Examining Uranus’ $\zeta$ ring in Voyager 2 Wide-Angle-Camera Observations:
Quantifying the Ring’s Structure in 1986 and its Modifications prior to the Year 2007

The $\zeta$ ring is a relatively broad and diffuse ring that was first clearly seen in a single image obtained by the Voyager 2 spacecraft as it flew past Uranus in 1986 (Smith et al., 1986), at which time it was designated 1986/U2R. The peak brightness of this ring was situated at a distance of around 37,500 km from Uranus’ center in this image (Smith et al., 1986; Esposito et al., 1991), making it the innermost known component of the Uranian ring system.

Interest in the $\zeta$ ring increased when Earth passed through Uranus’ ring-plane in 2007, enabling this ring to be clearly observed with Earth-based telescopes. These observations confirmed earlier Earth-based detections of this ring (de Pater et al., 2006a), and clearly demonstrated that the location of the ring’s peak brightness had shifted outwards from 37,500 km to around 39,000 km from Uranus’ center (de Pater et al., 2007). However, a major challenge for understanding these dramatic changes in the $\zeta$-ring’s morphology is that the Earth-based images were all obtained at low phase angles, while the single known Voyager 2 image was obtained at a phase angle of around 90^∘. Since tenuous rings like the $\zeta$ ring are typically composed primarily of dust-sized particles less than 100 microns across, and the brightness of such particle populations can vary significantly with phase angle, it was not straightforward to directly compare the observed brightness of the ring at specific distances from Uranus. Hence it was unclear whether the change in the peak location was accompanied with a change in the overall brightness of the ring. In fact, one could potentially attribute the change in the ring’s appearance to variations in its particle size distribution that made it more efficient at scattering light in one or the other geometry at different locations.

Understanding the distribution of material in the $\zeta$ ring has recently become of practical interest to potential missions to the Uranus system. At least one Uranus mission concept has the spacecraft pass between the main rings and the planet during orbit insertion (Simon et al., 2021). During this time, the spacecraft could potentially pass through the inner flank of the $\zeta$ ring, and so collisions with $\zeta$ ring particles may pose a risk. Quantifying this risk has been challenging because the Voyager image only captured the region exterior to 33,000 km from the planet’s center, which leaves the remaining 8,000 km above Uranus’ atmosphere unconstrained. Analyses of the more recent Earth-based images have suggested that there could be material extending all the way in to Uranus’ atmosphere (de Pater et al., 2006a; Dunn et al., 2010; de Pater et al., 2013). However, for these low-phase observations it is difficult to completely remove the signal from the planet itself, making the real signal levels close to the planet highly uncertain.

An opportunity to address both of these issues appeared in October 2021, when Ian Regan provided evidence that the Voyager 2 observations of the $\zeta$ ring were not confined to a single image. He co-added almost one hundred high-phase, low-resolution images obtained by the Voyager 2 spacecraft as it flew away from Uranus, and discovered that these images contained clear signatures of the $\zeta$ ring¹¹1These discoveries were first announced in a pair of tweets: https://twitter.com/IanARegan/status/ 1446983843886600195,
and https://twitter.com/IanARegan/status/ 1447332465127923721 (see Figure 1). These high-phase images provide the highest signal-to-noise profiles of the $\zeta$ ring currently available, and also provide constraints on the ring’s brightness down to within 3,000 km of Uranus’ atmosphere. Furthermore, close examination of similarly low-resolution images taken on approach reveal potential signals from the peak of the $\zeta$ ring at phase angles similar to those observed from Earth in 2007.

This paper analyzes all the previously known and recently discovered Voyager 2 images of the $\zeta$ ring in order to robustly estimate the brightness of ring material within 40,000 km of Uranus’ center during the Voyager encounter over a range of lighting geometries. Section 2 describes the relevant Voyager 2 observations and how we use these data to generate three radial brightness profiles. Section 3 then discusses the expected relationships between the observed ring brightness and parameters like particle number density. This section identifies the model-dependent factors in this relationship that complicate translating observed brightness into reliable estimates of particle number density, and thus motivate more empirical comparisons among different observations and rings. Section 4 then compares the observed properties of the $\zeta$ ring with those of other dusty rings, demonstrating that this ring has a similar phase function to Saturn’s G and D rings, Jupiter’s Main Ring and Uranus’ $\nu$ ring. More detailed comparisons of the $\zeta$-ring profile with those of Saturn’s G and D rings obtained at similar lighting geometries are used to clarify the $\zeta$-ring’s potential risk to spacecraft. The brightness trends with radius in the innermost parts of Saturn’s and Uranus’ ring system also suggest that the inner flank of the $\zeta$ ring probably consists of particles spiraling inwards under the influence of atmospheric drag. Finally, Section 5 compares these Voyager profiles to previously published high signal-to-noise profiles obtained from Earth-based telescopic observations in 2007. These comparisons confirm that the distribution of dust interior to the main rings changed substantially between these two epochs, and indicate that the previously-identified radial shift in the ring’s location was accompanied by a substantial increase in the ring’s brightness between 38,000 km and 40,000 km.

This study analyzes the calibrated, geometrically-corrected versions of the Voyager 2 images available via the Outer Planets Unified Search on the Ring-Moon Systems Node of the Planetary Data System. These images provide brightness estimates measured in terms of $I/F$, a standardized measure of reflectance that is normalized to unity for a Lambertian surface illuminated and viewed exactly face-on. In order to facilitate comparisons among the different observations, the observed $I/F$ is multiplied by the cosine of the observed emission angle $\mu$ to obtain the so-called “normal I/F” (or $\mu I/F$). For tenuous rings like the $\zeta$ ring this corresponds to the $I/F$ that would have been observed if the ring was viewed from exactly face-on, and so removes any dependence on the observed ring opening angle. Variations in the observed ring brightness due to other aspects of the lighting geometry like phase angle are more dependent on ring particle properties and so will be considered in more detail in subsequent sections.

Each image was geometrically navigated using the SPICE kernels listed in Table 1, with the camera pointing being refined based on the positions of bright stars in the field of view. After these corrections, it was possible to compute both the radius and the inertial longitude (relative to the intersection of the J2000 equatorial plane) in Uranus’ ringplane for every pixel in each image.

Voyager 2 was able to observe the $\zeta$ ring at three different phase angles during its flyby of Uranus. Each of the following subsections describes how the data from one of these observations were processed to produce a radial brightness profile of the $\zeta$ ring. First, Section 2.1 describes the single high signal-to-noise image of the ring obtained at low ring opening angles and moderate phase angles. Next, Section 2.2 discusses the sequence of high-phase, low-resolution images that provide the most complete information about the radial brightness profile of the $\zeta$ ring. Finally, Section 2.3 describes a set of low-phase, low-resolution images taken as the spacecraft approached the planet, where the signal from the $\zeta$ ring appears to be barely detectable.

In principle, one can translate the brightness profiles derived in the previous section into estimates of the particle number density within the ring. However, in practice this conversion involves multiple model-dependent parameters. Given the potential interest in using these profiles to assess risks to future missions, this section will discuss those parameters in some depth.

All rings consist of particles with a range of sizes that can be quantified in terms of the particle size distribution function $n(s)$, which is the differential number of particles per unit area and per unit size bin. The physical parameter most relevant for assessing risk to a spacecraft passing through a dusty ring is $N_{t}$, the surface number density of particles with a radius larger than some threshold size $s_{t}$ (i.e. the number of particles per unit area with a radius larger than $s_{t}$), which is given by the integral:

By contrast, for a tenuous ring like the $\zeta$ ring (where particles are far enough apart that they do not shadow each other significantly) the observable normal $I/F$ of a ring is given by the following integral over the size distribution:

where $Q_{scat}$ is the scattering efficiency coefficient and $f$ is the appropriately normalized phase function of the particles. Both these terms depend on the observation wavelength $\lambda$ and the structure/composition of the particles. The phase function also depends on the observed phase angle $\alpha$. For perfectly spherical particles with specified optical constants, these parameters can be computed using Mie scattering theory, while more complex calculations are needed for irregularly-shaped particles.

In practice, the above expression is often written as follows:

where $s_{min}$ is a minimum particle size and $A_{n}(\alpha)$ is the appropriately weighted average scattering efficiency factor $Q_{scat}f$. The remaining integral over the particle size distribution corresponds to the fractional area of the ring covered by particles with size above $s_{min}$, which (provided $s_{min}$ is sufficiently small) is also called the geometric optical depth $\tau_{g}$.

Since both $Q_{scat}$ and $f$ depend on particle size, in principle $A_{n}$ could be strongly dependent on the particle size distribution. However, in practice, the dominant challenge for estimating $A_{n}$ arise from uncertainties in the particles’ composition and structure.

For homogeneous spherical particles with specified indices of refraction, both $Q_{scat}$ and $f$ can be calculated using Mie theory. Figure 9 shows the product $Q_{scat}f$ as a function of particle size computed using the IDL code mie_single²²2http://eodg.atm.ox.ac.uk/MIE/mie_single.html (renormalized to align with the expressions given above) and assuming an observing wavelength of 0.46 $\mu$m, appropriate for the clear-filter Voyager images (Danielson et al., 1981). These curves were also smoothed over a range corresponding to 20% in the observed wavelength in order to roughly account for the finite bandpass of the camera³³3https://pds-rings.seti.org/voyager/iss/ inst_cal/vg1_wa_clear.html. These curves are also computed for three different phase angles that match the phase angles of the available Voyager observations of the $\zeta$ ring, and for three different values for the refractive index. The top panel shows $Q_{scat}f$ values assuming refractive indices appropriate for water ice (Mastrapa et al., 2009) in order to facilitate comparisons with Saturn’s rings. However, the main Uranian rings are known to be very dark (Karkoschka, 2001; de Kleer et al., 2013; Nicholson et al., 2018), so the bottom panel shows this factor assuming refractive indices appropriate for amorphous carbon (Rouleau and Martin, 1991), which is probably more appropriate for these rings. Finally, the middle panel shows values for a material with a refractive index intermediate between water ice and amorphous carbon in order to demonstrate that $Q_{scat}f$ is not extremely sensitive to the exact values of the optical constants.

While the detailed shapes of these curves differ, they all show some similar features. First of all, we can note that regardless of composition and phase angle, these curves all drop rapidly below 0.1 $\mu$m. This means that the scattering efficiency of particles smaller than 0.1 $\mu$m in radius is very low at optical wavelengths, and so these particles can be neglected when computing the ring’s brightness. In practice, this means that it is reasonable to assume $s_{min}=0.1\mu$m for these specific observations. Of course, particles smaller than this could still exist. While such small grains may not pose significant risks to spacecraft, constraints on their spatial distribution (which might be relevant to certain investigations or instruments) would require observations at other wavelengths.

Next, note that all the curves have a peak in the range between 0.1 and 1 $\mu$m, and above 1 $\mu$m these functions are fairly constant. The exception to this is the water-ice curve at 90^∘, which falls a factor of 2-3 between 1 $\mu$m and 100 $\mu$m. However, this particular behavior is related to the specific properties of spherical low-loss grains that lead to the formation of rainbows on Earth, and is not relevant for the absorbing, irregular particles in the Uranian rings.

This analysis would therefore suggest that it is reasonable to assume $A_{n}\sim 0.02-0.03$ for phase angles below 90^∘ and $A_{n}\sim 0.06-0.07$ at phase angles around 146^∘. For spherical particles, these numbers would even be underestimates since they neglect the peaks in scattering efficiency for sub-micron grains. However, it is important to note that this assumes that Mie scattering theory provides a reasonably accurate estimate of the scattering efficiency for real ring particles, which are likely irregularly shaped and perhaps somewhat porous.

For example, de Pater et al. (2013) assumed $A_{n}=0.001$ in their analysis of the low-phase Keck observations of the Uranian rings. They chose this number based on more empirical models of the ring-particle scattering properties. In essence, they assumed the single-scattering albedo of the ring particles was around 0.08 and the phase function could be approximated with a Henyey-Greenstein function. They do note that their value of $A_{n}$ is an order of magnitude lower than what Mie Theory would predict, and argue that the difference is because the particles have finite porosity that allows for more internal scattering, leading to a higher fraction of the incident light being absorbed rather than scattered. This is a reasonable argument for large ($s>$ 1 cm) ring particles found in the main rings. However, the light scattered by dusty rings like the $\zeta$ ring comes primarily from particles in the 1-100 $\mu$m size range, and it is less likely that such small particles would have enough internal scattering to change the ratio of scattered and absorbed radiation so dramatically. However, more detailed physical optics modeling would be needed to ascertain how much the structure and porosity of the ring particles could influence $A_{n}$.

Leaving aside this uncertainty in the appropriate value of $A_{n}$, we can write the above expression for $N_{t}$ as follows:

where

is a unitless factor that depends on the assumed shape of the particle size distribution. One common choice is to assume that the size distribution follows a power law with index $q$ so that $n(s)\propto s^{-q}$ for all $s$ between $s_{min}$ and some $s_{max}>s_{t}$. In that case, so long as $s_{min}<<s_{t}<<s_{max}$ then we can write out explicit expressions for $C_{n}$:

Note that if we assume $s_{max}$ is between 1 mm and 10 cm and $q=3$, then $C_{n}$ ranges between 0.036 and 0.054, so it is reasonable to use 0.05 as a nominal value for that constant. However, it is important to note that if the size distribution is allowed to be anything other than a power law, $C_{n}$ can have a much broader range of values.

Inserting nominal values for $A_{n}$ and $C_{n}$, the conversion between number density and brightness can be written in the following form:

Note that besides the dependence on the uncertain parameters $A_{n}$ and $C_{n}$, this expression also has a steep dependence on the threshold size $s_{t}$, which depends on the details of the spacecraft construction and operations. Hence any proper risk assessment would require careful consideration of all three of these parameters.

Given the substantial uncertainties involved in converting between observed brightness and particle number density, it is worthwhile to consider more empirical comparisons with other dusty rings. Of course, different rings can have different particle compositions and size distributions, so care must be taken when making these comparisons. Fortunately, the rings’ light-scattering properties enable us to identify the rings that are most likely to provide useful bases for comparison.

Section 4.1 below identifies the best analogs of the $\zeta$ ring among the faint rings of Jupiter, Saturn and Uranus based on their spectral and photometric properties. Section 4.2 provides more detailed comparisons between the $\zeta$ ring and Saturn’s G and D rings, two rings that not only have light-scattering properties similar to those of the $\zeta$ ring, but have also had spacecraft fly through them. Finally, Section 4.3 discusses aspects of the radial brightness profiles in the $\zeta$ and D rings that could be due to the particles in both rings experiencing significant amounts of atmospheric drag.

These new data also allow the structure of the $\zeta$ ring during the Voyager flyby to be better compared with the observations made by Earth-based telescopes around 2007 (de Pater et al., 2006a, 2007, 2013). The cleanest observations of the $\zeta$ ring made during this time occurred while the Earth passed through Uranus’ equatorial plane, which meant the ring system was viewed nearly exactly edge on(de Pater et al., 2007). This both suppressed the brightness of the main rings and made the signal from low optical depth rings like the $\zeta$ ring more detectable. de Pater et al. (2007) found that in these more recent images, the peak of the $\zeta$ ring was shifted outwards relative to its location in the Voyager moderate-phase image. However, there was no attempt to directly compare the brightness profiles of the two observations.

Since the rings were observed from nearly edge-on in 2007, the observable aspect of the ring is not normal I/F, but instead a vertically-integrated brightness of the ring versus radius. This vertically-integrated brightness is a convolution of all the rings exterior to the observed radius. However, these data can be converted into an estimate of the normal I/F versus radius using an onion-peeling algorithm that recursively removes signals from more distant ring material. This algorithm was performed on the high-quality Keck data in de Pater et al. (2007). However, de Pater et al. (2013) later performed a more comprehensive analysis of the Keck and VLT observations of the Uranian rings. In this paper, the authors did not onion-peel the edge-on-profiles to generate normal I/F profiles. Instead, they fit the observed edge-on-profiles to forward models assuming a specified dust distribution. This was done in part to account for the point spread function in the images, but also because it could handle situations where the main rings could block light from the dusty rings. The authors found some variations in the required brightness of the $\zeta$ ring in different observations, which they attributed to the finite vertical extent of the ring, but they also present a best-fitting model for all the observations with a uniform brightness in the three ranges 26,837-34,888 km, 34,888-37,850 km and 37,850-41,350 km. The $\mu I/F$ values (a parameter they call $A\tau_{0}$) for this model were $8\times 10^{-8}$, $4\times 10^{-7}$ and $3.7\times 10^{-6}$ in these three regions, on top of a $2.2\times 10^{-6}$ uniform dust sheet extending from the planet out through the main rings. This means that the core of the $\zeta$ ring in this model is about 1/2 its brightness in the onion-peeled profile from de Pater et al. (2007). Furthermore, the proposed signal levels interior to 37,850 km are comparable to the fluctuations seen in the onion-peeled profile, implying that the uncertainties in these parameters are probably substantial. Because of these potential issues, along with the relatively coarse spatial resolution of the model dust distribution, we chose to compare the Voyager profiles with only the onion-peeled profile from de Pater et al. (2007) here.

In order to compare this profile with the Voyager profiles, we need to correct for the differences in the observed phase angles and wavelengths. More specifically, we compute “corrected I/F” profiles for the Voyager profiles obtained at phase angles of 90^∘ and 146^∘ by multiplying each of these profiles by scaling factors of 0.78 and 0.26, respectively. These factors correspond to the predicted ratio of the ring’s brightness in each of these observations to its brightness in the low-phase (phase angle = $16^{\circ}$) observation, assuming that its brightness follows the same three-component Henyey-Greenstein function that best fit the Cassini data for Saturn’s G ring from Hedman and Stark (2015). Note that while the peak brightness of the ring does not exactly follow this trend (see Figure 10), the three corrected Voyager profiles are reasonably consistent with each other using these scaling factors (see Figure 12), so they should be sufficiently accurate for the present purposes.

The Keck profile comes from data obtained at a phase angle of around 3^∘ (de Pater et al., 2013), and since the brightness of dusty rings are relatively weak functions of phase angle at low phase angles (see Figure 10), we can neglect any phase correction for this profile. However, the Keck observations were also made at wavelengths around 2 $\mu$m, while the Voyager images were obtained at wavelengths around 0.46 $\mu$m (0.28-0.64 $\mu$m Danielson et al., 1981), and the dusty rings with similar phase functions as the $\zeta$ ring are also substantially brighter at longer wavelengths. Uranus’ $\nu$ ring has an integrated brightness or normal equivalent width of 1.30$\pm$0.13 m around 2.2 microns and 0.63$\pm$0.06 m between 0.3 and 0.7 micron (de Pater et al., 2006b), which gives a ratio of 2.06$\pm$0.28. Similarly, Saturn’s G ring has a normal equivalent width at 2.2 microns of 1.64$\pm$0.13 m (de Pater et al., 2004), while in the range between 0.3 and 0.7 microns its normal equivalent width is 0.62$\pm$0.12 m (0.30 micron), 0.65$\pm$0.16 m (0.45 micron), 0.74$\pm$0.16 m (0.54 micron) and 0.73$\pm$0.11 m (0.67 micron), which average together to give 0.68$\pm$0.07 m (Nicholson et al., 1996), yielding a ratio of 2.41$\pm$0.40. The situation for the Jovian ring is a bit more complicated, with Throop et al. (2004) indicating an average brightness ratio of around 3.5 between Voyager visible and Keck images. However, de Pater et al. (2008) found that profiles of the main ring obtained by Galileo and Keck matched well with a scaling factor of around 2.5. For this analysis, we will assume a nominal scaling factor of 2.5 between the Keck and Voyager wavelengths, and so multiply the Keck profile by $1/2.5=0.4$ in Figure 12 to produce a corrected brightness profile that can be compared with the corrected Voyager profiles.

The corrected brightness profiles shown in Figure 12 confirm that the location of the peak brightness shifted outwards from 37,500 km to around 39,000 km between 1986 and 2007. However, they also reveal that this outward shift is primarily due to a large increase in ring brightness between 38,000 and 40,000 km. Interior to 37,000 km the Voyager 2 and Keck profiles are actually fairly comparable to each other (at least to within the noise levels in these profiles). This implies that while the amount of dust exterior to 38,000 km has varied substantially with time, the brightness of material interior to 37,000 km may be more stable. Note that both these findings merit further investigation to ascertain what might have introduced additional dust into the system between 1986 and 2007 and to determine whether the similarities interior to 37,000 km are consistent with the observed cooling in the planet’s upper atmosphere (Melin et al., 2019).

The main results of the above analysis can be summarized as follows:

• •

  Uranus’ $\zeta$ ring was observed by Voyager 2 at high, moderate and low phase angles. In all these profiles the ring’s peak brightness falls around 37,500 km from Uranus’ center, and the relative brightness of the ring in these different lighting geometries are similar to Jupiter’s Main ring, Saturn’s D and G rings, and Uranus’ $\nu$ ring.

• •

  The $\zeta$ ring’s peak brightness at high phase angles is roughly an order of magnitude higher than the brightness of the part of Saturn’s G ring encountered by Voyager 2 and is roughly 40 times higher than the brightness of the region of Saturn’s D ring encountered by Cassini. The brightness of the ring decreases closer to the planet with a scale length of 2100$\pm$300 km, but the uncertainties in the profile are too large to establish whether the ring’s brightness ever reaches levels similar to those regions already encountered by spacecraft.

• •

  Comparisons between the Voyager observations from 1986 and the Earth-based observations from 2007 indicate that the brightness of the ring increased substantially between 38,000 km and 40,000 km from Uranus’ center, implying that additional dust was introduced into this region. Interior to 37,000 km, the differences between the profiles appear to be less dramatic.

We have also identified some productive avenues of future work to better understand the structure and evolution of the $\zeta$ ring, as well as its potential risks to spacecraft. Additional spectral and photometric analyses of both the Voyager and Earth-based data could better constrain the parameters needed to translate the observed ring brightness into estimates of the particle number densities. At the same time, realistic models of the interactions between the ring particles and Uranus’ upper atmosphere would clarify how the particle number density could vary with altitude, and perhaps even constrain trends in the local particle size distribution. Both of these tasks will be relevant to any effort to properly quantify risks posed to spacecraft passing between Uranus and the $\zeta$ ring. Finally, the $\zeta$ ring is visible in images of Uranus recently obtained by JWST⁴⁴4https://www.nasa.gov/feature/goddard/2023/ nasa-s-webb-scores-another-ringed-world -with-new-image-of-uranus, and these new data should reveal how this ring has changed since 2007.

This work was partially supported by the Jet Propulsion Laboratory (JPL) under consulting services agreement contract number 1676167. The authors thank Julie Castillo-Rogez, Erin Leonard and Benjamin Weiss for help with initiating this work. A portion of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004).This work also made use of the PDS Ring-Moon Systems Node’s OPUS search service. We also thank the reviewers for their helpful comments about an earlier version of this manuscript.