# 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

M. M. Hedman  Department of Physics, University of Idaho, Moscow, ID, USA
83843  I. Regan  Citizen Scientist, UK  T. Becker  Southwest Research
Institute, 6220 Culera Rd. San Antonio, TX, USA 78238  S. M. Brooks  Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA, 91109, USA  I. de Pater  University of California,
Berkeley, CA, USA, 94720  M. Showalter  SETI Institute, Mountain View CA, USA,
94043

###### Abstract

The $\zeta$ ring is the innermost component of the Uranian ring system. It is
of scientific interest because its morphology changed significantly between
the Voyager 2 encounter in 1986 and subsequent Earth-based observations around
2007. It is also of practical interest because some Uranus mission concepts
have the spacecraft pass through the inner flank of this ring. Recent re-
examinations of the Voyager 2 images have revealed additional information
about this ring that provide a more complete picture of the ring’s radial
brightness profile and phase function. These data reveal that this ring’s
brightness varies with phase angle in a manner similar to other tenuous rings,
consistent with it being composed primarily of sub-millimeter-sized particles.
The total cross section of particles within this ring can also be estimated
from these data, but translating that number into the actual risk to a
spacecraft flying through this region depends on a number of model-dependent
parameters. Fortunately, comparisons with Saturn’s G and D rings allows the
$\zeta$-ring’s particle number density to be compared with regions previously
encountered by the Voyager and Cassini spacecraft. Finally, these data
indicate that the observed changes in the $\zeta$-ring’s structure between
1986 and 2007 are primarily due to a substantial increase in the amount of
dust at distances between 38,000 km and 40,000 km from Uranus’ center.

##  1 Introduction

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.

Figure 1: Image of Uranus and its rings generated by Ian Regan from a series
of low-resolution Voyager 2 images obtained after the spacecraft passed by the
planet, along with a line drawing showing the locations of various rings based
on renderings provided by the Uranus Viewer on the Ring-Moon Systems Node of
the Planetary Data System. The innermost ring in the image is the $\zeta$
ring, and the outermost ring is the $\mu$ ring. The middle feature corresponds
to a sheet of dust extending between the $\delta$ and $\lambda$ rings, along
with the $\epsilon$ ring. The $\nu$ ring is not obvious in this particular
image. Streaks correspond to background stars. Note that this image has been
heavily processed to improve the visibility of the faint rings.

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.

##  2 Voyager 2 $\zeta$ ring observations

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.

Table 1: SPICE kernels used in this study vg2_v02.tf
---
vg2_issna_c02.ti
vg2_isswa_c02.ti
naif0010.tls
vg200022.tsc
pck0010.tpc
ura091-rocks-merge.bsp
vgr2_ura083.bsp
ura111.bsp
vgr2_super.bc
vg2_ura_version1_type1_iss_sedr.bc

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.

###  2.1 Moderate phase image

Figure 2: Voyager image C2684650 of the $\zeta$ ring at moderate (90∘) phase
angles near ring-plane crossing. The left panel shows the image, with the box
indicating the region used the derive the ring brightness profile. The
locations of the $\zeta$ and $\epsilon$ rings within this region are indicated
by the letters above the box. The right panel shows the resulting profile of
the region around the $\zeta$ ring derived from this portion of the image. The
$\zeta$ ring produces the prominent spike centered around 37,500 km, while the
peak around 42,000 km is likely due to the combined signal from the 4,5 and 6
rings. The grey points are sampled every 50 km, while the solid line is binned
over 500 km. Table 2: Brightness profile derived from the moderate-phase image
C2684650. Note the error bars correspond to statistical uncertainties based on
the scatter in brightness values within each radius bin.

Radius (km) $\mu I/F$ (10-6) 33500 0.48$\pm$0.05 34000 0.73$\pm$0.03 34500
0.72$\pm$0.05 35000 1.14$\pm$0.06 35500 1.34$\pm$0.07 36000 1.48$\pm$0.09
36500 1.97$\pm$0.09 37000 2.47$\pm$0.09 37500 2.34$\pm$0.08 38000
2.12$\pm$0.09 38500 1.91$\pm$0.10 39000 1.37$\pm$0.10 39500 1.11$\pm$0.11
40000 0.75$\pm$0.10 40500 0.66$\pm$0.12 41000 0.89$\pm$0.11 41500
0.49$\pm$0.11 42000 1.04$\pm$0.11

Prior to 2021, only one Voyager 2 image was known to contain a signal from the
$\zeta$ ring (Smith et al., 1986; Esposito et al., 1991; de Pater et al.,
2007). This wide-angle camera image is designated C2684650 in the PDS
database, and was obtained through the clear filter at a wavelength of 0.46
$\mu$m (Danielson et al., 1981), an emission angle of $88^{\circ}$ and a phase
angle of 90∘. Figure 2 shows this image, and the $\zeta$ ring can be seen as a
diffuse arc on the right side of the field of view (other bright bands
correspond to the main Uranian rings). In addition to the rings, this image
also contains a number of instrumental stray light patterns that form broad
arcs sweeping horizontally across the field of view.

These stray light signals can introduce spurious trends in the rings’ radial
brightness profile, so to construct the profile shown in the right panel of
Figure 2 we only used data from a small region near the bottom of the image
that is below the obvious stray-light streaks (marked by the rectangle in the
left panel of Figure 2). This profile was constructed by computing the radius
and emission angle at each point in the image and averaging the normal I/F
values within bins that were either 50-km or 500-km wide. Table 2 provides
these data in tabular form, along with statistical uncertainties in the
brightness based on the scatter in the brightness values within each radial
bin.

The profile clearly shows a peak in normal I/F corresponding to the $\zeta$
ring. In order to quantify the shape and brightness of this feature, we fit
the data between 34,000 km and 41,000 km to a Gaussian function plus linear
background. This fit yields a peak centered at 37,400 km with a Full Width at
Half-Maximum of 2920 km. This is consistent with the shape of the profile
derived from these same data presented in Figure 4c of de Pater et al. (2007).
The peak amplitude has a normal $I/F$ of $(1.7\pm 0.1)\times 10^{-6}$ above
the level seen around 35,000 km and 40,000 km of (0.7-0.8) $\times 10^{-6}$.
This peak brightness is compatible with the prior analysis of this image by
Smith et al. (1986), who say the observed $I/F$ of this ring is “in the range
of $10^{-4}$” (since for this particular image $\mu\sim 0.03$, this would
imply a normal $I/F$ of order $3\times 10^{-6}$). However, this brightness
estimate is roughly an order of magnitude lower than the profile generated
from the same image shown in Figure 4c of de Pater et al. (2007), which has a
peak normal $I/F$ of around $1.4\times 10^{-5}$ The reason for this
discrepancy is unclear, but could in part be because the de Pater et al.
(2007) profile was computed using data from the ansa, which is in a region
containing much larger instrumental backgrounds. Indeed, profiles constructed
along the ansa line show that the normal $I/F$ can increase by around 10-5
between 34,000 km and 37,000 km.

Table 3: Table of High-Phase Voyager Images of the $\zeta$ ring Image Set 1 | Images | Phase Angle | Emission Angle
---|---|---|---
Summary | C2690842-C2691447 | 146.96∘-147.18∘ | 145.45-145.66∘
Scanned Images | C2690842, C2690852, C2690902, C2690912, C2690922, C2690932
| C2690942, C2690952, C2691022, C2691032, C2691042, C2691052
| C2691102, C2691227, C2691237, C2691247, C2691257, C2691307
| C2691317, C2691327, C2691337, C2691407, C2691417, C2691427
| C2691437, C2691447
Background Images | C2691002, C2691012, C2691347, C2691357
Image Set 2 | Images | Phase Angle | Emission Angle
Summary | C2693634-C2694439 | 146.10∘-147.28∘ | 144.62-144.80∘
Scanned Images | C2693634, C2693644, C2693654, C2693704, C2693714, C2693724
| C2693734, C2693842, C2693852, C2693902, C2693912, C2693922
| C2693932, C2693942, C2693952, C2694022, C2694032, C2694042
| C2694052, C2694102, C2694229, C2694239, C2694249, C2694259
| C2694309, C2694319, C2694329, C2694359, C2694409, C2694419
| C2694429, C2694439
Background Images | C2694002, C2694012, C2694339, C2691357
Image Set 3 | Images | Phase Angle | Emission Angle
Summary | C2696644-C2697344 | 145.67∘-145.84∘ | 144.22-144.38∘
Scanned Images | C2696644, C2696654, C2696704, C2696714, C2696724, C2696734
| C2696744, C2696754, C2696804, C2696814, C2696824, C2696834
| C2696844, C2696854, C2696904, C2696934, C2696944, C2696954
| C2697004, C2697014, C2697024, C2697034, C2697044, C2697054
| C2697104, C2697114, C2697124, C2697134, C2697144, C2697154
| C2697204, C2697214, C2697244, C2697254, C2697304, C2697314
| C2697324, C2697334, C2697344
Background Images | C26946914, C2696924, C2697224, C2697234

Figure 3: Example images from the high-phase observations of Uranus containing
the $\zeta$ ring. The left panels show individual images from each of the
three image sets. The only visible object in each of these images is the
narrow lit crescent of Uranus (compare Figure 1). The middle panels show the
average image used to remove background stray-light patterns. In these images
Uranus itself falls near the top of the frame, but there are also faint
crescents in the center of these images. These are artifacts corresponding to
bright regions of earlier images. The right panel shows the left panels after
subtracting the background image. Note the greatly reduced stray-light
patterns over most of the image. The $\zeta$ ring is still too faint to be
seen in most of these individual images.

###  2.2 High-phase observations

The high-phase Voyager 2 observations used to create Figure 1 include three
image sets with roughly constant observing geometry (phase angles
$145.7^{\circ}$-147.2∘ and emission angles 144.2∘-145.7∘). Table 3 provides an
overview of these images, and Figure 3 shows representative examples of
individual images. All the relevant images from this sequence are clear-
filter, wide-angle camera images obtained after closest approach to Uranus.
While the apparent size of Uranus did decrease noticeably over the course of
the observations, the lighting and observation geometry (phase and emission
angles) were relatively constant. For practical reasons, we divide the images
into three groups based on the largest temporal gaps in the observation
sequence. This yields three image sets with 26, 32 and 39 images targeted at
Uranus. Outside of the crescent Uranus, there are significant brightness
variations in all these images that can be attributed to stray light from the
Sun.

These stray light patterns can be greatly reduced by using 4 images obtained
during each set that were targeted so that Uranus fell near the top edge of
the field of view. The average of these four images for each set are shown in
the central column of Figure 3. The background brightness variations are
clearly similar to those of the other images, so they provide a good template
for these backgrounds. Also note these images do contain a faint crescent at
the center of the image. This is because when the Voyager camera looks at a
bright target, subsequent images retain an after-image (also called a “ghost
image” Showalter, 1996) of that target for a finite time. Fortunately, this is
not a major issue for this particular analysis since the locations of the
crescent are close to each other in all the relevant images, and we avoid this
region in this particular study.

The right panels of Figure 3 shows the difference between each of the
individual images and the average background image template for the set. The
background brightness variations are clearly reduced over most of the image.
In particular, we can see the region to the right of the crescent
(corresponding to where the $\zeta$ ring can be most clearly seen in Figure 1)
is particularly clear. This is where illuminated rings should be visible past
the dark limb of Uranus, and so is the best place to look for faint ring
material. This region will therefore be the focus of the rest of this
analysis.

After removing the backgrounds, we re-projected the brightness data from each
image into maps of brightness versus ringplane radius and inertial longitude
(see Figure 4). The maps generated from all of the data in each image set were
then co-added together to further improve signal-to-noise, and the standard
deviation of the measurements among the different images were used to estimate
the uncertainties in the map. Close inspection of these combined maps revealed
longitudinal variations in the apparent position of the $\zeta$ ring peak with
amplitudes between 500 and 1000 km. These offsets were largest at longitudes
aligned with the horizontal direction in the images, and were overall
consistent with small (on the order of two pixels) errors in the pointing of
the individual images.

Since the $\zeta$-ring signal was too weak to be clearly seen in individual
images, improving the raw pointing of the original images was not practical.
Instead we applied an empirical correction to the geometry that would force
the $\zeta$ ring to appear at a more consistent radius. This was done by first
fitting the brightness profile in each longitude bin (each bin being 10∘ wide)
to a Gaussian to obtain an estimate of the $\zeta$-ring’s peak location
$R_{obs}$. Those radii were then fit to a sinusoidal function of inertial
longitude $\lambda$:

$R_{obs}=R_{c}\cos(\lambda-\lambda_{0})+R_{0}$ (1)

Where the amplitude of the position variations $R_{c}$, the mean ring position
$R_{0}$ and the phase $\lambda_{0}$ were all fit parameters. The best-fit
values for these parameters derived from the maps of each of the image sets
are provided in the Table 4. Note the $\lambda_{0}$ values being near 0∘ or
180∘ are consistent with horizontal offsets in the image pointing. Also these
numbers are different for the different image sets, which is inconsistent with
a real variation in the ring position. We therefore applied a simple radial
shift to each profile equivalent to the appropriate value of
$-R_{c}\cos(\lambda-\lambda_{0})$ to remove these longitudinal trends in the
peak position.

Table 4: Fit parameters for the location of the $\zeta$-ring’s peak brightness | $R_{c}$ | $\lambda_{0}$ | $R_{0}$
---|---|---|---
Image Set 1 | 475 km | 183∘ | 37730 km
Image Set 2 | 648 km | 8∘ | 37830 km
Image Set 3 | 943 km | -9∘ | 37670 km

Figure 4: Maps of the inner ring system derived from the high-phase Voyager 2
observations. Each map shows the brightness versus ringplane radius and
inertial longitude, and the green dotted line indicates the projected location
of the planet’s limb. The limb position varies with longitude because the
system was observed at a ring opening angle of around 55∘, so the spherical
planet is projected into an oval shape on the ringplane. Note the images have
all been stretched to best show the regions beyond the planet’s dark limb (the
ring actually extends all the way around the planet, see Figure 1). The
$\zeta$ ring is the bright band centered around 37,500 km in all the maps.
There is no obvious ring material interior to the $\zeta$ ring in these
images.

Figure 4 shows the resulting corrected maps of ring brightness versus radius
and inertial longitude derived from these observations, along with a map
created by co-adding all the data from these maps. These maps use a common
stretch where the regions near the planet’s lit limb are saturated, but the
region near the dark limb is clearly visible. The $\zeta$ ring is clearly
visible in all these maps around 37,500 km. Slight variations in the ring’s
appearance with longitude in these images are probably primarily due to noise
and variable trends in the residual instrumental backgrounds. There is no
obvious ring signal between the $\zeta$ ring and the planet’s limb. Also note
that there is not an obvious change in brightness across the predicted
location of the planet’s limb. Since the dark side of the planet should not
produce any light (unlike Saturn, reflected light from Uranus’ rings should be
a negligible source of illumination), the lack of a clear brightness change
near the limb implies that the brightness of any ring material in this region
must be very low.

Figure 5: Constraints on the absolute brightness of the $\zeta$-ring’s inner
flank based on the observed brightness differences between locations just
beyond Uranus’ dark limb and the dark side of Uranus itself. Each point in
this plot gives the difference in the average $\mu I/F$ values between regions
500-1500 km on either side of the predicted position of the planet’s limb for
a 10∘-wide longitude bin, with positive values indicating the ring region is
brighter than the dark side of Uranus. The horizontal location of each point
corresponds to the mean radial position of the selected region outside the
limb, and the error bars correspond to 1 standard deviation uncertainties.
Note the nonzero differences probably primarily reflect residual instrumental
backgrounds rather than real astronomical signals.

In order to quantify any potential ring signals interior to the $\zeta$ ring,
we first considered the brightness differences across the limb from all the
radial brightness profiles that contained no obvious bright limb signal (i.e.
inertial longitudes between -80∘ and +50∘). For each of these profiles, we
computed the average brightness within regions between 500 and 1500 km on
either side of the predicted position of the planet’s limb and took the
difference between the average value outside the limb and the average value
inside the limb. Any detectable ring signal within 500-1500 km of the limb
would result in the region outside the limb being brighter than the dark side
of Uranus, causing this difference to be significantly positive.

Figure 5 shows the observed values of these differences as a function of the
average radius of the region outside the limb. For the combined map, many of
these differences are between 1 and 2 standard deviations from zero, but these
offsets are more likely to be due to residual instrumental artifacts than real
astronomical signals. For example, the positive differences interior to 31,000
km may be related to a faint diagonal band extending from the tip of the
illuminated limb that can be seen in the bottom two panels of Figure 4 between
radii/longitudes of 28,000 km/-60∘ and 45,000 km/-40∘. This structure probably
represents residual stray light from the lit limb of the planet rather than a
real feature in the ring plane. Meanwhile, all of the points beyond 31,000 km
are negative, which implies that the dark side of Uranus is brighter than the
ringplane. While it may be that the unilluminated side of Uranus is not
completely dark due to some sort of atmospheric emission, artifacts associated
with the Voyager camera can produce negative excursions as well as positive
ones (note particularly the dark regions beyond -120∘), so these negative
differences could also be due to instrumental artifacts. In any case, these
data imply that these residual instrumental artifacts correspond to normal
$I/F$ values of only around 1$\times$10-6.

Figure 6: Radial brightness profile of the $\zeta$ ring derived from the high-
phase Voyager 2 images. Grey diamonds show the profiles from individual
columns in the combined map that are beyond Uranus’ limb. The solid line is
the average of all data beyond the planet’s limb. Note the profile has been
arbitrarily offset to have an average of zero between 28,000 and 32,000 km.
The brightness fluctuations interior to 32,000 km are likely due to
instrumental artifacts. Table 5: Brightness profile derived from high-phase
observations.

Radius (km) $\mu I/F$ (10-6) Radius (km) $\mu I/F$ (10-6) 36100 3.43$\pm$0.24
27300 0.64$\pm$0.92 36300 3.84$\pm$0.24 27500 0.84$\pm$0.95 36500
4.37$\pm$0.24 27700 0.89$\pm$0.97 36700 5.13$\pm$0.24 27900 0.24$\pm$0.98
36900 5.80$\pm$0.24 28100 -0.51$\pm$0.71 37100 6.30$\pm$0.25 28300
-0.22$\pm$0.65 37300 6.75$\pm$0.25 28500 -0.22$\pm$0.62 37500 7.20$\pm$0.25
28700 -0.56$\pm$0.65 37700 7.33$\pm$0.24 28900 -0.46$\pm$0.52 37900
7.00$\pm$0.24 29100 0.12$\pm$0.51 38100 6.60$\pm$0.24 29300 0.29$\pm$0.49
38300 6.40$\pm$0.24 29500 0.47$\pm$0.48 38500 6.37$\pm$0.24 29700
0.26$\pm$0.42 38700 6.03$\pm$0.24 29900 0.22$\pm$0.38 38900 5.52$\pm$0.24
30100 0.53$\pm$0.38 39100 5.06$\pm$0.24 30300 0.46$\pm$0.34 39300
4.48$\pm$0.23 30500 0.53$\pm$0.30 39500 3.89$\pm$0.23 30700 0.25$\pm$0.30
39700 3.08$\pm$0.23 30900 0.35$\pm$0.28 39900 1.85$\pm$0.24 31100
0.06$\pm$0.26 40100 1.11$\pm$0.24 31300 -0.18$\pm$0.26 40300 0.90$\pm$0.22
31500 -0.25$\pm$0.25 40500 0.79$\pm$0.22 31700 -0.40$\pm$0.25 40700
0.53$\pm$0.23 31900 -0.73$\pm$0.26 40900 0.47$\pm$0.23 32100 -0.63$\pm$0.26
41100 0.21$\pm$0.23 32300 -0.34$\pm$0.27 41300 0.08$\pm$0.23 32500
-0.31$\pm$0.27 41500 0.02$\pm$0.23 32700 -0.25$\pm$0.26 41700 0.12$\pm$0.22
32900 -0.29$\pm$0.24 41900 0.17$\pm$0.23 33100 -0.15$\pm$0.24 42100
0.15$\pm$0.23 33300 0.14$\pm$0.24 42300 0.13$\pm$0.22 33500 0.19$\pm$0.24
42500 0.14$\pm$0.23 33700 0.28$\pm$0.25 42700 0.33$\pm$0.22 33900
0.48$\pm$0.25 42900 0.54$\pm$0.22 34100 0.83$\pm$0.25 43100 0.76$\pm$0.22
34300 1.04$\pm$0.25 43300 0.93$\pm$0.23 34500 1.12$\pm$0.25 43500
0.93$\pm$0.22 34700 1.41$\pm$0.25 43700 0.65$\pm$0.23 34900 1.82$\pm$0.25
43900 0.49$\pm$0.23 35100 2.17$\pm$0.25 44100 0.69$\pm$0.22 35300
2.48$\pm$0.24 44300 1.07$\pm$0.22 35500 2.77$\pm$0.24 44500 1.35$\pm$0.22
35700 2.98$\pm$0.25 44700 1.46$\pm$0.22 35900 3.12$\pm$0.25 44900
1.57$\pm$0.22

With these limitations in mind, we constructed a radial brightness profile of
the ring. For this profile we again considered the brightness profiles in the
combined map that contained no hint of an illuminated limb. We then removed
residual background signals by fitting a linear trend versus radius to the
data in the ranges 20,000-25,000 km and 55,000-60,000 km (that is, obscured by
the dark side of the planet and beyond the detectable rings). We then took the
average of the profiles to construct a single high signal-to-noise profile,
and computed the uncertainty based on the scatter in the brightness
measurements within each radius bin. Finally, we offset this profile by
$2\times 10^{-6}$ so that the mean signal between 28,000 and 32,000 km was
zero (again, the signal from the unlit side of Uranus appeared to be on
average higher than the region interior to the $\zeta$ ring). The resulting
profile is shown in Figure 6 and Table 5. Note the profile does not extend
interior to 27,000 km because this region was always obscured by the planet
(see Figure 4).

This profile is consistent with Figure 5 in showing fluctuations of order $\pm
1\times 10^{-6}$ between 28,000 and 32,000 km, which are again likely due to
instrumental artifacts. Also, this profile is consistent with the profile
shown in Figure 2 in that the peak brightness in the $\zeta$ ring is around
37,500 km. Note, however, the peak brightness here is $(7.3\pm 0.3)\times
10^{-6}$, which is roughly 4 times higher than the peak signal in the moderate
phase image. This is consistent with the observed behavior of other diffuse,
dusty rings (see Section 4).

###  2.3 Low-phase observations

Figure 7: Two of the low-phase Voyager 2 images that might contain a weak
signal from the $\zeta$ ring. In both images, Uranus itself is positioned at
one corner of the field of view, and the dense main rings can be seen
extending towards the center of the frame. The right image contains after-
images of the planet’s bright disk in all four corners from previous
exposures. The $\zeta$ ring would fall between the visible rings and the
planet, and is not clearly visible in either of these individual images.

Finally, we examined a set of 17 clear-filter wide-angle camera images
(C2679143, C2679149, C2679207, C2679213, C2679307, C2679313, C2679319,
C2679325, C2679331, C2679337, C2679343, C2679349, C2679355, C2679401,
C2679407, C2679413, C2679419) that were obtained as Voyager 2 approached
Uranus. These images all had phase angles between 14∘ and 18∘ and emission
angles between 18∘ and 21∘. Each of these images had the center of Uranus
located near one corner of the field of view and so contained the entire ring
system extending towards the center of the frame (see Figure 7). Since these
images were all obtained at low phase angles, the planet itself is very
bright, so these data cannot provide strong constraints on the amounts of
faint material located close to the planet, but they do contain a signal that
can help constrain the location and brightness of the $\zeta$ ring’s peak.

As with the high-phase images, the data from each of the low-phase images were
re-projected onto regular grids of brightness versus radius and inertial
longitude. These data were then processed to produce a series of average
brightness profiles for 10∘-wide bins in inertial longitude for each image.
Since each image covered a limited range of inertial longitudes, we then co-
added the data from all the relevant images to create a complete map of the
ring system. Prior to co-adding data from the different images together, we
fit the data between 31,000 and 41,000 km to a fourth-order polynomial to
remove the strong background variations due to the nearby planet.

Figure 8: Evidence for the $\zeta$ ring in the low-resolution, low-phase Voyager images. The points come from the background-subtracted, co-added radial brightness profiles with 500 km radial sampling and covering 10∘ in inertial longitude. The line shows the average of all these profiles. While the signal-to-noise is rather low, there is a weak peak centered around 37,500 km with an amplitude around $3\times 10^{-6}$, consistent with the moderate-phase profile shown in Figure 2. Table 6: Brightness profile derived from low-phase observations Radius (km) |  $\mu I/F$ (10-6)
---|---
30000 | 7.04$\pm$4.89
30500 | 1.06$\pm$2.21
31000 | 0.89$\pm$0.92
31500 | -0.07$\pm$0.50
32000 | -0.42$\pm$0.59
32500 | 1.12$\pm$0.53
33000 | 2.00$\pm$0.54
33500 | 1.11$\pm$0.50
34000 | 1.14$\pm$0.47
34500 | -0.19$\pm$0.66
35000 | -0.18$\pm$0.55
35500 | 0.36$\pm$0.59
36000 | -0.94$\pm$0.52
36500 | 0.80$\pm$0.42
37000 | 0.96$\pm$0.52
37500 | 2.76$\pm$0.51
38000 | 2.81$\pm$0.51
38500 | 0.66$\pm$0.52
39000 | -0.70$\pm$0.39
39500 | 0.27$\pm$0.41
40000 | -0.73$\pm$0.38
40500 | 1.30$\pm$0.28
41000 | 1.62$\pm$0.62
41500 | 5.43$\pm$1.50
42000 | 25.63$\pm$2.72

Even after co-adding all the data together in each 10∘-wide longitude bin,
each profile still did not have sufficient signal-to-noise to clearly reveal
the $\zeta$ ring. Indeed, the signal from the $\zeta$ ring was only detectable
when all these profiles were averaged together. Figure 8 and Table 6 show the
resulting average brightness profile, with error bars based on the scatter in
the brightness for the different longitudes. Given both the aggressive
background subtraction used in this analysis (which would remove any broad
brightness trends) and the low signal-to-noise of the feature, this can only
be considered a marginal detection of the $\zeta$ ring. Still, it is a useful
confirmation of the other observations because the peak brightness is still
located around 37,500 km, consistent with the moderate-phase and high-phase
observations discussed above. Also, the peak normal $I/F$ is $(2.8\pm
0.6)\times 10^{-6}$, which is close to the $(1.7\pm 0.1)\times 10^{-6}$ found
in the moderate phase image. Thus this observation provides evidence that the
brightness of the ring does not vary much for phase angles below $90^{\circ}$,
which is consistent with the observed properties of other dusty rings and the
predictions from Mie theory (see Sections 3 and 4).

##  3 Some relevant Light-scattering theory

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:

$N_{t}=\int_{s_{t}}^{\infty}n(s)ds$ (2)

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:

$\mu I/F=\int_{0}^{\infty}\pi
s^{2}n(s)Q_{scat}(s,\lambda)f(s,\lambda,\alpha)ds,$ (3)

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:

$\mu I/F=A_{n}(\alpha)\int_{s_{min}}^{\infty}\pi
s^{2}n(s)ds=A_{n}(\alpha)\tau_{g},$ (4)

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 $\tau_{g}$.

Figure 9: Plots of $Q_{scat}f$ as functions of particle size $s$ computed from
Mie theory assuming a wavelength of 0.46 $\mu$m for three phase angles
relevant to the Voyager 2 Uranus observations and for three different optical
constants. These curves were computed using the IDL code mie_single and the
phase functions from that code were divided by 4 to ensure proper
normalization. These parameters have also been smoothed to account for the
finite bandpass of the camera. Note that the optical constants in the top
panel correspond to those of pure water ice at optical wavelengths, while
those in the bottom panel are for amorphous carbon. The middle panel uses
intermediate optical constants between these two extremes.

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:

$N_{t}=\frac{\mu
I/F}{A_{n}(\alpha)}\frac{\int_{s_{t}}^{\infty}n(s)ds}{\int_{s_{min}}^{\infty}\pi
s^{2}n(s)ds}=\frac{\mu I/F}{A_{n}(\alpha)}\frac{C_{n}}{\pi s_{t}^{2}}$ (5)

where

$C_{n}=\frac{s_{t}^{2}\int_{s_{t}}^{\infty}n(s)ds}{\int_{s_{min}}^{\infty}s^{2}n(s)ds}$
(6)

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
$C_{n}$:

$C_{n}=\begin{dcases}\frac{3-q}{q-1}\left(\frac{s_{t}}{s_{max}}\right)^{3-q}&q<3\\\
\frac{1}{2\ln(s_{max}/s_{min})}&q=3\\\
\frac{q-3}{q-1}\left(\frac{s_{min}}{s_{t}}\right)^{q-3}&q>3\end{dcases}$ (7)

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:

$\footnotesize N_{t}=\mu I/F\left(\frac{6\times
10^{7}}{m^{2}}\right)\left(\frac{100\mu
m}{s_{t}}\right)^{2}\left(\frac{0.025}{A_{n}}\right)\left(\frac{C_{n}}{0.05}\right)$
(8)

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.

##  4 Comparisons with other dusty rings

Figure 10: Phase curves showing the peak brightness of the $\zeta$ ring and
other diffuse rings as a function of observed phase angle. The $\zeta$-ring
data are compared with data from Jupiter’s Main ring (Throop et al., 2004),
Saturn’s G ring (Hedman and Stark, 2015) and Uranus’ $\nu$ ring (Showalter and
Lissauer, 2006). The curve through the G-ring data corresponds to the best-fit
three-component Henyey-Greenstein function from (Hedman and Stark, 2015), and
the dotted curve shows the same theoretical phase function scaled by a factor
5.5 to best fit the three $\zeta$ ring data points.

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.

###  4.1 Dusty rings with light-scattering properties similar to the $\zeta$
ring

Among the well-characterized dusty rings, Saturn’s E ring and Uranus’ $\mu$
ring are clear outliers in terms of their spectral and photometric properties.
In particular, both these rings have a blue color at low phase angles, while
most other dusty rings are red (de Pater et al., 2004, 2006b). This implies
that the particle size distributions for these two rings has a large fraction
of particles less than a couple microns in radius, a result that has been
confirmed for the E ring by in-situ sampling of the ring particles (Kurth et
al., 2007; Wang et al., 2006; Kempf et al., 2008; Ye et al., 2014, 2016). Most
other dusty rings are not only red at low phase angles, but also exhibit
similar trends in their brightness with wavelength and phase angle that likely
reflect commonalities in their particle properties (Hedman et al., 2018). More
specifically, it is particularly useful to compare the $\zeta$ ring’s
photometric properties with those of Jupiter’s Main ring, Saturn’s G and D
rings, and Uranus’ own $\nu$ ring. Not only are these a representative sample
of tenuous dusty rings from several different planets, but the G and D rings
were also encountered by the Voyager and Cassini spacecraft, and so provide a
valuable baseline for any risk assessment.

Figure 10 shows the peak brightness of the $\zeta$ ring from the three Voyager
images as a function of the observed phase angle. These data are compared with
the peak brightnesses of Jupiter’s Main ring (Throop et al., 2004), Saturn’s G
ring (Hedman and Stark, 2015) and Uranus’ $\nu$ ring (Showalter and Lissauer,
2006), along with the best-fit three-component Henyey-Greenstein function for
the G-ring observations provided by Hedman and Stark (2015). Note that the
brightness trends for the latter three rings are rather similar. More
specifically, the two $\nu$ ring measurements fall fairly close to the trend
defined by Saturn’s G ring. Meanwhile, even though the Jovian Main ring and
the G ring data have different slopes at phase angles below 120∘, the two
rings follow similar trends at high phase angles and so have comparable
brightness ratios between high and low phase angles. Similar trends are also
observed in the phase curves of several narrow dusty rings like Saturn’s F
ring and the innermost narrow feature in Saturn’s D ring (French et al., 2012;
Hedman and Stark, 2015; Hedman et al., 2018).

The peak brightness of the $\zeta$ ring follows a similar trend to these other
rings, as can be seen in Figure 10, which also includes a version of the
G-ring phase function scaled by a factor of 5.5. This trend does not perfectly
match the three data points, but the deviations between the data and the curve
are comparable to the scatter in the data from the Jovian Main ring. This
implies that the light-scattering properties of the $\zeta$-ring particles are
not that different from those of these other dusty rings (and that the peak
particle number density for the $\zeta$ ring could be roughly 5 times higher
than those of Saturn’s G ring). Furthermore, the brightness trends for all the
rings shown in Figure 10 are compatible with those predicted for large
particles not composed of pure water ice shown in Figure 9. This is actually
somewhat surprising for Saturn’s dusty rings because Saturn’s Main rings are
well known to be composed of very pure water ice (Esposito et al., 1991; Cuzzi
et al., 2009; Zhang et al., 2017). However, there is evidence that the dusty G
and D rings both might contain substantial amounts of non-icy material.
Saturn’s G ring contains the small moon Aegaeon, which is unusually dark for
the Saturn system (Hedman et al., 2020) and so its albedo is more like Uranus’
rings and moons. Since the G ring is probably derived from material knocked
off of Aegaeon by impacts (Hedman et al., 2007, 2010), this means that the
material in this ring may be similarly dark. At the same time, in-situ
sampling of material between Saturn and the D ring implies that ring material
flowing into the planet could be rather carbon-rich (Waite et al., 2018;
Miller et al., 2020).

Figure 11: Profiles of Saturn’s G and D rings derived from Cassini images at
phase angles similar to the high-phase Voyager 2 images of the $\zeta$ ring.
The top panel shows a profile derived from Figure 12 of (Hedman et al., 2012),
which was also obtained at a phase angle of around 146∘. The vertical line
marks the location where the Voyager 2 spacecraft passed through Saturn’s
ringplane. The bottom two panels show a background-subtracted profile of
Saturn’s inner D ring derived from Cassini images derived from an image
sequence containing 62 images obtained at phase angles between 145∘ and 149∘.
The two panels have different vertical scales to better show the signal at the
location of the outermost ring-plane crossing during Cassini’s Grand Finale
(marked with a dashed line). The profiles for the $\zeta$ and D ring are also
compared with an exponential model fit. Note that data for both these rings
are vertically shifted so that the exponential trend asymptotes to zero. The
$\zeta$ ring profile is offset by $1\times 10^{-6}$ relative to the version
shown in Figure 6, while the D ring profile is offset by $0.1\times 10^{-6}$.

###  4.2 Controlled comparisons with Saturn’s G and D rings

The above considerations suggest that we can use the in-situ observations of
the G and D rings to roughly approximate conditions in different parts of the
$\zeta$ ring. Of course, more detailed modeling would be needed to quantify
how much parameters like $A_{n}$ and $C_{n}$ are likely to vary among these
different systems, which must be left to future work.

With the above caveats in mind, Figure 11 shows brightness profiles of
Saturn’s G and D rings derived from Cassini observations at similar phase
angles as the high-phase Voyager 2 observations of the $\zeta$ ring. The G
ring profile is derived from a Cassini observation of the ring region near
Saturn’s shadow at a phase angle near 146∘ (Hedman et al., 2012). The nearby
regions in Saturn’s shadow allow instrumental backgrounds to be removed,
leaving a high-quality brightness profile showing a clear asymmetric peak near
167,500 km. Of particular interest here is the brightness around 172,124 km,
which is where Voyager 2 flew through Saturn’s ringplane during its encounter
with Saturn (Showalter and Cuzzi, 1993). At this location the Normal I/F of
the ring observed by Cassini is around $0.7\times 10^{-6}$ at 146∘ phase
angle. The brightness of the G-ring region encountered by Voyager 2 is
therefore an order of magnitude lower than the peak of the $\zeta$ ring and is
roughly 2-3 times lower than the background brightness variations seen
interior to 34,000 km at Uranus. Thus, if the $A_{n}$ and $C_{n}$ values are
similar for both these rings, the number density of particles around Uranus
interior to 34,000 km during the Voyager 2 encounter is probably less than 2-3
times the number density of particles encountered by Voyager 2 when it passed
through Saturn’s ringplane.

We can also compare the brightness of the $\zeta$ ring at 146∘ phase to the
brightness of the regions in the inner D ring encountered by the Cassini
spacecraft during its Grand Finale. Good-quality profiles of the inner D ring
at phase angles between 145.2∘ and 149.4∘ were derived from 62 Cassini images
(N1765071135-N1765102855). These data were co-added together to obtain a
single high signal-to-noise profile of the ring shown in the lower panels of
Figure 11. Since the region interior to 63,500 km was found to be largely
empty of debris (Ye et al., 2018), we modeled any remaining instrumental
background by fitting the data between 63,000 and 63,500 km to a linear trend
and subtracting this from the profile. The inner D-ring clearly increases in
brightness with distance from the planet, and Cassini passed through the inner
part of this region at a range of radii. The largest radius where the
spacecraft passed through the ringplane was around 64,300 km. At this
location, the D-ring’s Normal $I/F$ is around 0.1$\times 10^{-6}$ above the
average brightness between 63,000 and 63,500 km (or 0.2$\times 10^{-6}$ above
the asymptotic value of the exponential model discussed in the following
subsection). This is well below the Normal $I/F$ of the part of the G ring
encountered by Voyager 2. The portions of the D ring encountered by Cassini
are therefore roughly one order of magnitude less bright than the background
variations in the high-phase Voyager 2 observations of the $\zeta$ ring.

###  4.3 Radial trends close to the giant planets and the potential role of
atmospheric drag

The brightnesses of both the $\zeta$ and D rings smoothly increase with
distance from the planet, and the shapes of both brightness profiles can be
approximated as exponentials. Figure 11 shows fits to the brightness profiles
of the D and $\zeta$ rings. Each of these fits assumed the data followed an
exponential plus offset function ($y=A+Be^{-r/H}$) and uniform errors based on
the scatter of data points within a region interior to the ring. For the D
ring the data was fit to the region between 63,000 and 67,000 km and the error
was based on the scatter of data points between 63,000 and 63,500 km. For the
$\zeta$ ring the data was fit to the region between 31,000 and 37,000 km and
the error was based on the scatter in the data between 28,000 and 32,000 km.
For the D ring, we find the best-fit scale-height $H$ is 882$\pm$7 km, while
for the $\zeta$ ring it is between 1700 km and 2400 km.

Of course, it would not be appropriate to simply assume that the brightness of
the $\zeta$ ring strictly follows this exponential trend all the way down to
Uranus’ cloudtops. However, it is worth noting that these trends are roughly
consistent with simple models where atmospheric drag is the dominant
perturbation force acting on the ring particles (Broadfoot et al., 1986;
Goldreich and Porco, 1987; Burns et al., 2001; Perry et al., 2018; Mitchell et
al., 2018). Hence coupled models of the ring and atmosphere could potentially
help clarify some aspects of the potential dust populations close to the
planet. Such models are beyond the scope of this study, but we can provide a
brief sketch of the relevant physics that would underlie those models.

Both these rings lie well interior to the planet’s co-rotation radius, so the
orbiting ring particles are moving faster than any ions or molecules tracking
the planet’s rotation. Momentum exchange between the ring particles and the
planet’s upper ionosphere will therefore tend to slow the ring particles down
and cause them to fall into the planet. For particles on nearly circular
orbits, atmospheric drag causes a particle of size $s$ to slowly drift inwards
at the following rate (adapted from Broadfoot et al., 1986):

$\frac{dr}{dt}=-\frac{3}{2}C_{D}\frac{\rho_{a}}{\rho_{p}}\frac{w^{2}}{s\Omega}$
(9)

where $C_{D}$ is the particle’s drag coefficient $\rho_{a}$ is the local
atmospheric mass density, $\rho_{p}$ is the particle’s mass density, $\Omega$
is the particle’s mean motion around the planet, and $w$ is the speed of the
particle relative to the atmosphere. Note that this drift rate is directly
proportional to the density of the atmosphere, and so the drift rate increases
closer to the planet.

If there is a source of dust well above the atmosphere and the particles’
sizes remain approximately constant (i.e. particle erosion can be neglected),
then the radial flux of material across any given radial distance $r$ is just
the local number density of particles $\mathcal{N}(r)$ times the above drift
rate:

$\mathcal{F}(r)=-\mathcal{N}(r)\frac{3}{2}C_{D}\frac{\rho_{a}}{\rho_{p}}\frac{w^{2}}{s\Omega}$
(10)

If we consider two nearby radii $r_{1}$ and $r_{2}=r_{1}+\Delta r$ the net
flux of material into this region will be:

$\mathcal{F}_{net}=\mathcal{F}(r_{1})-\mathcal{F}(r_{2})$ (11)
$\footnotesize\mathcal{F}_{net}=\frac{3C_{D}}{2\rho_{p}s}\left(\frac{\mathcal{N}(r_{1})\rho_{a}(r_{1})w(r_{1})^{2}}{\Omega(r_{1})}-\frac{\mathcal{N}(r_{2})\rho_{a}(r_{2})w(r_{2})^{2}}{\Omega(r_{2})}\right)$
(12)

For this preliminary calculation, we may assume the density of the atmosphere
and the number density of the ring particles are much stronger functions of
distance from the planet than $w$ or $\Omega$, in which case we have the
simpler expression:

$\mathcal{F}_{net}=\frac{3C_{D}w^{2}}{2\rho_{p}\Omega
s}\left(\mathcal{N}(r_{1})\rho_{a}(r_{1})-\mathcal{N}(r_{2})\rho_{a}(r_{2})\right)$
(13)

This expression has a steady-state solution where the amount of material
between $r_{1}$ and $r_{2}$ remains roughly constant over time. In that
specific case, we have the relationship

$\mathcal{N}(r_{1})\rho_{a}(r_{1})=\mathcal{N}(r_{2})\rho_{a}(r_{2})$ (14)

or

$\mathcal{N}(r_{1})/\mathcal{N}(r_{2})=\rho_{a}(r_{2})/\rho_{a}(r_{1})$ (15)

Variations in the ring particle number density with altitude in this steady-
state situation should therefore be the inverse of the trends in the
atmospheric density. For example, if the density of the atmosphere follows an
exponential trend with scale height $H$:

$\rho_{a}(r)=\rho_{a0}e^{-r/H},$ (16)

then the number density and brightness of the ring would also be an
exponential function with the same scale height:

$\mathcal{N}(r)=\mathcal{N}_{0}e^{+r/H}$ (17)

For the $\zeta$ ring, this simple model might be a reasonable approximation
for the actual system, since the scale height of Uranus’ ionosphere measured
by the Voyager 2 radio occultations is around 2400 km at radii above 30,000 km
(Lindal et al., 1987; Strobel et al., 1991), which matches the observed scale
height of the $\zeta$ ring. However, it is worth noting that the D ring’s
scale height of around 900 km is not so clearly compatible with this sort of
simple model because the scale height of the ionosphere appears to vary over
the range of altitudes containing the inner D ring. Below 64,000 km from
Saturn’s center the scale height for the D ring is comparable to Saturn’s
ionospheric scale height reported in various publications. Analyses of radio
occultations found ionospheric scale heights between 500 and 1200 km below
64,000 km (Nagy et al., 2006), while analyses of RPWS data from the Grand
Finale found that below 64,500 km the scale height was 530-840 km (Hadid et
al., 2018) or 545-575 km (Persoon et al., 2019). However, the latter work also
found the ionospheric scale heights were much larger (2360-4780 km) at higher
altitudes. Thus more work is needed to ascertain what might be responsible for
these discrepancies. In addition, there is evidence that the temperature of
Uranus’ ionosphere has been steadily decreasing since 1993 (Melin et al.,
2019). These changes in the ionosphere’s temperature could potentially affect
its scale height and thus the distribution of dust in the innermost ring
system. Thus much more sophisticated models of Uranus’ atmosphere and how it
changes over time are needed to properly constrain dust populations close to
the planet.

##  5 Comparing Voyager and Keck observations of the $\zeta$ ring

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.

Figure 12: Plot showing the available brightness profiles of the $\zeta$ ring,
with lighter lines showing the uncertainties in each profile. Note that all
the Voyager profiles are scaled by different factors to account for the
varying phase angles assuming the brightness of the ring follows the three-
component Henyey-Greenstein function shown in Figure 10, while the Keck
observation is scaled by a factor consistent with the red color of many dusty
rings.

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).

##  6 Summary and Conclusions

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.

##  7 Acknowledgements

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.

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