<figure><img src="image_34.png" /><figcaption>**Figure 32.** Uranus ring apse and node precession rates. The top panel shows the observed apse rates of the eccentric rings, overplotted with the predicted radial variation in apse rate computed from the adopted gravity model (Fit 15 in Table 17). The middle panel shows the differences between the observed and predicted apse rates $\Delta\dot{\omega}$ for each ring's center of opacity (COO), (unlike the Table 5 values, which apply to the ring's geometric centers (COR)), plotted as filled symbols. The open symbols show the estimated anomalous precession based on separate fits for $J_2$ and $J_4$ that excluded the contribution of individual rings to the gravity field solution. These are listed in Table 18. Rings $\alpha$, $\beta$, and possibly $\gamma$ appear to have anomalously high precession rates compared to the predictions from the gravity model. For the $\gamma$ ring orbit fit that excluded the $m=3$ mode, the anomalous precession is even higher (shown as a filled and open squares). The bottom panel shows the node rate difference $\Delta\dot{\Omega}$ for the inclined rings.</figcaption></figure>
previous gravity field determinations and estimate the influence of these error sources on the derived gravity harmonics. Although we cannot obtain an independent estimate of $J_6$, we derive a set of observationally constrained linear relations between $J_2, J_4$, and $J_6$ based on their strong mutual correlations in the ring orbit fits.

### 7.2.1. Previous determinations of the Uranus gravity field

French et al. (1988) determined $J_2$ and $J_4$ from orbit fits to ring occultation observations from 1977–1986, with the results listed as Fit 1 in Table 17, converted from their assumed reference radius of $R = 26200$ km to $R = 25559$ km. Jacobson (2014) solved for $J_2$ and $J_4$ from a comprehensive analysis of all available *Voyager* navigation data and selected ring occultation measurements, and in a separate solution from the ring data alone. These results are listed as Fits 2 and 3, respectively.<fnref n="11" /> Jacobson (2014) assumed that $J_6 = 0$, but the quoted uncertainties in $J_2$ and $J_4$ incorporated an *a priori* uncertainty $\sigma(J_6) = 1.0 \times 10^{-6}$, as well as *a priori* uncertainties on other parameters of the fit including the absolute radius scale and the pole direction. The resulting “rings only” solution (Fit 3) is shown in Fig. 33 as the large solid-line brown ellipse.

More recently, Jacobson (2023) estimated $J_2$ and $J_4$ as part of a global solution for the Uranus ephemeris and system properties, based on a comprehensive analysis of historical astrometry, navigation data, and ring occultation observations. He set $J_6 = (0.58 \pm 0.12) \times 10^{-6}$, based on a suite of Uranus interior models by Neuenschwander and Helled (2022), and the satellite contribution to the precession of the rings was included for Ariel, Umbriel, Titania, Oberon, Miranda, and Puck only, ignoring the minor satellites. (The assumed masses of these six satellites are included in Table 2.) The Jacobson (2023) results are shown Fig. 33 as Fit 4, where the error bars include estimated uncertainties in the astrometry and navigation data used for the fit.

<table><caption>Table 17. Uranus gravity parameters</caption><thead><tr><th>Fit</th><th>J<sub>2</sub> × 10<sup>-6</sup></th><th>J<sub>4</sub> × 10<sup>-6</sup></th><th>J<sub>6</sub> × 10<sup>-6</sup></th><th>ρ(J<sub>2</sub>, J<sub>4</sub>)</th><th>Note</th></tr></thead><tbody><tr><td>1</td><td>3513.23 ± 0.34</td><td>-30.32 ± 4.73</td><td>-</td><td>-</td><td>French et al. (1988) (converted to R = 25559 km)</td></tr><tr><td>2</td><td>3510.7 ± 0.7</td><td>-34.2 ± 1.3</td><td>0.0 ± 1.0</td><td>0.9784</td><td>Jacobson (2014) adopted solution</td></tr><tr><td>3</td><td>3510.5 ± 1.3</td><td>-34.4 ± 1.3</td><td>0.0 ± 1.0</td><td>0.9784</td><td>Jacobson (2014) “rings only” solution</td></tr><tr><td>4</td><td>3510.465 ± 0.058</td><td>-34.145 ± 0.082</td><td>0.58 ± 0.12</td><td>0.9887</td><td>Jacobson (2023) (ura178 ephemeris)</td></tr><tr><td>5</td><td>3510.464 ± 0.066</td><td>-34.158 ± 0.092</td><td>[0.58]</td><td>0.9384</td><td>COR - 6 satellites (AUTOMP)</td></tr><tr><td>6</td><td>3510.474 ± 0.056</td><td>-34.135 ± 0.081</td><td>[0.58]</td><td>0.9853</td><td>COR wtd fit to apse/node rates (AUTOMP)</td></tr><tr><td>7</td><td>3511.175 ± 0.065</td><td>-33.492 ± 0.092</td><td>[0.50]</td><td>0.9384</td><td>COR + COO offsets (Table 16) - 6 satellites (AUTOMP)</td></tr><tr><td>8</td><td>3510.975 ± 0.065</td><td>-34.030 ± 0.092</td><td>[0.0]</td><td>0.9384</td><td>Fit 7 but J<sub>6</sub> = 0 × 10<sup>-6</sup></td></tr><tr><td>9</td><td>3511.374 ± 0.065</td><td>-32.954 ± 0.092</td><td>[1.0]</td><td>0.9384</td><td>Fit 7 but J<sub>6</sub> = 1 × 10<sup>-6</sup></td></tr><tr><td>10</td><td>3511.201 ± 0.065</td><td>-33.528 ± 0.092</td><td>[0.50]</td><td>0.9384</td><td>Fit 7 with Δa = +0.2 km for all rings</td></tr><tr><td>11</td><td>3510.723 ± 0.065</td><td>-33.957 ± 0.092</td><td>[0.50]</td><td>0.9384</td><td>Fit 7 including all but Cordelia and Ophelia</td></tr><tr><td>12</td><td>3509.291 ± 0.067</td><td>-35.522 ± 0.094</td><td>[0.50]</td><td>0.9383</td><td>Fit 7 incl. all sats, basis for Monte Carlo Fit 15</td></tr><tr><td>13</td><td>3513.217 ± 0.065</td><td>-31.669 ± 0.091</td><td>[0.50]</td><td>0.9384</td><td>Fit 7 excluding all satellites</td></tr><tr><td>14</td><td>3509.337 ± 0.065</td><td>-35.438 ± 0.091</td><td>[0.50]</td><td>0.9387</td><td>COO, all satellites, rings 6,5,4 ϵ for J<sub>n</sub></td></tr><tr><td>15</td><td>3509.291 ± 0.412</td><td>-35.522 ± 0.466</td><td>[0.50]</td><td>0.9861</td><td>Adopted solution: Fit 12 w/ composite error estimates</td></tr><tr><td>16</td><td>3509.291 ± 0.385</td><td>-35.522 ± 0.433</td><td>[0.50]</td><td>0.9957</td><td>Monte Carlo errors from COO fits to apse/node rates only</td></tr><tr><td>17</td><td>3509.291 ± 0.134</td><td>-35.522 ± 0.147</td><td>[0.50]</td><td>0.9997</td><td>Monte Carlo errors from satellite mass uncertainties only</td></tr><tr><td>18</td><td>3509.291 ± 0.026</td><td>-35.522 ± 0.036</td><td>[0.50]</td><td>0.1412</td><td>Monte Carlo errors from radius scale uncertainty only</td></tr><tr><th>Quantity</th><th>Value</th><th></th><th></th><th></th><th>Note</th></tr><tr><td>GM<sub>U</sub></td><td>5793950.300 km<sup>3</sup> s<sup>-2</sup></td><td></td><td></td><td></td><td>Jacobson (2023)</td></tr><tr><td>R</td><td>25559 km</td><td></td><td></td><td></td><td>Reference radius for J<sub>n</sub></td></tr><tr><td>dJ<sub>4</sub>/dJ<sub>2</sub></td><td>+1.134</td><td></td><td></td><td></td><td>slope of error ellipse of adopted solution</td></tr><tr><td>dJ<sub>4</sub>/dJ<sub>2</sub></td><td>+2.69540</td><td></td><td></td><td></td><td>slope of line connecting COO J<sub>6</sub> = (0, 1) × 10<sup>-6</sup></td></tr><tr><td>dJ<sub>2</sub>/dJ<sub>6</sub></td><td>+0.39909</td><td></td><td></td><td></td><td>over range J<sub>6</sub> = (0, 1) × 10<sup>-6</sup></td></tr><tr><td>dJ<sub>4</sub>/dJ<sub>6</sub></td><td>+1.07570</td><td></td><td></td><td></td><td>over range J<sub>6</sub> = (0, 1) × 10<sup>-6</sup></td></tr><tr><td>dJ<sub>2</sub>/dΔa</td><td>+0.130 × 10<sup>-6</sup> km<sup>-1</sup></td><td></td><td></td><td></td><td>from Fits 7 and 10</td></tr><tr><td>dJ<sub>4</sub>/dΔa</td><td>-0.180 × 10<sup>-6</sup> km<sup>-1</sup></td><td></td><td></td><td></td><td>from Fits 7 and 10</td></tr></tbody></table>

<footnote n="11">¹¹ The correlation coefficient $\rho(J_2, J_4)$ for these fits was provided by R. Jacobson, personal communication.</footnote>
<figure><img src="image_35.png" /><figcaption>**Figure 33.** Results of a suite of fits for *J*₂ and *J*₄, under a variety of assumptions about the value of *J*₆, the contributions of major and minor satellites to the ring precession rates, the differences between the ring COR and COO, and the absolute radius scale of the rings. The fit numbers correspond to the results in Table 17. The adopted fit (Fit 15) includes error estimates derived from the Monte Carlo set of solutions marked by the gray dots, representing the range of variation in *J*₂ and *J*₄ associated with estimated error sources. See text for details.</figcaption></figure>

### 7.2.2. Uranus gravity field from COR ring orbital elements

To provide an initial estimate of *J*₂ and *J*₄ from the complete set of COR measurements used to derive the orbital elements in Tables 5 and 6, we performed Fit 5, using the same fit parameters as our adopted COR fit except that the ring apse and node rates were calculated from the best-fitting values of *J*₂ and *J*₄, a fixed value for *J*₆, and the secular precession forced by the satellites included only Ariel, Umbriel, Titania, Oberon, Miranda, and Puck, omitting the estimated contributions of the smaller moons displayed in Fig. 4. To provide a direct comparison with Jacobson (2023), we let *J*₆ = 0.58 × 10⁻⁶. This solution is plotted as the red error ellipse in Fig. 33. The fitted values for *J*₂ and *J*₄ are very similar to the Jacobson (2023) result (Fit 4), showing that these two independent solutions based largely on the same set of observations yield nearly identical results.

Next, in Fit 6, we determined *J*₂ and *J*₄ from a weighted fit to the apse and node rates in Table 5, again assuming *J*₆ = 0.58 × 10⁻⁶. As seen in Table 17, the results are similar to Fit 5, but with slightly smaller error bars and a larger correlation coefficient. The fitted values and error bars are quite close to the Jacobson (2023) results in Fit 4 as well. This confirms that the primary constraints on *J*₂ and *J*₄ in the full RINGFIT results are the apse and node rates, while the other free parameters have less influence on the fitted values and uncertainties of the gravity parameters.

In a separate fit, not included in Table 17, we allowed *J*₆ to be an additional free parameter, but the resulting formal errors in all three gravity parameters were very large, the retrieved value of *J*₆ was implausibly large, and the correlations between the parameters were nearly singular.

### 7.2.3. Uranus gravity field from COO ring orbital elements
To explore the effects on the derived gravity field of applying the ring radius corrections associated with the COO measurements, in Fit 7 we applied the offsets $\Delta a_{COO}$ from Table 16 to the fitted ring semimajor axes when computing the precession rates and let $J_6 = 0.5 \times 10^{-6}$, with all other fit conditions as in Fit 5. The result is shown as a black error ellipse and is significantly displaced from the COR solution (Fit 5), demonstrating the sensitivity of $J_2$ and $J_4$ to the differences between the geometric ring centers (COR) and their dynamical centers (COO).

### 7.2.4. Sensitivity of $J_2$ and $J_4$ to the assumed value of $J_6$

Although our experiment with Fit 5 described above demonstrates that we cannot fit simultaneously for $J_2$, $J_4$, and $J_6$, we can nevertheless quantify the sensitivity of the results to assumed (fixed) values of $J_6$. In Fits 8 and 9, we let $J_6 = (0.0, 1.0) \times 10^{-6}$ respectively, applied the COO offsets as before, and solved for $J_2$ and $J_4$. The results are shown as the labeled gray error ellipses in Fig. 33, bracketing the COO Fit 7 that assumed $J_6 = 0.5 \times 10^{-6}$. From this pair of fits, we find the partial derivatives $dJ_2/dJ_6$ and $dJ_4/dJ_6$ listed at the bottom of Table 17. The ratio of these gives the slope of the line connecting the COO $J_6 = (0.0, 1.0) \times 10^{-6}$ solutions, shown as the gray dashed line in the figure and listed in Table 17.

### 7.2.5. Sensitivity of $J_2$ and $J_4$ to the absolute radius scale

To quantify the effect of a possible systematic error in the overall absolute ring radius scale, in Fit 10 we repeated Fit 7, but assumed a systematic radius scale error of $\Delta a = +0.2$ km (our $2-\sigma$ estimate, as shown in Section 5.2). The result of the fit is shown as the orange error ellipse in Fig. 33, only slightly displaced from Fit 7, indicating that a radius scale error at this level has only a modest effect on the derived gravity field. This is evident in the small magnitude of the differentials $dJ_2/d\Delta a$ and $dJ_4/d\Delta a$ included at the bottom of Table 17.

### 7.2.6. Effects of minor satellites on the derived gravity field

Previous published determinations of $J_2$ and $J_4$ have taken into account the secular precession due to Ariel, Umbriel, Titania, Oberon, Miranda, and Puck, but have ignored the contributions of the smaller moonlets. As shown in Fig. 4, several of these minor satellites each have predicted secular precession rates for several of the eccentric rings that are larger than those due to Oberon and Puck, and in particular Cordelia and Ophelia are estimated to contribute jointly to precession rate of the $\epsilon$ ring at a level of $\dot{\epsilon} > 10^{-4} \, \text{d}^{-1}$, greater than that of any of the other satellites except Ariel. To quantify the effect of the minor satellites, we first performed Fit 11, similar to Fit 7 except for the inclusion of all moons except Cordelia and Ophelia. The fitted values of $J_2$ and $J_4$ are significantly offset from the Fit 7 values, with the results shown as the blue error ellipse in Fig. 33. Then, in Fit 12 we added the predicted precession due to tiny Cordelia and Ophelia, with the resultant values of $J_2$ and $J_4$ shown as the magenta error ellipse, shifted even further from the Fit 7 values that used only the six most massive moons. For comparison, in Fit 13 we ignored the effects of all satellites when computing the ring precession rates. The resulting values are off-scale to the upper right of the figure. The differences between Fits 7, 11, 12 and 13 demonstrate the sensitivity of the derived values of $J_2$ and $J_4$ to the assumed minor satellite masses and the uncertainties in the masses of both major and minor satellites. We evaluate these effects below.

### 7.3. The influence of the $\alpha, \beta$, and $\gamma$ rings on the derived gravity field

Figure 32 shows that the apse rates of the $\alpha, \beta$, and $\gamma$ rings in the COR fit for $J_2$ and $J_4$ are systematically faster than predicted from the gravity solution. To quantify the influence of these three rings on the derived gravity field, we performed Fit 14, in which we applied COO corrections, included the effects of all major and minor satellites, but allowed the apse and node rates of the $\alpha, \beta$, and $\gamma$ rings to be free parameters, rather than being constrained by $J_2$ and $J_4$. Only rings 6, 5, 4, and $\epsilon$ were so constrained. As shown in Table 17, the fitted values for $J_2$ and $J_4$ differed only slightly from the Fit 12 results, indicating that the possibly anomalous precession rates of the $\alpha, \beta$, and $\gamma$ rings do not produce a significant systematic shift in the derived gravity field.
### 7.3.1. Effects of random and systematic errors on the derived gravity field

So far, we have separately estimated the sensitivity of the derived values of $J_2$ and $J_4$ to the assumed value of $J_6$, to differences between ring centerlines and centers of opacity (COR and COO, respectively), to possible systematic errors in the absolute radius scale, and to the contributions of small moons to the ring precession rates. All of these systematic effects have associated uncertainties that affect the robustness of the fitted gravity field. To quantify the effects of both random and systematic errors on the derived gravity field, we performed a series of Monte Carlo fits that sampled a random distribution of estimated uncertainties in key quantities that affect the best-fitting solution for $J_2$ and $J_4$. We used the conditions of Fit 12 (including all satellites and the COO offset) as our nominal best fit for $J_2$ and $J_4$. Then, for each of 50,000 fits, we applied separate offsets to the semimajor axes of each ring, drawn randomly from a normal distribution with the corresponding individual standard deviation in the COO semimajor axis $\sigma(\Delta a_{COO})$. To account for the uncertainty in the absolute radius scale, we applied an additional single random systematic offset $\Delta a$ to the semimajor axes of all of the rings, drawn from a normal distribution with a standard deviation of 0.2 km. Finally, to account for uncertainties in the masses of both major and minor satellites when computing their contributions to each ring's secular precession rate, we applied offsets to the individual satellite masses drawn randomly from a normal distribution with a standard deviation for each major satellite taken from Jacobson (2023) and for the minor satellites from Table 3, assuming our derived masses for Cressida, Cordelia, and Ophelia.

We determined $J_2$ and $J_4$ for each Monte Carlo instance from a least squares fit to the entire system geometry, with the apse and node rates of the eccentric and inclined rings being determined from the combined effects of the modeled gravity field and satellite contributions, assuming $J_6 = 0.5 \times 10^{-6}$ for all fits. The ensemble of fitted values of $J_2$ and $J_4$ represents the probability density function (PDF) associated with the applied random errors. To determine the corresponding error ellipse, we first determined the mean values of $J_2$ and $J_4$ and their standard deviations from the distribution of their fitted values. Next, we estimated the correlation coefficient $\rho(J_2, J_4)$ by determining its value such that the corresponding $1-\sigma$ error ellipse for $\sigma(J_2)$ and $\sigma(J_4)$ enclosed $e^{-1/2} \approx 0.6065$ of the fitted values, as appropriate for the complementary cumulative probability distribution of a bivariate normal distribution (see Appendix C). The results are given in Table 17 as Fit 15, our adopted solution and error bars. The mean values for $J_2$ and $J_4$ match those from Fit 12, as expected, but with much larger uncertainties, owing to the incorporation in the Monte Carlo runs of the estimated random errors in $\Delta a_{COO}$, $\Delta a$, and the satellite contributions to the ring precession rates.

To quantify the various contributions to the total error budget, we performed three separate Monte Carlo runs. For Fit 16, we included only the estimated errors due to standard deviations in the COO semimajor axes $\sigma(\Delta a_{COO})$. For Fit 17, we included only the contributions of the uncertainty in satellite masses to the error budget. Finally, in Fit 18, we included only the uncertainty in the absolute radius scale. The results are given in Table 17. In all cases, the mean values of $J_2$ and $J_4$ matched those of our adopted solution, as expected, and the quoted error bars correspond to the standard deviations in the fitted values $J_2$ and $J_4$ from the 50,000 individual random samples for each series. These fits show that the dominant source of error in the Monte Carlo runs is associated with the uncertainty in the locations of the ring centers of opacity (i.e., the random errors in $\Delta a_{COO}$). On the other hand, the substantial displacement of the best-fitting $J_2$ and $J_4$ that included only the AUTOMP satellites from our adopted solution accentuates the importance of including the effects of the minor satellites when computing the predicted precession rates of the rings.

### 7.3.2. Adopted gravity field solution and uncertainties

Our adopted solution for the Uranus gravity field is from Fit 15: $J_2 = (3509.291 \pm 0.412) \times 10^{-6}$, $J_4 = (-35.522 \pm 0.466) \times 10^{-6}$, and $\rho(J_2, J_4) = 0.9861$. These represent the best-fitting values of $J_2$ and $J_4$ from Fit 12, which used the entire set of ring observations, corrected for the estimated differences between COO and COR values, including the contribution to ring precession of both major and minor satellites, and assume $J_6 = 0.5 \times 10^{-6}$. The formal errors $\sigma(J_2)$ and $\sigma(J_4)$ and correlation $\rho(J_2, J_4)$ for Fit 12 account for random errors of the observations alone, whereas the Monte Carlo results summarized in Fit 15 account as well for the estimated uncertainties in $\Delta a_{COO}$, $\Delta a$, and the satellite contributions to the ring precession rates.
A striking characteristic of the results in Table 17 and evident in Fig. 33 is the small size of formal error ellipses in $J_2$ and $J_4$ from the individual fits, compared to the much larger differences in $J_2$ and $J_4$ that stem from a range of assumptions about the value of $J_6$, the satellites included in the calculation of secular precession, the choice of COR or COO for ring semimajor axes, the absolute radius scale, and realistic uncertainties in these quantities. Some of these differences can be predicted on the basis of the observed strong correlations between $J_2$, $J_4$, and $J_6$, as we show below. Nevertheless, the large size of the error ellipse of our final solution compared to the formal errors of the individual fits underscores the consequences of our limited knowledge of the internal radial structure of the rings that determines their dynamical semimajor axes, as well as uncertainties in the masses of the minor satellites, especially Cordelia and Ophelia, which primarily affect the $\epsilon$ ring's predicted precession rate, a key constraint on the gravity field determination.

### 7.3.3. Correlations between $J_2$, $J_4$, and $J_6$

The strong correlations between $J_2$, $J_4$, and $J_6$ can be used to restrict the range of models for the interior structure and differential rotation of Uranus that are consistent with the ring occultation solution for the planet's gravity field. In Appendix C, we evaluate these results numerically and provide a prescription to compute the difference in standard deviations between a model gravity field described by the first three gravity harmonic coefficients {$J'_2, J'_4, J'_6$} and our adopted solution.

## 8. ANOMALOUS PRECESSION RATES

Our determination of the gravity field of Uranus relies on a model of the radial dependence of ring apsidal precession and nodal regression rates that includes the effects of the planet and the known satellites. From an examination of the post-fit residuals in apse rate for individual rings, we can set limits on possible unseen moonlets, such as those that have been proposed to account for quasi-periodic optical depth variations in the $\alpha$ and $\beta$ ring *Voyager* RSS optical depth profiles (Chancia and Hedman 2016). Figure 32 shows small positive residuals for the apse rate of these two rings, and also for the $\gamma$ ring. Here, we determine the range of moonlet masses and orbital radii that could account for these residuals, and set limits on the anomalous precession in the vicinity of the eccentric rings.

We estimated the detection limits for anomalous precession rates of the eccentric rings from a series of fits in which we solved for full set of geometric and orbital parameters as in our adopted solution, except that we let each eccentric and possibly inclined ring in turn have its apse and node rates fitted as free parameters but solved for $J_2$ and $J_4$ from the other rings. The differences between the fitted values and the apse rates computed from Fit 12 in Table 17 (accounting for $\Delta a_{COO}$) are listed in Table 18, which summarizes our measured values and upper limits for anomalous precession. These are plotted as open circles in the middle panel of Fig. 32.

<table><caption>**Table 18.** Limits on anomalous precession rates</caption><thead><tr><td>Ring</td><td>Δω̇ (10<sup>-5</sup> ° d<sup>-1</sup>)</td><td>Description</td></tr></thead><tbody><tr><td>6</td><td>-9.07 ± 5.31</td><td>No detected anomalous precession</td></tr><tr><td>5</td><td>-1.04 ± 2.30</td><td>No detected anomalous precession</td></tr><tr><td>4</td><td>+0.02 ± 4.48</td><td>No detected anomalous precession</td></tr><tr><td>α</td><td>27.66 ± 4.99</td><td>Consistent with Chancia and Hedman (2016) wake model</td></tr><tr><td>β</td><td>12.79 ± 8.74</td><td>Consistent with Chancia and Hedman (2016) wake model</td></tr><tr><td>γ</td><td>31.21 ± 43.84</td><td>m = 3 mode included in orbit model</td></tr><tr><td>γ</td><td>123.16 ± 36.54</td><td>m = 3 mode not included in orbit model</td></tr><tr><td>ε</td><td>—</td><td>Not determinable from orbit model</td></tr></tbody></table>

### 8.1. Rings 6, 5, and 4
The precession rates of rings 6, 5, and 4 all contributed to the determination of the gravity field, so it is not surprising that there are no detected anomalous precession rates for these rings, although a single outlier among them would perhaps have been detectable.

## 8.2. The α ring

Chancia and Hedman (2016) proposed that wakes observed in the *Voyager* RSS α ring radial profiles could be produced by a moonlet with orbital radius $a_s = 44825_{-12}^{+22}$ km (compared to the α ring COR semimajor axis $a = 44718.470 \pm 0.086$ km) and mass $M_{sI} = (3_{-2}^{+4}) \times 10^{14}$ kg (or $M_{sI}/M_{Ur} = (3.46_{-2.30}^{+4.61}) \times 10^{-12}$) from the ingress profile and $M_{sE} = (1.0_{-0.6}^{+18}) \times 10^{14}$ kg (or $M_{sE}/M_{Ur} = (1.15_{-2.65}^{+20.7}) \times 10^{-12}$) from the egress profile. In Fig. 34, the solid diagonal line indicates the satellite mass, as a function of the orbital distance exterior to the ring COR, that would be required to produce the estimated anomalous precession rate $\Delta\dot{\tau} = (27.66 \pm 4.99) \times 10^{-5} \, \text{d}^{-1}$ of the α ring, from Table 18.

The dotted lines bound the uncertainty. The dashed diagonal line (visible at upper right) gives the corresponding results for a moonlet interior to the ring. The relative locations of the rings exterior to the α ring are shown as vertical dashed lines, and the masses of representative satellites are plotted as horizontal dotted lines, along with an estimate for the $\epsilon$ ring mass ratio $M_\epsilon/M_{Ur} = 7.57 \times 10^{-11}$ (Nicholson et al. 2014b). Selected moonlet radii $r$ are shown for an assumed average density $\rho = 1 \, \text{g cm}^{-3}$. For example, a Cordelia-sized moon orbiting 500 km from the α ring could produce the observed anomalous precession rate.

For comparison, the Chancia and Hedman (2016) estimates of the hypothetical moonlet's mass and distance from the α ring (using the more reliable RSS ingress profile) are shown by the labeled dot, with error bars that overlap the range of masses inferred from the measured anomalous precession rate of the α ring. This provides supportive evidence for an unseen icy moonlet with a radius $\gtrsim 4 \, \text{km} \simeq 100 \, \text{km}$ exterior to the α ring.

## 8.3. The β ring

Similarly, Chancia and Hedman (2016) proposed that wakes observed in the β ring radial profiles could be produced by a moonlet with orbital radius $a_s = 45738_{-4}^{+8}$ km (compared to the β ring COR semimajor axis $a = 45661.051 \pm 0.087$ km) and mass $M_{sI} = (0.5_{-0.2}^{+0.3}) \times 10^{14}$ kg (or $M_{sI}/M_{Ur} = (0.58_{-0.23}^{+0.35}) \times 10^{-12}$) from the ingress RSS profile and $M_{sE} = (0.7_{-0.2}^{+0.4}) \times 10^{14}$ kg (or $M_{sE}/M_{Ur} = (0.81_{-0.23}^{+0.46}) \times 10^{-12}$) from the egress RSS profile. In Fig. 35, the solid diagonal line indicates the satellite mass, as a function of the orbital distance exterior to the ring COR, that would be required to produce the estimated anomalous precession rate $\Delta\dot{\tau} = (12.79 \pm 8.74) \times 10^{-5} \, \text{d}^{-1}$ of the β ring. The dotted lines bound the uncertainty. The dashed diagonal line (visible at upper right) gives the corresponding results for a moonlet orbiting interior to the ring. The relative locations of the rings exterior to the β ring are shown as vertical dashed lines.

For comparison, the Chancia and Hedman (2016) estimate of the hypothetical moonlet mass and distance from the β ring from the more reliable RSS ingress profile is shown by the labeled dot, with error bars that overlap the range of masses inferred from the estimated anomalous precession rate of the ring. As with the α ring, this provides supportive evidence for an unseen moonlet produced the wake signature in the ring profile, in this case with a radius $\gtrsim 3 \, \text{km} \sim 80 \, \text{km}$ exterior to the β ring.

## 8.4. The γ ring

Finally, we examine the evidence for possible unseen moonlets in the vicinity of the γ ring. The putative $m = 3$ normal mode resulted in a small positive but uncertain anomalous precession rate $\Delta\dot{\tau} = (14.48 \pm 42.10) \times 10^{-5} \, \text{d}^{-1}$ (Table 5), consistent with zero. When the $m = 3$ mode is excluded from the fit, the anomalous precession has a substantially larger value: from Table 18, $\Delta\dot{\tau} = (123.16 \pm 36.54) \times 10^{-5} \, \text{d}^{-1}$, plotted as a filled square in Fig. 32. Figure 36 shows the satellite masses required to account for both of these possibilities. The upper solid and dashed
<figure><img src="image_36.png" /></figure>

**Figure 34.** Satellite size and mass required to match the observed anomalous precession rate of the α ring, plotted as a function of orbital distance from the semimajor axis of the ring's COR and shown as a solid diagonal line, bounded by dotted lines indicating its uncertainty. The dashed black line that deviates from the solid line at the upper right corresponds to satellite orbits interior to the α ring's semimajor axis. Representative masses and radii of moonlets and the ϵ ring are shown, assuming $ρ = 1 \text{ g cm}^{-3}$. The locations of the rings exterior to the α ring are marked by vertical dotted lines. The black dot with error bars marks the hypothetical satellite mass and orbital radius proposed by Chancia and Hedman (2016) to account for wakes observed in the *Voyager* RSS α ring radial profiles. The predictions from the observed anomalous precession rate match within the error bars.

diagonal lines correspond to the fit that excluded the *m* = 3 mode, giving the inferred satellite mass as a function of distance exterior to or interior to the γ ring COR, respectively. When the *m* = 3 mode is included in the fit, the predicted satellite masses are smaller, with the nominal estimates based on the measured anomalous precession rate of the *m* = 1 mode being shown as the red solid and dashed lines, defined as before, with the upper limit shown as the red dotted line. (There is no corresponding lower limit, since $\sigma(\Delta\dot{\omega})$ exceeds its measured value.)

The amplitude and location of the putative *m* = 3 mode provide an independent constraint on the mass and location of a hypothetical satellite that forces the mode. The resonance location for the *m* = 3 mode is ~1.101 km exterior to the outer edge of the ring, and if the mode is forced by a first-order ILR with an unseen satellite, its mean motion would be $\Omega_P(\text{OER}) = n = 765.3977 \text{ }^\circ \text{ d}^{-1}$, with a corresponding semimajor axis *a* = 62366.66 km, a few hundred km interior to the orbit of Desdemona. Based on the amplitude of the COR mode $A_3 = 0.577 \pm 0.073$ km and the distance from resonance $\Delta a_P = 2.706 \pm 0.048$ km, the required satellite mass would be $m_{\text{sat}}/M_{\text{Ur}} = 12.5 \times 10^{-10}$, or about twice the mass of Cordelia (see Section 9). This is too large to have escaped previous detection in *Voyager* images and would probably produce a detectable perturbation in Cordelia's orbit. Furthermore, it is too small to account for the *m* = 1 anomalous precession. On this basis, we reject the hypothesis that the *m* = 3 mode is forced by an external satellite. If instead the *m* = 3 mode is unforced or entirely absent, and we assume that the lower radius limit of detectable satellite is ~10 km, the inferred *m* = 1 anomalous precession rate could be provided by a hidden moonlet orbiting within 150–250 km on either side of the γ ring, assuming $ρ = 1 \text{ g cm}^{-3}$.
<figure><img src="image_37.png" /></figure>

**Figure 35.** Satellite size and mass required to match the observed anomalous precession rate of the *β* ring, plotted as a function of orbital distance from the semimajor axis of the *β* ring's COR and shown as a solid diagonal line, bounded by dotted lines indicating its uncertainty. The dashed black line that deviates from the solid line at the upper right corresponds to satellite orbits interior to the *β* ring's semimajor axis. Representative masses and radii of moonlets and the *ϵ* ring are shown, assuming *ρ* = 1 g cm⁻³. The locations of the rings exterior to the *β* ring are marked by vertical dotted lines. The black dot with error bars marks the hypothetical satellite mass and orbital radius proposed by Chancia and Hedman (2016) to account for wakes observed in the *Voyager* RSS *β* ring radial profiles. The predictions from the observed anomalous precession rate match within the error bars.

The $\epsilon$ ring precession rate was the main tie point for the inferred gravity field, and so no measurable anomalous precession was detectable.

## 9. MOONLET ORBITS, MASSES, AND DENSITIES

Five of the detected normal modes in the Uranian rings appear to be forced by nearby satellite resonances: the $\eta$ ring 3:2 ILR with Cressida, the $\gamma$ ring 6:5 ILR and the $\epsilon$ OER 14:13 ILR with Ophelia, and the $\delta$ ring 23:22 ILR and the $\epsilon$ IER 24:25 OLR with Cordelia. In this section, we examine the satellite-driven modes detected at ring midlines or edges, compare their pattern speeds and phases with the mean motions and orbital longitudes at epoch of the three satellites, and estimate the masses and densities of the satellites from the amplitudes of the modes.

### 9.1. Satellite-driven normal modes

First-order Lindblad resonances of wavenumber *m* located near narrow rings produce radial forces that excite normal modes: *m*-lobed distortions in the ring particle orbits that rotate with a pattern speed matching the satellite's mean motion. In the idealized case where ring material can be viewed as independent test particles, the periapse of the distortion is aligned with the mean longitude of the satellite for orbits interior to the resonant radius, with the apoapse being so aligned for particles orbiting exterior to the resonance (Murray and Dermott 1999). Examples in the Saturn system include the Titan 1:0 ILR in the C ring, where the orientations of a host eccentric features on either side of the resonance match these expectations (see Fig. 19 Nicholson et al. (2014b)), and the Mimas 2:1 ILR near the outer edge of the B ring, where eccentric features in the Cassini Division exterior to the resonance all have their apoapses closely aligned with Mimas (see Fig. 25 French et al. (2016b)).
<figure><img src="image_38.png" /></figure>

**Figure 36.** Satellite size and mass required to match the observed anomalous precession rate of the γ ring, plotted as a function of orbital distance from the ring's COR. The orbit fit that excluded the *m* = 3 mode had a very high anomalous precession rate, resulting in corresponding large satellite masses, as shown by the solid black diagonal line, bounded by dotted lines denoting its uncertainty. For the alternative case that included the *m* = 3 mode, the anomalous precession rate was lower, resulting in smaller predicted satellites, as indicated by the solid red diagonal line, bounded by its upper uncertainty as a red dotted line. The dashed black and red lines that deviate from the solid lines at the upper right correspond to satellite orbits interior to the γ ring's semimajor axis. Representative masses and radii of moonlets and the ε ring are shown, assuming ρ = 1 g cm⁻³. The locations of the rings exterior to the γ ring are marked by vertical dotted lines.

The situation is more complicated for an eccentric narrow ring affected by nearby resonances when the ring's self-gravity is taken into account (see Longaretti (2018) for an extensive review). In this case, streamline models show that the torque balance at the ring edges can result in shepherding, which serves to account for both the sharp edges of a narrow ring and the absence of differential precession across the ring. As applied to the ε ring, this model predicts that the outer edge is confined by the Ophelia 14:13 ILR and the inner edge is confined by the Cordelia 24:25 OLR, each producing an *m*-lobed edge wave. In the absence of dissipation, the forced mode (periapse/apoapse) is predicted to be aligned with the satellite mean longitude for an (ILR/OLR), respectively, and when dissipation is present, the relevant apse (lags/leads) the satellite mean longitude in angle for an (ILR/OLR), respectively. (In Appendix D, we justify these statements on dynamical grounds.)

Finally, both theoretical and numerical models show that under special circumstances a single shepherd satellite can confine a narrow ring (Goldreich et al. 1995; Hänninen and Salo 1994, 1995; Lewis et al. 2011), but the applicability of these results to the Uranian rings has not been explored in detail.

The characteristics of the forced normal modes identified in the Uranian rings are summarized in Table 19. For each mode, we include from Tables 5 and 6 the fitted semimajor axis *a* of the feature, the wavenumber *m*, its amplitude *A*<sub>m</sub>, the pattern speed Ω<sub>m</sub>, and the distance of the predicted resonance radius from the observed ring feature Δ*a*<sub>P</sub> = *a*<sub>res</sub> − *a*. Also listed are the mean satellite longitude at epoch λ<sub>sat</sub> from the ura111 ephemeris and the longitude difference Δλ = δ'<sub>m</sub> − λ<sub>sat</sub>, where δ'<sub>m</sub> is defined modulo 360°/*m* as the inertial longitude at epoch relative to nearest apse, as identified in the final column.<sup>12</sup> Error bars reflect the uncertainties in δ'<sub>m</sub> and do not include the

<footnote n="12">¹² For an *m*-lobed normal mode, the phase offset of the mode apse and the satellite longitude is given by *m*Δλ.</footnote>
very small uncertainties in the satellite longitude at epoch (see Table 4, Jacobson (1998)). Following Chancia et al. (2017), we estimate the satellite mass $M_{sat}/M_{Ur}$ from the mode amplitude $A_m$ in the vicinity of a Lindblad resonance and the distance from resonance $|\Delta a_P|$ using Eq. (10.22) from Murray and Dermott (1999):

$$A_m = \frac{2\alpha a^2 (M_{sat}/M_{Ur}) |f_d|}{3(m-1)|\Delta a_P|}, \quad (40)$$

where $\alpha = a/a_s$ (the ratio of the ring and satellite semimajor axes) and $f_d$ is the Laplace factor, given by Murray and Dermott (1999). (As noted by Chancia et al. (2017), $2\alpha|f_d|/m-1$ varies between 1.5 and 1.6, depending on $m$.) This relationship is based on test-particle approximation that is strictly applicable only in the evanescent region of the mode and at least 1/4 wavelength from the resonance, and so it is probably best thought of as an approximate scaling estimate, rather than as a dynamically rigorous result for all of the forced modes considered here.

<table><caption>Table 19. Satellite-driven normal modes</caption><thead><tr><th>Satellite</th><th>Ring</th><th>a<br>km</th><th>Δa<sub>p</sub><br>km</th><th>m</th><th>A<sub>m</sub><br>km</th><th>m<sub>sat</sub>/M<sub>Ur</sub><br>×10<sup>-10</sup></th><th>Ω<sub>p</sub><br>° d<sup>-1</sup></th><th>δ'<sub>m</sub><sup>(a)</sup><br>deg</th><th>λ<sub>sat</sub><sup>(b)</sup><br>deg</th><th>Δλ<sup>(c)</sup><br>deg</th><th>apse</th></tr></thead><tbody><tr><td rowspan="3">Cressida</td><td>η (COR)</td><td>47176.009</td><td>-4.595</td><td>3</td><td>0.600 ± 0.069</td><td>21.18 ± 2.44</td><td>776.584048 ± 0.001164</td><td>15.90 ± 2.36</td><td>17.44</td><td>-1.54 ± 2.36</td><td>apoapse</td></tr><tr><td>γ (IER)</td><td>47624.606</td><td>0.754</td><td>6</td><td>0.590 ± 0.110</td><td></td><td>956.418079 ± 0.000877</td><td>303.40 ± 2.21</td><td>298.08</td><td>5.33 ± 2.21</td><td>apoapse</td></tr><tr><td>γ (COR)</td><td>47626.170</td><td>-0.861</td><td>6</td><td>0.637 ± 0.063</td><td>4.23 ± 0.42</td><td>956.419622 ± 0.000474</td><td>302.52 ± 1.17</td><td>298.08</td><td>4.44 ± 1.17</td><td>apoapse</td></tr><tr><td rowspan="3">Ophelia</td><td>γ (OER)</td><td>47627.865</td><td>-2.552</td><td>6</td><td>0.892 ± 0.117</td><td></td><td>956.419529 ± 0.000560</td><td>298.03 ± 1.36</td><td>298.08</td><td>-0.05 ± 1.36</td><td>apoapse</td></tr><tr><td>ε (COR)</td><td>51149.279</td><td>28.425</td><td>14</td><td>0.383 ± 0.071</td><td></td><td>956.418015 ± 0.000269</td><td>296.96 ± 0.63</td><td>298.08</td><td>-1.12 ± 0.63</td><td>periapse</td></tr><tr><td>ε (OER)</td><td>51178.588</td><td>-0.887</td><td>14</td><td>0.590 ± 0.130</td><td>3.67 ± 0.81</td><td>956.418119 ± 0.000298</td><td>297.76 ± 0.70</td><td>298.08</td><td>-0.32 ± 0.70</td><td>periapse</td></tr><tr><td rowspan="3">Cordelia</td><td>δ (COR)</td><td>48300.227</td><td>2.173</td><td>23</td><td>0.339 ± 0.063</td><td>5.58 ± 1.09</td><td>1074.523021 ± 0.000189</td><td>69.45 ± 0.47</td><td>70.00</td><td>-0.55 ± 0.47</td><td>periapse</td></tr><tr><td>ε (IER)</td><td>51120.014</td><td>1.082</td><td>-24</td><td>1.011 ± 0.107</td><td>7.83 ± 0.83</td><td>1074.522889 ± 0.000099</td><td>70.42 ± 0.26</td><td>70.00</td><td>0.41 ± 0.26</td><td>apoapse</td></tr><tr><td>ε (COR)</td><td>51149.279</td><td>-28.177</td><td>-24</td><td>0.443 ± 0.061</td><td></td><td>1074.522703 ± 0.000142</td><td>70.33 ± 0.35</td><td>70.00</td><td>0.33 ± 0.35</td><td>apoapse</td></tr></tbody></table>

a The epoch is 1986 Jan 19 12:00 (TDB). $\delta'_m$ is the observed longitude of the mode's periapse or apoapse, as specified in the final column. It is ambiguous by additive multiples of 360°/m, adjusted to minimize the magnitude of $\Delta\lambda$.

b Mean longitude of the satellite at the epoch, from the urs111 ephemeris.

c $\Delta\lambda = \delta'_m - \lambda_{sat}$. Error bars reflect the uncertainties in $\delta'_m$ and do not include the uncertainties in the satellite ephemerides (see Table 4, Jacobson (1998)).

All of the normal modes in Table 19 have pattern speeds that are very close to those of the satellites associated in the resonances. Table 20 lists the weighted average pattern speeds of all identified modes associated with Cressida, Ophelia, and Cordelia, as well as satellite mean motions from published and unpublished sources.

### 9.1.1. Cressida

The $m=3$ ILR forced by Cressida in the $\eta$ ring was first identified by Chancia et al. (2017), based on a ring orbit fit to a subset of the data used in our current analysis. The Cressida resonance lies $\Delta a_P = -4.595$ km interior to the ring COR. We find $A_m = 0.600 \pm 0.069$ km, $\Omega_P = 776.584048 \pm 0.001164$° d$^{-1}$, $\delta'_m = 15.90 \pm 2.36$°, and $\Delta\lambda = -1.54 \pm 2.36$° from 60 $\eta$ ring COR measurements, comparable to but with smaller error bars than the Chancia et al. (2017) values of $A_m = 0.667 \pm 0.113$ km, $\Omega_P = 776.58208 \pm 0.00169$° d$^{-1}$, and $\Delta\lambda = -2.72 \pm 6.12$° from 49 measurements.

<table><caption>Table 20. Satellite mean motions</caption><thead><tr><th>Satellite</th><th>Source</th><th>Ω<sub>P</sub> (° d<sup>-1</sup>)</th></tr></thead><tbody><tr><td rowspan="4">Cressida</td><td>This work</td><td>776.584048 ± 0.001164</td></tr><tr><td>Jacobson (1998)</td><td>776.582414 ± 0.000022</td></tr><tr><td>Showalter and Lissauer (2006)</td><td>776.582789 ± 0.000059</td></tr><tr><td>Robert French (pers. comm.)</td><td>776.5826393 ± 0.0000053</td></tr><tr><td rowspan="3">Ophelia</td><td>This work</td><td>956.418404 ± 0.000171</td></tr><tr><td>Jacobson (1998)</td><td>956.42833 ± 0.00908</td></tr><tr><td>Robert French (pers. comm.)</td><td>956.418293 ± 0.000037</td></tr><tr><td rowspan="2">Cordelia</td><td>This work</td><td>1074.522858 ± 0.000075</td></tr><tr><td>Jacobson (1998)</td><td>1074.51832 ± 0.00187</td></tr></tbody></table>
The pattern speed of the 3:2 ILR in the η ring is reasonably consistent with previous values for Cressida's mean motion, differing by 1.4σ from the Jacobson (1998) value, by 1.1σ from the Showalter and Lissauer (2006) value, and by 1.2σ from the unpublished value (Robert French, pers. comm) based on an analysis of HST images.

These results are consistent with the test particle interpretation of the response of the η ring to the Cressida resonance, which is too distant to produce single-sided shepherding. Since the resonance is interior to the ringlet, the normal mode apoapse is predicted to be aligned with the satellite, as observed, with no statistically significant lead or lag in longitude.

### 9.1.2. Cordelia and Ophelia

The *m* = 6 ILR Ophelia resonance has long been recognized as being located near to the γ ring (Porco and Goldreich 1987), and as described in Section 4.1 and tabulated in Table 5, we detected the *m* = 6 mode at the ring midline and both edges, with the measured amplitude *A*<sub>m</sub> increasing systematically from 0.591 ± 0.110 km at the IER, to 0.637 ± 0.063 km at the COR, and finally to 0.891 ± 0.117 km at the OER. The resonance radius lies Δ*a*<sub>P</sub> = -2.552 km interior to the ring OER, where the mode amplitude is greatest. The mode apoapse leads Ophelia by Δλ = +5.33 ± 2.21° at the ring's IER, by +4.44 ± 1.17° at the ring's COR, and at the OER, where the *m* = 6 mode amplitude the largest, the apoapse is nearly aligned with Ophelia: Δλ = -0.05 ± 1.36°.

The dynamics of the γ ring are complex, with the simultaneous presence of five free normal modes and the Ophelia 6:5 ILR forced mode that is located within the ring itself (Fig. 28). The near-alignment of the *m* = 6 mode apoapse with Ophelia is as expected if the ring material responds as isolated test particles to the satellite's forcing, but this neglects the possibly significant role of self-gravity for this narrow ringlet of high average optical depth. The apparently statistically-significant phase offsets of the mode at the ring's inner edge and midline have no obvious explanation.

As shown previously in Sections 4.1.4 and 4.1.5, normal mode scans revealed the presence of the *m* = 14 ILR with Ophelia in the ε OER and COR, and the *m* = -24 OLR with Cordelia in the ϵ IER and COR. The final values for the mode amplitudes, phases, and pattern speeds when included in our adopted orbit fit are given in Table 5 and in more detail in Table 19. The periapse of the *m* = 14 OER mode lags behind Ophelia by Δλ = -0.55 ± 0.47°, as expected for a normal mode at the outer edge of the ring, and the apoapse of the *m* = -24 IER mode leads Cordelia by Δλ = 0.41 ± 0.26°, again in keeping with the dynamical arguments presented in Appendix D for a forced normal mode at the inner edge of a ring in the presence of dissipation.

The *m* = 23 ILR Cordelia resonance was first associated with the δ ring by Porco and Goldreich (1987). We detected the signature of this resonance in our normal mode scans of the δ ring COR, although the fitted mode amplitude is only *A*<sub>23</sub> = 0.339 ± 0.063 km, too small to be detectable in either of the noisier ring edge (IER/OER) orbit fit residuals. The resonance is located slightly outside of the ring COR (Δ*a*<sub>P</sub> = 2.173 km). The test-particle model predicts that the mode periapse should be aligned with Cordelia, as observed. There is a marginally-detectable lag in the orientation of periapse behind Cordelia at the 1-σ level: Δλ = -0.55 ± 0.54°, perhaps due to dissipation in the ring.

Comparisons of the pattern speeds and orbital longitudes of the normal modes and the ura115 ephemeris values for Ophelia and Cordelia are shown in Fig. 37. The top left panel shows the fitted pattern speeds and their error bars for the γ ring IER/COR/OER *m* = 6 mode measurements and the ϵ ring COR and OER *m* = 14 results, compared to the Jacobson (1998) value. The weighted mean value of our measurements is shown by the solid green line, bracketed by the estimated uncertainty. The dashed blue lines show the rather large 1-σ uncertainty in Ophelia's mean motion from Jacobson (1998). All of the measured pattern speeds are systematically below the ura115 ephemeris value by ≈ 0.01 ° d⁻¹. The weighted mean pattern speed for the normal modes of Ω<sub>P</sub> = 956.418404 ± 0.000171 ° d⁻¹ is just ~ 0.65σ from the unpublished value of Ω<sub>P</sub> = 956.418293 ± 0.000037 ° d⁻¹ shown in red, derived from HST observations (Robert French, personal communication). The lower left panel shows the longitude differences Δλ between the measured phases of the detected modes and the mean longitude of Ophelia at the epoch. The γ ring *m* = 6 normal mode apoapse leads the satellite for the IER and COR, but aligns with the satellite at the OER. On the other hand, the periapse of the ϵ ring *m* = 14 ILR at the ring's outer edge lags behind Ophelia, as expected in the presence of dissipation.
<figure><img src="image_39.png" /><figcaption>**Figure 37.** Pattern speed and phase comparison for the observed normal modes forced by Ophelia and Cordelia. The upper left panel shows the observed pattern speeds $\Omega_m$ for each of five labeled modes associated with Ophelia, plotted as filled circles with error bars. The solid horizontal lines mark the mean motions of Ophelia (bounded by their uncertainties, shown dashed) reported by Jacobson (1998) (shown in blue and labeled Jac98) and Robert French (pers. comm.) shown in red and labeled RF23, compared to the weighted average of the mode pattern speeds from this work, shown in green. The upper right panel compares the Jacobson (1998) and our results for Cordelia. The bottom panels show the longitude differences $\Delta\lambda$ between the fitted periapse or apoapse longitudes and the satellite mean longitudes at epoch from the ura115 satellite ephemeris (from Table 19).</figcaption></figure>

The right column of Fig. 37 shows the comparable results for Cordelia. In this case, the best fitting pattern speeds for the $\delta$ ring COR $m = 23$ mode and the $\epsilon$ ring $m = -24$ mode values for the IER and COR are in excellent mutual agreement, although they lie above the Jacobson (1998) values, which have much larger error bars. In the lower right panel, the longitude of the apoapse of the $\delta$ ring $m = 23$ normal mode lags slightly behind Cordelia, while the $\epsilon$ ring's inner edge mode associated with the Cordelia 24:25 OLR leads the satellite, as expected in the presence of dissipation.

## 9.2. Moonlet masses and densities

The densities of Cressida, Ophelia, and Cordelia can be estimated from the ratio of their masses given in Table 19 to their volumes, calculated from the prolate satellite dimensions from Table IV of Karkoschka (2001a). These can be compared to the Roche critical density $\rho_{\text{Roche}}$

$$ \rho_{\text{Roche}} = \frac{4\pi\rho_p}{\gamma(a/R_p)^3} = \frac{3M_p}{\gamma a^3}, \qquad (41) $$

where $\rho_p$, $R_p$ and $M_p$ are the planet's density, radius, and mass, $a$ is the distance from the planet, and $\gamma$ is a dimensionless parameter associated with the mass distribution and shape of the satellite (Tiscareno et al. 2013). An object without material strength and density less than $\rho_{\text{Roche}}$ would be pulled apart, and thus the critical density curve represents a tidally governed lower limit to density of gravitationally bound aggregates. Following Porco et al. (2007) and Tiscareno et al. (2013), we assume $\gamma = 1.6$, corresponding to the case when the satellite's mass is assumed to be uniform in density and distributed into the shape of its Roche lobe.
<figure><img src="image_40.png" /></figure>

**Figure 38.** Densities of Cordelia, Ophelia, and Cressida computed from fitted normal mode amplitudes $A_m$, distances from resonance $\Delta a_p$, and satellite dimensions from Karkoschka (2001b). Here, we use the COR amplitudes for the $\eta, \delta$ and $\gamma$ ring resonances, since the ring midline fits have smaller formal errors than the edge amplitudes that are based on noisier data. The error bars on the individually marked resonances reflect only the contribution of the fractional errors of $A_m$ to the overall error budget, which is dominated by the satellite volume uncertainties. The gray dots are the weighted mean densities for each satellite and the dotted error bars correspond to the combined uncertainties in the mode amplitudes and the satellite volumes (Table 3). The Roche critical density is plotted as a solid line for shape parameter $\gamma = 1.6$ and $R_p = 25559$ km, with filled circles at the orbital radii of the satellites. Dashed vertical lines are labeled with the locations of the ten narrow rings.

The computed satellite densities are plotted as a function of orbital radius in Fig. 38. Cordelia and Ophelia each have two separate resonances that yield mutually-consistent densities with overlapping error bars computed from the uncertainties in the fitted mode amplitudes alone, although the realistic uncertainty in their average value (plotted in gray) is dominated by the contribution of the much larger satellite volume uncertainty. Even taking these larger error estimates into account, the calculated satellite densities appear to decrease with orbital radius by about a factor of two from innermost Cordelia to outermost Cressida. For comparison, the Roche critical density is plotted as a solid line, assuming that the shape parameter $\gamma = 1.6$. It increases inward quite rapidly, from $0.5 \text{ g cm}^{-3}$ at $2.6 R_p$ to over $2.0 \text{ g cm}^{-3}$ at ring 6, supporting the view that the Uranian rings are more rocky and less porous than Saturn's rings (Tiscareno et al. 2013). For smaller values of $\gamma$, the critical density curve would lie above the measured densities and require that the moonlets have material strength. The observed densities follow the overall trend of the Roche critical density curve for $\gamma = 1.6$, which may reflect the formation process and possible orbital migration of these small moons.

# 10. DISCUSSION

## 10.1. Ring orbits, width variations, and masses

The most secure results from nearly three decades of occultation observations of the narrow rings of Uranus are their orbital shapes and width variations. Although the number of individual occultation observations for a given ring is at most just 89 (well below the ~150–260 for Saturn’s rings from Cassini occultation observations), the long time-baseline and remarkable geometric accuracy of the orbit reconstructions enables the detection of normal modes and width variations of the rings on a scale of ~0.2 km for the Uranian rings. The internal structure of the narrowest of the rings is largely unresolved in Earth-based observations, preventing the accurate reconstruction of streamline
models that could provide detailed information about self-gravity wakes, ring confinement mechanisms, and signatures of unseen satellites.

Standard self-gravity (SSG) models predict unrealistically small ring masses and surface densities and fail to match the actual variations in $q_e(r)$ observed in the rings, but they successfully predict the slopes of the observed width-radius relations for nearly all of the inner and outer Lindblad resonances detected in the rings. Collisional self-gravity models (CSG) predict much larger ringlet masses and surface densities more consistent with expectations based on particle size estimates, but they sequester the bulk of the mass in narrow regions near the ring edges, predicting U-shaped radial optical depth profiles that are a poor match to high-resolution *Voyager* occultation observations. More work is needed to ascertain whether the differences between the various predicted profiles and the observed profiles can be explained as the result of variations in the effective opacity of the ring material across each ring, or if they instead require a more complex dynamical model that produces a more uniform optical depth profile. Here, it might prove fruitful to compare the Uranus results with surface densities of Saturn's rings in regions of comparable optical depth.

The accurate determination of the orbits and width-radius relations for the detected normal modes provides strong evidence for the overall balance between differential precession due to planetary oblateness and to self-gravity in preserving the apse alignment of the inner and outer edges of the eccentric ringlets. Only a subset of the Uranus rings have excited normal modes, but the $\gamma$ ring's five free modes puts it in the class of several of Saturn's rings and ring edges that have as many or more normal modes. The excitation mechanism behind the observed modes is poorly understood, although it seems likely that non-linear coupling between modes might be responsible for the simultaneous presence of many modes, as seems to be the case for the complex shape of the outer edge of the B ring (French et al. 2023b).

While we found no statistically significant evidence for normal modes driven by internal planetary oscillations, it is important to note that the patterns generated by satellite resonances on the $\epsilon$, $\delta$ and $\gamma$ rings are relatively subtle, and finding resonances with planetary normal modes is more challenging because their pattern speeds cannot be accurately predicted in advance. This lack of detections therefore does not rule out the possibility that planetary normal modes are responsible for confining some of these rings. Still, these data, combined with more detailed analyses of the internal structures and longitudinal brightness variations within each of these rings, should help constrain the properties of the planet's internal normal modes.

## 10.2. Ring-moonlet connections

Among the host of narrow rings in the solar system, only the $\epsilon$ ring is securely established as being confined by shepherd satellites, with the inner satellite Cordelia having a 24:25 OLR with the ring's inner edge and the outer satellite Ophelia having a 14:13 ILR with the outer edge (Porco and Goldreich 1987). The amplitudes, pattern speeds, and phases of the edge waves match dynamical expectations and provide estimates of the masses of the confining satellites. The resonance radii of both edge modes are located within the ring, ~1 km from the corresponding ring edge. In the future, it will be interesting to compare these values with numerical simulations.

In addition to their roles as shepherd satellites for the $\epsilon$ ring, Cordelia and Ophelia each are the source of an additional forced normal mode in the rings: the 23:22 ILR of Cordelia with the $\delta$ ring and the 6:5 ILR of Ophelia with the $\gamma$ ring. Cressida's 3:2 ILR with the $\eta$ ring provides a third example. The orientations of the excited normal modes in all three cases match a simple test-particle interpretation of the ring response to the satellite forcing (although the complex dynamics of the $\gamma$ ring remain to be investigated in detail). An example of this mechanism in the Saturn system is provided by the Barnard Gap outer edge, which is perturbed by a Prometheus 5:4 ILR. Here, the apoapse of the mode is aligned with the satellite, as expected from the test-particle model (French et al. 2016b).

The internal structure of several narrow ringlets has been interpreted as evidence for satellite wakes generated by unseen moonlets near the $\delta$ ring (Horn et al. 1988) and the $\alpha$ and $\beta$ rings (Chancia and Hedman 2016). Although our study offers no additional supporting evidence for a $\delta$ ring moonlet, our measurements of anomalous precession rates in both the $\alpha$ and $\beta$ rings are consistent with the mass estimates for the proposed unseen moonlets near $\alpha$ and $\beta$ rings, making them just below the level of detectability in *Voyager* or *HST* images. Direct detections of these moons may
have to await a future mission to the Uranus system, but these small moons could produce localized disturbances in
nearby rings that could be detectable in *HST* or *JWST* images.

A provocative result of our study is the striking trend of moonlet densities with orbital distance from Uranus. Taken at face value, the densities of Cordelia, Ophelia, and Cressida follow the systematic trend of the Roche critical density for $\gamma = 1.6$, rising from $0.71 \text{ g cm}^{-3}$ for outermost Cressida to double that value for innermost Cordelia. This finding is consistent with these moons having relatively little internal strength, which is compatible with the overall architecture of the Uranian ring-moon system (Tiscareno et al. 2013). This strikes an interesting contrast with the Neptune system, whose innermost moons must have sufficient internal strength to survive at their current locations (Tiscareno et al. 2013; Brozović et al. 2020).

Cordelia's high density is particularly interesting because it is comparable to the densities of the much larger moons Ariel (1.54 g cm⁻³), Umbriel (1.52 g cm⁻³), Titania (1.65 g cm⁻³) and Oberon (1.67 g cm⁻³), and is actually larger than the density of Miranda (1.18 g cm⁻³) (Thomas 1988; Jacobson 2014). This exceptionally high density likely has implications for the origins and evolution of these small moons. For example, according to standard tidal theory, Cordelia should be migrating inwards, so perhaps Cordelia represents the dense core of a larger moon whose outer layers have been stripped away by tidal forces. In this scenario, the $\epsilon$ ring would be composed of the last bits of material lost from Cordelia's surface.

### 10.3. *Uranus gravity field, internal structure, and winds*

Accurate measurements of the gravity fields of giant planets can provide powerful constraints on their internal structure and the depth and nature of their wind profiles, as has been shown recently for Saturn and Jupiter from Cassini and Juno measurements of high-order gravitational moments (Iess et al. 2018, 2019; Dietrich et al. 2021; Kulowski et al. 2021; Militzer et al. 2022; Galanti et al. 2022). Uranus has very strong winds at the cloud level, with features at $\pm 60^\circ$ latitude moving around the planet over 200 m s⁻¹ faster than the those near the equator (Hammel et al. 2001, 2005; Sromovsky and Fry 2005). If these winds penetrate far enough into Uranus' interior, they could significantly influence the planet's gravity field. Kaspi et al. (2013) concluded that the difference between the observed value of $J_4$ and the predicted value based on relatively simple interior models was sufficient to constrain the dynamics to the outermost 0.15% of the total mass of Uranus, corresponding to a relatively thin weather layer no more than about 1,000 kilometers deep. More recently, Movshovitz and Fortney (2022) showed that the moment of inertia (MoI) is tightly correlated with the gravity field beyond $J_2$, and that to improve interior models and the MoI determination, the planet's rotation period must be known to comparable precision. Neuenschwander and Helled (2022) demonstrated that knowledge of $J_6$ to 10% accuracy is required to constrain wind depths on Uranus: relatively high values of $J_6 = (59.9-69.04) \times 10^{-8}$ can only be explained by the existence of deep winds with a penetration depth of more than 250 km, whereas lower values $J_6 = (46.12-53.76) \times 10^{-8}$ do not allow for deep winds with a penetration depth as large as 1100 km. Significantly, they also demonstrated that more accurate values of $J_2$ and $J_4$ can further constrain the density distribution and narrow the range of predicted solutions in $J_6, J_8$, and MoI of Uranus and Neptune. Finally, Soyuer et al. (2023) conclude from model studies that the scale height of the zonal winds on Uranus is below $\sim 0.03R_U$ (770 km), somewhat shallower than found by Kaspi et al. (2013).

Nearly all of these modeling efforts for the interior structure and dynamics of Uranus have made use of the Jacobson (2014) values for $J_2$ and $J_4$ listed under Fits 2 and 3 in Table 17, but they have not yet exploited the strong degree of correlation between the measured values of $J_2$ and $J_4$ as an additional constraint. Although we are unable to set useful limits on $J_6$ alone, as we have shown in Table 17 and Fig. 33, our adopted gravity field solution based on a more complete set of ring observations provides tighter limits not only on $J_2$ and $J_4$, but also on allowed combinations of values of $J_2, J_4$, and $J_6$, after accounting for systematic uncertainties in the center of opacity of the rings, the absolute radius scale, and the masses of the major and minor satellites. The prescription in Appendix C can be used to restrict the range of density distributions and wind models for Uranus that are consistent with the ring occultation results, although significant advances in our understanding of the planet's internal structure will probably require high-precision gravity measurements possible only from an orbiting spacecraft (Movshovitz and Fortney 2022).
10.4. *Open questions and future prospects*

Decades after their discovery, the Uranian rings continue to raise vexing dynamical puzzles. What confines the narrow rings? How can they maintain their eccentricities and inclinations in the presence of differential apsidal and nodal precession rates? What are their connections with the planet's host of small moons? How recently were they formed, and what is the time scale for significant evolution of ring systems? The findings presented in this paper demonstrate the richness and complexity of the Uranian ring system, and its potential to illuminate other aspects of the Uranus system. Already, these rings have provided constraints of the gravity field of Uranus itself, estimates the mass densities of several known satellites, and evidence for a number of still-unseen moons. Future observations by earth-based telescopes or missions to the Uranus system could therefore provide key insights into multiple aspects of Ice Giant Systems (National Academies of Sciences, Engineering, and Medicine 2022).

On the one hand, detailed studies of the structure and dynamics of the Uranian rings will provide additional insights into the interiors of Ice Giants. Continued monitoring of the rings' orbital evolution will yield increasingly tight constraints on the higher-order components of the planet's gravitational field. At the same time, if any structures in the rings can be attributed to resonances with oscillations or asymmetries in the planetary normal modes (A'Hearn et al. 2022), those structures will provide novel information about the internal structures of Ice Giants, which are still largely unconstrained (Helled 2022).

Meanwhile, further studies of these rings should clarify whether small moons do in fact exist near the rings. The sizes, locations and densities of such moons will constrain the conditions under which solid material is more likely to accrete or fragment, which is relevant to not only the Uranian rings and moons, but also the formation of moons and planets more generally.

11. CONCLUSIONS

Our main results are summarized below:

* From an analysis of 31 Earth-based stellar occultations and three *Voyager 2* occultations spanning 1977–2006 (Paper 1), we have determined 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 limited secure detections.

* We characterized a host of free and forced normal modes in several of the ring centerlines and inner and outer edges (IER/OER). In addition to the previously-known free modes $m=0$ in the $\gamma$ ring and $m=2$ in the $\delta$ ring, we have identified two additional OLR modes ($m=-1$ and $-2$) and a possible $m=3$ mode in the $\gamma$ ring. No normal modes were detected for rings 6, 5, 4, $\alpha$, or $\beta$.

* The origin of the $\gamma$ ring's putative $m=3$ mode is mysterious. Its pattern speed is much slower than oscillation frequencies expected from internal oscillations of the planet, and it is unlikely to be due to an unseen satellite, whose predicted size would be too large to have avoided prior detection. If it is an unforced mode, it is unusual in having its resonance radius more than 1 km exterior to the ring's outer edge. If the mode is omitted from the orbit solution, the $\gamma$ ring has a very large anomalous apsidal precession rate of unknown origin.

* Five separate normal modes are forced by small moonlets: the 3:2 inner Lindblad resonance (ILR) of Cressida with the $\eta$ ring, the 6:5 ILR of Ophelia with the $\gamma$ ring, the 23:22 ILR of Cordelia with the $\delta$ ring, the 14:13 ILR of Ophelia with the outer edge (OER) of the $\epsilon$ 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.

* From separate orbit fits to the ring edge measurements obtained from square-well model fits to individual ring profiles, we determined 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, as predicted by standard self-gravity models. Ring mass and surface density estimates based on standard self-gravity and collisional self-gravity models differ by orders magnitude and predict radically different radial optical depth profiles. These differences are unresolved at present, and warrant additional investigation. We found 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 $\delta_P = 15.172795 \pm 0.000618^\circ$, where the error bars take into account possible systematic errors in the *Voyager* ephemeris. The slight pole precession predicted by Jacobson (2023) was 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 determined the zonal gravitational coefficients $J_2 = (3509.291 \pm 0.412) \times 10^{-6}$ and $J_4 = (-35.522 \pm 0.466) \times 10^{-6}$ with a correlation coefficient $\rho(J_2, J_4) = 0.9861$ and $J_6$ held fixed at= $0.5 \times 10^{-6}$, for a reference radius $R=25559$ km. This result is significantly displaced from previous results (Jacobson 2014), owing to the inclusion of previously neglected systematic effects. The quoted errors account for systematic differences (and their uncertainties) between the fitted semimajor axes of the ring centerlines (COR) and their estimated centers of opacity (COO), a surrogate for the center of mass (COM) semimajor axes that should be used when computing the radial gradient in the apse and node rates. The minor satellites Cordelia and Ophelia contribute significantly to the precession rate of the *ε* ring, affecting the final solution for $J_2$ and $J_4$. 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 measured anomalous apsidal and nodal precession rates of the $\alpha$ and $\beta$ rings are consistent with the presence of unseen moonlets with masses and orbital radii predicted by Chancia and Hedman (2016).

* From the amplitudes and resonance radii of normal modes forced by moonlets, we determined the masses of Cressida, Cordelia, and Ophelia. Their estimated densities vary systematically with orbital radius and generally follow the radial trend of the Roche critical density for a shape parameter $\gamma = 1.6$. This may provide an important clue to the formation process and possible orbital migration of these small moons.

## 12. ACKNOWLEDGEMENTS

We are grateful to two anonymous reviewers for their helpful suggestions. We also thank Robert Jacobson and Ryan Park of JPL for incorporating the Uranus ring occultation data into the updated planet, satellite, and Voyager ephemerides used for our orbit fits. Robert French generously shared unpublished orbital elements for Cressida and Ophelia from *HST* satellite astrometry. Naor Movshovitz and Benno Neuenschwander provided helpful assessments of the implications for Uranus internal models of the new constraints on $J_2, J_4$, and $J_6$. Matthew Tiscareno provided helpful insight into the interpretation of the trend of small satellite densities with orbital radius. This work was supported by NASA PDART grant NNX15AJ60G: “Restoration and submission of Uranus ring occultation observations to the Planetary Data System” and NASA Solar System Workings grant NNX15AH45G: “Uranian ring dynamics and constraints on Uranus’ internal structure from occultation data.” This work has made use of data from the European Space Agency (ESA) mission *Gaia* (https://www.cosmos.esa.int/gaia), processed by the *Gaia* Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the *Gaia* Multilateral Agreement.
<figure><img src="image_41.png" /><figcaption>**Figure A1.** Raw intensity profile of *Voyager 2* γ ring egress profile, uncorrected for diffraction. The best-fitting square-well model is overplotted, and the box function square-well model is shown in red, with a width of 1.37 km.</figcaption></figure>

A. *VOYAGER 2* OCCULTATIONS AND SQUARE-WELL MODEL EVALUATION

The *Voyager 2* radio and stellar occultations provide our current best view of the fine-scale radial internal structure of the narrow Uranian rings. (For a detailed description of the *Voyager* observations and of the shapes and observed structure of each ring, see French et al. (1991).) The high-SNR σ Sgr occultation chord intersected only the δ, λ, and ε rings, while the lower-SNR β Per event spanned the entire ring system on ingress and egress. The viewing geometry of the two stellar occultations was quite different from all other Uranus ring occultations, providing additional constraints on the Uranus pole direction.

The *Voyager* RSS data present an excellent opportunity to assess the accuracy of the square-well model approximation to ring structure because the raw observations are affected by diffraction with a Fresnel scale *F* ~ 1.6 – 2.3 km, comparable to typical values for Earth-based stellar occultations, while the diffraction-reconstructed data reveal the true ring structure at 50 m resolution (Gresh et al. 1989). **Figure A1** shows the diffraction pattern visible in the uncorrected RSS observations of the egress γ ring profile, along with the best-fitting square-well model to the measured intensity. It closely resembles the diffraction pattern in Fig. 3. In this instance, the best fitting square-well model is 1.37 km wide and nearly opaque, with a corresponding diffraction pattern computed for the event geometry that matches the observations for about seven fringes on either side.

Unlike stellar occultation observations, however, the availability of phase information in the recorded RSS signal enables the diffraction effects to be greatly reduced using Fresnel inversion (Marouf et al. 1986). The diffraction-corrected profile for this profile is shown in **Fig. A2**, at a processing resolution of 50 m. The upper panel shows the optical depth profile, with sharp edges and perhaps some evidence of noise-limited internal structure. The lower panel shows the reconstructed intensity profile, which is a more appropriate representation for comparison with a square-well model. In this instance, the ring does indeed resemble a square-well with sharp edges and a measured width of 1.43 km, determined from Eq. (A1) below (shown dashed), very close to the 1.37 km width of the square-well fit to the diffraction-limited data. Note that the radius scale is from the data in the PDS archive file, and not from the current orbit determination.¹³

The close agreement between the measured width and the width obtained from the square-well model demonstrates
its accuracy and applicability for actual observations of an intrinsically sharp-edged ring with an intensity profile that
<figure><img src="image_42.png" /><figcaption>**Figure A2.** Diffraction-reconstructed RSS egress γ ring profile at an effective resolution of 50 m, plotted as a function of normal optical depth (upper panel) and normalized intensity (lower panel). The width of the reconstructed profile determined from Eq. (A1) is 1.427 km, shown by the vertical dashed lines, very similar to the 1.370 km width obtained from the square-well model fit to the diffraction-limited profile shown in Fig. A1.</figcaption></figure>

resembles a box function. As shown by Gresh et al. (1989), however, several of the Uranian rings have radially variable internal structure, and not all of them have sharp edges. We repeated the exercise described above and compared the ring widths determined from the square-well fits of the diffraction-limited RSS observations to the observed ring widths of the diffraction-corrected profiles. To provide an objective measure of the ring width, we define the edges as the locations where

$$I = 1 - f(1 - \min I), \qquad (A1)$$

where *I* is the normalized diffraction-corrected signal intensity and *f* = 0.364, a value chosen to minimize the mean offset between the square-well model widths obtained from fits to the complete set of diffraction-limited RSS ring profiles and the actual ring widths estimated from the diffraction-corrected profiles as given by Eq. (A1). Here, *f* corresponds to the chosen fractional drop in the signal (from unit intensity to min *I*, the minimum observed intensity of a ring profile) that formally defines the edge of the ring. The results of these comparisons are given in **Table A1**, which lists the ingress and egress width estimates and their differences for each ring. The mean difference in width
$dW$ = 0.002 km between the square-well fits and the measured widths is very small, and the standard deviation $\sigma(dW)$ = 0.27 km.

<table><caption>**Table A1.** Comparison of square-well and *Voyager* ring widths</caption><thead><tr><th>Ring</th><th>Dir<sup>a</sup></th><th>W (km)<sup>b</sup><br>(square-well)</th><th>W (km)<sup>c</sup><br><i>Voyager</i></th><th>dW (km)</th></tr></thead><tbody><tr><td>6</td><td>I</td><td>0.899</td><td>1.141</td><td>-0.242</td></tr><tr><td>6</td><td>E</td><td>1.408</td><td>1.337</td><td>0.071</td></tr><tr><td>5</td><td>I</td><td>2.615</td><td>2.376</td><td>0.239</td></tr><tr><td>5</td><td>E</td><td>1.169</td><td>1.279</td><td>-0.111</td></tr><tr><td>4</td><td>I</td><td>1.405</td><td>1.295</td><td>0.110</td></tr><tr><td>4</td><td>E</td><td>1.880</td><td>1.663</td><td>0.217</td></tr><tr><td>α</td><td>I</td><td>9.423</td><td>8.676</td><td>0.747</td></tr><tr><td>α</td><td>E</td><td>3.866</td><td>4.120</td><td>-0.254</td></tr><tr><td>β</td><td>I</td><td>5.069</td><td>5.081</td><td>-0.012</td></tr><tr><td>β</td><td>E</td><td>8.334</td><td>8.099</td><td>0.235</td></tr><tr><td>η</td><td>I</td><td>1.371</td><td>1.479</td><td>-0.108</td></tr><tr><td>η</td><td>E</td><td>1.105</td><td>1.220</td><td>-0.115</td></tr><tr><td>γ</td><td>I</td><td>3.648</td><td>3.745</td><td>-0.097</td></tr><tr><td>γ</td><td>E</td><td>1.371</td><td>1.427</td><td>-0.056</td></tr><tr><td>δ</td><td>I</td><td>5.182</td><td>5.777</td><td>-0.595</td></tr><tr><td>δ</td><td>E</td><td>2.229</td><td>2.272</td><td>-0.044</td></tr><tr><td>ε</td><td>I</td><td>22.247</td><td>22.315</td><td>-0.068</td></tr><tr><td>ε</td><td>E</td><td>74.855</td><td>74.776</td><td>0.079</td></tr></tbody></table>

Shown graphically in **Fig. A3**, it is evident from these results that, except for two outliers (the ingress $\alpha$ and $\delta$ ring profiles), all of the measured width differences $dW$ are within $1-\sigma$ of zero. Of particular relevance for the measured normal mode distortions of selected ring edges in Section 4.1, we find that the RSS $\eta$ ring profiles have measured width differences $|dW| \le 0.115$ km, the $\gamma$ ring profiles have $|dW| \le 0.097$ km, and $\epsilon$ ring profiles have $|dW| \le 0.079$ km, suggesting that the square-well fitted widths to these intrinsically sharp-edged rings are reliable, at least for ring profiles that are not significantly smoothed by the finite angular diameters of the occultation stars or by instrumental time constants.

The two exceptions noted above for the $\alpha$ and $\delta$ rings illustrate some of the challenges of defining a metric for ring widths that reflects their actual internal structure. **Figure A4** shows the RSS diffraction-reconstructed $\alpha$ ingress ring profile, with the vertical dashed lines marking the ring edges as defined by Eq. (A1). The marked outer edge reasonably matches the moderately sharp outer edge of the ring, but the marked inner edge is clearly a compromise between the abrupt drop in intensity near 44734 km and the gradual variation in ring opacity between 44729–44734 km. (Note that the radius scale is from the data in the PDS archive file, and not from the current orbit determination.<fnref n="14" />) The best-fitting square-well model fit is shown in **Fig. A5**, where the underlying structure of the $\alpha$ ring’s inner edge (seen at right in this time-ordered ingress profile) is resolved at the level of a few km and resembles the observed structure near the inner edge of the *Voyager* RSS diffraction-corrected profile. The square-well model accurately matches the outer edge but not the inner edge, resulting in likely systematic errors in the fitted midtimes and widths for at least some Earth-based $\alpha$ ring profiles.

A similar situation prevails for the $\delta$ ingress event, whose RSS diffraction-reconstructed profile is shown in **Fig. A6**. Again, the outer edge of the ring is seen to be intrinsically quite sharp, but the inner edge is indistinct, with the ring’s opacity increasing gradually over a radial range of ~3 km before reaching a maximum near mid-ring. The vertical dashed lines mark the ring edges as defined by Eq. (A1). (Note that the radius scale is from the data in the PDS archive file, and not from the current orbit determination.<fnref n="15" />) In this instance, the square-well model fit, shown in

<footnote><sup>a</sup> I: ingress E:egress</footnote>

<footnote><sup>b</sup> Square-well fits provide the COR midtimes of *Voyager* RSS profiles used in ring orbit fits.</footnote>

<footnote><sup>c</sup> From Eq. (A1) with $f = 0.364$.</footnote>

<footnote n="14">14 VG_2803/U_RINGS/EASYDATA/KMOO_025/RU1P2XAI.TAB</footnote>

<footnote n="15">15 VG_2803/U_RINGS/EASYDATA/KMOO_025/RU1P2XDI.TAB</footnote>
<figure><img src="image_43.png" /></figure>

**Figure A3.** Differences *dW* between the widths of *Voyager* RSS ring profiles obtained from square-well model fits to the uncorrected diffraction patterns in the raw data and the measured widths of the diffraction-corrected profiles using Eq. (A1) with *f* = 0.364. The dotted lines mark the ±1-σ values computed from the full set of residuals, including the two outlier points (αI and δI). The barely visible dashed line shows the mean offset in *dW* of the measurements.

<figure><img src="image_44.png" /></figure>

**Figure A4.** Radial profile of the diffraction-corrected *Voyager* RSS observations of the α ring during ingress. The dashed vertical lines mark the midline and the inner and outer edges of the ring as estimated using Eq. (A1) with *f* = 0.364, corresponding to the flux level marked by the solid horizontal line. The solid vertical lines mark the corresponding radii as determined from the square-well model fit to the diffraction-limited RSS profile, based on the radius scale of the archived profiles on the PDS, which differs slightly from the radius scale of our adopted orbit solution.
<figure><img src="image_45.png" /><figcaption>**Figure A5.** Square-well model fit to the uncorrected diffraction-limited *Voyager* RSS α ring ingress profile. Note the good fit of the square-well model to the sharp outer edge of the ring (at left) and the poor fit to the more gradual inner edge at right.</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>**Figure A6.** Radial profile of the diffraction-corrected *Voyager* RSS observations of the δ ring during ingress. The dashed vertical lines mark the midline and the inner and outer edges of the ring as estimated using Eq. (A1) with *f* = 0.364, corresponding to the flux level marked by the solid horizontal line. The solid vertical lines mark the corresponding radii as determined from the square-well model fit to the diffraction-limited RSS profile, based on the radius scale of the archived profiles on the PDS, which differs slightly from the radius scale of our adopted orbit solution.</figcaption></figure>

**Fig. A7**, underestimates the measured with of the high-resolution profile. Notice that the diffraction fringes at the left (earlier in time, corresponding to the outer edge of this ingress profile) match the data for several fringes, whereas the diffraction-limited RSS observations of the inner edge show much more disorder, resulting from the lack of a sharp inner ring edge.

Given this level of disagreement between the square-well results and the actual ring observations for rings with non-sharp edges, one could envision a generalization of the simple square-well picture to a multi-well model that fitted the data to a more realistic radial opacity profile. Unfortunately, the radial optical depth profiles of the narrow Uranian
<figure><img src="image_47.png" /></figure>

**Figure A7.** Square-well model fit to the uncorrected diffraction-limited *Voyager* RSS *δ* ring ingress profile. The sharp outer edge of the ring at left is well-matched by the square-well model, but the modeled diffraction fringes for the inner edge at right are a poor match to the observed diffraction pattern of the gradual inner edge.

rings at different ring longitudes do not follow a simple, discernible pattern, and thus it has not proven possible to
construct an *N*-streamline model for any of the rings that accurately matches the observed structure at all longitudes.

B. CENTER OF OPACITY (COO) FROM VOYAGER AND EARTH-BASED RING OCCULTATION PROFILES

In Section 7.1, we distinguished between the observed radius of a ring’s centerline (COR) and its radius derived from its radially-integrated optical depth (COO), using the *Voyager* ring 6 occultation profiles as an example in Fig. 31. We apply this same method to *Voyager* ring occultation profiles for rings 5, 4, α, β, γ, and ε in Figs. B1–B6. For the ε ring, we have made additional use of the *Voyager* σ Sgr optical depth profiles, in part because the RSS ingress profile sampled the ring near periapse (f = 30.0°) with noise-affected high optical depths. (Note that RSS optical depths are twice as large as comparable spacecraft or Earth-based stellar occultation profiles, due to diffraction scattering signal loss.) The four *Voyager* profiles yield values of Δ*a*<sub>COR</sub> ranging from 0.46 to 1.53 km. The ring is also radially resolved in several of the Earth-based stellar occultations. We have selected thirteen of the highest SNR profiles from this set. To minimize the effects of uncertain normalization that afflict estimates of high optical depth when the observed normalized stellar flux is near zero, we limited our selection to profiles more than 90° from periapse, shown in Fig. B7. They yielded values of Δ*a*<sub>COR</sub> ranging from 0.54 to 1.95 km, comparable in average and range to the *Voyager* results. The mean ε ring value for Δ*a*<sub>COR</sub> = 1.134 ± 0.488 km.

The derived centers of opacity for all the eccentric ringlets are given in **Table 16**, which lists *a*<sub>COR</sub>, the unweighted average values for Δ*a*<sub>COO</sub> and their RMS variation, the statistical significance Δ*a*<sub>COO</sub>/σ(Δ*a*<sub>COO</sub>), and finally of *a*<sub>COO</sub>, with an uncertainty given by the sum in quadrature of σ(*a*<sub>COR</sub>) and σ(Δ*a*<sub>COO</sub>).

C. THE COMPATIBILITY OF A MODEL GRAVITY FIELD {J'<sub>2</sub>, J'<sub>4</sub>, J'<sub>6</sub>} WITH THE ADOPTED SOLUTION
FROM RING OCCULTATIONS

In Section 7, we noted the strong correlations between *J*₂, *J*₄, and *J*₆ in the gravity field determination from the Uranus ring occultation data. Here, we use the observed correlations to provide a prescription for determining the compatibility of model values {*J*'<sub>2</sub>, *J*'<sub>4</sub>, *J*'<sub>6</sub>} with the adopted gravity field.

First, we quantify the approximate relationship between *J*₂ and *J*₄. Given the strong correlation between *J*₂ and *J*₄ (reflected in the narrowness of the error ellipse for Fit 15 shown in Fig. 33), it is possible to construct an approximate
<figure><img src="image_48.png" /><figcaption>Figure B1. Ring 5 center of opacity from *Voyager* RSS optical depth profiles.</figcaption></figure>

<figure><img src="image_49.png" /><figcaption>Figure B2. Ring 4 center of opacity from *Voyager* RSS optical depth profiles.</figcaption></figure>

relationship between the two leading terms in the gravitational field. Under the restricted circumstances that $J_6 = 0$, the local value of $dJ_4/dJ_2$ can be estimated by setting the differential of the leading terms in $\dot{\omega}$ to zero:

$$d\dot{\omega} = \left(\frac{GM_{\text{Ur}}}{a^3}\right)^{1/2} \left[\frac{3}{2}dJ_2\left(\frac{R}{a}\right)^2 - \frac{15}{4}dJ_4\left(\frac{R}{a}\right)^2\right] = 0, \quad (C1)$$
<figure><img src="image_50.png" /><figcaption>Figure B3. Ring α center of opacity from *Voyager* RSS optical depth profiles.</figcaption></figure>

<figure><img src="image_51.png" /><figcaption>Figure B4. Ring β center of opacity from *Voyager* R/coS optical depth profiles.</figcaption></figure>

from which we find

$$dJ_4/dJ_2 = \frac{2}{5} \left(\frac{a}{R}\right)^2 = 1.24 \left(\frac{a}{45000 \text{ km}}\right)^2, \qquad (C2)$$
<figure><img src="image_52.png" /><figcaption>**Figure B5.** Ring $\gamma$ center of opacity from *Voyager* RSS optical depth profiles.</figcaption></figure>

<figure><img src="image_53.png" /><figcaption>**Figure B6.** Ring $\epsilon$ center of opacity from *Voyager* RSS and $\sigma$ Sgr optical depth profiles. Note that stellar occultation geometric optical depths should be multiplied by a factor of two before comparison with RSS normal optical depths.</figcaption></figure>
<figure><img src="image_54.png" /><figcaption>**Figure B7.** Estimates of the $\epsilon$ ring center of opacity (COO) from Earth-based occultation profiles. Note that stellar occultation geometric optical depths should be multiplied by a factor of two before comparison with RSS normal optical depths. See text for details.</figcaption></figure>

Setting $a$ in Eq. (C2) to 44695 km, the mean of the semimajor axes of rings 6, 5, 4, $\alpha$, $\beta$, and $\epsilon$ (those with accurate apohe rates), and letting $R = 25559$ km, we obtain $dJ_4/dJ_2 \simeq 1.223$. This is close to the slope of the long axis of the error ellipse for our adopted fit in Fig. 33, given by $dJ_4/dJ_2 = \tan\theta = 1.134$, where $\theta$ is defined in Eq. (C20) below. Under a linear approximation, a model value $J'_4$ that is consistent with the observed values of $J_2$ and $J_4$ and an assumed model value $J'_2$ is given by

$$J'_4(J'_2) \simeq J_4 + \frac{dJ_4}{dJ_2}(J'_2 - J_2). \qquad (C3)$$

Evaluated numerically:

$$J'_4(J'_2) \simeq [-35.533 + 1.134 \times (J'_2 - 3509.281)] \times 10^{-6}. \qquad (C4)$$

Next, we assume that the partial derivatives in Table 17 accurately represent the sensitivity of the fitted values for $J_2(J_6, \Delta a)$ and $J_4(J_6, \Delta a)$ in our adopted solution (Fit 15) to small differences from our assumed values of $J_6 = 0.5 \times 10^{-6}$ and $\Delta a = 0$. Then the following relations specify the values of $J_2$ and $J_4$ predicted to result from a fit in which $J_6$ and the systematic radius scale offset have the fixed a priori values $J'_6$ and $\Delta a$:

$$J_2(J'_6, \Delta a) = J_2(0.5 \times 10^{-6}, 0) + \frac{dJ_2}{dJ_6}(J'_6 - 0.5 \times 10^{-6}) + \frac{dJ_2}{d\Delta a}\Delta a \qquad (C5)$$
$$J_4(J'_6, \Delta a) = J_4(0.5 \times 10^{-6}, 0) + \frac{dJ_4}{dJ_6}(J'_6 - 0.5 \times 10^{-6}) + \frac{dJ_4}{d\Delta a}\Delta a. \quad (C6)$$

Numerically, our final values for $J_2$ and $J_4$ map onto the corresponding values for non-zero $J'_6$ and $\Delta a$ according to:

$$J_2(J'_6, \Delta a) = 3509.281 \times 10^{-6} + 0.39909 \times (J'_6 - 0.5 \times 10^{-6}) + 0.130 \times 10^{-6} \Delta a(\text{km}) \quad (C7)$$

$$J_4(J'_6, \Delta a) = -35.533 \times 10^{-6} + 1.07570 \times (J'_6 - 0.5 \times 10^{-6}) - 0.180 \times 10^{-6} \Delta a(\text{km}). \quad (C8)$$

To determine whether any planetary gravity model {$J'_2, J'_4, J'_6, \Delta a$} falls within the 1-$\sigma$ error ellipse of our adopted solution, the Mahalanobis distance $r$ must be $\le 1$, where

$$r = \sqrt{\frac{x'^2}{\lambda_1} + \frac{y'^2}{\lambda_2}}, \quad (C9)$$

and $\lambda_1$ and $\lambda_2$ are the eigenvalues of the $(J_2, J_4)$ covariance matrix:

$$\operatorname{covar}(J_2, J_4) = \begin{bmatrix} a & b \\ b & c \end{bmatrix}, \quad (C10)$$

with

$$a = \sigma^2(J_2), \quad (C11)$$

$$b = \rho(J_2, J_4)\sigma(J_2)\sigma(J_4), \quad (C12)$$

and

$$c = \sigma^2(J_4), \quad (C13)$$

and $x'$ and $y'$ are defined below.<fnref n="16" /> The halfwidths of the two axes of the error ellipse are given by $\sqrt{\lambda_1}$ and $\sqrt{\lambda_2}$, where

$$\lambda_1 = \frac{a+c}{2} + \sqrt{\left(\frac{a-c}{2}\right)^2 + b^2}, \quad (C14)$$

and

$$\lambda_2 = \frac{a+c}{2} - \sqrt{\left(\frac{a-c}{2}\right)^2 + b^2}. \quad (C15)$$

The following transformation rotates the error ellipse by $\theta$:

$$x' = x \cos \theta + y \sin \theta \quad (C16)$$

$$y' = -x \sin \theta + y \cos \theta, \quad (C17)$$

where

$$x = J'_2 - J_2(J'_6, \Delta a), \quad (C18)$$

$$y = J'_4 - J_4(J'_6, \Delta a), \quad (C19)$$

and the rotation angle $\theta$ is specified by

$$\theta = \tan^{-1}(\lambda_1, b), \quad (C20)$$

where the arguments of the arctangent are the column entries of the covariance matrix eigenvectors, simplified by multiplying by their common divisor.

From our adopted gravity parameters (Fit 15 in Table 17), $\sigma(J_2) = 0.412 \times 10^{-6}$, $\sigma(J_4) = 0.466 \times 10^{-6}$, and $\rho(J_2, J_4) = 0.9861$, yielding the numerical values $\lambda_1 = 3.84252 \times 10^{-13}$, $\lambda_2 = 2.64830 \times 10^{-15}$, $a = 1.697440 \times 10^{-13}$, $b = 1.893233 \times 10^{-13}$, $c = 2.17156 \times 10^{-13}$, and $\theta = 48.5686^\circ$.

<footnote n="16">¹⁶ https://cookierobotics.com/007/</footnote>
The complementary cumulative probability CCP(r) that a given sample has a standard deviation ≥ r by chance for a given (J'₂, J'₄, J'₆, Δa) is given by

$$CCP(r) = e^{-r^2(J'_2, J'_4, J'_6, \Delta a)/2}, \qquad (C21)$$

appropriate for a two-dimensional gaussian distribution, where r is given by Eq. (C9). Note that CCP(1) = 0.6065, CCP(2) = 0.1353, CCP(3) = 0.0111, and CCP(4) = 0.0003. In comparison, the complementary cumulative probability for a one-dimensional normal distribution is given by erfc(r/√2), yielding the more familiar corresponding values 1σ = 0.3173, 2σ = 0.0455, 3σ = 0.0027, and 4σ = 0.0001.

The inverse relationship

$$r = \sqrt{-2 \ln \text{CCP}(r)} \qquad (C22)$$

provides the factor r for a given confidence level CCP(r). For example, the values of r corresponding to CCP=30%, 10% and 1% cutoff values for acceptable interior models that are consistent with our gravity measurements are 1.55, 2.15 and 3.03, respectively.

The prescription for computing the difference in standard deviations r(J'₂, J'₄, J'₆, Δa) between a model with assumed values {J'₂, J'₄, J'₆, Δa} and our adopted solution is:

* Specify values for J'₂, J'₄, J'₆, and Δa.

* Compute J₂(J'₂, Δa) and J₄(J'₄, Δa) from Eqs. (C7) and (C8).

* Compute x and y from Eqs. (C18) and (C19).

* Compute x' and y' from Eqs. (C16) and (C17), using the given numerical value for θ.

* Compute r from Eq. (C9) and the given numerical values for λ₁ and λ₂.

## D. ALIGNMENT OF ILR AND OLR RING EDGE MODE APSES RELATIVE TO FORCING SATELLITE

In the main body of the paper, we have assumed the following properties of ring edge modes force by a satellite:

* In the absence of dissipation, the forced mode (periapse/apoapse) is aligned with the satellite mean longitude for an (ILR/OLR), respectively.

* In the presence of dissipation, the relevant apse is (lagging/leading) the satellite mean longitude in angle for an (ILR/OLR), respectively.

Here, we justify these assumptions on dynamical grounds by drawing on well-known results for ILR edge modes and making use of an exact formal symmetry between inner and outer edge modes to determine the corresponding OLR edge mode properties.

ILR edge modes are to date much more extensively studied than OLR ones; in particular, detailed numerical solutions are available only for ILR edge modes. It turns out that a few properties, including simple geometric features, can only be obtained through detailed solutions of the dynamical equations. OLR edge mode properties must therefore also be obtained from such detailed solutions.

In the large |m| limit, the dynamical equations display an exact formal symmetry between inner and outer edge modes. This symmetry allows us to deduce a number of properties of inner edge modes from outer edge ones.<fnref n="17" />

<footnote n="17">¹⁷ The large m limit applies quantitatively as soon as |m| is larger than a few; the approximation is also semi-quantitatively relevant even for |m| = 2.</footnote>
Let us show this explicitly. In the streamline formalism, the mode structure is controlled by two dynamical equations, one for the mode eccentricity profile and one for its apsidal shift profile. In a discrete, N-streamlines formulation, these equations read (see Longaretti 2018 and Longaretti 2023 for details):

$$ \sum_{j \neq i} \frac{\bar{n} m_j}{\pi M_p} a^2 H(q_{ij}^2) \frac{\epsilon_i - \epsilon_j}{(\Delta a_{ij})^2} - [\kappa_i - m(\Omega_i - \Omega_P)]\epsilon_i = -\bar{n} \frac{a \Psi_{m,k}}{2GM_p}, \quad (D1) $$

$$ \sum_{j \neq i} \frac{\bar{n} m_j}{\pi M_p} a^2 H(q_{ij}^2) \frac{\epsilon_i \sin m\Delta_i - \epsilon_j \sin m\Delta_j}{(\Delta a_{ij})^2} - [\kappa_i - m(\Omega_i - \Omega_P)]\epsilon_i \sin m\Delta_i = \frac{2\pi}{\bar{n}m_i} \Delta^{\pm} \left( \frac{a}{q} \frac{d\epsilon}{da} t_1 \right). \quad (D2) $$

In these equations, m is the azimuthal wavenumber (number of lobes of the edge mode), $a_i$, $\epsilon_i$ and $\Delta_i$ are the semi-major axis, eccentricity and apsidal shift of streamline i ($\Delta_i = -\delta_i$ in the notation of Eq. 8 of the present paper), $m_j$ the mass of streamline j, $\Delta a_{ij} = a_j - a_i$, $\bar{n}$ is the mean motion at the mode resonance and a the resonance location, $q_{ij} = a(\epsilon_j - \epsilon_i)/\Delta a_{ij}$, $\kappa_i$ and $\Omega_i$ the epicyclic radial frequency and angular rotation velocity of streamline i, $M_p$ the planet mass (here Uranus), $\Omega_P$ the mode pattern speed, $\Psi_{m,k}$ the satellite forcing term, $H(q_{ij})$ a dimensionless function of order unity which arises from the nonlinearity of the ring self-gravity, and $t_1$ is an effective stress tensor quantity, characterizing the ring viscous dissipation. The viscous coefficient $t_1$ is a function of the nonlinearity parameter $q = |ade/da|$; $t_1 < 0$ in the absence of viscous overstability, and $|t_1| \to 0$ when $q \to 0$. Finally, $\Delta^{\pm}X$ is the variation of X across the current streamline (outer boundary minus inner boundary value of X). The azimuthal wavenumber m is positive at an inner Lindblad resonance and negative at an outer Lindblad resonance.

The first term on the left-hand side of both equations represents the effect of the ring self-gravity, the second term the frequency drift with respect to the pattern speed, as by definition $\kappa - m(\Omega - \Omega_P) = 0$ at the resonance. On the right-hand side, one finds the satellite forcing term in the first equation and the ring internal (collisional) dissipation in the second one. The two equations are very similar; in particular the left-hand sides are formally identical under the substitution $\epsilon \leftrightarrow \epsilon \sin m\Delta$.

Furthermore, one can write

$$ \kappa_i - m(\Omega_i - \Omega_P) = \frac{3\Omega(m-1)}{2a}(a_i - a) \approx \frac{3\Omega m}{2a}(a_i - a), \quad (D3) $$

for $|m| > 1$. The mode exists mostly outside the resonance at an ILR and inside the resonance at an OLR. The second equality applies in the large $|m|$ limit.

<figure><img src="image_55.png" /><figcaption>**Figure D1.** Sketch of the ILR/OLR streamline symmetry: streamline semimajor axes $a_i$ are symmetrized with respect to the resonance location a, while keeping their index order. As a consequence, they are ordered in increasing semimajor axis order at ILR but decreasing semimajor axis at OLR.</figcaption></figure>
This large $|m|$ limit reveals an interesting symmetry property between an ILR and an OLR equations with the same but opposite $m$ wavenumbers (see Fig. D1). To see this, let us make a formal correspondence between an ILR and an OLR. We assume for now that the two resonance radii are equal.<fnref n="18" /> We specify the streamlines semi-major axes of an ILR edge mode by a '+' superscript as $a_i^+$. We also define a mirror set of streamlines with respect to the resonance location, with semi-major axes $a_i^- = a - (a_i^+ - a)$; by construction, $a_i^-$ is the symmetric location of $a_{N+1-i}^+$, with respect to the resonance location $a$. Calling $m^+ > 0$ the ILR azimuthal wavenumber and $m^- = -m^+ < 0$ the OLR one, in the large $|m|$ limit,

$$ \kappa_i^+ - m^+(\Omega_i^+ - \Omega_P) = \kappa_i^- - m^-(\Omega_i^- - \Omega_P), \qquad (D4) $$

with the same meaning for the superscripts + and - on the frequencies, e.g. $\kappa_i^+ = \kappa(a_i^+)$. Defining similarly $\epsilon_i^\pm$ and $\Delta_i^\pm$, this symmetry implies that the left-hand sides of Eqs. (D1) and (D2) are unchanged under the substitution $m^+ \leftrightarrow m^-$, $a_i^+ \leftrightarrow a_i^-$ if one assumes $\epsilon_i^- = \epsilon_i^+$ and $\Delta_i^- = \Delta_i^+$.

This would also provide the correct OLR mode solution from the ILR one if the right-hand sides were unchanged. However, this is not the case, due to changes in the right-hand sides under the ILR/OLR symmetry.

Let us start with Eq. (D1). In the large $m$ limit, and for leading order inner Lindblad resonances ($k=0$, the only ones of interest here):

$$ \begin{aligned} \Psi_{m,0}(a) &\approx -\frac{GM_s}{a_s} \frac{2m}{\pi} [2K_0(2/3) + K_1(2/3)] \\ &\approx -\frac{GM_s}{a_s} \frac{5.04m}{\pi} \approx -\frac{GM_s}{a_s} \frac{5m}{3}, \end{aligned} \qquad (D5) $$

where $M_s$ is the satellite mass, $a_s$ its semimajor axis, and $K_0$ and $K_1$ are modified Bessel functions (see Goldreich and Tremaine 1980 for this approximation). The same expression holds for a leading order OLR of same $|m|$. Note that $\Psi_{m,k} < 0$ at an ILR and $> 0$ at an OLR due to the change of sign of $m$.

These remarks have the following implication: under the substitution of an ILR by an OLR as described above and assuming for simplicity that $\Psi_{m-,k} = -\Psi_{m+,k}$ — which is nearly true in the $|m| \gg 1$ limit — the ILR solution of Eq. (D1) is also a solution of the OLR form of the equation under the substitution $\epsilon^- = -\epsilon^+$. In other words, the eccentricity profile of an ILR forced mode is also the eccentricity profile of the associated OLR (with the meaning specified above for this association), once this profile is antisymmetrized with respect to the resonance location.

This sign change has a notable consequence. From known numerical solutions of the forced mode equations at an ILR (Longaretti 2023), dissipationless ILR modes have their periapse aligned with the origin of phases of the forcing potential (i.e., the mean longitude of the satellite for leading order Lindblad resonances). The change of sign implies that, at an OLR, the apoapse is aligned with the satellite mean longitude instead, as $a(1 - \epsilon)$ (periapse) becomes $a(1 + \epsilon)$ (apoapse).

Let us now consider Eq. (D2). The symmetry just described leaves the nonlinearity parameter profile $q = |ad\epsilon/da|$ unchanged but symmetrized with respect to the resonance location. Relatedly $ad\epsilon/da$ is also symmetric with respect to the resonance location (due to the change of sign of the eccentricity, the derivative remains positive). This implies that $(a/q)(d\epsilon/da)t_1$ is symmetric as well, so that, for any given streamline, the right-hand side of Eq. (D2) changes sign under the symmetry discussed here, but is otherwise left unchanged. Therefore, $\Delta_i^- = -\Delta_i^+$ is the correct solution at an OLR under the same symmetry (as $\epsilon$ has changed sign according to the preceding discussion of the eccentricity profile and $m$ changes sign as well).

To sum up: under the ILR/OLR symmetry described above, the ILR solution for a forced edge mode is also a solution at an OLR under antisymmetrization of the eccentricity $\epsilon$ and apsidal shift $\Delta$ profiles with respect to the resonance location. As a consequence, the mode apoapse is aligned with the satellite mean longitude in the absence of

<footnote n="18">¹⁸ This is hypothetical, and only for the sake of displaying the OLR/ILR edge mode symmetry. This does not imply that such a configuration of degenerate ILR and OLR resonances actually exists in rings.</footnote>
dissipation. When dissipation is added, the apoapse is expected to be leading (in angle) with respect to the satellite mean longitude because the mode global phase $m\phi + m\Delta$ vanishes for some positive $\phi$ ($\phi$ is the azimuthal angle of the mode in the rotating frame). At an ILR, the mode periapse is nearly aligned with and lagging with respect to the satellite instead.

The conclusions just stated on the nature of the apse aligned with the satellite for an ILR/OLR forced mode are robust, but the sign of the lag applies for the simplest physical context only, and is therefore subject to various qualifications. These are discussed in the body of the text.

Note finally that the sign of the lag is consistent with the requirement that angular momentum exchanges with the satellite flow from the inside to the outside, in order for the satellite to confine a ring edge. This conclusion follows very simply from the following generic torque expression

$$T_s = \int da \pi m a \sigma_0 \Psi_{m,k} \epsilon \sin m \Delta, \qquad (D6)$$

which is valid for inner and outer resonances (Longaretti 1992, 2018). For an ILR edge mode, $m > 0$, $\Psi_{m,k} < 0$ and $\epsilon \sin m\Delta > 0$ so that $T_s < 0$, as expected (see Longaretti 2023 for actual eccentricity and apsidal shift profiles to substantiate this statement). For an OLR edge mode, and because of the ILR/OLR symmetry shown in this Appendix, $m < 0$, $\Psi_{m,k} > 0$ and $\epsilon \sin m\Delta < 0$ so that $T_s > 0$, again in line with expectations.

The arguments presented here are exact for identical ILR/OLR resonance locations and identical but opposite satellite forcing terms for both an ILR and an OLR. These requirements are not satisfied in reality, but the difference is only quantitative, not qualitative. In particular, for the large $|m|$ resonances of interest in the Uranian rings context, the relative quantitative difference introduced with respect to actual resonance locations and satellite forcing magnitudes is a small fraction of unity. Such a small difference will produce barely noticeable deviations in the OLR eccentricity and apsidal shift profiles from their symmetric ILR counterpart, for any given satellite. Furthermore, the qualitative issues examined here (which apse is nearly aligned with the satellite, and sign of their small lags) are independent of the quantitative details, such as the satellite mass, the azimuthal wavenumber, and the resonance location.

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