Uranus ring occultation observations: 1977–2006

The Uranian rings were discovered serendipitously on 10 March 1977 during a stellar occultation (Elliot et al., 1977a; Millis et al., 1977), and a rich set of subsequent Earth-based occultations revealed that these narrow and sharp-edged rings were eccentric and inclined, precessing under the gravitational influence of the oblate central planet. Considerable progress has been made in understanding the observed characteristics of narrow rings and sharp edges (Nicholson et al., 2018) and their associated dynamics (Longaretti, 2018), but ever since their discovery, the Uranian rings have posed dynamical puzzles that resist simple explanations. The observational basis to address these questions for the Uranus system rests largely on occultation measurements of the narrow rings spanning nearly 30 years, beginning in 1977 and concluding most recently in 2006. Nearly all of these occultation data sets are available in digital form on NASA’s Planetary Data System (PDS) Ring-Moon Systems node, but many of them have not been previously published or described in detail. This paper serves as a guide to the PDS archive and provides essential information about the observations and the methods used to determine the ring widths, mean optical depths, and occultation event times from individual occultation profiles. Additional detail is provided in the Supplementary Online Material accompanying this publication. In a companion paper (French et al., 2023b), we make use of these observations to determine the Uranus ring orbits, pole direction, and gravity field, and the orbital characteristics and masses of three small Uranian moons – Cressida, Ophelia, and Cordelia – from their forced normal modes on the rings. ∗ Corresponding author. E-mail address: rfrench@wellesley.edu (R.G. French). vailable online 19 February 2023 019-1035/© 2023 The Author(s). Published by Elsevier Inc. This is an open access a https://doi.org/10.1016/j.icarus.2023.115474 Received 4 November 2022; Received in revised form 6 February 2023; Accepted 8 rticle under the CC BY license (http://creativecommons.org/licenses/by/4.0/). February 2023 Icarus 395 (2023) 115474 R.G. French et al. Fig. 1. Map of the sky showing the apparent path of Uranus as viewed from Earth between 1975 and 2050, with + symbols at five year intervals as labeled. The brightness of the background is proportional to the logarithmic areal density of stars in the 2MASS catalog (Skrutskie and 30 colleagues, 2006) with apparent magnitude K < 15. The Milky Way appears as a wavy band. Between 1985 and 1990, Uranus traversed the rich star fields in the direction of the galactic center, resulting in many opportunities for high-SNR occultation events with ring elevation angle |B| > 70◦. For subsequent observations between 1990 and 2006, |B| decreased roughly linearly, approaching ring plane crossing in 2007. The next traversal of the Milky Way by Uranus will not take place until 2030–2035, by which time the ring elevation will again be in a favorable range (|B| > 50◦) for Earth-based occultation observations. Until then, the frequency of high-SNR stellar occultations by Uranus and its rings will be at a typical cadence of one every few years.


A B S T R A C T
The Uranian rings were discovered serendipitously on 10 March 1977 during a stellar occultation (Elliot et al., 1977a;Millis et al., 1977), and a rich set of subsequent Earth-based occultations revealed that these narrow and sharp-edged rings were eccentric and inclined, precessing under the gravitational influence of the oblate central planet. Considerable progress has been made in understanding the observed characteristics of narrow rings and sharp edges (Nicholson et al., 2018) and their associated dynamics (Longaretti, 2018), but ever since their discovery, the Uranian rings have posed dynamical puzzles that resist simple explanations. The observational basis to address these questions for the Uranus system rests largely on occultation measurements of the narrow rings spanning nearly 30 years, beginning in 1977 and concluding most recently in 2006. Nearly all of these occultation data sets are available in digital form on NASA's Planetary Data System (PDS) Ring-Moon Systems node, but many of them have not been previously published or described in detail. This paper serves as a guide to the PDS archive and provides essential information about the observations and the methods used to determine the ring widths, mean optical depths, and occultation event times from individual occultation profiles. Additional detail is provided in the Supplementary Online Material accompanying this publication. In a companion paper (French et al., 2023b), we make use of these observations to determine the Uranus ring orbits, pole direction, and gravity field, and the orbital characteristics and masses of three small Uranian moons -Cressida, Ophelia, and Cordelia -from their forced normal modes on the rings.

Introduction
The Uranian rings were discovered serendipitously on 10 March 1977 during a stellar occultation Millis et al., 1977), and a rich set of subsequent Earth-based occultations revealed that these narrow and sharp-edged rings were eccentric and inclined, precessing under the gravitational influence of the oblate central planet (Elliot et al., 1978;Elliot and 6 colleagues, 1981;Elliot et al., 1981;Elliot and 9 colleagues, 1983;Nicholson et al., 1978Nicholson et al., , 1981Nicholson et al., , 1982French et al., 1982French et al., , 1986. The structure of the newly-discovered Uranus ring system was in striking contrast to the expectation supported by pre-Voyager low-resolution Saturn observations that rings must be broad and diffuse sheets of material in the planet's equatorial plane, smoothed by the diffusive effects of interparticle collisions. With the discoveries of the dusty rings of Jupiter and the incomplete rings of Neptune (see De Pater et al., 2018b,a for recent reviews), the dizzying variety of giant planet ring systems became evident. Stellar and radio occultations during the Voyager 1 and 2 encounters of Saturn in 1980 and 1981 provided high-resolution views of the complex structure of its classical rings, and the Voyager 2 Uranus encounter in 1986 greatly expanded our detailed understanding of its rings from a combination of wide angle camera (WAC) and narrow angle camera (NAC) images and radio and stellar occultations. Looking back at Earth after the Uranus flyby, the forward-scattered images revealed an extensive sheet of tenuous dusty material with rich radial structure, intermingled with ten narrow, sharp-edged rings detectable in stellar and radio occultations, named in order of increasing radius 6, 5, 4, , , , , , , and . Subsequent exquisite HST imaging observations (Showalter and Lissauer, 2006) and adaptive optics images from large Earth-based telescopes (de Pater et al., 2006;de Pater and 9 colleagues, 2013) revealed the broad and tenuous , , and rings. (For a post-Voyager overview of the Uranus ring system, see French et al., 1991; for a more recent broad review, see Nicholson et al., 2018a.) Considerable progress has been made in understanding the observed characteristics of narrow rings and sharp edges (Nicholson et al., and ′ (as observed from Earth and sun, respectively), over the course of the Uranus ring occultation observations presented in this paper. The annual small amplitude sinusoidal modulations are due to the Earth's orbital motion, and the longer term overall variation is due to Uranus's orbital motion. Open symbols mark previously published ring observations; filled symbols mark previously unpublished ring occultations that are described in detail in this paper. The Voyager 2 encounter occurred on 24 Jan 1986, when = −85 • . 2018b) and their associated dynamics (Longaretti, 2018), but ever since their discovery, the Uranian rings have posed dynamical puzzles that stubbornly resist simple explanations: 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? What lessons can we apply from Cassini's extensive investigations of Saturn's rings from multi-wavelength occultations, imaging, dust detections, and particles and fields measurements (see Colwell et al., 2009 andCuzzi et al., 2018 for recent reviews)?
The observational basis to address these questions for the Uranus system relies significantly on occultation measurements of the narrow rings spanning nearly 30 years, beginning in 1977 and concluding most recently in 2006. Three factors have conspired to restrict their observability to this time period. First, the path of Uranus across the sky moved out of the dense star fields of the Milky Way in the early 1990s, limiting the number of candidate occultations of bright stars. Second, and somewhat perversely, the high-speed IR aperture photometers with 1-10 msec integration times that were widely available at large telescopes around the world during the 1980s were replaced by IR arrays that often did not support rapid sub-array readouts, thereby limiting the radial resolution of the observations. Finally, the favorable nearly face-on aspect of the Uranus ring plane as viewed from Earth in the 1980s resulted in slow radial ring plane velocities and thus high SNR at diffraction-limited resolution, but as the rings became more edge-on, high-SNR occultation opportunities became increasingly rare, with the most recent set of known successful occultation observations being taken in 2006, shortly before 2007 ring plane crossing as viewed from Earth. 1 Nearly all of these occultation data sets are available in digital form on NASA's Planetary Data System (PDS) Ring-Moon Systems node, 2 but many of them have not been previously published or described in detail. This paper provides essential information about these archived observations, and is organized as follows. We begin in Section 2 with an 1 In recent years, the rings have returned to a more favorable open configuration, but during this period Uranus traversed a fallow region for bright stars, and few occultation opportunities have arisen. The prospects for future occultations are presented in the prediction list of French (2023  overview of the occultation observations and the changing orientation of the Uranian rings as viewed from Earth over nearly 30 years. Next, in Section 3, we describe the methods used to determine the ring widths, mean optical depths, and occultation event times from individual occultation profiles. In Section 4, we describe each of the Earth-based occultation events, providing updated information of observed ring events for previously published data sets, and more detailed descriptions of previously unpublished occultations. We also include gallery plots of individual ring profiles for selected occultations that illustrate important characteristics of the data quality or processing methods. In Section 5, we review the key characteristics of the Voyager 2 occultation observations. In the final section, we summarize our main results and present our conclusions. There are two appendices: the first summarizes the contents of the PDS archive of Earth-based Uranus ring occultation data, and the second summarizes the contents of the Supplementary Online Material (SOM). In a companion paper (French et al., 2023b), we make use of these observations to determine the Uranus ring orbits, pole direction, and gravity field, and the orbital characteristics and masses of three small moons from their forced normal modes on the rings.

Overview of uranus occultation observations
The complete set of Earth-based Uranus occultations presented here includes 31 separate stellar occultations, often viewed with multiple telescopes and at multiple wavelengths, including two events observed R.G. French et al. Fig. 3. Sky plane views of Uranus occultation events U0 through U23. The apparent path of the occulted star over the time interval recorded in the data is shown as a solid line for each observing site included in this paper, labeled by PDS bundle observatory name, with the earliest time for each occultation chord marked by a filled circle. from the Hubble Space Telescope (HST). Table 1 provides a chronological list of the 54 separate telescope observations of these 31 events, identified by their PDS ''bundle" ID (defined below), occultation date, occulted star ID, observatory name, telescope diameter, observed wavelength, and ring opening angle (also referred to as the observed ring elevation) as seen from Earth. These are supplemented by three Voyager 2 occultations in 1986: one by the Radio Science Subsystem (RSS; Tyler and 9 colleagues, 1986;Gresh et al., 1989) and stellar occultations of Sgr and Per observed by the Photopolarimeter (PPS; Lane and 10 colleagues, 1986;Colwell and 12 colleagues, 1990) and the Ultraviolet Spectrometer (UVS; Holberg et al., 1987).

Characteristics of observed occultation events
The majority of the Earth-based events were observed between 1980-1990, when Uranus was traversing the Milky Way and high-SNR occultation opportunities were frequent, as shown in Fig. 1. After the initial discovery observations, which were made at visual wavelengths, it was quickly recognized that the SNR could be greatly enhanced by observing in the infrared K band ( ∼ 2.2 μm), where very strong absorption of sunlight by methane in the planet's atmosphere would minimize the contribution of scattered sunlight by the disk of Uranus to the total signal within the photometric aperture centered on the occultation star. Nearly all of the post-1977 events were observed in the IR using high-speed InSb aperture photometers. The earliest such events (U5, U9, and U11) were recorded using strip charts, rather than in digital form, and as we describe below, we extracted both the time signals and occultation lightcurves from scanned digital images of the chart recordings to obtain one-dimensional time series lightcurves for selected individual ring profiles for these occultations. Later IR observations were recorded digitally at time resolutions of a few msec to ∼ 0.1 sec. In many cases, the IR observations were recorded continuously in what is known as DC mode. In other cases, especially for faint stars or events with a strong background signal, observations were conducted in chopping mode, where the secondary mirror of the telescope rapidly nodded between the event star and the nearby sky, with filtering electronics recording the difference between the brightness within the aperture at these two alternate positions. The tradeoffs between these two approaches will be more clearly evident in Section 3, when we present the results of individual observations. A small number of events were observed using imaging arrays at visual and IR wavelengths, with aperture photometry performed digitally on individual image frames.
As Uranus moved along its heliocentric orbit between 1977 and 2006, the observed ring elevation as observed from Earth ( ) and from the sun ( ′ ) gradually changed from being partially open in 1977 to nearly face on in 1985, and then eventually to nearly edge-on in 2006 (the year of the most recent observations), as illustrated in Fig. 2. 3 When the rings are nearly face-on, the radial velocity of the occultation ray in the ring plane is roughly comparable to the sky plane velocity of the occultation star relative to the center of the planet, which varies from a few to tens of km s −1 , but as the rings become more edge-on, the radial velocity can be many tens of km s −1 , reducing the SNR and the spatial resolution of the ring profile for a given time resolution. The intrinsic resolution of Earth-based occultations is diffraction-limited by the Fresnel scale ∼ √ ∕2, where is the observed wavelength and is the distance between the observer (or spacecraft) and the ring plane. For typical IR Uranus ring observations, ∼ 1.75 km. As described in more detail below, the resolution is further limited by the projected diameter of the occulted star, by time constants associated with electronic filters, and by the intrinsic sampling interval of the recorded data.
Figs. 3-5 show the sky plane view of Uranus and the rings for all separate occultations in Table 1. The apparent path of the occulted star from each station over the time interval recorded in the data is shown as a solid line, with the earliest time for each occultation chord marked by a filled circle.

NASA planetary data system archive
Nearly all of the data sets described in this paper are available on the PDS Ring-Moon Systems node. 4 The archive includes a User Guide and a comprehensive set of high-SNR digitally-recorded Earthbased Uranus occultations and individual high-resolution ring profiles registered on an accurate radius scale based on the least-squares fit to the occultation data for the orbits of the ten classical Uranian rings and the Uranus pole direction described in French et al. (2023b). Browse products provide a quick overview of each occultation, normalized and geometrically registered lightcurves of the entire recorded event (including atmospheric occultations, when present), and detailed model fits to individual ring profiles for the ten classical narrow Uranian rings (6, 5, 4, , , , , , , and ). Each occultation set contains a digital table listing details of each observed ring event, such as fitted values  of the ring width and optical depth, the geometry of each individual profile, a quality index to provide a shorthand assessment of the quality of each profile, and predicted ring event times for rings that were not detected in the observations. 5 The PDS Uranus ring occultation archive is contained in a set of bundles of two types: observation bundles that each contain detailed information about a single occultation observation, and a single support bundle that contains information applicable to all observation bundles, including the User Guide, the global ring orbit fit used to determine the geometry for each occultation, NAIF frame kernels 6 to compute the Uranus ring geometry at any given time, based on this orbit fit, and tables listing the quality scores and radial profiles for all archived ring occultation observations. For convenient reference, Appendix A contains a summary of the contents of a representative observation bundle and of the support bundle.

Absolute timing and photometric calibration
A first step in the data analysis is to determine the absolute timescale of the occultation lightcurve, which is especially important for events observed from multiple stations. In some cases, accurate WWVB time signals were recorded directly into the digital data stream. For many of the occultations observed in the 1980s, reliable absolute time signals 5 These are especially useful for investigating the azimuthally incomplete structure of the ring. 6 NASA's Navigation and Ancillary Information Facility (NAIF) uses frames to enable users to compute the geometry of an observation in a variety of reference frames. In this instance, the individual ring planes of the ten classical Uranian rings are defined in a special-purpose frame kernel file that is available as part of the PDS archive. See https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/ C/req/frames.html. R.G. French et al. Fig. 7. Representative ring profiles with quality index QI = 1 through 4. (a) QI = 1 U36 AAT egress; (b) QI = 2 U17B SAAO egress; (c) QI = 3 U65 IRTF egress, showing a noisy but convincing detection; (d) QI = 4 U65 IRTF egress, showing a marginal detection that is nevertheless very close to the expected location based on our final ring orbit model. were not universally available at observatories, and portable calibrated clocks were sometimes carried around the world to provide absolute timing. Even with these efforts there remain occasional systematic differences in absolute timing between stations that need to be included as fitted parameters in the final ring orbit model. A second important step is photometric calibration to account for the background sky signal and reflected sunlight from the rings present in the photometric aperture, amounting in some cases to a few percent of the full intensity of the occultation star. In practice, this has proved difficult to remove accurately in the absence of well-documented calibration scans for most data sets. Final normalization of the lightcurve by the brightness of the occultation star is often challenging over the course of an hours-long occultation observed under non-photometric sky conditions, especially in the case of aperture photometers used in many of the early observations, although this is mitigated to some degree by imaging detectors that allow for differential photometry.
Once the observations have been calibrated, the next step in the analysis is to determine the midtimes and widths of each observed ring event.

Square-well model fits to individual ring profiles
The ring orbit model used to determine the occultation geometry and the ring orbital elements is based on a non-linear least squares fit to the set of estimated midtimes of individual ring profiles from a large set of occultations. In most cases, Earth-based Uranus stellar ring occultation profiles are diffraction-limited, smoothed by the finite angular diameter of the occulted star, and affected by time constants associated with the recording electronics and/or chopping of the telescope secondary. We model the instrumental response function as a single-or double-pole filter, as defined by . Following past practice and for simplicity and consistency, we determine the midtimes of individual ring profiles using a diffraction-based square-well model  that accounts for stellar and instrumental smoothing, as well as the detector response over the wavelength range of the filter used for the observations. Fig. 6 illustrates a sequence of calculations of the square-well model for the U23 CTIO ring egress profile. In each row, the observed ring profile is shown in red, the square-well itself is shown in green with R.G. French et al. its midline marked in gray, the convolution kernel of the star and instrument response is shown in blue, and the model calculation in black. All of the calculations are performed as a function of time along the lightcurve at a resolution of 0.005 s, with the conversion factors between time and distance being provided by the geometry of the final ring orbit model. The five-step sequence is as follows: • The initial square-well model at the top assumes a monochromatic point source with a central wavelength of = 2.2 μm, with no instrumental time constant. The computed diffraction pattern shows much more detailed structure than the observations, which are systematically delayed in time relative to the assumed square-well shown in green. • In the second row, the finite bandpass of the K-band filter ∕ = 0.4 μm∕2.2 μm= 0.18 is modeled as the sum of 41 unweighted monochromatic point source curves, uniformly spaced over the range = 2.0 − 2.4 μm. The diffraction fringes in the resultant black model curve are now more muted, but still contain more structure than the observed ring profile. • The square-well model in the third row includes the smoothing effects of the finite angular diameter of the occulted star, here modeled as a uniform disk with a projected diameter at Uranus of 1.50 km, or 0.12 s in units of time. (In Section 3.3.2 below, we describe how we estimate the stellar diameter from the color and magnitude of the star, as well as from individual ring profile fits, and in Section 3.3.3 we consider the effects of possible limb darkening of the stellar disk.) The convolution kernel of the stellar disk is shown in blue. The model diffraction fringes have been further muted, but there is still a systematic offset in time between the observations and the model curve.
• The fourth row illustrates the contribution of the known finite response time of the IR electronics. Here, we model the star once more as a point source so that the effects of the time constant can be seen in isolation. Again, the blue curve shows the convolution kernel of the response function, and black model curve and the red observations no longer have a pronounced systematic offset. • The final model includes the finite bandwidth of the filter, the angular diameter of the star, and the instrumental time constant. The blue curve shows the joint convolution kernel for both the stellar diameter and the instrumental time constant. This high-SNR ring observation is nicely matched by the final square-well model, including the muted fringes at each edge of the profile.
The fitted ring event times from all square-well fits are tabulated in the PDS support bundle and details of individual square-well fits for each occultation data set are provided in the corresponding PDS data bundle.

Error estimates for square-well model parameters
The fitted square-well itself is charactered by three model parameters: the midtime 0 , the width 0 , and the fractional transmission 0 (zero for an opaque ring and one for a transparent ring). The leastsquares fit provides formal errors for each of these quantities, but in practice these are unrealistically small because they fail to account for possible systematic errors in auxiliary model parameters, including the angular diameter * of the occulted star, the instrumental time constant , the incident starlight̄ * , and the background signal̄, following the notation of . As part of our standard processing, we estimate the uncertainties in these quantities and compute separate R.G. French et al.  Table 3. Our adopted values d * and associated uncertainties are plotted vertically, with error bars, and the values d * using the Kervella et al. (2004) and van Belle and 15 colleagues (1999)  square-well fits with each of these auxiliary parameters fixed at ± of their nominal values. We record the resultant values of 0 , 0 , and 0 and their final uncertainties. We compute these by adding in quadrature the formal error from the least-squares square-well fit and the estimated contributions from the uncertainties in the projected diameter at Uranus of the occulted star, in the full stellar signal at the time of the ring event, and in the instrumental time constant (if non-zero). We also fit for the equivalent width of the square-well model and the equivalent depth (the radially-integrated normal optical depth) and their propagated uncertainties -see  for detailed descriptions of these alternate measures of the fractional transmission of the ring.

Quality index (QI)
The full set of occultation profiles varies widely in quality, from very high-SNR events that are nearly perfectly matched by the model fits, to individual ring events that are noise-limited and perhaps undetectable. We capture this information in the PDS archive and the Supplementary Online Material (SOM) by assigning a subjective quality index (QI) to each observed/predicted ring occultation. (Given the non-Gaussian characteristics of the noise and the significant correlations between adjacent data points for data with large instrumental time constants, we were not able to devise a more quantitative SNR estimate that accurately reflected the relative quality of different observations.) Possible QI values are: 0: Not observable -observations at the predicted occultation event time for this ring were either not recorded (for example, the star was below the horizon) or the data were too noisy to provide useful results (for example, during sunrise). 1: High SNR profile with sharp edges matched by square-well model fit. 2: High to moderate SNR profile with well-defined midtime and edge estimates from square-well model fit but possible systematic deviations of observed ring profile from model fit due to ring structure. 3: Low SNR profile with clear ring detection but less-reliable ring width and or mid-time due to noise or substantial convolution by star diameter and/or instrumental time constant. 4: Unreliable detection -some hint of a ring occultation, fitted by square-well model, but ∼50% chance that it is just noise. 5: No detection -High SNR signal level but no evidence of a ring occultation. This usually applies to the ring, which is known to be azimuthally incomplete.
Examples of ring events with QI = 1 through 4 are shown in Fig. 7. The complete set of QI values for all rings and all observations is tabulated below in Section 4.31 below.

Occultation stars
The original predictions for most of the Earth-based occultations in this paper are from Klemola and Marsden (1977), Klemola et al. (1981), Mink and Klemola (1985), Nicholson et al. (1988), Klemola and Mink (1991), the plate catalog of J. Mink and A. Klemola (personal R.G. French et al. Fig. 11. Gallery of ring occultations U0 observations from KAO plotted as raw counts as a function of ring plane radius (upper axis) and time (lower axis). The ring was not detected during either ingress or egress. communication), and A. Bosh, 7 providing stellar positions, photometric measurements, and the geometric circumstances of each event. We have updated this information to make use of considerably improved star positions from the Gaia DR3 catalog (Gaia Collaboration, 2022) in our ring orbit fits. We also use more extensive absolute and differential IR photometry to estimate the angular diameters of the stars, a key parameter in the square-well model fits used to determine ring widths and midtimes.

Star positions
Until recently, the typical astrometric uncertainty in the parallaxand proper motion-corrected star positions was substantially larger than the uncertainty in the Uranus ephemeris, and orbit fits to the rings included fitted corrections to the predicted star positions at the epoch of each occultation. With the release of the Gaia EDR3 and DR3 catalogs (Gaia Collaboration and 425 colleagues, 2021;Gaia Collaboration, 2022), the situation has reversed, with small but measurable systematic drifts in the Uranus ephemeris that greatly exceed the star position uncertainties. We have cross-referenced the predicted star positions for all Earth-based Uranus ring occultations, using VizieR (Ochsenbein et al., 2000), to obtain the J2000 Gaia DR3 star positions given in Table 2 for a geocentric observer at the given epoch for each occultation, corrected for proper motion and parallax. (As noted in French et al., 2023b, in our actual orbit fits we solve for the topocentric coordinates of the star, corrected for parallax and proper motion at the time of each individual ring event.) Note that several of the star positions have Gaia RUWE (renormalized unit weight error) greater than 2, indicating an uncertain astrometric solution, and two of the events (U36 and U102) were revealed to be multiple star systems from the observed pattern of individual ring occultation profiles. We include a magnitudedependent correction for the proper motion derived from the EDR3 catalog (Cantat-Gaudin and Brandt, 2021), but equally applicable to the DR3 proper motions as well (personal communication T. Contat-Gaudin). In the final two columns of the table, we include propagated position error estimates, using the prescription of Butkevich and Lindegren (2014), computed using the SORA occultation data analysis package (Gomes-Júnior and 8 colleagues, 2022).
The Voyager 2 stellar occultation stars Sgr and Per are too bright to provide accurate Gaia DR3 positions, and for these stars we make use of the Hipparcos catalog positions (Perryman and 19 colleagues, 1997).

Stellar photometry and angular diameters
As noted previously, the stellar angular diameter * is a key parameter in the square-well model used to fit individual ring profiles. By combining V-K differential photometry as a function of dwarf spectral type (Glass, 2005) with the corresponding stellar radii and absolute V magnitudes (Lang, 2012), we obtained an empirical calibration function for the angular diameter of the star * as a function of V-K and V, as shown in Fig. 8. (An equivalent result could be obtained by fitting with respect to V-K and K, but the resulting curve has more curvature R.G. French et al. and is not monotonic.) The solid line is a third-order polynomial fit to calibration values of the function: where the angular diameter of the star * is expressed in mas. The calibration values are shown as asterisks; the filled red circles show values for the Uranus occultation stars, which are mostly late spectral types. The fit is of the form where coefficients of the fit are 0 = 0.48278390, 1 = 0.26532447, 2 = 0.023803480, and 3 = −0.0049626961. The projected stellar diameter * in km is given by * = * (radian), where is the range from the observer or spacecraft to the ring intercept point of the occultation ray in the ring plane and * (radian) = 2 360 × 3600 × 1000 * (mas).
(4) Table 3 lists the stellar photometry and the derived and adopted projected star diameters for the Earth-based Uranus ring occultations. The apparent V and K magnitudes were obtained either from photometric measurements at the time of each occultation or from the VizieR catalog. The adopted projected angular diameter * and the associated uncertainty were determined for each event from a combination of the predicted values given by Eq. (2) and from a series of square-well fits to the narrowest rings that included the star size as a fitted parameter. For comparison, we include in the table the computed stellar diameters d * from the more recent calibration models of Kervella et al. (2004), based on V-K and B-K color indices, and by van Belle and 15 colleagues (1999), van Belle and 10 colleagues (2021), as implemented in SORA. The quoted value and uncertainty for each measurement are the mean and standard deviation of the set of derived diameters using both V-K and B-K colors and the Kervella et al. (2004) and van Belle and 15 colleagues (1999) models. Finally, we include stellar diameter estimates d 2 * computed from the Gaia DR2 catalog (Andrae and 15 colleagues, 2018), when available. Unlike the other diameter estimates, these include corrections for interstellar extinction, but are dependent on the accuracy of the parallax for each star. The quoted error bars for these positions reflect the estimated average 10% uncertainty in the star diameters (Andrae and 15 colleagues, 2018), and are not individual error estimates for each star.
In Fig. 9 we compare our adopted stellar diameters (on the vertical axis) with the corresponding predictions for each occultation using the Kervella et al. (2004) and van Belle and 15 colleagues (1999), van Belle and 10 colleagues (2021) models, implemented using SORA, and with the Gaia DR2 estimates. Our results are in quite good agreement with the SORA ensemble, and are systematically a bit larger than the Gaia DR2 evaluations, most likely because these take into account interstellar extinction, which systematically reduces the derived stellar diameter. Since the list of Gaia DR2 estimates is incomplete and because our estimated diameters are based on the best fits to the actual shape of individual ring profiles (perhaps including the effects of instrumental time constants as well), we have chosen to use our calculated diameters and estimated uncertainties. We will return to the question of possible systematic effects of this choice in French et al. (2023b).

Limb darkening
In , the square-well model is generalized to include the effects of stellar limb darkening, assuming a linear limb darkening law: where = cos and is the angle between the surface normal of the star and the observer. From detailed comparisons with stellar atmospheric models, Claret et al. (1995) find best-fitting K band values of the linear limb darkening parameter = 0.25 − 0.35 for spectral types K through M, typical of the Uranus occultation stars. Hestroffer and Magnan (1998) fit an exponential form to the observed intensity distribution of the solar spectrum over a range of wavelengths: where is a wavelength-dependent fitted parameter. For the sun (spectral type G2) they find ∼ 0.28 near the CH 4 890 nm absorption band sometimes used for occultation observations, and ∼ 0.13 near the CH 4 2.3 μm band within the K-band IR filter used in most IR occultation observations. Fig. 10 shows examples of these two limb-darkening laws, plotted as a function of radius = (1 − 2 ) 1∕2 . The upper panel shows ( ) and the lower panel shows the corresponding strip brightness distribution ( ), normalized by (0), which is the convolution kernel used in the squarewell model computations of the smoothing due to the finite size of the occulted star. In all instances, the limb darkening is rather modest. This is consistent with the results of Kervella et al. (2004), who show that at the K band, the derived angular diameters including limb darkening differ by at most a few percent from those assuming a uniform stellar disk.
We compared the results of square-well fits to narrow rings (excluding , , and ) using a simple uniform disk model ( = 0) and more computationally intensive models using = 0.25 − 0.35 for several representative high-SNR ring profiles, and found that the fitted ring widths differed by less than 1% from the uniform disk models. Based on these tests, we have adopted a uniform disk model for all square-well fits presented here. (Note that French et al., 1986; French and 10 colleagues, 1988 assumed a limb-darkening parameter = 1 for most observations, which in retrospect is far too large. All of our square-well fits for the present work have been recomputed using revised angular stellar diameters and = 0.) R.G. French et al. Upper right: Image of strip chart, processed to remove grid lines and extraneous marks. The labeled times 6:57:57 and 6:58:07 are offset to the right of the corresponding timing tick marks -the midtime of the ring occurs very nearly at 6:58:00, as was found by Millis and Wasserman (1978). Bottom: extracted time series of the ring profile.

Fig. 14.
Extraction of U5 egress ring profile from strip chart. Left: strip chart image, with data in red and 1 Hz timing marks in green. There is a horizontal pen offset between the data and timing marks that was taken into account when digitizing the data. The line labeled ''zero" does not correspond to the zero level of stellar flux, which was estimated from other portions of the recorded data. Note that time increases from right to left. Right: Extracted time series profile. The measured midtime of the profile is close to the originally published estimate of 6:21:21.3 UTC (Nicholson et al., 1978). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Earth-based uranus ring occultation observations
In this section, we present essential information about every Earthbased occultation data set listed in Table 1. Extensive figures and tables for each data set are available in the PDS archive, including detailed results of individual square-well fits for each ring profile, gallery plots of all ring events, convenient browse products that provide an overview of each event, and calibrated digital tables of every occultation lightcurve. In Section 4.31, we tabulate the complete set of individual ring events for all observations, including the likely identifications of the elusive ring. To avoid unnecessary duplication in the main body of this paper, we include only a few representative results for previously published observations, highlighting any differences in our reanalysis. We provide additional essential information for occultations that have not been R.G. French et al.

SAO 158687 (U0) 1977 Mar 10
The Uranus rings were first detected during the widely observed 1977 Mar 10 occultation of the bright star SAO 158687 (Bhattacharyya and Bappu, 1977;Brahic, 1977;Chen and 10 colleagues, 1978;Elliot et al., 1977;Hubbard et al., 1977;Millis et al., 1977;Morrisby et al., 1977;Tomita, 1977). (We assign the shorthand name of U0 to this occultation.) The unexpected and intermittent brief dips in starlight before and after the planetary occultation were initially attributed to a swarm of small satellites (Elliot and 9 colleagues, 1977), but eventually nine rings were convincingly identified. In order of increasing radius, these are 6, 5, 4, , , , , , and -the mixed notation and order are reminders of the initial uncertainty and confusion about the nature of the occulting material.

Kuiper Airborne Observatory (KAO)
The highest-SNR digitally recorded observations of the U0 occultation were obtained with the 0.91 m telescope onboard the Kuiper Airborne Observatory (KAO), using high-speed aperture photometers at three visual wavelengths.  describe the observations and the square-well model used to determine the ring widths and the adopted event midtimes for an early ring orbit model, and French et al. (1986) provide details about additional processing steps used in their later orbit determination. For this work, we reprocessed the entire KAO data set to produce a single normalized lightcurve from the weighted sum of the three separate data channels, with a central wavelength = 734 nm (included in the SOM). No details are available about the source or uncertainty of the absolute time scale of the observations, but they are likely to be small compared to the ∼ 1 km uncertainty in the location of the KAO as a function of time. Fig. 11 shows a gallery of the predicted ring regions for all ten narrow rings; the ring was not detected during ingress or egress. We fitted square-well models to all detected ring profiles (nine each on ingress and egress), adopting a projected stellar diameter at Uranus of 7.5 ± 1.0 km ( Table 3). This is the largest projected stellar size of any of the occultations in this study, a reflection of the unusual brightness and early spectral type of the U0 star. The fitted ring event times for this and all other occultations are included in the PDS support bundle contents described in Appendix A.
The flight path of the KAO during the U0 occultation is shown in Fig. 12. We converted the KAO navigation file (smoothed to correct for truncation errors) to a NAIF ephemeris file urkao_v1.bsp to enable the use of the NAIF SPICE toolkit (Acton, 1996) to compute the occultation geometry for this moving observatory. This enables the aircraft velocity to be taken into account when computing the apparent velocity of the star perpendicular to the ring as projected in the sky plane.

Other U0 observations
The U0 occultation was observed from six sites, as summarized in Table I of French et al. (1986), but four of these were recorded only on strip charts and timing was uncertain for the digitally recorded emersion observations by Hubbard and Zellner (1980). While these additional observations were useful for early identification of individual rings, only the KAO observations had sufficient SNR and accurate time resolution for use in our current ring orbit analysis.

U2 Observatorio del Teide 1977 Dec 23
Shortly after the discovery of the Uranus ring system, predictions of future events were produced by Klemola and Marsden (1977). Successful observations of the occultation of the second star on their list (identified here as U2) were obtained from Tenerife on the Canary Islands from the Observatorio del Teide 1.5 m telescope using a highspeed aperture photometer at = 880 nm, centered near a methane absorption band that dimmed the reflected sunlight from the planet and enhanced the SNR of the occultation .
Analog data were recorded on tape for later digitization, but we had access only to the original strip charts. The only visible detections      were the , , and immersion events. The estimated relative accuracy of initial measurements of the three events was ±0.1 s, with a corresponding ring plane radius uncertainty of about 3 km. In an effort to improve the measurement accuracy, we scanned the original strip chart and extracted both the raw lightcurve and the time ticks to produce a one-dimensional time series of the occultation. Fig. 13 shows the ring profile as recorded on the strip chart and in final time series form, and gives a sense of the SNR of the event.
From photometry of the bright U2 star (K = 6.992), we estimated a projected stellar diameter of 3.4 ± 0.5 km at Uranus. We fitted squarewell models to the three ring events, which we included in our final ring orbit model. Since two degrees of freedom were used to fit for offsets between the predicted and fitted sky plane positions of Uranus (or, equivalently, corrections to the catalog position of the U2 star), the three data points do not add materially to the final orbit model, but the results are useful as estimates of Uranus ephemeris errors, as discussed in French et al. (2023b).

U5 Las Campanas Observatory 1978 Apr 10
The occultation of star U5 (K = 9.971), the fifth star on the prediction list of Klemola and Marsden (1977), was observed by Nicholson et al. (1978) from the Las Campanas Observatory du Pont 2.5 m telescope using an InSb aperture photometer with a K filter ( = 2.2 μm). The data were recorded on a high-speed strip chart and the originally published ring event mid-times (Table II Nicholson et al., 1978) had a relative uncertainty of 0.1 s. For the characteristic ring plane radial velocity of the event of 16 km s −1 , the corresponding radius uncertainty is 1.6 km. In an effort to improve on this, we followed the same procedure as for U2 by extracting individual ring event lightcurves from scans of the strip chart. We corrected for the uneven rate of the strip chart, using polynomial fits to the unevenly-space 1 Hz tick marks, taking into account the horizontal offset between the timing and data strip chart pens.  Nicholson et al. (1978) for the corresponding strip chart images. Note that the labeled ring 6 egress event in this figure is ∼ 10 s from the expected time of 6:12:18 UTC and is likely to be a noise feature.
We fitted square-well models to the extracted ring profiles, assuming a projected stellar diameter of 0.60 ± 0.09 km and an instrumental time constant of 0.1 s for a single-pole filter. The post-fit RMS residual for U5 in our final ring orbit fit, using the model fits to the digitized strip chart profiles, is 0.36 km, a considerable improvement over the RMS of 1.11 km when using the original published event times (Nicholson et al., 1978).

U9 Las Campanas Observatory 1979 Jun 10
The occultation of star U9 (K = 11.200), the ninth star on the prediction list of Klemola and Marsden (1977), was observed by Nicholson et al. (1981) from the Las Campanas Observatory du Pont 2.5 m telescope using an InSb aperture photometer with a K filter ( = 2.2 μm). As with the U5 occultation, this event was recorded only on a strip chart, which we digitized in a similar fashion. Fig. 15 shows the egress profile to give sense of the data quality; the other events are included in Fig. 1 of Nicholson et al. (1981).

Table 3
Star photometry and diameters. We fitted square-well models to the extracted ring profiles, assuming a projected stellar diameter of 0.4 ± 0.08 km and an instrumental time constant of 0.1 s for a single-pole filter. The post-fit RMS residual for U9 in our final ring orbit fit, using the model fits to the digitized strip chart profiles, is 0.45 km, corresponding to ±0.02 s in time, considerably below the original estimated timing uncertainty of 0.1 s (Nicholson et al., 1981).

U11 CTIO 1980 Mar 20
The occultation of U11 (number 11 in the list of Klemola and Marsden, 1977) was successfully observed with the 4 m CTIO telescope at = 2.2 μm with an InSb aperture photometer. The observations are described in detail by Elliot and 6 colleagues (1981), and previous square-well fits to the six detected ring events are given in Table VI of French et al. (1986). For this work, we refitted these ring profiles using a projected stellar diameter of 0.75±0.15 km, assuming a uniform disk (the French et al., 1986 results used a projected diameter of 0.75 km as well, but assumed a limb darkening parameter = 1). A gallery of our final square-well fits and other informative plots are included in the SOM. The RMS residual for the seven U11 CTIO measurements included in our final orbit fit is 0.30 km. In addition, there is a weak feature near the expected location of the ring during egress.

U12 1980 Aug 15
The occultation of U12 (number 12 in the list of Klemola and Marsden, 1977) was successfully observed from three telescopes in South America, as described by Elliot and 9 colleagues (1983) and summarized by French et al. (1986). For our current analysis, we renormalized all of the observations in a uniform manner and adopted a projected stellar diameter of 1.60±0.25 km, assuming no limb darkening (in the previous analysis, the assumed stellar diameter was 1.50 ± 0.23 km). Detailed results are included in the SOM. Other differences from previously published results are noted below.

U12 CTIO
For our current ring orbit solution, we used square-well fits to the original U12 CTIO 4 m telescope observations that included the effects an instrumental time constant, rather than fits to the deconvolved data described by French et al. (1986).

U12 ESO
As with the U12 CTIO observations, we used square-well fits to the original U12 ESO 3.6 m telescope observations that included the effects of an instrumental time constant, rather than fits to the deconvolved data described by French et al. (1986).

U12 Las Campanas Observatory
The U12 Las Campanas 2.5 m du Pont telescope = 2.2 μm observations were recorded only on strip charts. For our present analysis, we fitted square-well models to the digitized ring event lightcurves described by French et al. (1986). A small systematic time offset between ingress and egress of unknown origin was clearly present, which we included as a free parameter in our final ring orbit fit (French et al., 2023b).

U13 Siding Spring Observatory 1982 Apr 26
The occultation of U13 (star 13 on the list of Klemola et al. (1981)) was successfully observed from the Anglo-Australian Telescope (AAT) at the Siding Spring Observatory. The observations are described briefly by French et al. (1982) and previous data processing steps are described by French et al. (1986). Ingress was observed in chopping mode at a time resolution of 0.1 s. After the atmospheric occultation was complete, the egress ring region was observed in DC mode with a time resolution of 0.11 s. We renormalized the entire lightcurve and adopted a projected stellar diameter of 2.35 ± 0.35 km, compared to 2.45 ± 0.32 km as in French et al. (1986). Plots of the raw and normalized event lightcurves and square-well fits are included in the SOM.

U14 1982 Apr 22
The high-SNR occultation of U14 (star 14 on the list of Klemola et al., 1981) was observed from five stations and at multiple wavelengths.  provide extensive details about the observations, and French et al. (1986) include additional information about data processing. For this work, we have renormalized every lightcurve and assumed a projected stellar diameter of 4.750 ± 1.188 km ( = 0), compared to the previously adopted limb-darkened ( = 1.0) value of 7.06 ± 0.55 km (French et al., 1986), which we now judge to be too large, based on radiative transfer models of stellar atmospheres (Claret et al., 1995) and interferometric measurements (Kervella et al., 2004). (Note the relatively large scatter in the estimates of the diameter of this star in Table 3, which prompted the 25% error bar assigned to our adopted value.) Plots of the raw and normalized event lightcurves and square-well fits are included in the SOM.

U14 CTIO
The U14 event was observed in the IR ( = 2.2 μm) using the CTIO 1.5 m telescope, and at = 880 nm with the 4 m telescope. The instrumental time constant for the IR observations was substantial (estimated to be 0.1535 s from step-function response measurements), resulting in considerable smoothing of the observations. We explored a variety of deconvolution techniques for the IR data, but in the end decided to use only the 4 m data in our final orbit fits, since the data were of comparable SNR and did not require deconvolution. The orbit fit RMS residual is 0.56 km for the U14 CTIO data.

U14 ESO
In our present orbit fit, we include both ingress and egress ring event times for U14 ESO 1.04 m telescope observations, whereas French et al. (1986) used only the higher SNR egress ring events for these observations. The orbit fit RMS residual is 0.20 km for 13 data points, which compares favorably with results for the other U14 stations.

U14 Las Campanas Observatory
The U14 occultation was observed from Las Campanas Observatory with the 1 m Swope telescope at = 880 nm, and with the 2.5 m telescope at = 2.2 μm. In our previous analysis (French et al., 1986), we used only the IR data, but for the present work we have included the somewhat lower-SNR 1 m telescope results as well. In the end, we used 16 IR ring measurements with an RMS error of 0.34 km and 10 = 880 nm measurements with an RMS error of 0.50 km in the adopted ring orbit model.

U14 Observatoire du Pic du Midi et de Toulouse (OPMT)
In our previous analysis (French et al., 1986), we excluded the U14 OPMT measurements from our adopted ring orbit solution. For the PDS archive, we processed both the = 2.2 μm observations from the 2 m telescope and the = 880 nm observations from the 1.06 m telescope, and these results are included in the SOM as well. For our final orbit fit, we included nine IR ring events with an RMS of 9.45 km and four = 880 nm measurements with an RMS of 0.20 km. Significant timing offsets of uncertain origin were included as free parameters in the adopted orbit fit: +3.73 s for the 2 m data and +0.78 s for the 1.06 m data.

U14 Observatorio del Teide
The U14 occultation was observed from the Observatorio del Teide 1.55 m telescope with a two-channel photometer at = 0.499 μm and = 0.880 μm (Millis et al., 1987). The event occurred under conditions of variable transparency, but by subtracting the sky contributions from each channel and taking the ratio of the two signals, the effects of intermittent clouds were greatly reduced. This ratio technique was used for the ring ingress profile and the atmospheric immersion event; for the other ring events and atmospheric emersion, atmospheric conditions were clear and we used just the red channel ( = 880 nm) observations that were centered on a methane band. All observations were recalibrated and normalized to account for sky background and variable transmission. In the end, 12 reliable ring event times were used in the orbit model (five for ingress and seven for egress), with separate fitted offset times for ingress and egress. Previous versions of these results were first included in the ring orbit model of French et al. (1987).

U15 Mt. Stromlo Observatory 1982 May 1
The 1 May 1982 occultation of U15 (Klemola et al., 1981) by Uranus and its rings was observed at = 2.2 μm using the 1.9 m telescope of the Mt. Stromlo Observatory . In our reanalysis of these data, we renormalized the lightcurve, corrected small errors in the absolute timing, used a time constant of 0.05 s for a double-pole filter, and adopted a projected stellar diameter of 3.100±0.775 km, with no limb darkening. In the end, we included 17 ring events (all but ring 6 ingress) in the orbit model, with an RMS residual of 0.42 km. The ring was not detected during this occultation. Fig. 25. Lightcurve of the ingress U36 occultation from the IRTF, showing paired occultation events for the , , and rings. The occultations of stars U36A and U36B occurred virtually simultaneously, followed several minutes later by the U36C occultation. The signal drop from the combination of U36A and U36B, compared to that from U36C, is consistent with the relative brightness of the three stars inferred from egress ring events such as those shown in Fig. 24.

U16 Palomar Observatory 1982 Jun 4
The U16 occultation (Klemola et al., 1981) by Uranus and the rings was observed from Palomar Observatory with the 5 m Hale Telescope in K band (2.0-2.4 μm). Uranus was at an altitude of 30-35 • at the time of the occultation. The occulted star U16 is relatively bright (K = 9.4) and the ring plane radial velocity averaged 20 km s −1 . This, combined with the nearly face-on aspect of the rings ( = 73 • ), resulting in excellent SNR and stable baselines. The entire lightcurve is included in the SOM.
Preliminary results from the U16 Palomar Observatory results were included in previous orbit models (French et al., 1991), but the details of the observations have not been previously published. The occultation was observed with a single-channel InSb photometer (D-68) at the f/70 focus of the low-background chopping secondary. The timing was done by recording time signals from a WWVB clock as part of the data stream. The beam for all observations was 7.5 arcsec diameter. The chopper throw was 15 arcsec north-south, and the waveform was an equal-duty-cycle square wave at 50 Hz. An offset autoguider was used to guide in the visual on a relatively bright star out of the field. The guider corrected both for differential refraction between the guide star in the visual and occultation star in the infrared, as well as for image motion by feedback to the secondary, which has fast two-orthogonal-axis control.
The signal was synchronously demodulated with a lockin amplifier. From square-well model fits to the ring 5 ingress profile, we modeled the time constant as a double-pole filter with a time constant of 0.055 s during ingress, which was reduced prior to atmospheric egress, once it was recognized that the ingress ring events were significantly smoothed by the long time constant. From fits to narrow ring profiles, we estimated the egress time constant to be only 0.015 s, with a substantial uncertainty, and in the end we did not include an instrumental time constant in our egress square-well fits. The demodulated signal was sent to a strip chart recorder and also to a voltage-to-frequency converter (10 MHz). The output of the VtoF was sent to a high speed counter system, counted in 5 ms intervals and recorded by computer onto disk. Absolute timing was providing by recording time signals from a WWVB clock as a separate channel in the digital data stream.
Our final square-well fits are shown in Fig. 16, for which we assumed a projected stellar diameter of 1.2±0.4 km, with no limb darkening. The ingress events show the smoothing effects of the instrumental R.G. French et al. time constant, which as noted above was reduced prior to the egress occultations. The ring egress profile was affected by signal instability, accounting for the partial profile in this figure. The ring was not detected during either ingress or egress. The fitted ring event times are tabulated in the PDS support bundle and details of individual squarewell fits are provided in the uranus_occ_u16_palomar_508 cm PDS data bundle. The U16 Palomar orbit fit RMS residual was 0.33 km for the 18 ring measurements.

U17B SAAO 1983 Mar 24
The occultation of U17B (Klemola et al., 1981) by Uranus and its rings was observed with the 1.88-m telescope at the South African Astronomical Observatory (SAAO) through a K filter with an InSb detector . Immersion of the nine main rings and the emersion of rings, 5, 4, , and were recorded with the approach of sunrise. For our reanalysis of the observations, we renormalized the lightcurve and adopted a projected angular diameter of 1.05 ± 0.15 km for U17, with no limb darkening. (The previous analysis by Elliot et al. (1987) assumed a projected diameter of 1.3 km, with an unrealistically large limb-darkening parameter = 1.) The U17B SAAO orbit fit RMS residual was 0.18 km for the 13 ring measurements.
Extensive details of the data processing for this occultation are contained in the PDS Uranus ring occultation support bundle User Guide (see Appendix B), which uses this event as an example to document the structure and contents of the archive. Additional information, including plots and ring event times, is provided in the SOM and in the uranus_occ_u17b_saao_188 cm PDS data bundle.

U23 1985 May 4
The occultation of the K = 9.244 magnitude star U23 (Mink and Klemola, 1985) was successfully observed from CTIO, McDonald Observatory, and the Observatorio del Teide. We adopted a projected stellar diameter of 1.500 ± 0.375 km (limb darkening parameter = 0), and reanalyzed all of the data sets for this work. We revised the normalization of the lightcurves and the instrumental time constants as needed.

U23 CTIO
The U23 CTIO 4 m telescope observations at = 2.2 μm were obtained at high elevation angle under partially clear skies. All nine main rings were observed at high SNR during both ingress and egress, and a possible ring event was observed during ingress as well. Additional details are provided by French and 10 colleagues (1988). The RMS residual from the orbit model for the 18 CTIO events was 0.32 km.

U23 McDonald Observatory
The egress ring occultations of U23 were observed from the 2.7 m McDonald Observatory telescope at = 2.2 μm at relatively high airmass (elevation angle 20 • ), resulting in moderate-SNR detections of the nine main rings; the ring was not detected. Additional details are provided by French and 10 colleagues (1988). The RMS residual from the orbit model for the nine McDonald ring events was 0.42 km.

U23 Observatorio del Teide
The ingress ring occultations of U23 were observed under poor conditions with the 1.5 m Observatorio del Teide telescope at = 2.2 μm, resulting in relatively low-SNR detections of the , , , and rings; the ring was not detected. Additional details are provided by French and 10 colleagues (1988). The RMS residual from the orbit model for the four secure ring events was 0.64 km.

U25 1985 May 24
The 24 May 1985 occultation of the bright star (K = 6.4) U25 (Mink and Klemola, 1985) was observed from CTIO, the IRTF, McDonald Observatory, and Palomar Observatory. French and 10 colleagues (1988) describe the first three of these data sets, which they incorporated in the latest published ring orbit model. We adopted a projected stellar diameter of 3.7 ± 0.5 km (limb darkening parameter = 0), and reanalyzed all of the data sets for this work. We revisited the normalization of the lightcurves and the instrumental time constants as needed. To these, we add the high-SNR Mt. Palomar observations for our present analysis.

U25 CTIO
The CTIO 4 m telescope ring occultation observations are among the highest SNR Earth-based Uranus ring profiles: all 18 main ring events earned a quality index rating Q = 1. (The ring was not detected during either ingress or egress.) The RMS residual from the orbit model for the 18 ring events was 0.36 km.

U25 IRTF
The U25 IRTF observations were carried out under cloudy conditions, and only the ring egress event was detected, with low SNR. We exclude this point from our present ring orbit analysis.

U25 McDonald Observatory
U25 observations from McDonald Observatory's 2.7 m telescope were carried out under clear conditions, with high humidity. French and 10 colleagues (1988) deconvolved the occultation data to reconstruct the input signal prior to filtering by the electronics, based on a measured impulse response function, but in our present analysis we fitted square-well models to the normalized lightcurve itself, including a single-pole filter with a time constant of 0.02 s, which yielded very similar results. Just as for the CTIO observations, all 18 main ring events from the McDonald Observatory earned a quality index rating Q = 1. (The ring was not detected during either ingress or egress). The RMS residual from the orbit model for the 18 ring events was 0.55 km.

U25 Palomar Observatory
The U25 occultation was observed from Palomar Observatory with the 5 m Hale Telescope in the K band (2.0-2.4 μm) with a single channel InSb photometer (D-68) at the f/70 focus of the low-background chopping secondary. The observing strategy was similar to that used for U16 -see Section 4.10.
The sky was generally clear. Uranus was at an altitude of 30-35 • at the time of the occultation. The occulted star U25 was quite bright and      the ring plane radial velocity was 11-16 km s −1 . This, combined with the nearly face-on aspect of the rings ( = −82 • ), resulting in excellent SNR and stable baselines. The entire lightcurve is included in the SOM.
A gallery of square-well fits to the ring events is shown in Fig. 17. The ring was not detected during either ingress or egress. The U25 Palomar orbit fit RMS residual was 0.38 km for the 18 ring measurements.

U28 IRTF 1986 Apr 25
The 25 Apr 1986 occultation of the moderately bright star (K = 8.732) U28 (Mink and Klemola, 1985) was observed at high SNR from the IRTF at = 2.2 μm, as described in detail by French and 10 colleagues (1988). We adopted a projected stellar diameter of 1.5 ± 0.2 km (limb darkening parameter = 0), and revised the normalization of the lightcurve for this analysis. All nine of the major rings were observed during both ingress and egress, except for the ingress event, which occurred while the star was being recentered in the aperture. Tentative ring events were observed near the expected locations, but we caution that there were several other ''false'' events present in the lightcurve that French and 10 colleagues (1988) attributed to a nearby infrared source about 5 arcsec away from the occultation star that may have drifted in and out of the 10 arcsec photometer aperture. The U25 IRTF orbit fit RMS residual was 0.39 km for the 17 secure ring measurements. A gallery of the fitted square-well model profiles is contained in the SOM.

U34 IRTF 1987 Feb 26
The 26 Feb 1987 occultation of the relatively faint star (K = 11.09) U34 (Mink and Klemola, 1985) by Uranus and the rings was observed from the IRTF at = 2.2m. Data were recorded in DC mode (unchopped), using the GRABBER system. Accurate 1-Hz time ticks were recorded on a separate channel as part of the digital data stream, providing an absolute time reference. We applied an optimized sinc (sin( )∕ ) filter to minimize extraneous 60-Hz pickup and harmonics present in the raw data. The ingress ring region was observed at relatively high airmass (altitude ∼ 20 • ), but conditions improved over time as Uranus rose in the sky (altitude ∼ 40 • for the egress ring events). The observed lightcurve is included in the SOM. We adopted a projected diameter of U34 of 0.500 ± 0.125 km (limb darkening parameter = 0). Square-well fits to the observed ring events are shown in Fig. 18. The SNR is highly variable, as reflected by our assigned quality index values that ranged between QI = 1 for the egress event to QI = 4 for four marginal detections. The ring was not detected during either ingress or egress. The post-fit RMS residual for the U34 IRTF observations was 0.41 km, with the ring measurements being the biggest outliers.

U36 1987 Mar 30 -April 2
The occultation of U36 (Mink and Klemola, 1985) by Uranus and the rings was a remarkable multi-day event, lasting from 1987 Mar 30 through Apr 2 and occurring while Uranus was at the end of its retrograde loop (Fig. 4). Ring immersions were observed from the IRTF and UKIRT, and ring emersions were observed from CTIO, the IRTF, Mount Stromlo Observatory (MSO), the Anglo-Australian Telescope (AAT) at Siding Spring, and the ANU telescope at Siding Spring Observatory (Elliot and 13 colleagues, 1987). 8 The occultation by the planet's atmosphere was recorded at SAAO in DC mode with highly variable background level, and we have not included these data in the PDS archive or in the present analysis. Ring occultation events from four separate stars, possibly members of a single gravitationally bound system, were identified in the collective data set. The slow event velocity and small projected diameter of the primary star (estimated to be 0.65 ± 0.10 km) resulted in several diffraction-limited high-SNR ring profiles.
In this section, we describe the circumstances of each of the observations, identify the individual ring occultations by the brightest component of the U36 system (U36A), and present examples of the secondary events by components U36B, U36C, and U36D. In French et al. (2023b), we use the complete set of U36 observations to determine the relative positions of stars U36A, U36B, and U36C. (For additional observational details and a previous analysis of the U36 event using a subset of the data presented here, see Kangas, 1989.)

U36 CTIO
The U36 CTIO observations were carried out from the 4 m telescope at = 2.2 μm using a high-speed InSb aperture photometer. On 2 Apr 1987, the egress atmosphere and ring occultations were observed under clear skies over the course of eight hours. Uranus was at high elevation during the ring events, but sunrise occurred shortly after the ring occultation, resulting in degraded SNR for the outer rings. Nevertheless, occultations of the primary star U36A by all nine main rings were detected. A gallery plot of square-well model fits to these profiles is shown in Fig. 19.

U36 IRTF
The U36 IRTF observations were carried out at = 2.2 μm in DC mode using the RC2 high-speed InSb aperture photometer. The , , , and rings observed during ingress on 30 Mar 1987 under generally clear skies, beginning at 20 • elevation for the ring, increasing to 55 • by the time of sunrise, which prevented detection of the ingress ring and rings 4, 5, and 6. The egress ring events were observed three days later on 2 Apr 1987 under dark and generally clear skies, over a range of elevations 30-40 • . The observing system crashed twice during the egress ring events, resulting in a loss of data and absolute timing, but the , , , and rings were observed. Kangas (1989) describes the process we used to reconstruct an accurate time scale for the data. A gallery plot of square-well model fits to the ring occultations of the primary star U36A observed from the IRTF is shown in Fig. 20 (the egress profile is omitted here because it was contaminated by a simultaneous occultation by a secondary star within the U36 system).

U36 UKirt
The U36 observations from the 3.8 m United Kingdom Infrared Telescope (UKIRT) were carried out at = 2.2 μm in DC mode using a high-speed InSb aperture photometer. The ring system was observed during ingress on 30 Mar 1987 under generally clear skies.
The observing system first acquired Uranus just after the predicted ring event, and the , , , and rings were seen in succession at high SNR before sunrise obscured detection of the ring event. The egress ring events were observed three days later on 2 Apr 1987 under dark and generally clear skies, over a range of elevations 30-40 • . The , , and rings were observed during egress, although as with the IRTF, the egress ring profile was corrupted by an overlapping occultation of a second component of the U36 multiple-star system, as we show below in Section 4.16.6. A gallery plot of square-well model fits to the ring occultations of the primary star U36A observed from UKIRT is shown in Fig. 21.

U36 AAT
The U36 occultation was observed from the AAT at = 2.2 μm in DC mode using a high-speed InSb aperture photometer. The ring system was observed during egress as Uranus was rising. Reliable data recording commenced shortly before the ring event, and high-SNR egress profiles were obtained of rings , , , and , although once again the ring profile was corrupted by an overlapping secondary event. A gallery plot of square-well model fits to these features is shown in Fig. 22. The profile is notable as perhaps the best example of the effects of diffraction on Earth-based ring occultation observations.

U36 Siding Spring Observatory -ANU
The U36 observations of the egress ring region from the 2.3 m Australian National University telescope at Siding Spring Observatory were carried out on 2 April 1987 at = 2.2 μm in DC mode using a high-speed InSb aperture photometer. Only the and ring events were observed, and the latter profile was affected by the overlapping occultation of star U36B, as described below. Plots of square-well model fits to these features are shown in Fig. 23.

U36 secondary events
In addition to the occultations of the primary star U36A shown above, secondary ring occultation events from three additional stars were observed. During the egress ring observations, the highest-SNR events showed the presence of three adjacent or overlapping ring profiles. An example is shown in the upper panel of Fig. 24 for the AAT egress ring events. The occultation profiles of U36A and U36B overlap in the case of the relatively wide ring, but are distinct events for the narrower rings, with the U36B event following the primary event by  about 16 s for all egress ring observations. (The sharp, narrow drop shortly after the ring events is the result of a brief sky-level check.) The occultation of the dimmer star U36C is seen at left, occurring 79-87 s prior to U36A for all rings seen during egress from a variety of stations.
The AAT observations revealed an occultation of a fourth star (U36D) by the ring during egress, as shown in the bottom panel of Fig. 24, occurring 17 min after the three previous events. The duration of the U36D ring profile is similar to that for the other three stars, supporting the identification of this event as caused by the ring at a true anomaly near to that U36A, U36B, and 36C, but no other detections of U36D events have been identified, preventing us from determining the position of U36D relative to the other stars.
The U36 ingress ring events do not show the same pattern of three nearby profiles associated with separate occultations of U36A, U36B, and U36C. Instead, at most two distinct profiles are observed, even for the high-SNR IRTF ingress events. Remarkably, as we demonstrate below, the relative positions of U36A and U36B in the skyplane resulted in virtually simultaneous occultations of these two stars during ingress, while the occultation of U36C followed by 292 − 412 s. The pairs of detected profiles are shown in Fig. 25 for the U36 IRTF ingress , , and events.
Two lines of evidence support the conclusion that the U36A and U36B ingress ring events occurred simultaneously. First, the observed signal drop of the main ingress ring profile is consistent with that expected from a combination of U36A and U36B, compared to that from U36C. This is evident from a comparison of Figs. 24 and 25. Second, we solved for the predicted ingress ring event times for U36B based on the skyplane coordinates of U36B relative to Uranus obtained by fitting the egress U36B events, and in each case, the predicted times were within 1 s of the observed times for U36A, which for the very slow skyplane velocities of the occultation differed by at most 0.4 km in the ring plane radius. By adding appropriately weighted square-well diffraction models for the combination of the putative U36A+ U36B events, shifted in time by up to 1 s, we confirmed that the resultant profiles were virtually indistinguishable from the fitted square-well models assuming a single star.
The U36 ring event times for all four stars and the derived ring plane geometry for all observations are given in Table 4.

U1052 IRTF 1988 May 12
The 12 May 1988 occultation of the K = 9.974 magnitude star U1052 (Klemola and Mink, 1991) was observed from the IRTF at = 2.2 μm. It was a grazing event, with the occultation chord missing rings 6, 5, 4, and but capturing rings , , , and at low ring plane radial velocity (3.3 − 8.4 km s −1 ), resulting in excellent SNR. Data were recorded in DC mode (unchopped), using the GRABBER system. Accurate 1 Hz time ticks were recorded on a separate channel as part of the digital data stream, providing an absolute time reference. Uranus was ∼ 50 • above the horizon throughout the occultation, and skies were clear with excellent photometric conditions. The observed lightcurve is included in the SOM. We adopted a projected stellar diameter of 0.87±0.20 km, with limbdarkening parameter = 0, for our square-well model fits to the ring R.G. French et al. profiles, shown in Fig. 26. The ring was not detected during ingress or egress. The RMS residual for the ten measured ring events used in our ring orbit fit was 0.14 km.

U65 IRTF 1990 Jun 21
The 21 Jun 1990 occultation of U65 (Mink and Klemola, 1985) was observed in DC mode from the IRTF using an InSb aperture photometer at = 2.2 μm. The occulted star was relatively bright (K = 7.26) with an estimated projected angular diameter of 3.00 ± 0.75 km, with a limb darkening parameter = 0. Observing conditions were fair, with some cirrus and a bright moon nearby that prevented the use of the autoguider, resulting in some drifting of the occultation star to the edge of the aperture. This is evident in the variable baselines in the lightcurves of the ingress and egress ring occultation regions. (These are included in the SOM.) A gallery plot of square-well model fits to the detected ring events from U65 IRTF is shown in Fig. 27. Ring 6 was hidden in the noise, and the ring was not detectable. The ring orbit fit residuals for the 15 U65 IRTF measurements had an RMS error of 0.28 km.

U83 IRTF 1991 Jun 25
The 25 Jun 1991 occultation of U83 (Klemola and Mink, 1991) was observed from the IRTF using the RC2 InSb dewar in DC mode aperture photometer at = 2.2 μm (French et al., 1991). Observing conditions were favorable, with a partial moon and no visible haze or cirrus. The complete occultation lightcurve is included in the SOM, with variable signal levels during ingress likely due to drifting of the star to the edge of the aperture. The occulted star was moderately bright (K = 9.29), with an estimated projected angular diameter of 1.15 ± 0.20 km, assuming a limb darkening parameter = 0. High-SNR ingress and egress profiles were obtained for the nine main rings (the ring was not observed during either ingress or egress), with the main source of smoothing being diffraction and the finite filter bandpass, rather than due to the angular size of the star or the instrumental time constant, as can be seen in the gallery of square-well model fits shown in Fig. 28. The ring orbit fit residuals for the 18 U83 IRTF measurements had an RMS error of 0.43 km.

U84 IRTF 1991 Jun 28
The 28 Jun 1991 occultation of U84 (Klemola and Mink, 1991) took place just three days after the U83 event, and was observed from the IRTF in the same fashion as U83 (French et al., 1991). Observing conditions were again favorable. The complete occultation lightcurve is included in the SOM, with variable signal levels during ingress again likely due to drifting of the star to the edge of the aperture. The occulted star was moderately bright (K = 9.66), with an estimated projected angular diameter of 0.90±0.20 km, assuming a limb darkening parameter = 0. High-SNR ingress and egress profiles were obtained for the nine main rings (the ring was not observed during either ingress R.G. French et al. Fig. 35. Skyplane view of the HST observations of the U138 occultation. Observations began at left, and at the end of the first HST orbit, the ingress ring was observed. The atmospheric occultation occurred while Uranus was blocked from view by the earth. The apparent path of the star during the second HST orbit is seen at right, terminated by the red dot, just prior to the egress ring occultation, which was not observed. or egress). A gallery of square-well model fits shown in Fig. 29. The ring orbit fit residuals for the 18 U84 IRTF measurements had an RMS error of 0.33 km.

U102A/B IRTF 1992 Jul 8
The 8 Jul 1992 occultation of the U102 (Klemola and Mink, 1991) by Uranus and the rings was observed from the IRTF 3.2 m telescope using high-speed aperture photometry at = 2.2 μm (French et al., 1992). Ring ingress and the ingress/egress atmosphere events were observed in chopping mode with the Dual Dewar Mount (DDM) using two K-band InSb photometers. The egress ring region was observed in DC mode. The event lightcurve is included in the SOM. The star was first centered in the two photometric apertures just prior to the ring ingress event. Ring events occurred in pairs separated by ∼ 0.9 s, indicating that U102 is a double star. We label the brighter/fainter binary components U102 A and U102B, with estimated projected diameters of 0.55 ± 0.15 and 0.45 ± 0.15 km, respectively. The instrumental time constant for these observations was rather long (0.125 s), resulting in considerable smoothing of the ring profiles.
Gallery plots of square-well models of the detected ring events for U102 A and U102B are include in the SOM. We identified six individual U102 A ring events to include in our ring orbit model, with a post-fit RMS residual of 0.31 km. The five detected fainter U102B ring events showed more scatter (post-fit RMS residual ∼ 1 km), and were excluded from our adopted ring solution. Instead, we adopted the final orbit model and geometry of the ring system to solve for the skyplane offset of U102B relative to U102 A, based on the five measured ring events.

U103 1992 Jul 11
The 11 Jul 1992 occultation of U103 (Klemola and Mink, 1991) was observed from CTIO, ESO, and Palomar Observatory. Details of the observations are provided by French and colleagues (1996) and overview and gallery plots of the observations are included in the SOM. For our reanalysis of the observations, we renormalized all of the observations and adopted a projected stellar diameter for U103 (K = 10.1) of 0.86 ± 0.12 km, with no limb darkening.

U103 CTIO
As noted in French and colleagues (1996), instrument failure prevented digital recording of the CTIO observations of U103, but digitized scans of the high-speed strip chart recording revealed the likely detection of the azimuthally variable ring. We adopted the ring event time for these observations as provided by French and colleagues (1996). These were not included in our overall orbit fit for the nine main rings, but were used to determine the geometry of the ring detection.

U103 ESO
For the U103 ESO observations, we renormalized the occultation lightcurve and refitted the ring profiles using the updated uniform disk projected stellar diameter. The post-fit RMS of the 15 U103 ESO observations in the ring orbit model was 0.28 km.

U103 Palomar Observatory
The occultation chord from Palomar Observatory missed rings 6, 5, and 4. We renormalized the occultation lightcurve and refitted the ring profiles using the updated uniform disk projected stellar diameter. The post-fit RMS of the 12 U103 Palomar observations in the ring orbit model was 0.29 km.

U9539 CTIO 1993 Jun 30
The 30 Jun 1993 occultation of U9539 (from the plate catalog of Klemola and Mink, personal communication) by Uranus and the rings was observed with the CTIO 4 m telescope using an InSb aperture photometer at = 2.2 μm (French et al., 1993). Sky conditions were clear throughout the occultation. The complete occultation lightcurve is included in the SOM. The ingress and egress ring regions were observed in DC mode to preserve high time and spatial resolution; the atmosphere occultation was observed in chopping mode to provide stable baselines. The occulted star was relatively faint (K = 11.8), with an estimated projected angular diameter of 0.30 ± 0.05 km, assuming a limb darkening parameter = 0.
Eighteen likely ring events were observed, with a range of SNR reflected in the quality index attributions shown in the gallery of square-well fits shown in Fig. 30. Once again, the ring proved elusive and was not detected during ingress or egress. The ring orbit fit residuals for the 18 U9539 CTIO measurements had an RMS error of 0.26 km.

U134 SAAO 1995 Sep 9
The 9 Sept 1995 occultation of U134 (Klemola and Mink, 1991) by Uranus and the rings was observed with the South African Astronomical Observatory (SAAO) 1.88-m telescope through a K filter with an InSb aperture photometer. Observing conditions were good throughout the occultation. We observed in DC mode, rather than chopping, and the baselines were very stable. The MILLIE data acquisition system was used to record the data digitally at 10 ms time resolution. The complete occultation lightcurve is included in the SOM.
All of the usual nine main rings were observed at high SNR during ingress and egress, with the ring being absent once again. The estimated projected diameter of U134 was 1.75±0.25 km, assuming no limb darkening. The ring profiles show prominent diffraction signatures, owing to the small star and short instrumental time constant. All 18 profiles garnered a quality index Q = 1, as shown in the gallery of square-well fits in Fig. 31. We grouped the ingress and egress rings separately in our ring orbit fit, and fitted for the offset time between the two data subsets to correct for absolute timing uncertainties in the MILLIE system. Each set of nine U134 SAAO measurements had an RMS error of 0.4 km.

U137 1996 Mar 16
The 16 Mar 1996 occultation of U137 (Klemola and Mink, 1991) was observed from the Hubble Space Telescope (HST) and from the IRTF. The star was relatively bright (K = 7.0, V = 8.5), but by this date the rings were no longer nearly face-on as seen from Earth ( = −43.48 • ; see Table 1 and Fig. 5) and the ring plane radial velocity of the occultation was high. This resulted in lower SNR and more limited spatial resolution than in earlier occultations of comparably bright stars. We adopted a projected stellar diameter of U137 of 2.5 ± 0.5 km, assuming no limb darkening.

U137 HST
Both the U137 and U138 occultations were observed with the HST's Faint Object Spectrograph (FOS) under Program ID 5823 (PI A. Bosh). In each case, an offset acquisition star was observed well before the start of the occultation to ensure accurate placement of the target star within the science aperture. The U137 HST observations began just prior to the egress atmospheric occultation and lasted for ∼ 17 min.
Observations from HST data file y34y0101t_c1f.fits were coadded over the range = 0.362 − 0.705 μm of the G650L/RD FOS element to form the occultation lightcurve shown in Fig. 32. The top panel shows the raw lightcurve, exhibiting an overall gradual decrease in the count rate as the occultation proceeded and the aperture contained less background reflected light from Uranus. The lower panel shows the normalized curve, beginning with the egress atmosphere event and followed by several visible sharp dips due to the narrow rings.
The time resolution was ∼ 0.095s per sample, and for the ring opening angle = −43.48 • , the typical ring plane radial velocity was quite rapid (̇= 27 − 35 km s −1 ), resulting in a corresponding sampled radial resolution of 2.6 − 3.3 km. Although this is wider than the projected width in the sky plane of some of the narrow rings themselves, the diffraction pattern of the ring extends beyond the geometrical shadow of the ring edges. Fig. 33 shows the observed egress ring profile and the overplotted best-fitting square-well model in the upper panel, and the computed square-well model at higher resolution, prior to averaging at the observed time resolution, in the bottom panel. The observed lightcurve is at too low a time resolution to reveal the details of the underlying diffraction pattern, but the averaged squarewell model matches the observations quite well. We fitted square-well models to five detectable egress ring events: rings , , , , and . All but the ring had QI = 3; the ring detection is marginal and has QI = 4. A gallery plot of square-well model fits to both tentative and secure ring identifications for the U137 HST data is included in the SOM. The ring orbit fit yielded an RMS residual of 0.22 km for the five included HST measurements.

U137 IRTF
The U137 occultation was observed from the IRTF using the Multi-Object Photometer (MOP) at J and K bands ( = 1.25 and 2.2 μm, R.G. French et al. respectively) in chopping mode. For this analysis, we make use of the K-band data only. The J band data have lower SNR and more variable background, and have not been included in the PDS archive. The event occurred under clear conditions at high airmass, with an elevation angle of about 15 • for the ingress ring occultation, gradually rising to about 25 • by the time of the egress ring event. The K-band lightcurve is included in the SOM.
We detected all nine main rings during both ingress and egress. We modeled the instrumental response as a double-pole filter with a time constant of 0.03 s. A gallery plot of square-well model fits to the U137 IRTF ring events (Fig. 34) clearly shows the smoothing and time offset effects of the instrumental time constant. The ring was not detected. The ring orbit fit yielded an RMS residual of 0.37 km for the 18 included IRTF measurements.

U138 1996 Apr 10
The 10 Apr 1996 occultation of U138 (Klemola and Mink, 1991) was observed from the Hubble Space Telescope (HST) and from Palomar Observatory. The star was comparable in brightness to U137 in the IR (K = 7.0) but fainter at visual wavelengths (V = 9.2). We assumed a projected stellar diameter of 2.5 ± 0.5 km for U138, with no limb darkening.

U138 HST
The HST observations of U138 were carried out with the same instrumentation as U137, over two HST orbits. The skyplane view of the event is shown in Fig. 35. The timing of the event was such that only the ring was detected during the first (ingress) HST orbit, and all but the ring were observed during the second (egress) orbit. The ingress and egress lightcurves shown in Fig. 36 were formed by coadding the signal over the range = 0.362-0.705 nm of the G650L/RD FOS element from HST data files y34y0401t_c1f.fits and y34y0402t_c1f.fits, respectively, with a time resolution ∼ 0.11 s. A gallery of square-well fits to the ten observed ring events is shown in Fig. 37. The ring orbit fit yielded an RMS residual of 0.27 km for the nine main ring measurements, excluding the ring candidate.

U138 Palomar Observatory
The egress U138 occultation by Uranus and the rings was observed under clear conditions from Palomar Observatory's 5 m Hale Telescope in K band (2.0-2.4 μm) with a single channel InSb photometer (D-68) at the f/70 focus of the low-background chopping secondary. The observing strategy was similar to that used for U16 -see Section 4.10. The occultation lightcurve is included in the SOM. Fig. 38 shows a gallery of square-well model fits to the nine detected ring events. The ring orbit fit yielded an RMS residual of 0.30 km for these measurements,

U144 Centro Astronómico Hispano-Alemán (CAHA)
Observations from the CAHA f/8 1.23 m telescope were carried out with the BlackMagic NICMOS3 256 × 256 IR camera in the K band ( = 2.2 μm). Subframes 16 × 16 in size were recorded at ∼10 Hz.  Limitations on the duration of a single sequence of exposures required that ingress and egress be recorded separately, with a significant time gap in between. Based on pre-event predictions, the recording intervals were selected to ensure that the ring was recorded during both ingress and egress. The result is that rings 4, 5, and 6 were not captured on either side. Uranus was low in the sky: the elevation angle was ∼ 28 • for the ingress event, decreasing to 15 • by the time of egress. The resultant lightcurves (not shown, but included in the SOM) were obtained using digital aperture photometry on the two sequences of images. Fig. 39 shows a gallery of the detected ring events, along with best-fitting square-well models. The ring orbit fit yielded an RMS residual of 0.34 km for the ingress measurements and 0.45 km for the egress measurements.

U144 South African Astronomical Observatory (SAAO)
Observations from SAAO of the U144 occultation of the rings during ingress were carried out using the same instrumentation as for the U134 event -see Section 4.24 for a description. Sky conditions were partly cloudy and the background signal was highly variable, resulting in relatively low SNR ring detections, in spite of the brightness of the star. The complete lightcurve is contained in the SOM. Fig. 40 shows a gallery of the six detected ring events, along with best-fitting squarewell models. Each had a quality index QI = 3, reflective of the poor observing conditions. The ring orbit fit yielded an RMS residual of 0.25 km for these measurements.

U149 IRTF
The U149 IRTF observations were carried out at = 2.2 μm using NSFCAM. The ingress ring and atmosphere occultations were not observed because of clouds, but the sky cleared before the emersion ring occultations. Additional details of the observations are described by Young et al. (2001), including approximate ring event times. Images were recorded with a time resolution of 0.42 s, the typical ring plane radial velocity waṡ= 13 km s −1 , and the skyplane velocity of the star perpendicular to the rings was ∼ 8−9 km s −1 , resulting in low-resolution ring profiles with characteristics similar to those shown in Fig. 33 for the U137 HST observations. A gallery plot of square-well model fits to the six detected U149 IRTF ring events is shown in Fig. 41. The ring orbit fit yielded an RMS residual of 0.25 km for these measurements.

U149 Lowell Observatory
The observations of the U149 atmosphere and ring occultations from Lowell Observatory's 1.8 m Perkins Telescope were carried out R.G. French et al. Fig. 44. Square-well model fit to the U0602 IRTF egress ring profile. The upper panel shows data and the best-fitting square-well model, averaged to the same time resolution as the data, and the corresponding square-well model for the assumed ring structure (red). The lower panel shows the modeled diffraction pattern of the square well at higher time resolution, including the averaging effects of the finite width of the K-band filter and convolution with the strip brightness distribution of the occulted star (shown in violet). Notice that diffraction pattern is much wider than the narrow square-well and extends over several data points. at = 0.89 μm using the PCCD camera (Buie and 9 colleagues, 1993), with a time resolution of 0.25 s. Skies were clear but the event was observed at very low altitudes: the atmospheric observations were at airmass = 2.0 − 6.5. Data were collected from 3:30 UTC (Uranus at an altitude of 27 • ) to 5:47 UTC, with the last useful data taken at ∼ 5:30 UTC, shortly after the ring egress event, when Uranus was at an altitude of 7.5 • . Additional details of the observations are described by Young et al. (2001). We fitted square-well models to the six ring events shown in Fig. 42. The ring orbit fit yielded an RMS residual of 0.18 km for these measurements.

U0201 Palomar Observatory 2002 Jul 29
The U0201 occultation was predicted by A. Bosh 9 and was observed from Palomar Observatory with the 5 m Hale Telescope in K band (2.0-2.4 μm) with a single channel InSb photometer (D-68) at the f/70 focus 9 http://www2.lowell.edu/users/amanda/occs2000/UTable3.6b.html of the low-background chopping secondary. The observing strategy was similar to that used for U16 -see Section 4.10.
Uranus was at an altitude of about 45 • at the time of the occultation. The occultation track missed the inner three rings (Fig. 5), but captured the rest of the ring system on both ingress and egress. The occulted star U0201 was comparatively faint (K = 11.4) and the ring plane radial velocity reached 33 km s −1 for the outer rings, owing to the low ring opening angle ( = −19.69 • ), resulting in only moderate SNR in spite of the large telescope aperture. The complete occultation lightcurve is included in the SOM.
We adopted a projected diameter of U0201 of 0.30 ± 0.05 km, with no limb darkening, and modeled the instrumental response function as a double-pole filter with a time constant of 0.015 s. Fig. 43 shows a gallery of the detected ring events, along with best-fitting square-well models. The ring was not detected during ingress or egress.
The RMS of the orbit fit residuals for the 12 U0201 Palomar Observatory ring measurements was 0.33 km.

U0602 IRTF 2006 Sep 20
The 20 Sep 2006 occultation of star U0602 (predicted by A. Bosh, personal communication) was observed from the IRTF at = 2.2 μm using the Old SpeX medium-resolution spectrograph (Rayner and colleagues, 2003). From the photometry of the star (V = 10.746, K = 8.408 obtained from VizieR) we estimate a projected stellar diameter of 1.50 ± 0.25 km, assuming no limb darkening. The ring opening angle = −5.42 • and the ring plane radial velocity was quite high: = 99 − 159 km s −1 . The apparent velocity of the star perpendicular to the ring edges in the sky plane was also high: ⟂ ≈ 22 km s −1 . Subframes were recorded using the Guidedog interface at an average time resolution over the course of the observations determined to be 0.1127 s. The corresponding ring plane resolution of 11-18 km per data point relates a timing uncertainty in the ring event time to the corresponding uncertainty in the ring plane radius. On the other hand, the skyplane resolution is 2.5 km per data point, which provides a measure of the averaging over the diffraction pattern of the projected ring shadow, with a characteristic broadening given by the Fresnel scale ∼ 1.77 km for this event. As a result, the predicted signal varies over a time span longer than the actual width of the ring, enabling us to determine the midpoint of the ring to sub-sample accuracy. This is illustrated in Fig. 44, which shows a square-well model fit to the egress ring event. The star diameter is 1.5 km, a bit larger than the fitted square-well width, but smaller than the sky plane distance spanned by a single data point. In comparison, the modeled diffraction pattern spans several data points.
The ingress and egress occultation lightcurves of the ring regions shown in Fig. 45 were obtained from frame-by-frame aperture photometry. Predicted locations of the rings are marked by vertical dashed lines, based on the adopted ring orbit model. Fig. 46 shows the observations in more detail for each of the predicted locations of the rings. Because of the relative orientations of the ascending nodes of inclined rings 5 and 6, the egress ring 5 occultation preceded the ring 6 event. A slight dip in the signal occurs in the vicinity of the predicted location of the ring during ingress; a more convincing narrower feature is present in the egress lightcurve. The ring profiles are quite broad, owing to the detection at the highly oblique viewing angle of the broad sheet of optically thin material that extends ∼ 55 km exterior to the sharp narrow core of the ring, as discussed in detail by Elliot and Nicholson (1984). On the basis of post-fit ring orbit residuals, the ring detections do not appear to be similarly affected by the qualitatively similar ∼ 10 km-wide low optical depth companion just interior to the main ring described by French et al. (1991), presumably because the radially integrated slant-path optical depth was dominated by the core of the ring itself.
We determined the ring event times from square-well model fits, shown in Fig. 47. Table 5 lists the measured ring event times, the R.G. French et al. Fig. 45. Ingress and egress lightcurves for U0602 IRTF observations. Predicted locations of all ten rings are marked by the vertical dashed lines, based on the adopted ring orbit model. corresponding observed radius in the ring plane, the model radius, and the residual between these two. Because of the viewing geometry of the U0602 event and the oblique orientation of the ring plane, the event times of U0602 ring occultations are especially sensitive to the right ascension of the direction of the Uranus pole and the inclinations of the rings. The RMS residual for this event is 0.44 km.

Summary of observations and quality index (QI)
We summarize the entire set of Earth-based ring occultation data bundles in the PDS archive in Table 6. For each bundle, we include the Fresnel scale , the adopted projected stellar diameter * , the effective smoothing distance perpendicular to the edge of the ring in the R.G. French et al. Fig. 46. Observations of the U0602 IRTF occultation in the vicinity of the predicted event times for all ten rings during ingress and egress. skyplane (given by ∕ ⟂ , where is the instrumental time constant and ⟂ is the average perpendicular velocity of the star in the skyplane relative to the ring edge), and the quality index QI for each ring. For each data bundle, we assign an overall average rating according to the following scale: • E (excellent) -rounded mean QI for observed rings = 1, excluding the ring • VG (very good) -rounded mean QI for observed rings = 2, excluding the ring • G (good) -rounded mean QI for observed rings = 3, excluding the ring • F (fair) -rounded mean QI for observed rings = 4, excluding the ring

Voyager 2 Uranus ring occultation observations
During the Voyager 2 flyby of Uranus on 1986 Jan 24, two stellar occultations by the rings were observed simultaneously with the Photopolarimeter (PPS) at = 265 nm (Lane and 10 colleagues, 1986;Colwell and 12 colleagues, 1990) and the Ultraviolet Spectrometer (UVS) at = 110 nm (Holberg et al., 1987). The chord of the high-SNR occultation of Sgr passed just interior to the ring, with an average ring opening angle of = −62.9 • ; the lower-SNR Per occultation spanned the entire ring system on ingress and egress with an average ring opening angle of = −53.2 • . The rings were also observed by radio occultation with the RSS at X-and S-band (3.6 and 13 cm, respectively), with a nearly face-on average ring opening angle of = −81.5 • (Tyler and 9 colleagues, 1986;Gresh et al., 1989). For a detailed review of these Voyager ring observations, see French et al. (1991).
All three Voyager 2 occultations were incorporated in the French et al. (1986) ring orbit fit, and here we briefly summarize the observations used in our current solution (French et al., 2023b), highlighting the few differences from our previous approach. For the stellar occultations, we used only the PPS observations, which are at higher time resolution and SNR than the UVS data. Our standard orbit solution fits simultaneously for the ring orbital elements of the original nine narrow rings and the Uranus pole direction, and then uses this geometric solution to fit separately for both secure and tentative ring detections (identified as 1986U1R in French et al., 1986. We used the PPS event times from French et al. (1986) (Tables XI and XII) for both stellar occultations. We used Hipparcos positions for both stars (they are too bright to be included in the Gaia DR3 catalog), corrected for parallax and proper motion. For the RSS occultation, we again used the square-well diffraction model fits to the diffraction-limited RSS 3.6 cm observations (Table X, French et al., 1986), in preference to the estimated midtimes of the diffraction-corrected ring profiles, both because this more closely resembles our approach to diffraction-limited Earth-based observations and because several of the narrow rings have at least one edge that is not intrinsically sharp (Gresh et al., 1989), reducing the reliability and objectivity of midtime determinations from the high-resolution profiles.
Largely because of the unique viewing geometry of the rings during the Voyager 2 stellar occultations, these observations provide constraints on the Uranus pole direction that are complementary to those from Earth-based stellar occultations and the Voyager 2 RSS occultation.   French et al. (2023b) show that the adopted Voyager 2 trajectory results in a ring plane radius scale and pole direction that are consistent with determinations from Earth-based data alone.

Summary and conclusions
Since their discovery in 1977, the Uranian rings have provided the archetypal example of a system of narrow and sharp-edged rings, most only a few km wide. The complete set of ring occultation observations span nearly thirty years, and from detailed studies of the many hundred individual occultation ring profiles, it has proven possible to set tight constraints on the orbits of the inclined and elliptical rings, to detect normal modes in the shapes of the rings and ring edges, some of them forced by small moons, to determine the direction of the planet's rotation axis, and to estimate its gravitational field, as we describe in a companion paper (French et al., 2023b). They are a counterpart to the extensive Cassini observations of Saturn's rings, providing opportunities for fruitful comparisons of the structure and dynamics of these two very different ring systems. The occultation observations documented here are permanently available on NASA's PDS, and are a legacy for ongoing detailed dynamical studies and for future spacecraft explorations of Uranus and the other outer planets.

Data availability
The data are available on the PDS and as part of Supplementary Online Material.

Acknowledgments
We are grateful to Joshua Colwell for reviewing this paperhis careful reading and helpful suggestions significantly improved the manuscript. James Elliot preserved detailed documentation of many of the early occultation observations reported here, without which this work could not have been completed. We are grateful to the telescope operators and support staff at observatories around the world who made these observations successful. 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, 10 processed by the Gaia Data Processing and Analysis Consortium (DPAC). 11 Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. We are also grateful to the authors of SORA: Stellar Occultation Reduction and Analysis package (Gomes-Júnior and 8 colleagues, 2022).  -3  3  3  3  3  3  3  3  4  3  3  3  3  3  3  3  3  3  3 3 G Column definitions: BundleID: the complete PDS bundle ID has prefix uranus_occ_; : Fresnel scale = √ ∕2; d*: projected stellar diameter (km); ∕ ⟂ : is the instrumental time constant and ⟂ is the average perpendicular velocity of the star in the skyplane relative to the ring edge; QI: Quality Index (see Section 3.2.2); Rating: see Section 4.31.

Appendix A. Planetary data system uranus earth-based occultation archive
The PDS archive of Uranus Earth-based occultations has two components. The first is a set of individual data ''bundles'' for each observed occultation, identified by bundleID uranus_occ_u*_obs_*cm, where u* is the occultation star, obs is the observatory, and *cm is the telescope aperture in cm. A separate single ''support bundle'' contains information applicable to all data bundles, including a comprehensive User Guide that contains detailed information about the archive contents and how to make use of them. The User Guide is the authoritative source of information about the current archive contents, and it will be updated whenever revisions or additions are made to the PDS Uranus ring archive.

Representative data bundle contents
The directory structure of the PDS Uranus Earth-based occultation archive is annotated below for a representative data bundle: uranus_occ_ u17b_saao_188 cm. Every file in the bundle has an associated XML header file (*.xml) that defines individual columns for data files and includes additional information applicable to the entire file as well as internal references used by PDS for cross-referencing files. Text files have the suffix *.txt. Data files (*.tab,*.csv) are human-readable tables; all figures are in PDF format (*.pdf).

Support bundle contents
The directory structure of the PDS Uranus Earth-based occultation archive support bundle is annotated below: uranus_occ_support: bundle.xml -internal PDS use readme.txt -contents listed below: This support bundle contains supporting documentation and the underlying Uranian ring models used in the preparation of a large set of occultation data bundles.
The document collection of this bundle contains a detailed, comprehensive User Guide which provides an overview of the Earth-based Uranus stellar occultations in the PDS.
The data collection contains the tables representing the underlying Uranian ring models and the associated input files.
. The SOM contents are in subdirectories that match the structure of the PDS archive on the Ring-Moon Systems node for ''Earth-based observations of stellar occultations of the Uranus system": https://pds-rings.seti.org/pds4/bundles/uranus_occs_Earth-based/.
The PDS archive is organized by bundleID uranus_occ_u*_obs_*cm, where u* is the occultation star, obs is the observatory, and *cm is the telescope aperture in cm. Each SOM bundleID directory includes key figures from the PDS archive browse/ products and a PDF file named [bundleID]_plots.pdf that contains plots of most of the ring data files in the PDS archive of that bundleID (these plots are not included in the PDS archive itself).
For additional information about the PDS archive, users should download the current version of the User Guide that is part of the PDS uranus_occ support bundle. 12 The User Guide that was current as of the date of the preparation of the SOM is included in the top-level SOM directory.
The directory structure of the SOM is shown below for a representative bundleID uranus_occ_u17b_saao_188 cm: README.txt Earth-based-uranus-stellar-occultation-user-guide.pdf The SOM contents are in subdirectories that match the structure of the Planetary Data System (PDS) archive on the Ring-Moon Systems node for "Earth-based Observations of Stellar Occultations of the Uranus System": URL: https://pds-rings.seti.org/pds4/bundles/uranus_occs_Earth-based/ The PDS archive is organized by bundleID. Each SOM bundleID directory includes key figures from the PDS archive "browse" products and a PDF file named [bundleID]_plots.pdf that contains plots of most of the ring data files in the PDS archive of that bundleID.
For additional information about the PDS archive, download the current version of "Earth-based-uranus-stellar-occultation-user-guide.pdf" that is part of the "uranus_occ" support bundle: https://pds-rings.seti.org/pds4/bundles/uranus_occs_Earth-based/uranus_occ_support/ document/user_guide/Earth-based-uranus-stellar-occultation-user-guide.pdf The user guide that was current as of the date of the preparation of the SOM is included in this top-level SOM directory.
The complete contents of the SOM directory are listed below: ...  Each PDS bundle contains data tables (but not plots) of ingress and egress ring occultation regions averaged at 100 m, 500 m, and 1000 m radial resolution. The SOM file uranus_occ_u17b_saao_188 cm_plots.pdf includes plots of each of these regional files, as illustrated in