, width $\bar{W} = \delta a$, wavenumber $m$, mode amplitude $A_m$, difference in mode amplitudes between the inner and outer edges $a\delta e_m = A_m(\text{OER}) - A_m(\text{IER})$, pericenter longitude difference between the outer and inner edges $\delta\varpi_{0m}$, gradients $q_{em}$ and $q_{\varpi m}$, width-radius relation $dW_m/dr = a\delta e_m/\bar{a}e_m$, and $\delta\Omega_P$, the observed difference in pattern speeds between the outer and ring edges, defined for $m = 1$ as

$$ \delta\Omega_P = \delta\dot{\varpi} = \dot{\varpi}(\text{OER}) - \dot{\varpi}(\text{IER}) \qquad (27) $$

and for $m \neq 1$ as

$$ \delta\Omega_P = \Omega_P(\text{OER}) - \Omega_P(\text{IER}). \qquad (28) $$
<table><caption>Table 14. Width-radius results</caption><thead><tr><th>Ring</th><th>ā<br>km</th><th>W&#773;<br>km</th><th>m</th><th>A<sub>m</sub><br>km</th><th>aδe<sub>m</sub><br>km</th><th>δω<sub>0m</sub><br>deg</th><th>q<sub>em</sub></th><th>q<sub>ωm</sub></th><th>dW<sub>m</sub>/dr</th><th>δΩ<sub>p</sub><br>° yr<sup>-1</sup></th></tr></thead><tbody><tr><td>6</td><td>41837.092</td><td>2.316</td><td>1</td><td>42.499</td><td>0.862</td><td>-0.207</td><td>0.372</td><td>0.077</td><td>0.020</td><td>-0.035</td></tr><tr><td>5</td><td>42234.893</td><td>2.684</td><td>1</td><td>80.237</td><td>0.412</td><td>0.028</td><td>0.153</td><td>-0.004</td><td>0.005</td><td>-0.020</td></tr><tr><td>4</td><td>42571.124</td><td>3.231</td><td>1</td><td>45.347</td><td>0.810</td><td>-1.103</td><td>0.251</td><td>0.277</td><td>0.018</td><td>0.125</td></tr><tr><td>α</td><td>44718.473</td><td>7.277</td><td>1</td><td>33.916</td><td>2.823</td><td>0.359</td><td>0.388</td><td>-0.139</td><td>0.083</td><td>0.237</td></tr><tr><td>β</td><td>45661.056</td><td>8.516</td><td>1</td><td>20.106</td><td>3.172</td><td>-2.967</td><td>0.372</td><td>1.105</td><td>0.158</td><td>-0.043</td></tr><tr><td>η</td><td>47176.009</td><td>2.227</td><td>3</td><td>0.600</td><td>0.224</td><td>-1.262</td><td>0.101</td><td>0.127</td><td>0.374</td><td>-0.540</td></tr><tr><td>γ</td><td>47626.170</td><td>3.258</td><td>1</td><td>5.306</td><td>0.454</td><td>-0.129</td><td>0.139</td><td>0.018</td><td>0.086</td><td>0.280</td></tr><tr><td></td><td></td><td></td><td>0</td><td>5.509</td><td>-1.435</td><td>3.605</td><td>-0.440</td><td>1.587</td><td>-0.261</td><td>0.135</td></tr><tr><td></td><td></td><td></td><td>6</td><td>0.637</td><td>0.302</td><td>-5.378</td><td>0.093</td><td>0.499</td><td>0.475</td><td>0.530</td></tr><tr><td></td><td></td><td></td><td>-1</td><td>1.822</td><td>0.299</td><td>-6.051</td><td>0.092</td><td>0.556</td><td>0.164</td><td>—</td></tr><tr><td></td><td></td><td></td><td>-2</td><td>0.690</td><td>-1.099</td><td>—</td><td>-0.337</td><td>—</td><td>-1.592</td><td>4.742</td></tr><tr><td></td><td></td><td></td><td>3</td><td>0.577</td><td>0.935</td><td>—</td><td>0.287</td><td>—</td><td>1.622</td><td>-0.790</td></tr><tr><td>δ</td><td>48300.227</td><td>4.977</td><td>2</td><td>3.169</td><td>1.686</td><td>-0.438</td><td>0.339</td><td>0.148</td><td>0.532</td><td>-0.146</td></tr><tr><td></td><td></td><td></td><td>23</td><td>0.339</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>ε</td><td>51149.279</td><td>58.574</td><td>1</td><td>405.894</td><td>38.712</td><td>-0.028</td><td>0.661</td><td>-0.019</td><td>0.095</td><td>-0.003</td></tr></tbody></table>

### 6.1. Rings 6, 5, 4, α, and β

We begin with the five inner rings, each of which has a measurable eccentricity and inclination but no detected free or forced normal modes. In Fig. 26, each column shows key characteristics for a given ring. In the top row, $\Delta r(\lambda) = r(\lambda) - a_{COR}$ is the difference between the model ring radius $r(\lambda)$ of the $m=1$ mode (computed from Eq. (1)) at the epoch of the orbit fit) and the mean semimajor axis $a_{COR}$, plotted as a function of inertial longitude $\lambda$ at epoch for each ring's inner edge (blue), midline (green), and outer edge (red). In the second row, the ring widths $W(\Delta r)$ are plotted as a function of $\Delta r = r_{COR} - a$. When the longitudes of the periapses of the two edges are aligned, $W(r)$ is a straight line with slope $dW/dr = \delta e/e$, but for misaligned apses, the shape is an ellipse. All five inner rings show a positive slope, as predicted for a self-gravitating eccentric ring by Eq. (20). The apses are most closely aligned for rings 6, 5, and $\alpha$, with more significant misalignments for rings 4 and $\beta$, as indicated by the elliptical shapes of the width-radius relations.

The third row of Fig. 26 shows the semimajor axes (dashed lines) calculated from the fitted pattern speeds of the normal modes (in this case, the periapse precession rates) at the ring edges and midlines, compared to the observed ring midline and edge semimajor axes themselves, shown as solid lines. The computed semimajor axes (generically referred to as “resonance radii” to include the case of modes forced by satellites) based on the combined precessional effects of the gravity field of the planet and the known satellites closely match the ring midlines for all but ring $\alpha$, which together with ring $\beta$ precess somewhat faster than predicted for the mean radii of these rings. This anomalous precession may be due to small unseen satellites, as discussed below in Section 8.

Finally, the fourth row compares the observed pattern speeds of the IER, COR, and OER relative to the predicted pattern speed of the COR, plotted as a function of semimajor axis. In the ideal case of a uniformly precessing ring, the precession rates $\dot{\omega}$ of the IER, COR, and OER should be equal. Systematic offsets between the observed and predicted COR precession rates are in some cases due to forced satellite modes, where the forcing frequency is close to but not equal to the COR precession rate, and is also affected in some cases by the difference between the geometric midline of a ring and its radial center of mass, an issue addressed in Section 7.1. Statistically significant differences in the pattern speeds at the ring edges such as those seen for rings 4 and $\alpha$ could be indications of libration or circulation of a normal mode, as has been observed at the outer edge of Saturn's B ring (Spitale and Porco 2010; Nicholson et al. 2014a; French et al. 2023b). We explore these possibilities in Section 6.5.

### 6.2. The $\eta$ and $\delta$ rings
<figure><img src="image_28.png" /><figcaption>**Figure 26.** Width variations of rings 6, 5, 4, $\alpha$, and $\beta$. The top row shows $\Delta r(\lambda)$, the variation of ring radius with inertial longitude $\lambda$ at epoch for the IER, COR, and OER (blue, green, and red, respectively. The second row shows the corresponding variation of ring width $W(\Delta r)$, and the third row shows the predicted resonance radii for each detected mode (shown dashed) relative to ring's semimajor axis $a_{COR}$, compared to the locations of the ring edges (shown as solid lines). The final row shows the differences in the observed pattern speeds of the inner and outer edges, and that of the COR, relative to the *predicted* COR value (based on the Uranus gravity field and satellite contributions to the precession rates), plotted as a function of semimajor axis relative to $a_{COR}$.</figcaption></figure>

**Figure 27** shows the corresponding results for the $\eta$ and $\delta$ rings. In the top panel, the edge and midline locations are plotted as a function of $m\lambda$, since there are $m$ complete normal mode cycles around the circumference of the ring. The $\eta$ ring is nearly constant in width, with a very small but positive $dW/dr$, and has nearly aligned mode pericenters. The $m=3$ normal mode is clearly driven by the 3:2 ILR with Cressida, as discussed by Chancia et al. (2017) and confirmed here by the resonance locations for the edges and midline being about 5 km interior to the ring and well-matched to the Cressida resonance radius calculated from the satellite mean motion of Showalter and Lissauer (2006), labeled as SL06. The fitted $m=3$ pattern speeds for the IER, COR, and OER agree within the error bars, but are controlled by the forcing frequency of Cressida, rather than by the gravity field of the planet, resulting in the large vertical offset in the left panel in the fourth row of the figure.

The $\delta$ ring $m=2$ mode has a substantial width variation, with nearly identical calculated resonance radii for the IER, COR, and OER falling very near to the ring midline. It has a positive width-radius relation, as expected from Eq. 21 for a self-gravitating ring with free normal mode $m > 1$, with nearly aligned pericenters at the ring midline
and edges. The fitted $m = 2$ pattern speeds for the IER, COR, and OER agree within the error bars, as shown in the middle panel in the fourth row of the figure.

The $\delta$ ring $m = 23$ mode is relatively weak, with $A_m = 0.342 \pm 0.063$ km, and is detectable only in the ring midline (see Table 6) – the outer and inner ring edges are shown as dashed lines in the upper right panel of the figure, and $W_m(\Delta r)$ is unobserved. The resonance location for the observed pattern speed matches the Jacobson (1998) value for the mean motion of Cordelia (labeled Jac98 in the figure), and lies just inside the outer edge of the $\delta$ ring. Once again, there is a large vertical offset between the observed pattern speed and the predicted COR pattern speed, since the $m = 23$ mode is controlled by the forcing frequency of Cordelia. We will make use of the measured amplitude and resonance location to estimate Cordelia’s mass in Section 9.

### 6.3. The $\gamma$ ring

The $\gamma$ ring has the most complex shape of all the Uranian rings, being distorted by six distinct normal modes. The contributions of each mode to the ring width and shape are shown in Figure 28, with the following notable features:

* The $m = 1$ mode has a large COR amplitude $A_1 = 5.306 \pm 0.071$ km and well-aligned apses, but it contributes rather little to the overall width variation of the ring itself, with a very small but positive slope $dW_m/dr = 0.086$. The outer edge of the ring shows an anomalously high $m = 1$ precession rate: from Table 5 (for the model including the $m = 3$ normal mode) the resonance radius computed from the fitted apse rate for the OER lies 8.591 km interior to the COR semimajor axis, as shown in the left panel of row three of Fig. 28.

* The $m = 0$ mode has a similarly large COR amplitude $A_0 = 5.511 \pm 0.076$ km, but unlike the $m = 1$ mode, it is narrowest at apoapse and widest at periapse, resulting in a negative value of $dW_m/dr$ and slightly misaligned periapse phases, producing the elliptical shape of the $W_m(\Delta r)$ plot. The resonance location closely matches the semimajor axis of the ring. The negative slope of the width-radius relation is consistent with the self-gravity model prediction for an outer Lindblad resonance (OLR) with $m \le 0$ (see Eq. (21)). The width of the $\gamma$ ring is primarily controlled by the $m = 0$ mode and not the $m = 1$ mode, even though both modes have comparable $ae \sim 5$ km. This observation supports the physical argument posited by Nicholson et al. (2018b) that self-gravity needs to overcome only the relatively slow differential precession for an $m = 1$ perturbation, whereas for $m \ne 1$ it is necessary to overcome the much greater gradient in mean motion.

* The weaker $m = 6$ mode is driven by the 6:5 ILR with Ophelia, with edge and midline pattern speeds matching the Jacobson (1998) value for the satellite mean motion within its error bars (labeled Jac98 in the figure). The more accurate recent Ophelia mean motion (Robert French, personal communication – labeled RF23) has a predicted resonance location that closely matches that of the observed $m = 6$ pattern speed, lying within the ring itself. The proximity of the Ophelia 6:5 resonance to the inner edge of the $\gamma$ ring is surprising and the opposite of standard theories of resonant shepherding. A potential analog to this structure is the ringlet ER1 in Saturn’s inner C ring. The outer edge of this ringlet falls near a resonance with a planetary normal mode (French et al. 2021), which is also the opposite of expectations since the pattern speed of this normal mode is faster than the ring’s orbital motion. Both these rings may have both their edges shepherded by their respective resonances, as predicted by Borderies et al. (1986).

* The $m = -1$ mode has a small positive $dW_m/dr = 0.164$, contradicting the expectation that for an isolated OLR, the slope should be negative. Given the multiple simultaneous modes in the $\gamma$ ring and the small amplitude of this mode, this apparent discrepancy may not be dynamically significant.

* The $m = -2$ mode is undetectable on the outer edge of the ring, and is strongest on the inner edge. In this case, $dW_m/dr = -1.592$, negative as expected for this OLR, and second in magnitude of all measured modes in any of the rings, with the exception of the $m = 3$ mode described next.

* The $m = 3$ mode was detected only on the outer edge and midline of the ring, with a positive $dW_m/dr = 1.622$, the largest value for any of the modes identified here. The fitted apses of the COR and OER $m = 3$ modes
<figure><img src="image_29.png" /><figcaption>**Figure 27.** Width variations of rings *η* and *δ*, with the same layout as for Fig. 26. The predicted resonance location for Cressida, assuming the mean motion given by Showalter and Lissauer (2006) (labeled SL06), is shown as a dashed line for the *η* ring *m* = 3 mode, and lies ~ 5 km interior to the ring's midline. The predicted resonance location for Cordelia, assuming the mean motion given by Jacobson (1998) (labeled Jac98), is shown as a dashed line for the *δ* ring *m* = 23 mode, and lies just interior to the outer edge of the ring. The *m* = 23 mode was not detected in the shape of the IER or OER of the *δ* ring.</figcaption></figure>
are well-aligned and match closely in pattern speed. There is no known satellite that forces this mode. On the other hand, if this is a free mode, its resonance radius is quite far removed from the ring midline, unlike the other free modes identified in the Uranus system. The pattern speed of the $m = 3$ structure is much less than that expected for fundamental planetary normal modes (A'Hearn et al. 2022) and is significantly larger than the planet's rotation rate (Desch et al. 1986; Warwick et al. 1986; Helled et al. 2010). Thus this pattern is unlikely to be generated by either a fundamental planetary normal mode or a persistent anomaly carried around by the planet's winds similar to those that produce waves in Saturn's rings (Hedman et al. 2022). If this signal turns out to be real and generated by the planet, it therefore would have to be due a more complex and possibly mixed planetary mode. A final possibility is that this putative detection is instead a random pattern emerging from the co-addition of five other modes to the shape of the ring. We will see below in Section 8, however, that exclusion of this mode from the ensemble of $\gamma$ ring modes results in a very large anomalous $m = 1$ precession rate with no identifiable source. Furthermore, the RMS residuals for the $\gamma$ ring COR fit when the mode is absent increase from 0.401 km to 0.490 km, the largest of any of the rings; for the OER, the increase is from 0.692 km to 0.895 km (Table 5). At this point, the origin of the $m = 3$ mode remains mysterious.

## 6.4. The $\epsilon$ ring

The $\epsilon$ ring is the widest and most eccentric of the Uranian rings, with a well-defined width-radius relation ($dW_m/dr = 0.095$) shown in Fig. 29. Its width varies by almost a factor of five, from 19.9 km at periapse to 97.3 km at apoapse. The periapses of the inner and outer edges are aligned to within $0.028^{\circ}$ and the resonance radius for the $m = 1$ mode closely matches the midline of the ring, which is not surprising since the $\epsilon$ ring establishes a tie point for the determination of the planet's gravity field that governs its precession rate. The very small systematic offset between the observed IER, COR, and OER pattern speeds and the predicted COR value is a consequence of the radial offset of the geometric center of the ring from the ring center of mass, discussed in Section 7.1. The resonance radius of the $m = -24$ OLR lies within the ring 1.082 km outside the ring's IER, near to the $m = -24$ resonance of Cordelia computed from the mean motion given by Jacobson (1998), as shown at upper right. Similarly, the resonance radius of the $m = 14$ ILR also lies within the ring 0.887 km inside the ring's OER, near to the mean motion resonance of Ophelia computed from the Jacobson (1998) mean motion, as shown at lower right. The more accurate recent Ophelia mean motion (Robert French, personal communication - labeled RF23) closely matches the observed $m = 14$ pattern speed.

## 6.5. Ring masses, surface densities, and a search for librations

In Section 6.1, we compared the pattern speeds of the inner and outer edges of the eccentric rings, and in several instances there appear to be significant differences that could be indications of librations, as has been observed at the outer edge of the B ring (Spitale and Porco 2010; Nicholson et al. 2014a; French et al. 2023b). Notably, $\Delta\bar{\omega}$ for ring 4 and the $\alpha$ ring, and $\Delta\bar{\omega}_0$ for the $\beta$ ring, appear to be significantly different from zero (see Table 5 and Fig. 26), although neither is statistically significant for the $\epsilon$ ring, where the experimental uncertainties are a factor of 10 smaller.

From the two-streamline self-gravity model of eccentric rings, Borderies et al. (1983) identified a characteristic libration frequency $\Omega_{sg}$ given by

$$ \Omega_{sg} = \frac{n M_r}{\pi M_{\text{Ur}}} \left(\frac{a}{\delta a'}\right)^2 H(q_e^2) = \frac{4n M_r}{\pi M_{\text{Ur}}} \left(\frac{a}{\delta a}\right)^2 H(q_e^2), \qquad (29) $$

where $n \approx \sqrt{GM_{\text{Ur}}/a^3}$ is the ring particle mean motion and the factor of 4 in the right hand expression arises from the difference between $\delta a'$ and $\delta a$ (Eq. (17)). The function $H(q_e^2)$ is defined in Eq. 14.

Assuming that self-gravity prevents differential precession due primarily to $J_2$ and combining Eqs. (13) and (29), the libration period for an eccentric ringlet $P_{\text{lib}} = 2\pi/\Omega_{sg}$ is given by

$$ P_{\text{lib}} = \frac{8\pi}{21} \frac{a^{9/2}}{J_2 R^2 (GM_{\text{Ur}})^{1/2}} \frac{q_e}{ae}. \qquad (30) $$
<figure><img src="image_30.png" /><figcaption>**Figure 28.** Width variations of the γ ring due to six normal modes, with the same layout as for Fig. 26. The predicted resonance location for Ophelia, assuming the mean motion given by Jacobson (1998) (labeled Jac98), is shown as a dashed line (bounded by its uncertainty) for the *m* = 6 mode, and lies within the ring, near its inner edge. The more accurate recent Ophelia mean motion (Robert French, personal communication – labeled RF23) has a predicted resonance location matching that of the observed *m* = 6 pattern speed. The *m* = -2 mode was not detected in the shape of the OER, and the *m* = 3 mode was not detected in the shape of the IER. The final row shows the differences in the observed pattern speeds of the inner and outer edges, and that of the COR, relative to the predicted COR value (based on the Uranus gravity field and satellite contributions to the precession rates), plotted as a function of semimajor axis relative to *a*<sub>COR</sub>. Note the different vertical axis ranges for the final row of plots.</figcaption></figure>

Alternatively, this can be written more simply as

$$P_{\text{lib}} = \left( \frac{2\pi}{\frac{d\dot{\varpi}}{da}\delta a} \right) \frac{\delta e}{e} = P_{\Delta\dot{\varpi}} \frac{\delta e}{e}, \qquad (31)$$

where $\frac{d\dot{\varpi}}{da}\delta a$ is just the differential precession frequency across the ring and the factor in parentheses is the corresponding differential precession period $P_{\Delta\dot{\varpi}}$ of the inner and outer ring edges. Since $\delta e/e \ll 1$ for most of the eccentric rings, the predicted libration periods are considerably shorter than the differential precession periods.

Substituting numerical values appropriate for Uranus into Eq. (30),

$$P_{\text{lib}} = 960 \left( \frac{a}{50,000 \text{ km}} \right)^{9/2} \left( \frac{10 \text{ km}}{ae} \right) q_e \text{ yr.} \qquad (32)$$
<figure><img src="image_31.png" /><figcaption>**Figure 29.** Width variations of the $\epsilon$ ring. Upper left: $\Delta r(\lambda)$, the variation of ring radius with inertial longitude $\lambda$ at epoch for the IER, COR, and OER (blue, green, and red, respectively. Lower left: the corresponding variation of ring width $W(\Delta r)$. Upper middle: the predicted resonance radii corresponding to the observed apse rates at each ring edge relative to ring's semimajor axis $a_{COR}$. Lower middle: the difference in the observed pattern speeds of the IER, COR, and IER relative to the predicted COR pattern speed, plotted as a function of semimajor axis relative to $a_{COR}$. Upper right: the resonance location of the $m = -24$ mode (shown as a dashed blue line) relative to the inner edge of the $\epsilon$ ring (solid blue line), computed from the fitted $m = -24$ pattern speed of the IER. The corresponding resonance location computed from the Jacobson (1998) mean motion of Cordelia (labeled Jac98) is shown as a black dashed line, bounded by its uncertainties shown as dotted lines. Lower right: the resonance location of the $m = 14$ mode (shown as a dashed red line) relative to the outer edge of the $\epsilon$ ring (solid red line), computed from the fitted $m = 14$ pattern speed of the OER. The corresponding resonance location computed from the Jacobson (1998) mean motion of Ophelia is shown as a black dashed line labeled Jac98, bounded by its uncertainties shown as dotted lines. The more accurate recent Ophelia mean motion (Robert French, personal communication – labeled RF23) has a predicted resonance location that closely matches that of the observed $m = 14$ pattern speed.</figcaption></figure>

From the two-streamline self-gravity model, the mass of an eccentric ringlet (i.e., with a free $m = 1$ mode) is given by

$$M_r = \frac{21\pi}{16} J_2 \left(\frac{R}{a}\right)^2 \left(\frac{e}{\delta e}\right) \left(\frac{\delta a}{a}\right)^3 \frac{M_{U_r}}{H(q_e^2)} \quad (33)$$

For a ringlet with a free mode with $m \neq 1$, the corresponding relations are

$$P_{\text{lib}} = \frac{4\pi}{3(m-1)n} \frac{q_e}{e_m} = 3.1 \frac{1}{(m-1)} \left(\frac{a}{50,000 \text{ km}}\right)^{5/2} \left(\frac{10 \text{ km}}{ae_m}\right) q_e \text{ yr} \quad (34)$$

and

$$M_r = \frac{3\pi(m-1)}{8} \left(\frac{e_m}{\delta e_m}\right) \left(\frac{\delta a}{a}\right)^3 \frac{M_{U_r}}{H(q_e^2)} \quad (35)$$

As noted by Goldreich and Tremaine (1979), the two-streamline self-gravity ring mass estimate differs from the results of more careful treatments that use a large number of streamlines $N \gg 2$. Assuming that $q_e$ is constant across
the ring and that $\delta e/e \ll 1$, numerical results show that the many-streamline ring mass from the standard self-gravity (SSG) model converges almost exactly to

$$M_{\text{SSG}} = \frac{1}{2} M_r \qquad (36)$$

for all of the eccentric Uranus rings.

The corresponding mean surface mass density from the many-streamline standard self-gravity model is given by

$$\Sigma_{\text{SSG}} = \frac{M_{\text{SSG}}}{2\pi a \delta a} = 3.18 \left( \frac{M_{\text{SSG}}}{10^{14} \text{ kg}} \right) \left( \frac{50,000 \text{ km}}{a} \right) \left( \frac{10 \text{ km}}{\delta a} \right) \text{g cm}^{-2}. \qquad (37)$$

The standard self-gravity model has been criticized on a number of grounds. For example, the derived surface densities for the $\alpha$ and $\beta$ rings are quite low, and inconsistent with the inferred large particle sizes for these rings (French et al. 1991). Furthermore, the observed variations in the radial optical depth profile of the $\epsilon$ ring over a range of true anomalies disagree with model predictions (Graps et al. 1995). Goldreich and Porco (1987) pointed out that gas drag due to the extended hydrogen atmosphere of Uranus poses a severe problem for shepherding of the $\alpha$ and $\beta$ rings unless their masses have been seriously underestimated. Finally, the model ignores potentially important dynamical effects, such as the requirement for ring shepherding that there be substantial collisional dissipation near ring edges (Borderies et al. 1982).

Motivated in part by these concerns, Chiang and Goldreich (2000) developed a heuristic collisional self-gravity (CSG) model for apse alignment of narrow eccentric rings that includes the pressure-induced accelerations resulting from interparticle collisions near ring edges. Their model assumes that $\delta e/e \ll 1$, that $q_e$ is constant across the ring, and that the collisional pressure gradient is confined to a localized edge zone of radial scale $\lambda \simeq c_b/n$ (with dispersion velocity $c_b$ and mean motion $n$) over which the ring surface density falls sharply to zero. The combined contributions to the precession rate across the ring due to planetary oblateness, ring self-gravity, and collisions uniquely determine the ring's radial surface density profile and total mass, for an assumed radial scale $\lambda$ near the ring edge. By including the estimated effect of collisions, Chiang and Goldreich (2000) found surface densities for the $\alpha$, $\beta$, and $\epsilon$ rings as large as $\Sigma = 75 - 100 \text{ g cm}^{-2}$, larger by more than a factor of 10 than the SSG values. Subsequently, Mosqueira and Estrada (2002) modified the treatment of the strong density gradients near ring edges in the Chiang and Goldreich (2000) model, but with generally similar results. Qualitatively, both of these CSG approaches result in large ring masses, concentrated near the ring edges, significantly exceeding the SSG values.

Using the width-radius results from Table 14, we have computed the masses and surface densities for each of the rings with a dominant free normal mode, using both the SSG and CSG models, as shown in Table 15, along with the libration periods $P_{\text{lib}}$ predicted from the two-streamline SSG model. For the SSG calculations, we used N=500 streamlines. We adopted the computationally simpler Chiang and Goldreich (2000) solution to illustrate the collisional self-gravity approach, for a representative dispersion velocity of $c_b = 1 \text{ cm s}^{-1}$, corresponding to $\lambda = 35 - 50 \text{ m}$ over the ring system. (For the CSG calculations, we assumed a very large number of streamlines - N=20,000 - to ensure that the narrow transition zone near the ring edge was adequately sampled to provide accurate surface density values.) As these authors note, both $M_{\text{CSG}}$ and $\Sigma_{\text{CSG}}$ depend rather sensitively on the width of the ring edge transition region (or, equivalently, on the dispersion velocity), scaling approximately as $c_b^{1.5}$ in their model. In contrast, the Mosqueira and Estrada (2002) results are more weakly dependent on $\lambda$ but nevertheless typically confine most of the ring mass to a small region near each edge.

To illustrate the general features of the SSG and CSG results, we show in Fig. 30 a set of radial surface density profiles computed for the $\alpha$ ring at quadrature, using N=500 streamlines. The surface density $\Sigma(a)$ is shown on a logarithmic scale at left. Under the assumption that the ring optical depth is proportional to the surface mass density, we show at right the corresponding radial optical depth profiles on a linear scale, normalized at the point of highest surface mass density. The legend at right applies to both panels. Notice first that the collisionless self-gravity profile (i.e., the SSG model) has a convex shape, peaking at the midline of the ring, with a very low maximum surface mass density $\Sigma_{\text{SSG}} \simeq 1 \text{ g cm}^{-2}$. The Chiang and Goldreich (2000) results for $c_b = 1, 2$ and $4 \text{ cm s}^{-1}$ (labeled CG2000) share the same general U-shape, but with substantially greater surface density than the SSG model, increasing as
<table><caption>Table 15. Ring masses, surface densities, and libration periods</caption><thead><tr><th>Ring</th><th>M<sub>SSG</sub> × 10<sup>14</sup> (kg)<sup>a</sup></th><th>Σ<sub>SSG</sub> (g cm<sup>-2</sup>)<sup>a</sup></th><th>P<sub>lib</sub> (yr)<sup>b</sup></th><th>M<sub>CSG</sub> × 10<sup>14</sup> (kg)<sup>c</sup></th><th>Σ<sub>CSG</sub> (g cm<sup>-2</sup>)<sup>d</sup></th></tr></thead><tbody><tr><td>6</td><td>0.04</td><td>0.58</td><td>37.6</td><td>6.78</td><td>111.37</td></tr><tr><td>5</td><td>0.23</td><td>3.20</td><td>8.5</td><td>7.51</td><td>105.47</td></tr><tr><td>4</td><td>0.11</td><td>1.23</td><td>25.6</td><td>8.13</td><td>94.05</td></tr><tr><td>α</td><td>0.19</td><td>0.93</td><td>66.3</td><td>12.46</td><td>60.96</td></tr><tr><td>β</td><td>0.15</td><td>0.60</td><td>118.1</td><td>13.52</td><td>55.33</td></tr><tr><td>δ<sup>d</sup></td><td>3.84</td><td>25.42</td><td>3.0</td><td></td><td></td></tr><tr><td>ε</td><td>32.70</td><td>17.37</td><td>17.1</td><td>69.89</td><td>37.13</td></tr></tbody></table>

a Computed using N= 500 streamlines in the standard self-gravity model (SSG), neglecting collisional effects.

b Computed from Eq. (32).

c Computed using N= 20,000 streamlines in the collisional self-gravity (CSG) model of Chiang and Goldreich (2000) for an assumed dispersion velocity $c_b = 1$ cm s$^{-1}$. As noted in the text, the ring mass and surface density scale as $c_b^{1.5}$.

d Computed from the two-streamline self-gravity solution in Eq. (35) for $m = 2$.

$c_b$ becomes larger and eventually reaching an average value $\Sigma_{CSG} > 100$ g cm$^{-2}$, but even larger values in excess of 1000 g cm$^{-2}$ near the ring edges. (In comparison, even the most opaque regions of Saturn's B ring have surface densities in the range 100-140 g cm$^{-2}$ (Hedman and Nicholson 2016).) The corresponding optical depth profiles at right are completely different from the bowl-shaped SSG profile, showing narrow opaque ring edge regions that increase in width with increasing $c_b$ (or, equivalently, increasing $\lambda$), with much lower optical depth in the central region of the ring. To illustrate the modifications introduced by Mosqueira and Estrada (2002), we used their Eq. (20) representation of the collisional precession term to compute their CSG model, using $N = 500$ streamlines and $c_b = 2$ cm s$^{-1}$ (labeled ME2002). The derived surface density profile shows more massive edge regions than the Chiang and Goldreich (2000) models, with lower mass density in the interior of the ring near the midline. All of the CSG results shown have optical depth profiles that differ substantially from both the SSG model and (perhaps more significantly) are clearly a poor match to the observed shapes of the Voyager optical depth profiles (see Fig. B3).

Reviewing the entries in Table 15, the width-radius results for rings 6, 5, and 4 provide the first opportunity to estimate the masses and surface densities of these very narrow and morphologically similar rings. The SSG results yield very low but disparate surface densities $\Sigma_{SSG}$ between 0.58–3.17 g cm$^{-2}$, roughly comparable to the $\alpha$ and $\beta$ ring values. The CSG surface densities are larger by factors of $\sim 100$, ranging from $\Sigma_{CSG} = 94$ to 111 g cm$^{-2}$, the largest of any of the rings shown. (We exclude the $\gamma$ ring from the table because of the confounding and uncertain influence of its many normal modes.) The CSG surface densities for the $\alpha$ and $\beta$ rings are in the range $\Sigma_{CSG} = 50–60$ g cm$^{-2}$, similar to the results obtained by Chiang and Goldreich (2000), although the detailed models share the same issue in producing predicted radial optical depth profiles that differ substantially from the observations. The $\delta$ ring SSG surface density is computed from the $m = 2$ normal mode properties, rather than from the keplerian ellipse term in its orbital elements, yielding a value in between the $\epsilon$ ring SSG and CSG surface density values. For this very broad ringlet, the SSG and CSG models give broadly similar results, but very different predicted shapes for the radial optical depth profile.

In assessing the importance of collisional self-gravity to the narrow Uranian rings, we note that the physical assumptions of the Chiang and Goldreich (2000) model have a direct influence on the magnitude of the surface density enhancement relative to the standard self-gravity model. The most relevant of these relates to the magnitude of the very narrow scale $\lambda$ over which the ring surface density drops to zero at the ring edge. More specifically, they assume that $\lambda \sim c_b/n$ where $c_b$ is the ring velocity dispersion in the region of width $w$ perturbed by the satellite responsible for the edge confinement, and $n$ is the local mean motion. There is no explicit justification for this choice of $\lambda$ in Chiang and Goldreich (2000), but it is consistent with the following arguments:

* On the one hand, $\lambda$ cannot be smaller than $c_b/n$, otherwise the Rayleigh stability criterion would be violated (the pressure gradient would be large enough to change the specific angular momentum distribution from stable
<figure><img src="image_32.png" /></figure>

**Figure 30.** Surface density profiles for the α ring at quadrature, shown on a logarithmic scale and left and normalized by their maximum values at right on a linear scale to show their predicted normal optical depth profiles. The collisionless self-gravity model (SSG) has a convex shape in both panels and a very low mean surface density $\Sigma_{SSG} < 1\text{g cm}^{-2}$. Three CSG profiles computed using the Chiang and Goldreich (2000) model with dispersion velocities $c_b = 1, 2$, and $4 \text{ cm s}^{-1}$ show much larger mean surface densities, increasing $\propto c_b^{1.5}$, with characteristic U-shaped optical depth profiles at right. The alternate collisional model of Mosqueira and Estrada (2002) is included for $c_b = 2 \text{ cm s}^{-1}$, showing qualitatively similar high surface densities confined to the ring edges.

to unstable (Lin and Papaloizou 1993). If for some reason the drop in surface density at the edge occurred on a yet smaller scale, the ring edge would become unstable and broaden again until stability is restored and this minimum scale satisfied.

* On the other hand, choosing the smallest possible scale $\lambda$ maximizes the pressure gradient at the edge and, as a consequence, the mass increase of a narrow ring at a given geometry with respect to the SSG model.

Maximizing the mass of the ring allowed Chiang and Goldreich (2000) to assess whether the CSG model is able to overcome the mass deficit problem pointed out by Goldreich and Porco (1987). But although explaining sharp ring edges was a puzzle when they were first discovered, making the edge as sharp as possible is not a dynamical necessity; ultimately, the scale $\lambda$ is determined by (unknown) detailed properties of the ring stress tensor. As seen in Figs. 31, B1, and B2, there is no observational reason to assume that the inner edges of rings 6, 5, and 4 are as sharp as assumed by Chiang and Goldreich (2000), and the outer edges, while generally sharper, may be gradual on the scale of tens of meters. Given our current state of understanding, these points indicate that it is probably best to view the SSG surface density as a minimum value and the CSG one as a maximum one.

Turning now to the predicted libration periods in Table 15, note that for most rings they are decades long, making any such libration challenging to detect over the rather sparsely populated 30-year span of the occultation observations. At the other extreme, the *m* = 2 mode of the *δ* ring has a predicted libration period of only 3 years, which is comparable to the typical interval between observations and thus susceptible to aliasing. Using the algorithm of Nicholson et al. (2014a), we performed a series of fits for each ring, similar to normal mode scans, in which we specified a grid of possible libration periods between 1 and 100 years, solved for the best-fitting amplitude and phase of the libration for each period, and tabulated the RMS residuals for each fit. None of the scans returned a convincing, unique fit with a minimum RMS for a single libration period, and there was considerable aliasing for libration periods shorter than $\sim 5$ years owing to the sparse sampling of the observations on this timescale.

Looking in more detail at specific examples, Fig. 26 shows that the inner and outer edges of rings 4 and α have fitted apse rates that differ by $\Omega_P(\text{OER}) - \Omega_P(\text{IER}) = \delta\Omega_P \sim 0.1 - 0.2^\circ \text{yr}^{-1}$ with OER and IER apses misaligned by $\delta\varpi = -1.10 \pm 0.26^\circ$ for ring 4 and by $+0.36 \pm 0.29^\circ$ for the α ring.
These can be compared to the apse misalignment expected over one quarter of a libration period $\Delta\varpi_{\text{lib}} = \Delta\Omega_P P_{\text{lib}}/4$. For ring 4, $\Delta\varpi_{\text{lib}} = 0.8^\circ$, of the same order as $\Delta\varpi$, meaning that the observed apse misalignment is of the same order as the range of excursions expected if the ring is librating. For the $\alpha$ ring, on the other hand, $\Delta\varpi_{\text{lib}} = 3.9^\circ$, more than an order of magnitude larger than the observed apse misalignment. If this ring is librating with a long period compared to the span of the observations, the probability of observing the apses so closely aligned would be rather small. If the ring is librating, $\Delta\Omega_P$ should be at its maximum when the apses are aligned. The $\epsilon$ ring has both well-aligned apses ($\Delta\varpi = -0.028^\circ$ from Table 5) and very small $\Delta\Omega_P < 0.0026^\circ \text{ yr}^{-1}$. Assuming simple harmonic motion, the amplitude of the libration $\Delta\phi \sim \Delta\Omega_P/\Omega_{\text{sg}} = \Delta\Omega_P P_{\text{lib}}/2\pi \sim 0.007^\circ$. That is, if the $\epsilon$ ring is indeed librating with a period of 17.1 yr, the fitted edge apses and apse rates imply that the amplitude of libration is very small ($\sim 0.01^\circ$), below the level of detectability in our libration scans.

In a more systematic attempt to set lower limits on librations, we make use of the fit results in Table 15 and proceed in the follow steps:

1. Assume that $\delta\dot{\varpi} = A_L \cos[2\pi(t-t_0)/P_L]$, where $P_L$ is the (unknown) libration period and $A_L$ is the (unknown) libration amplitude.

2. It follows that $\delta\dot{\varpi} = -\frac{2\pi A_L}{P_L} \sin[2\pi(t-t_0)/P_L]$.

3. Assume the nominal values of $P_L$ given in Table 15, based on the standard SSG model.

4. Use the observed value of either $\Delta\varpi$ or $\Delta\dot{\varpi}$ to estimate a lower limit on $A_L$. The larger amplitude $A_L$ wins.

Following this prescription, we find that for the $\alpha$ ring, $\delta\dot{\varpi}$ dominates and $A_L > 2.4^\circ$. For the $\beta, \delta$, and $\epsilon$ rings, $\delta\varpi$ dominates and $A_L > 3.0^\circ, 0.4^\circ$, and $0.03^\circ$, respectively. For rings 6, 5 and 4 the two limits are comparable (within a factor of 2), with $A_L > 0.2^\circ, 0.03^\circ$, and $1.1^\circ$, respectively.

In principle, more conclusive results could be obtained by subdividing the observations into several time intervals and mapping the variations of the fitted values of $\delta\Omega_P$ and $\delta\varpi$ over time to see if their phases matched the pattern expected for librations. This approach proved unsuccessful, however, owing to the uneven distribution of observations over time and the relatively small number of observations of each ring. We are left with the conclusion that any librations, if present, are at the margin of detectability.

# 7. URANUS GRAVITY FIELD

In the orbit fits for the eccentric and inclined rings given in Table 5, we treated the apsidal precession and nodal regression rates as kinematical parameters without any dynamical constraints. We now use the fitted apse and node rates to constrain the lowest-order zonal harmonic coefficients $J_2, J_4$, and $J_6$ of the Uranus gravity field. An immediate challenge is that the radial range of the measurements spans less than 10,000 km, or equivalently, $R/a$ in Eq. (2) varies only between 0.61 and 0.50 from the ring 6 to the $\epsilon$ ring, whereas to leading order the radial dependence of the $J_n$ terms is a sequence of steepening power laws: the $J_2$ term $\propto a^{-7/2}$, $J_4 \propto a^{-11/2}$, and $J_6 \propto a^{-15/2}$. As a result, $J_2, J_4$, and $J_6$ are strongly correlated, and in particular the ring occultation data alone cannot yield an independent estimate of $J_6$, although the fit results provide coupled constraints on the trio $J_2, J_4$, and $J_6$. In this section, we identify additional systematic and random errors that affect the determination of the gravity field, quantify their effects on the derived $J_n$, and provide a prescription to compute the difference in standard deviations between our best-fitting model and other model gravity fields described by the first three even gravity harmonics.

## 7.1. Ring midlines, centers of opacity, and centers of mass

In addition to the challenge of a restricted radial range of observed apse and node rates noted above, a second complication that has not been previously addressed is that the apse and node rates in Table 5 are usually assumed to
<figure><img src="image_33.png" /><figcaption>**Figure 31.** Ring 6 center of opacity (COO) estimated from *Voyager* RSS optical depth profiles. The derived values of $r_{COO}$ are shown by the solid vertical blue line near the middle of each profile, but on opposite sides of the geometric center (COR) for ingress and egress, marked by the vertical dashed red lines mid-way between the two edges.</figcaption></figure>

refer to the semimajor axes of the geometric ring midlines (COR), which may not correspond to the dynamical radius, or semimajor axis of the average center of mass (COM) of the rings. This is of particular concern for the $\epsilon$ ring, which has manifestly non-uniform internal radial structure, as seen in Fig. 2, but it applies to the other rings as well, as can be seen from the high-resolution *Voyager* RSS ring profiles. **Figure 31** shows the ingress and egress RSS optical depth profiles of ring 6. The ingress profile is near apoapse with true anomaly $f = 144.4^{\circ}$, while the egress profile is almost exactly at periapse, with $f = 2.7^{\circ}$. Under the simplifying approximation that optical depth is proportional to surface mass density, and thus using the center of opacity (COO) as a surrogate for the ring’s COM, we define the radii of the inner and outer ring edges $r_{IER}$ and $r_{OER}$ by the vertical dashed lines, chosen by eye, and determine the radius of the center of opacity $r_{COO}$ by integrating the radial normal optical depth profile $\tau(r)$ using

$$r_{COO} = \frac{\int_{r_{IER}}^{r_{OER}} \tau(r)rdr}{\int_{r_{IER}}^{r_{OER}} \tau(r)dr} \qquad (38)$$

The derived values of $r_{COO}$ for the two RSS ring 6 observations are shown by the solid vertical blue line near the middle of each radial profile. For both ingress and egress, this is offset from the midline of the ring, marked by the vertical dashed red line mid-way between the two edges. We then determine the equivalent offset of the ring semimajor axis center of opacity $\Delta a_{COO}$ from the relation

$$\Delta a_{COO} = \left( \frac{r_{COO} - r_{IER}}{r_{OER} - r_{IER}} - \frac{1}{2} \right) \overline{W} \qquad (39)$$

where $\overline{W}$ is the mean width of the ring defined by the difference in semimajor axes of the OER and IER fits in Table 5. Numerically, $\Delta a_{COO} = +0.18$ km and $-0.10$ km for the ring 6 ingress and egress profiles, respectively.

In this instance, it is clear that the internal structure of ring 6 is quite different at the two locations sampled by the RSS occultations, which may not be representative of the structure at other ring longitudes. In any event, there are no streamline models for the internal structure of any of the rings that accurately capture the ring structure at
all longitudes, and we use the RSS observations to estimate the uncertainty of the ring COO or COM as well as its systematic offset from the geometric ring center determined from square-well fits to the ensemble of ring profiles.

We proceed similarly in Appendix B for the other eccentric rings, making additional use of the best Earth-based radial profiles for the $\epsilon$ ring in instances where it is wide enough to reveal internal structure. The derived centers of opacity are given in Table 16, which lists $a_{COR}$, the unweighted average values for $\Delta a_{COO}$ and their RMS variation, the statistical significance $\Delta a_{COO}/\sigma(\Delta a_{COO})$, and finally of $a_{COO}$, with an uncertainty given by the sum in quadrature of $\sigma(a_{COR})$ and $\sigma(\Delta a_{COO})$.

<table><caption>**Table 16.** Ring center of opacity</caption><thead><tr><th>Ring</th><th>a<sub>COR</sub> (km)</th><th>Δa<sub>COO</sub> (km)</th><th>Δa<sub>COO</sub>/σ(Δa<sub>COO</sub>)</th><th>a<sub>COO</sub> (km)</th></tr></thead><tbody><tr><td>6</td><td>41837.092 ± 0.096</td><td>0.040 ± 0.193</td><td>0.208</td><td>41837.132 ± 0.215</td></tr><tr><td>5</td><td>42234.893 ± 0.091</td><td>0.169 ± 0.100</td><td>1.686</td><td>42235.061 ± 0.135</td></tr><tr><td>4</td><td>42571.124 ± 0.091</td><td>0.404 ± 0.087</td><td>4.620</td><td>42571.528 ± 0.126</td></tr><tr><td>α</td><td>44718.473 ± 0.086</td><td>0.172 ± 0.132</td><td>1.309</td><td>44718.645 ± 0.157</td></tr><tr><td>β</td><td>45661.056 ± 0.087</td><td>0.322 ± 0.274</td><td>1.175</td><td>45661.378 ± 0.288</td></tr><tr><td>γ</td><td>47626.170 ± 0.089</td><td>0.119 ± 0.125</td><td>0.954</td><td>47626.289 ± 0.153</td></tr><tr><td>ε</td><td>51149.279 ± 0.081</td><td>1.134 ± 0.488</td><td>2.324</td><td>51150.414 ± 0.495</td></tr></tbody></table>

The mean offset $\Delta a_{COO}$ is positive for all seven eccentric rings but statistically significant at the 2-$\sigma$ level only for ring 4 and the $\epsilon$ ring. We will make use of both the offsets and their uncertainties in our final determination of the Uranus gravity field.

As an illustration of the effect of the offset $\Delta a_{COO}$ on the computed precession rate, note that at the radius of the $\epsilon$ ring, $d\dot{\omega}/da = -9.355 \times 10^{-5} \circ d^{-1} \text{ km}^{-1}$, and the corresponding change in apse rate for $\Delta a_{COO} = 1.134$ km from Table 16 is $\Delta\dot{\omega} = -1.06 \times 10^{-4} \circ d^{-1}$. To put this in context, the formal error in the apse rate $\sigma(\dot{\omega})$ for the $\epsilon$ COR from Table 5 is just $4.2 \times 10^{-6} \circ d^{-1}$, and thus the radial change $\Delta a_{COO}$ corresponds to change of $\Delta\dot{\omega}/\sigma(\dot{\omega}) \sim -1.06 \times 10^{-4}/4.2 \times 10^{-6}$, or a 25$\sigma$ change in the predicted apse rate. Note that for the $\epsilon$ ring COR in Table 5, $\Delta a_{\dot{\omega}} = 1.154 \text{ km} \approx \Delta a_{COO} = 1.134 \text{ km}$, demonstrating that the adopted gravity field takes into account the systematic difference between the ring's COR and COO.

**Figure 32** (top panel) shows the observed ring apse and node rates from Table 5, plotted as solid dots, compared to the smooth curve computed from our adopted gravity model (Fit 18 in Table 17), described below in Section 7.2. The second panel shows the difference $\Delta\dot{\omega}$ between the individually fitted apse rates from Table 5 (with their error bars) and the rates predicted by the adopted gravity model (solid dots), and from individual alternative fits (shown as open circles) that ignored the apse and node rates of each ring in turn when fitting for $J_2$ and $J_4$. (Both series of fits account for the offsets $\Delta a_{COO}$.) The bottom panel shows $\Delta\dot{\Omega}$ for the node rates of the measurably inclined rings. The observed apse and node rates generally match the predicted values, with the possible exceptions of the $\alpha$, $\beta$, and $\gamma$ ring apse rates. In Section 8, we will examine the evidence for anomalous precession rates of these rings and evaluate possible explanations.

## 7.2. Determining the planetary gravity field from ring precession rates

Historically, the most accurate estimates of the Uranus gravity field have been derived from the observed precession rates of the rings (French et al. 1988; Jacobson 2014). With the inclusion of all ring occultation data between 1977–2006, the uncertainties in the ring apsidal and nodal rates have been substantially reduced compared to prior determinations. As we will show, however, both the mean values of $J_2$ and $J_4$ and their realistic uncertainties from ring occultation data are significantly affected by the assumed value of $J_6$ and by previously ignored systematic and random errors in: (1) the estimates of the ring centers of opacity (COO), (2) the absolute radius scale, and (3) the estimated contributions of both major and minor satellites to the ring precession rates. In this section, we review