Icarus 190 (2007) 127–144
www.elsevier.com/locate/icarus
Self-gravity wakes and radial structure of Saturn’s B ring
J.E. Colwell a,∗ , L.W. Esposito b , M. Sremčević b , G.R. Stewart b , W.E. McClintock b
a Department of Physics, University of Central Florida, Orlando, FL 32816-2385, USA
b LASP, University of Colorado, 392 UCB, Boulder, CO 80309-0392, USA
Received 19 July 2006; revised 2 March 2007
Available online 13 April 2007
Abstract
We analyze stellar occultations by Saturn’s rings observed with the Cassini Ultraviolet Imaging Spectrograph and find large variations in the
apparent normal optical depth of the B ring with viewing angle. The line-of-sight optical depth is roughly independent of the viewing angle
out of the ring plane so that optical depth is independent of the path length of the line-of-sight. This suggests the ring is composed of virtually
opaque clumps separated by nearly transparent gaps, with the relative abundance of clumps and gaps controlling the observed optical depth. The
observations can be explained with a model of self-gravity wakes like those observed in the A ring. These trailing spiral density enhancements
are due to the competing processes of self-gravitational accretion of ring particles and Kepler shear. The B ring wakes are flatter and more closely
packed than their neighbors in the A ring, with height-to-width ratios <0.1 for most of the ring. The self-gravity wakes are seen in all regions of
the B ring that are not opaque. The observed variation in total B ring optical depth is explained by the amount of relatively empty space between
the self-gravity wakes. Wakes are more tightly packed in regions where the apparent normal optical depth is high, and the wakes are more widely
spaced in lower optical depth regions. The normal optical depth of the gaps between the wakes is typically less than 0.5 and shows no correlation
with position or overall optical depth in the ring. The wake height-to-width ratio varies with the overall optical depth, with flatter, more tightly
packed wakes as the overall optical depth increases. The highly flattened profile of the wakes suggests that the self-gravity wakes in Saturn’s
B ring correspond to a monolayer of the largest particles in the ring. The wakes are canted to the orbital direction in the trailing sense, with a
trend of decreasing cant angle with increasing orbital radius in the B ring. We present self-gravity wake properties across the B ring that can be
used in radiative transfer modeling of the ring. A high radial resolution (∼10 m) scan of one part of the B ring during a grazing occultation shows
a dominant wavelength of 160 m due to structures that have zero cant angle. These structures are seen at the same radial wavelength on both
ingress and egress, but the individual peaks and troughs in optical depth do not match between ingress and egress. The structures are therefore not
continuous ringlets and may be a manifestation of viscous overstability.
© 2007 Elsevier Inc. All rights reserved.
Keywords: Saturn, rings; Planetary rings
1. Introduction
Saturn’s B ring is the broadest and most massive planetary
ring in the Solar System, extending from 92,000 to 117,500 km
from Saturn’s center and with an estimated mass of 3 × 1019 kg,
or ∼80% of the entire ring system (Cuzzi et al., 1984). Measurements of the optical depth from various occultation techniques can still place only a lower limit on the optical depth
in the core of the B ring, which lies between 99,000 and
* Corresponding author
E-mail address: jcolwell@physics.ucf.edu (J.E. Colwell).
0019-1035/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.icarus.2007.03.018
108,000 km. New measurements by the Cassini spacecraft show
the structure of the B ring in unprecedented detail.
A strong azimuthal brightness asymmetry in the A ring, exterior to the B ring, is due to the presence of aligned self-gravity
wakes in that ring (Camichel, 1958; Colombo et al., 1976;
Lumme and Irvine, 1976; Reitsema et al., 1976; Lumme et
al., 1977; Gehrels and Esposito, 1981; Thompson et al., 1981;
Dones and Porco, 1989; Dones et al., 1993; Dunn et al., 2004;
Nicholson et al., 2005). The geometric properties of the selfgravity wakes in the A ring have been determined from occultation measurements of the dependence of the ring transparency on viewing angle (Colwell et al., 2006; Hedman et al.,
2007). Here we examine the B ring where similar variations
128
J.E. Colwell et al. / Icarus 190 (2007) 127–144
in transparency are observed in Cassini occultation measurements. Asymmetries in the B ring have been seen in imaging
from Hubble Space Telescope (French et al., 2007) and radar
observations (Nicholson et al., 2005).
Unlike the A ring, which is characterized by broad regions of
slowly varying background optical depth, punctuated by density
and bending waves, the B ring is filled with poorly understood
structure. The B ring is more opaque than the A ring, making
the signals of occulted stars that much weaker and therefore
noisier. The outer few 100 km of the B ring is not circular due
to forcing from the moon Mimas (e.g., Cuzzi et al., 1984) so we
restrict our analysis to ring plane radii R < 117,000 m.
We find a strong dependence of apparent normal optical
depth on the occultation viewing angle that can be explained
by self-gravity wakes like those in the A ring, but with a much
larger width/height aspect ratio. We apply the model to the entire B ring where sufficient data exist to model the azimuthal
asymmetry. We use an analytic expression with the assumption
of opaque self-gravity wakes as well as the simple ray-trace
method of Colwell et al. (2006) with no assumptions about
the wake optical depth. We analyzed the data at both 10 and
50 km resolution, the latter incorporating solar occultations
which have lower inherent resolution than the stellar occultations. We found no significant differences in calculated wake
properties between the different techniques and resolutions.
In the next section we describe the observations and the
techniques for determining ring optical depths from the measurements. In Section 3 we present the resulting optical depth
profiles from the set of occultations used in this study. Our
model of self-gravity wakes is explained in Section 4 together
with our results on wake properties in the B ring. The α Leo
occultation minimum radius was in the outer B ring and provided a detailed look at the azimuthal and radial structure of
one location in the ring, revealing axisymmetric, but discontinuous wave-like structure with λ ∼ 160 m likely due to viscous
overstability. This is discussed in Section 4.3. In Section 5 we
summarize our results and discuss the implications for viscous
evolution of the B ring.
2. Observations
The Cassini Ultraviolet Imaging Spectrograph (UVIS) has
three channels for observing occultations. The High Speed Photometer (HSP) is used for stellar occultations and measures the
intensity of starlight between 110 and 190 nm with a 1–8 ms
sampling interval (Esposito et al., 2004). The Far Ultraviolet
Spectrograph (FUV) channel has the same spectral bandpass as
the HSP with 1024 spectral resolution elements and 64 spatial
resolution elements along the slit. At each integration period 64
spectra are obtained. A single FUV integration requires much
more data than the HSP which is typically operated in a 9-bit
compression mode, so the integration periods are much longer
with the FUV channel than the HSP channel. The imaging capabilities of the FUV allow simultaneous measurements of the
ring brightness in the same part of the spectrum that the HSP
measures. For some occultations this information is used in
Table 1
Occultation geometry
Occultation star (rev)
Date
(year-day)
|B|
φ
R (km)
δ Sco (Voyager)
ξ 2 Cet (A)
126 Tau (8)
δ Aqr (8)
α Leo (9, ingress)
α Leo (9, egress)
Solar (9)
126 Tau (10)
σ Sgr (11)
Solar (11)
α Sco B (13, ingress)
α Sco B (13, egress)
λ Cet (28)
α Sco B (29)
λ Sco (29)
α Vir (30)
1981-237
2004-281
2005-139
2005-141
2005-159
2005-159
2005-159
2005-175
2005-195/6
2005-196
2005-232
2005-232
2006-256
2006-269
2006-269
2006-285
28.7
14.9
21.1
12.2
9.5
9.5
21.4
21.1
29.1
21.1
32.2
32.2
15.3
32.2
41.7
17.2
104.3–107.3
81.2–79.2
73.1–85.0
59.6–54.3
112–126.8
112–97.1
78.2–78.1
155.8–150.2
133.3–119.9
74.9–75.2
25–23.5
25–26.4
78.2–78.1
57.3–75.1
6.9–32.8
122.7–132.9
72,000–142,500
109,000–135,550
70,400–141,300
60,700–169,900
114,100–131,500
114,100–176,200
66,100–168,600
103,200–144,200
86,000–146,900
54,100–116,000
101,200–155,800
101,200–146,600
74,300–144,000
79,900–149,400
88,500–143,800
64,000–151,600
Notes. Range in φ is for the B ring (92,000–118,000 km) or the fraction of
the B ring spanned by the occultation. Range in R is for the entire observation
rounded to 100 km. ξ 2 Ceti data are for the second of two observations of that
star. δ Sco values of R only refer to the data subset retrieved from the Planetary
Data System.
constructing a model of the background component of the HSP
signal.
The Extreme Ultraviolet Spectrograph (EUV) is coaligned
with the FUV and HSP, but it also includes a pick-off mirror
that allows observations of the Sun without aiming the Cassini
remote sensing pallet at the Sun. Because of data volume restrictions and the lower signal from stars in the EUV compared
to the FUV, the EUV channel is not used during stellar occultations. The instrument and observing techniques for stellar
occultations are described in Esposito et al. (1998, 2004). The
HSP observed seven occultations of stars by Saturn’s rings during the first phase of Cassini’s four-year tour of the Saturn
system (Esposito et al., 1998), and two solar occultations of
the rings were observed by the EUV. The solar occultations are
at a much lower resolution, limited by the size of the Sun on
the rings as seen from Cassini (∼100 km), than the stellar occultations which are limited by diffraction to ∼ 20 m. Stellar
occultations in the latter half of the Cassini nominal mission
are at higher elevation angles and are therefore less diagnostic
of the properties of aligned clumps in the rings, such as selfgravity wakes. We also use the Voyager PPS δ Sco occultation
(Lane et al., 1982). The geometric properties of the occultations
used in this study are presented in Table 1. These occultations
cut the rings at a variety of angles with respect to the ring plane
(B) and with respect to the local radial direction (φ, where
φ = ±π/2 when the line of sight is tangent to the rings at the
occultation point).
The transparency of the ring along the line of sight to the star
is
I −b
T=
(1)
,
I0
and the normal optical depth, τn , is
τn = μ ln T −1 ,
(2)
Self-gravity wakes in Saturn’s B ring
where I is the measured stellar intensity, I0 is the unocculted intensity of the star, b is the background signal, and μ = sin(B) is
the projection factor to convert from line of sight optical depth
to normal (vertical) optical depth. This standard expression for
normal optical depth is based on a model of the rings as a planar,
homogeneous medium with non-zero and finite transparency,
such as a cloud. For such a medium the factor μ in Eq. (2) is
necessary to correct for the length of the observed line of sight
which is always greater than the vertical thickness of the ring.
If the positions of the ring particles are uncorrelated and the
rings are azimuthally symmetric, and if the particles are small
compared to the area of the ring sampled in a single occultation integration period, then Eq. (2) gives the same value of the
ring optical depth for all values of B and φ. However, the clustering of ring particles into elongated and aligned clumps, or
self-gravity wakes, causes a variation of τn with viewing angle.
Calculations of optical depth through numerical N -body simulations (cf. Fig. 16 in Salo et al., 2004) show the same variation
of apparent normal optical depth, τn , with viewing angle that
we find with our simple model (Colwell et al., 2006) and that is
seen in the data described in this paper. This variation is diagnostic of the shape, spacing, and alignment of the self-gravity
wakes. We present values of τn using Eq. (2) for consistency
with previous discussions of ring optical depth and to illustrate
the magnitude of microstructure in the rings on the inferred
optical depth. The occultations presented here show large variations of τn as calculated from Eq. (2) in the B ring.
The background signal, b, is dominated by sunlight scattered
from the rings except in cases where the HSP field of view is
within the shadow of the planet on the rings or on the unlit face
of the rings. On the sunlit face of the rings (the south face during the observations described here), the signal from the rings
recorded by the HSP is <2000 counts/s, with the precise value
depending on the viewing geometry, the filling factor of ring
material in the HSP field of view, and the proximity of Saturn
to the field of view. On the unlit side of the rings the background
is usually dominated by Lyman-α emission from interplanetary
hydrogen which gives b ∼ 100 counts/s. The importance of the
background signal in determining optical depth depends on the
viewing geometry (amount of sunlit ring material in the field
of view and the angular proximity of the line of sight to Saturn), the star brightness, and the optical depth of the rings. In
general I0 and b must be determined by independent measurements of the signal I = I0 + b in ring gaps and I = b when
the star is behind fully opaque (to the occulted star) regions of
the rings. In some cases there are no opaque regions of the ring
covered by an occultation, and independent determinations of
the background must be made. In some cases this can be modeled with simultaneous measurements of the ring with the UVIS
FUV channel (Section 2.1, for example). All occultations analyzed here included views of the star through gaps in the rings,
and these observations have a background contribution from the
rings on the lit face of the rings, or primarily from interplanetary hydrogen on the unlit face.
The behavior of the UVIS HSP to very bright stars (I0 > 104
counts per second) complicates the determination of I0 . During
cruise calibration measurements of the star α Virginis (Spica)
129
Fig. 1. Total signal measured by the HSP of the star σ Sagittarii binned to
100-ms samples illustrating the ramp-up behavior of the HSP with bright stars.
After being partially obscured by the outer A ring from t = 0.25–0.32 h after the
start of the observation, the star re-emerges behind the Encke Gap at a slightly
lower count rate than when it was first occulted by the A ring. As the Encke
Gap drifts in front of the star the measured signal restarts its ramp-up behavior.
and other stars we noticed a complicated ramp-up behavior of
the instrument response to the star signal. When first exposed
to a bright star the instrument does not reach a constant count
rate. Instead the count rate reaches a value of ∼90% of the asymptotic value in the first sampling interval (usually 2 ms) and
then follows a non-linear ramp-up for a period of up to several minutes followed by a slow linear increase in the counting
rate. The long time scales for this response are unusual for a
photomultiplier tube. The precise shape and magnitude of the
ramp-up are different for each observation and are not a simple function of the star brightness or of the time that the star
has been shining on the detector. When the star has been occulted by the rings and then re-emerges into a gap, the signal
recovers most of the pre-occultation rate and then resumes the
ramp-up behavior (Fig. 1). This ramp-up response for the HSP
is not seen for fainter stars. In practice, and below, we model b
and I0 based on a combination of measurements of the occulted
and unocculted star and, when available, independent measurements of the ring brightness with the FUV channel.
Because the signal at any given time is a combination of
the star brightness as seen by the detector, I0 , the optical depth
of the ring at the location sampled at that time, and the background signal, there are several components to the uncertainty
in the measurement of the optical depth. The background is
not constant in general, and must be modeled from simultaneous measurements with the FUV and/or background component
of the overall signal is therefore a model result with an estimated error. The star brightness is never measured without
a background contribution, and the instrument behavior described above introduces an uncertainty in our determination
of I0 , so our determination of I0 is also a model result with an
estimated error. The estimated errors in the determination of b
and I0 are “probable errors” (Bevington, 1969) and are distinct
from the uncertainty due to counting statistics. These errors are
based on the differences between determinations of I0 (and b)
at different points in the occultation. Counting statistics alone
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J.E. Colwell et al. / Icarus 190 (2007) 127–144
√
give the following Zσ = Z I bounds on the measured optical
depth, τn :
τ± = μ ln I0 − μ ln I0 exp(−τn /μ) ∓ Z I0 exp(−τn /μ) + b ,
(3)
where Z is the number of standard deviations away from the
measurement and I0 and b are measured in counts. The probable errors in b and in I0 are usually smaller than the statistical
uncertainty, and they are discussed in the detailed summary of
each occultation below. Ultimately, when we apply our simple
geometric model of self-gravity wakes to the data, we introduce an additional error which is the deviation of the model
results due to the model imperfections in describing the actual
arrangement of ring particles. In these summaries we describe
our determination of b and I0 and the corresponding uncertainties in τn for each occultation included in this analysis. We
discuss the stellar occultations first, in chronological order, followed by the solar occultations.
2.1. ξ 2 Ceti occultation
Cassini observed the occultation of the star ξ 2 Ceti by the
rings on 2004-280 to 2004-281 (October 6–7) with the UVIS
HSP. The Far Ultraviolet Spectrograph channel (FUV) also observed the occultation which was the most distant stellar occultation to be observed by Cassini (Table 1). The large distance
of Cassini from Saturn and the foreshortening of the rings resulted in only 5–7 spatial elements of the FUV detector filled
with ring material, and the star signal in two of these detector
rows. This poor spatial coverage limits the utility of the FUV
observation in providing an independent measurement of the
background. The speed of the star across the rings projected in
the radial direction varied between 0.88 km/s at the inner edge
of the C ring and 1.03 km/s in the outer A ring where the observation ended. The total duration of the occultation from the
cloud tops to the F ring was 23.5 h. The occultation was observed in two parts separated by a downlink of spacecraft data.
The first observation covered the inner portion of the C ring;
the outer C ring data were lost due to downlink problems. The
second observation began while the star was behind the dense
core of the B ring at R = 109,000 km and continued to near the
outer edge of the A ring at R = 135,500 km.
The large distance of the spacecraft from Saturn for this occultation also meant that a large portion of the ring system was
within the HSP 6 mrad field of view. The star is north of Saturn’s ring plane so at the time of the occultation Cassini was
observing the lit face of the rings. Because the star is relatively
dim and the sampling intervals in the ring plane are closely
spaced we can estimate the star brightness from the change
in the signal at edges of optically thick rings and ringlets. The
Huygens ringlet at R = 117,800 km in the Cassini Division has
sharp edges and is optically thick, and the outer edge of the
B ring has a peak in optical depth that brings the measured signal down to the background level. The measured difference in
I between the Huygens ringlet and the Huygens gap interior to
the ringlet gives I0 = 1430 ± 10 counts/s.
With I0 fixed, the background in the known clear gaps in the
rings is given by b = I − I0 . The background level drops by one
third between the outermost gap in the Cassini Division at R =
120,300 km and the Encke Gap at R = 133,600 km. We set b
in the B ring at the minimum count level observed when the
star was behind the opaque core of the B ring at R = 105,000
to 110,000 km. We then modeled b with a linear decrease from
the outer Cassini Division across the A ring, with a value in the
Encke Gap set by I − I0 there. A constant value of b in the
B ring part of the observation is expected given that the HSP
field of view is filled by the illuminated optically thick B ring.
As a check on the linear model we calculated the fraction of the
HSP field of view that is filled by ring material, weighted by
optical depth using the Voyager PPS occultation profile (Lane
et al., 1982). We then tied the resulting curve to the gaps where
the background value can be directly measured. The resulting
curve differs from the linear model by less than 20 counts/s out
of 600–800 counts/s depending on the location.
In the multiple gaps of the Cassini Division the background
derived from the HSP field of view filling factor was a slightly
worse fit to the observations than the linear model because
the filling factor model had an increasing background where
the data were consistent with a constant or decreasing background. The constant value of b = 800 ± 10 counts/s in the
B ring core is consistent with the data out to R = 115,000 km
where I − b becomes negative. The reflected sunlight measured
with the FUV across the rings is consistent with this picture,
but is contaminated by starlight and therefore not used in refining the background model. For the B ring regions studied here
we therefore adopt a linear decrease in b from 800 counts/s at
R = 114,000 km to 772 counts/s at R = 118,000 km with a
probable error in b estimated at 10 counts/s across the B ring.
The data with this background model subtracted are shown in
Fig. 2 binned by 50 points (0.4 s) and smoothed to approximately 20 km resolution. Below we compare measurements
at 50 km resolution; for this occultation that resolution corresponds to about 50 s of data. We estimate the probable error in our model value of b at that resolution to be ±200
counts or comparable to the statistical uncertainty in the sig-
Fig. 2. HSP data from three occultations at 50 km resolution, with the background removed (see text).
Self-gravity wakes in Saturn’s B ring
√
nal of I . In the optically thick B ring the probable error in
I0 at this resolution is much smaller and does not contribute
significantly to our estimated error of the optical depth or ring
transparency.
We set the minimum measurable transmitted signal Imin =
I − b = 200 counts at 50 km resolution based on the probable
error in b; this corresponds to a maximum normal optical depth,
τmax = 1.51. At this resolution the statistical uncertainty in the
optical depth is τ < 0.03 at local optical depth minima and
τ < ±0.2 everywhere that τ < τmax . As τ approaches τmax
the upper error bar becomes infinite because the probable error in b prevents us from distinguishing that signal from zero
transparency.
2.2. 126 Tauri occultations
The star 126 Tauri (126 Tau) was observed by Cassini on
its orbital revolutions (revs) 8 and 10 of Saturn. The 126 Tau
rev 8 occultation (126Tau8 hereafter) was an egress occultation that spanned the entire ring system with a data dropout in
the middle of the C ring. The occultation was observed from
∼20 RS resulting in a 9-h track across the rings with a radial
sampling interval in the ring plane of 3 m in the inner C ring
to 5.5 m in the F ring. The occultation was also observed with
the FUV channel on the lit face of the rings, and the long duration of the occultation provided a high signal-to-noise map of
the ring reflectance in the FUV (Fig. 3). This map was used in
conjunction with HSP measurements of the signal in gaps and
opaque regions of the rings to derive a background model. The
data binned by 500 points to 1-s resolution in time are shown in
Fig. 4 with our background model.
With this background model we determine I0 = 4244
counts/s. We bin the data to 50 km resolution, or about 200 s in
the outer B ring. In the B ring, where the background is nearly
constant, we estimate the probable error of b to be 20 counts/s.
Setting Imin = 20 counts/s, τmax = 1.93, with uncertainties in
the optical depth of τ < 0.01.
The rev 10 occultation of 126 Tau (126Tau10) was an ingress
occultation. The star passed behind Saturn in the inner B ring
at R = 103,000 km. There is a data dropout in the A ring ramp
in the outer Cassini Division. Cassini revs 8 and 10 have the
same geometry, and the 126Tau10 occultation is the ingress
portion of the 126Tau8 egress occultation. Scattered light is
less in 126Tau10 because it is on the night side of the planet
while 126Tau8 egress was on the day side. The orientation of
the FUV spectrograph slit on the rings for the 126Tau10 occultation is more nearly radial across the rings than for 126Tau8
in which the detector rows immediately adjacent to those with
the star signal cut nearly radially across the rings. Because the
distance from Saturn was large for both occultations (∼23 RS
for 126Tau10), the resolution from row to row of the detector is
∼1400 km in 126Tau10.
The scattered light measurements of the rings with the FUV
in 126Tau10 are consistent with a nearly constant background
across the B ring and a linear decrease across the A ring. Because of the low resolution of the FUV measurements and
significant row-to-row variations in the scattered light mea-
131
surements due to the viewing geometry, we model the background from the HSP measurements of gaps and opaque regions in the rings. We find I0 = 4550 counts/s with a probable error of 30 counts/s. The difference in I0 between revs
8 and 10 is real. We could not find a single value for I0
that fit both occultations. This may be due to instrumental effects: we have observed different count rates of the same star
in different calibration observations suggesting long-term unmodeled variability in instrument sensitivity, perhaps related to
the ramp-up behavior described above. The background model
consists of 7 linear segments from the middle B ring to the
F ring. In the B ring region studied here, the background increases from b = 1730 counts/s at R = 103,213 km to b =
1920 counts/s at R = 108,000 km, followed by a decrease
to b = 1770 counts/s at R = 116,500 km and then another
increase to b = 1835 counts/s at R = 118,000 km. This is
followed by a monotonic decrease in b across the Cassini Division and A ring. We estimate the probable error of b to be
40 counts/s. Setting Imin = 40 counts/s, τmax = 1.7 with uncertainties in the optical depth of τ < 0.06 in the outer B ring.
2.3. δ Aquarii occultation
The star δ Aquarii (δ Aqr) was observed on 2005-141 (rev 8)
following the first 126 Tau occultation and an occultation of the
star α Virginis (Spica) that did not reach the B ring. This is
the faintest UV star occultation observed by UVIS in the initial
phase of the Cassini mission. However, the dependence of apparent optical depth on viewing angle discussed below is such
that light from δ Aqr at B = 12.2 degrees and φ ≈ 55 degrees
was able to penetrate all of the A ring as well as much of the
inner and outer portions of the B ring. Because of the relatively
low count rate from the star the accuracy of the background determination has a stronger effect on the derived optical depths
than in occultations of the brighter stars.
The FUV channel observed the δ Aqr occultation providing an independent measure of the background. The occultation was observed from the unlit face of the rings which kept
the background signal below even that of the faint star; the
background signal is dominated by Lyman-α emission from
interplanetary Hydrogen shining through the rings. Observations of dark sky as well as the shadowed rings with the HSP
have a mean counting rate due to Lyman-α of approximately
150 counts/s (Chambers et al., 2007). The image of the occultation in the FUV also demonstrates that starlight is passing
through the inner and outer regions of the B ring as well as
the A ring (Fig. 5). To model the background we averaged and
smoothed the signal in the rows of the FUV detector on either
side of the two rows with the star signal to get the dependence
of the background on radial position in the ring plane. We then
scaled this curve to the HSP signal, I , in the opaque part of
the B ring at 100,000–105,000 km where we assume the star
signal is completely attenuated. This is a safe assumption given
that occultations of much brighter stars observed by UVIS (Section 2.5) as well as Cassini Radio occultations (Marouf et al.,
2005) were unable to penetrate this dense core of the B ring. We
then determine the star signal I0 = 515 counts/s from the sig-
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J.E. Colwell et al. / Icarus 190 (2007) 127–144
Fig. 3. FUV image of the rings made during the 126Tau8 occultation. The star is
centered in the spectrograph slit which is oriented vertically and time increases
to the right. The color image was made by assigning red to Lyman-α emission,
and green and blue to longer wavelengths in the FUV where water ice does not
absorb. The rings are blue in this scheme with the sky red due to interplanetary
hydrogen. The star is visible in the central few rows. The vertical bands are due
to bleeding of the bright star signal across the spatial dimension of the FUV
detector. Scans of the ring brightness above and below the rows with star signal
were used to model the shape of the background in the HSP occultation data.
Fig. 4. HSP data from the 126Tau8 occultation binned to one second intervals
(1.5 to 2.8 km radial resolution) together with the background model (dashed
line) constructed from the scattered light measurements with the FUV and fits
to opaque and clear regions in the rings (see text). The increase in the background close to the planet is due to scattered light from Saturn. This is the most
complicated background model of the occultations discussed here due to the
relatively high b/I0 ratio and the changes in b during the occultation.
nal I = I0 + b in known gaps in the Cassini Division and A ring
and between the A and F rings.
Statistical fluctuations in I as well as probable errors in both
I0 and b mean that the quantity I − b can be negative. We
select a positive minimum detectable signal of Imin = I − b
that at 50 km resolution results in a maximum optical depth of
0.95. While smaller values of Imin give a nominally larger τmax ,
Fig. 5. FUV image of the δ Aqr occultation. The occultation slit is oriented
vertically and time increases from left to right. There are 164 records with a
duration of 60 s each in the observation. The star was centered in the slit and
the signal from the star is seen on the central two of 64 spatial elements of the
imaging spectrograph. The planet is the pink feature at left. The color scheme
is the same as in Fig. 2. The observation was of the unlit face of the rings, so the
opaque B ring is dark. The star signal is visible everywhere except the dense
core of the B ring. The faint blue line at the right is scattered sunlight from the
unresolved F ring. The C ring is also visible in scattered sunlight, shown in blue
in this image, just interior to the dark B ring.
Fig. 6. FUV image of the α Leo occultation. The occultation slit is oriented
vertically and time increases from left to right. There are 160 records with an
integration period of 60 s each. The color scheme is the same as in Fig. 2. The
star emerges from behind the planet at left before being occulted by the A ring
which is in the shadow of the planet. The occultation path crosses the shadow
boundary near the outer edge of the Cassini Division before making a grazing
cut across the outer B ring. The star signal is clearly seen through the B ring.
Also obvious is the asymmetry between the transmitted starlight in the A ring
on egress (right edge of image) and ingress (portion in shadow; Colwell et al.,
2006). The vertical bands are due to bleeding of the bright star signal across the
spatial dimension of the FUV detector.
the uncertainties at those larger values of tau are so big as to
render the measurement meaningless. At our selected value of
Imin = 5.5 counts/s, the uncertainty in optical depth at τ = τmax
is τ ∼ 0.4. However, most of the areas of the B ring chosen
for this study have a one-σ uncertainty in τn of τ = ±0.02
Self-gravity wakes in Saturn’s B ring
to 0.03, although some locations were at the maximum value of
τn = 0.95. Those measurements were not included in the analysis below.
2.4. α Leonis occultation
The α Leonis (α Leo, Regulus) occultation on Cassini orbital
rev 9 was a grazing occultation that penetrated to a minimum
ring plane radius of 114,150 km (Fig. 6). The egress (outbound)
portion of the occultation was observed out to a ring plane radius of 131,550 km. The mean signal, I , in the gaps in the
A ring and the Cassini Division varies between 4.5 × 104 and
4.8 × 104 counts/s. The gap signals increase with time indicating that the variation is mainly due to the instrument response
to a bright star and not background variations. These count
rates are significantly higher than the minimum count rates observed in the more optically thick regions of the A ring of 300
counts/s on ingress (when the occultation was in the shadow of
the planet, Fig. 6).
During both the ingress and egress cuts of the B ring the
occultation was not in shadow and the background is higher.
However, the B ring was not opaque to this star, and on egress
the A ring was particularly transparent due to the viewing angle with respect to the self-gravity wakes (Fig. 6; Colwell et
al., 2006). This prevents a direct measurement of the background signal for the latter part of the occultation. The signal,
I , is lower in the Cassini Division gaps in the egress portion
of the occultation than in the ingress portion, even though the
background signal must be higher due to the illumination geometry. We attribute this to the instrument response time variation
described above. In the outer Cassini Division in the egress portion of the occultation the signal, I , has recovered to nearly
the same level that it had at that location in the ingress portion. We adopt I0 = 4.6 × 104 counts/s with a probable error
of 1 × 103 counts/s for the B ring and egress portions of the
occultation, though I0 = 4.7 × 104 counts/s is more appropriate for the ingress A ring occultation. This makes a difference of
3.6 × 10−3 in τn . We estimate the background b = 300 counts/s
on ingress to a ring plane radius of R = 122,000 km followed
by a linear increase with decreasing radius to b = 1500 counts/s
at R = 117,600 km. We estimate the probable error in the background to be 100 counts/s in the shadow (b = 300 counts/s)
and 500 counts/s when b = 1500 counts/s, and we set Imin =
500 counts/s. At 50 km resolution the maximum statistical uncertainty in τn in the B ring is τ = 0.02 and τ < 0.01 across
most of the A ring.
2.5. σ Sagittarii occultation
This occultation spanned the ring system from beyond the
F ring across the A and B rings to the outer C ring at 85,955 km.
The star σ Sagittarii (σ Sgr) is south of the ring plane so
the unilluminated face of the rings was observed. The path
of the occultation passed behind the most optically thick region of the B ring between 105,000 and 110,000 km. The signal there provides a measurement of the background of b =
1000 ± 100 counts/s. The FUV channel recorded data for this
133
occultation at a time resolution of 60 s with the star near the
central row of the 64 spatial pixel FUV detector. The FUV signal is dominated by light from the star even in rows far from
the star due to bleeding of the star signal. Any scattered light
from the rings is not separable from the starlight in the FUV
data. The background may be less on the more transparent portions of the ring, but here we analyze the optical depths in the
optically thick B ring. The total signal is much greater than the
background for this occultation, so even changes as large as a
factor of two in the background have a small effect on the calculated optical depth in those more tenuous rings.
The mean signal, I , just before the star is occulted by the
outer edge of the A ring, is 1.200 × 105 counts/s, and in the
Huygens gap just prior to immersion behind the B ring the signal is 1.199 × 105 counts/s. With b = 103 ± 102 counts/s we
adopt I0 = 1.19 × 105 counts/s for this occultation. The maximum value of τn is τmax = 3.43, set by our assumption of the
uncertainty in the background and requiring I − b to be greater
than that uncertainty. With this large estimate of the uncertainty
in b, the 1-σ statistical uncertainty in τn is ∼0.08 at the maximum values of τn and is less than 0.01 throughout the rest of
the B ring. At the specific positions in the B ring we analyze in
detail below, the 1-σ error bars in τn are τ = 0.004 or less.
2.6. α Scorpii occultations
The α Scorpii revolution 13 (α Sco 13) occultation track
was a chord that traversed the outer portion of the ring system to a minimum ring plane radius of 101,172 km, just interior to the dense B ring core. The Cassini VIMS instrument
was the primary instrument for α Sco observations, and UVIS
took data as a “rider.” UVIS detects light from α Sco B, the
distant (3 arc-s) binary of α Scorpii (Antares), while Antares itself is detected by VIMS.1 We do not have FUV data of this
occultation. The star is relatively dim in the UV so the HSP
ramp-up effect is small. Like the σ Sgr occultation, this one
was observed from above the unlit face of the rings. The background is significantly smaller for α Sco, however, due to less
scattered light from Saturn. In the B ring core, the minimum
signals are b = 120 counts/s and b = 100 counts/s on ingress
and egress, respectively. We adopt b = 110 counts/s with a
probable error of 20 counts/s for the B ring portion of the
occultation. The greater transparency of the Cassini Division
and A ring allow Lyman-α emission from interplanetary Hydrogen to add to the background signal, possibly increasing
it by as much as another 100–200 counts/s. With this determination of the background we estimate the star brightness
from measurements in the Cassini Division and A ring gaps
of I0 = 3375 counts/s with a probable error of 200 counts/s.
We set Imin = 100 counts/s resulting in τmax = 1.87. The statistical error in the optical depth is τ < 0.11 in the B ring and
τ < 0.03 across most of the A ring at 50 km resolution. We
1 The identification of the star providing the signal in UVIS data as the secondary star in the binary α Scorpii system was not made in Colwell et al. (2006),
but the difference in ring plane radii of the tracks of the two stars (∼0.2 km) is
much less than the resolution of the data presented in that paper.
134
J.E. Colwell et al. / Icarus 190 (2007) 127–144
estimate the systematic error in τn due to the errors in the determination of b and I0 to be <0.02.
The α Scorpii revolution 29 (α Sco 29) occultation was an
ingress occultation that spanned the ring system from the F
ring to the inner C ring. The occultation was observed from
the unlit face of the rings, so the background level is low. FUV
data were obtained for this occultation. Serendipitously, another
hot star (HD 148605) was in the FUV field of view on spatial
pixel number 56 approximately 23 mrad from α Sco. α Sco
itself, as expected, was not in the FUV field of view due to imperfect alignment of the HSP, FUV, and VIMS fields of view.
Given the unlit-face viewing geometry of the occultation, the
FUV signal is dominated by interplanetary Lyman-α transmitted through the rings. In the optically thick core of the B ring the
signal reaches a mean minimum level of only 50–60 counts/s.
Although the background may be higher behind the Cassini Division and C rings, the signal in the outer C ring gaps is nearly
identical to that between the A and F rings. Rather than adopt
an uncertain radius-dependent model for both I0 and b whose
differences cancel to match the observed overall signal, we
adopt a constant b = 60 counts/s and I0 = 3600 counts/s, and
we estimate the probable errors as 20 and 50 counts/s respectively based on the variation in the total signal between C ring
gaps and Cassini Division gaps. At 50 km resolution we have
τmax = 2.77 with uncertainties in τ of ±0.3 at those large optical depths, and τ < 0.1 for τ < 2 and τ < 0.01 for τ < 1.
2.7. λ Ceti occultation
The λ Ceti (λ Cet) occultation track traversed the entire main
ring system from the F ring through the C ring. FUV data were
obtained for this occultation and they show a decrease in the
FUV signal to a minimum in the central part of the B ring that
is likely due to increased Lyman-α transmission through the
A and C rings than through the more opaque B ring. The star
is relatively dim in the UV so instrumental effects are small.
The star is north of the ring plane, so at the epoch of this observation the lit face of the rings was observed. The observation
is on the night side of Saturn, so there is not much scattered
light from Saturn itself. The HSP field of view is within the
shadow of the planet on the rings between the middle C ring
and the Encke Gap. In the B ring core, the minimum signals
are I = 46.5 counts/s between 105,000 and 107,000 km. This
average value is skewed upward by narrow regions of the core
that are not totally opaque. We adopt b = 45 counts/s between
100,000 and 108,000 km and have a linear increase in b to 105
counts/s at R = 85,000 km and then a steeper linear increase
to 190 counts/s at 74,300 km (the inner extent of the occultation). Exterior to the B ring core we adopt a linear increase
in the background to 109 counts/s at 140,000 km. With this
background model the ringlet at the Titan 1:0 nodal resonance
is totally opaque at 10 km resolution, but the gaps surrounding it show a higher unocculted star signal than in the outer
portion of the ring system. Part of this increase may be the
HSP instrumental response, and part of it may be due to errors
in the simplified background model presented here. We adopt
I0 = 2450 counts/s at R > 85,000 km, and a linear increase in
I0 to 2525 counts/s at the inner extent of the occultation. Based
on the spread of the signal in the opaque core of the B ring
and the Titan 1:0 ringlet we estimate the probable error in b to
be 20 counts/s. From a similar inspection of the signal in the
gaps across the ring system we estimate the probable error in
I0 also to be 20 counts/s. We adopt Imin = 20 counts/s which
gives τmax = 1.27 at 50 km resolution. With our estimates for
the probable error in b and I0 we get a systematic error uncertainty in normal optical depth of ±0.05 everywhere τn < 1,
with τ < 0.15 at higher optical depths. The statistical errors
are comparable (within a factor of two, at this 50 km resolution)
to the systematic errors for this occultation.
2.8. λ Scorpii occultation
The λ Scorpii revolution 29 (λ Sco 29) occultation was an
ingress occultation that spanned the ring system from the F ring
to the outer C ring. The occultation was observed from the unlit face of the rings, so the background level is low. FUV data
were obtained for this occultation, but the spectra all along the
slit are dominated by starlight because the star is particularly
bright and because the rings on the unlit face are quite dark.
Consequently the FUV data are not helpful in estimating the
background contribution to the HSP signal. The background is
likely higher in the optically thin regions of the ring where interplanetary Lyman-α can shine through. The optically thick core
of the B ring has a minimum mean count rate of 100 counts/s.
Dark sky measurements with the HSP show 100–150 counts/s
due to Lyman-α emission. We adopt b = 100 counts/s for this
occultation in the B ring, and b = 150 counts/s elsewhere, with
a probable error in b of 50 counts/s.
The high count rate for this star results in a varying baseline level for the unocculted star brightness, I0 . The magnitude of this variation is larger than the anticipated variation
in the background signal described above. We therefore adopt
a time-dependent value for I0 that is fit to the measured signal minus the background in the gaps in the rings. The model
value of I0 varies between 2.74 × 105 counts/s in the B ring
to 3.02 × 105 counts/s in the Cassini Division, with I0 =
2.87 × 105 counts/s in the A ring. We estimate the probable
error in I0 to be 1.5 × 103 counts/s. Setting a minimum detectable signal at 50 counts/s we have τmax = 5.79, and both
statistical and systematic uncertainties in τ of less than 0.2 in
the central B ring and less than 0.01 in the A and C rings and
lower optical depth regions of the B ring.
2.9. α Virginis occultation
The rev 30 α Virginis (Spica, α Vir 30) occultation track traversed the entire main ring system from the F ring through the C
ring. At the time of this occultation the star was on the lit side of
the rings so it was observed from the unlit side. This reduces the
background of scattered sunlight from the rings. Poor weather
over the Deep Space Network antennas in Goldstone, California, resulted in 104 data dropouts in this observation. Most of
the dropouts are in the data for the outer A ring and F ring re-
Self-gravity wakes in Saturn’s B ring
gion, the Cassini Division, and the outer B ring with a loss of
approximately 12% of the data.
FUV data were obtained for this occultation, but due to the
brightness of α Vir there is significant bleeding across the spatial pixels. Combined with the low background level on the unlit
face of the rings this prevents the FUV data from being useful
in determining a background model. The minimum mean signal
in the opaque parts of the B ring core is 140 counts/s. The background likely increases behind the more transparent regions of
the rings where interplanetary Lyman-α can shine through, but
b = 140 counts/s is in the range of typical dark sky count rates
for the HSP. In the optically thick region between 100,000 and
101,000 km the minimum mean count rate is 180 counts/s,
but this count rate could include some signal from the star. We
adopt b = 160 counts/s for the entire occultation with a probable error of 40 counts/s to allow for the possibility of enhanced
Lyman-α transmission through the A and C rings. In the central
B ring, adopting b > I simply means those regions are effectively opaque for this occultation. Inspection of the light curve
shows that significant signals are at levels much higher than the
minimum signals, so there is virtually no loss of information in
taking b = 160 counts/s instead of 140 counts/s.
The HSP instrumental response to bright stars (the rampup behavior) results in a changing baseline value of I0 across
the occultation. We fit linear segments between the clear gaps
in the rings to create a model of I0 as a function of ring
plane radius, R. The resulting model varies between 5.14 ×
105 counts/s in the Encke Gap and 5.44 × 105 counts/s in
the inner C ring. Because there are no gaps in the B ring a
constant value of I0 across the B ring is assumed with I0 =
5.34 × 105 counts/s. We adopt a probable error of 103 counts/s
for I0 , and τmax = 2.82 at 50 km resolution. With our estimates
for the probable error in b and I0 we get a systematic error
uncertainty in normal optical depth of less than ±0.008 everywhere τn < 1, with τ < 0.05 everywhere τn < 2. Statistical
uncertainties (1σ ) in τ are ±0.23 in the optically thick B ring
core and ∼0.01 elsewhere.
135
most of the disk of the Sun outside the EUV solar port field
of view. As a result the integrated signal over all EUV wavelengths was only 7500–8000 counts per 4-s integration period.
We assume a constant value of I0 = 1741.5 counts/s based on a
background determination in the B ring core, and then fit a third
order polynomial to the signal I0 + b measured beyond the A
ring, in gaps in the Cassini Division, and interior to the C ring.
We then construct an optical depth profile using Eq. (2).
The line of sight to the Sun moved 40–50 km in radius in
the ring plane during the 4-s integration periods. The distance
to the rings along the line of sight varied from 2.6 × 105 km in
the C ring to 2.1 × 105 km at the F ring. The Sun therefore was
a disk 210–260 km in diameter on the sky at the distance of the
rings. There was virtually no projection effect in the radial direction, so the projected size of the Sun in the radial direction
was also 210–260 km. However, most of the disk of the Sun was
outside the UVIS field of view due to the pointing error, reducing the effective size of the observed disk of the Sun. The flux in
the rev 11 solar occultation (solar11) from the entire solar disk
was approximately 105 counts/s. Assuming a uniformly bright
solar disk this gives an observed area of the Sun in solar9 of
approximately 1.7% of the full solar disk, so the integration period sets the resolution at ∼50 km. We set Imin = 10 counts/s
for the solar9 occultation resulting in τmax = 1.89. The statistical uncertainty τ = 0.19 at τn = τmax , and τ < 0.05 in the
optical depth minima in the outer B ring discussed below.
The solar11 occultation extended from the atmosphere to the
outer B ring at R = 116,000 km. This observation had the full
solar disk in the EUV solar port field of view. The limiting resolution is the projected size of the Sun in the ring plane, or
240–260 km. We determine I0 = 3.95 × 105 counts per 4-s integration period from the measurement of the unocculted Sun interior to the C ring and a background determination of b = 750
counts per 4-s integration from the signal in the B ring core at
R = 105,000–110,000 km. We set Imin = 100 counts/s based
on our estimated uncertainty in b. This results in τmax = 2.5.
Because of the high counting rate, the statistical uncertainty
τ < 0.004 at R > 111,500 km, and τ = 0.03 at τn = τmax .
2.10. Solar occultations
3. Radial structure of the B ring
Solar occultations are observed with the EUV channel of the
UVIS using a pick-off mirror 20 degrees away from the main
optical boresights to avoid direct solar pointing (Esposito et al.,
2004). The mirror is a section of a cylinder designed to disperse
the image of the Sun along the spatial direction of the detector
to prevent too high a flux of photons onto the detector. We include data from the first two ring solar occultations observed by
UVIS in revs 9 and 11. There are no strong spectral features in
the main rings in either solar occultation so we have summed
the signal at all wavelengths in the EUV channel. The angular diameter of the Sun is 1 mrad at Saturn, and this limits the
spatial resolution of these occultations.
The rev 9 solar occ (solar9 hereafter) spanned the entire ring
system from beyond the F ring to inside the C ring. The optical depth of the inner D ring is too low to be detected by UVIS
at the moderate value of B = 21.45 degrees of this occultation.
The solar9 occultation suffered from a pointing error that placed
The σ Sgr and λ Sco occultations probed the highest optical
depths of the stellar occultations discussed here because they
are the brightest stars at higher values of B observed in the first
part of the Cassini tour. The σ Sgr occultation incidence angle is
comparable to that of the Voyager PPS observation of the δ Sco
occultation. The angle between the line of sight and the radial
direction, φ, is also similar between these two occultations. The
calculated normal optical depths are virtually indistinguishable
between the Cassini σ Sgr occultation and the Voyager δ Sco
occultation (Esposito et al., 1983) across the entire ring system
to within measurement uncertainties. Maximum optical depths
are higher for σ Sgr due to the greater signal and therefore
smaller detectable transparencies. At the higher elevation angle B = 41.7 degrees of the λ Sco occultation, apparent normal
optical depths are higher, while occultations at lower values of
B show a reduced normal optical depth (Figs. 7–10).
136
J.E. Colwell et al. / Icarus 190 (2007) 127–144
Fig. 7. Normal optical depth profiles for the inner B ring from four Cassini
UVIS occultation observations at 10 km radial resolution. The apparent normal
optical depth increases with increasing incidence angle, B (see Table 1). At
lower optical depths the increase slows or stops at B ∼ 30◦ .
Fig. 8. Three occultation profiles at 10 km radial resolution showing rapidly
changing optical depth in the B ring core. The region is characterized by large
fluctuations in optical depth with maximum values that are opaque to occultations. In these regions the variation in optical depth with occultation angle B is
smaller than in the outer core where the valleys in optical depth are not as deep.
At R > 104,000 km the low optical depth valleys become more widely spaced
and have larger minimum optical depths than in the regions 99,000–100,000
and 101,000–104,000 km.
In the A ring, there is a strong correlation between optical
depth and φ indicating that the alignment of self-gravity wakes
controls the apparent transparency of the ring (Colwell et al.,
2006). This was seen most dramatically in the α Leo occultation which spanned most of the A ring at two different values
of φ (but identical B for the same star) with a factor of two difference in normal optical depth. The two occultations of 126
Tau at different φ also showed a strong correlation with the
transparency of the ring with alignment of the line of sight with
self-gravity wakes.
The B ring optical depth shows a weaker dependence on φ
and a strong correlation with B. This correlation is on top of
the large radial variations in optical depth across the B ring.
The inner B ring from R = 92,000 to R = 99,000 km is
characterized by optical depths of τn = 0.5–2.0. There is a
nearly featureless valley of relatively low optical depth at R =
94,450–95,350 km, and the long Janus/Epimetheus 2:1 density
wave train at R = 96,200–96,750 km (Fig. 7). There is an enhancement in the optical depth in the valley at R = 95,200 km
that is 50–100 km in width in the otherwise featureless valley
in this region. This enhancement is most clear in occultations
at higher values of B. This may indicate a change in the ring
particle properties at that location which affect the formation of
self-gravity wakes. However, the amplitude of the enhancement
(<0.05) is too small for differences in the wakes to be identified
with our model and the data sets presented here.
Beginning at R = 98,830 km the optical depth increases and
enters a regime of large fluctuations on a range of spatial scales.
Peak normal optical depths exceed 5 in this region, and transitions between the peaks and the neighboring valley optical
depths of ∼1 usually span 10–50 km but take less than 1 km in
some cases. This region extends out to R = 110,000 km (Figs. 8
and 9). At R = 104,000 km the number of valleys decreases and
the optical depth in the valleys increases out to R = 108,000 km
(Fig. 8). Horn and Cuzzi (1996) found quasi-periodic structures
across the B ring, including this region, in analyses of Voyager
imaging data, consistent with these optical depth fluctuations.
Local minima in the optical depth, although near the maximum
detectable values for some occultations, correlate between the
occultations (Figs. 8 and 9). Minima in the normal optical depth
in the σ Sgr and λ Sco profiles are τn = 2–3, and maxima exceed the measurement limit (Fig. 8). The mean optical depth begins to decline at R = 108,000 km as the valleys broaden, and
the maximum optical depths drop to τn ∼ 3 at R > 110,000 km
with the exception of the peak at R = 116,400 km. The outer
B ring (R > 110,000 km) has complicated variations in optical depth with more gradual transitions and lower peak optical
depths than the region at R = 99,000–110,000 km (Figs. 9
and 10). In this region the dependence of transparency on φ can
be seen by comparing the two occultations of 126 Tau which
had nearly orthogonal values of φ (Fig. 10). The ingress and
egress portions of the α Leo occultation show a smaller divergence as the difference in φ increases from zero at the turnaround point to ∼25 degrees at the outer edge of the B ring
(Fig. 10). The difference in φ between the ingress and egress
branches of the α Sco occultation is ∼65 degrees, but the apparent optical depth is high in both cases and the differences
between the two branches, where measurable, are small.
4. Aligned structures in the B ring
4.1. Self-gravity wake model
The strong dependence of apparent normal optical depth on
the inclination angle B is illustrated in Fig. 11 (see also Figs. 2,
7–10). The factor of μ in Eq. (2) corrects for the longer line
of sight of an observation through a semi-transparent medium
than would be seen from normal incidence (B = 90 degrees).
This correction assumes that the optical depth scales with path
length. If the medium is entirely opaque or entirely transparent, then path length is irrelevant and the factor of μ in Eq. (2)
introduces a trend like the one seen in Fig. 11. We find that
Self-gravity wakes in Saturn’s B ring
137
versely correlated with overall optical depth. High optical depth
regions have tightly packed wakes with little empty space between them, and lower optical depth regions have more widely
spaced self-gravity wakes.
We model the B ring observations with the same model that
was used to interpret the azimuthal transparency asymmetry
due to self-gravity wakes seen in occultations by the A ring
(Colwell et al., 2006). This model consists of regularly spaced
slabs of normal optical depth τwake , length L, and width W separated by relatively clear gaps of normal optical depth τ gap and
gap width S. Both the self-gravity wakes and the gaps are assumed to have a height H , and the orientation of the self-gravity
wakes is φwake as measured from the local radial direction. With
the simplification that τwake = ∞ and L = ∞, a simple analytic
expression for the transparency is given by
Fig. 9. Optical depth profiles at 10 km radial resolution showing the transition
from the optically thick core to the complex structure in the outer B ring at
about 110,000 km. In most occultations the ring remains mostly opaque out to
R = 110,000 km as in the δ Aqr, λ Cet, and 126 Tau occultation profiles shown
here. Gaps and plateaus in the profiles occur when there is no detectable signal
from the star. Occultations at higher elevation angles (B) and with bright stars
such as σ Sgr and λ Sco reveal complex structure within the core that resembles
the outer B ring region, but at a larger average optical depth.
T = exp(−τn /μ)
[S/W − H /W | sin(φ − φwake )| cot B]
exp(−τgap /μ). (4)
=
S/W + 1
We fit the data at both 50 and 10 km resolution using the analytic expression in Eq. (4) and the simplified ray tracing technique of Colwell et al. (2006) that allows for finite τwake . Solar
occultation data are only included in the 50 km resolution fits.
We fit the model parameters to the data by minimizing the
quantity D where
D=
N
1 (τn,i − τm,i )2 ,
N
(5)
i=1
Fig. 10. Six Cassini UVIS optical depth profiles of the outer B ring at 10 km
radial resolution showing variations in apparent normal optical depth with viewing angle. The α Leo occultation only penetrated to R = 114,150 km. The two
126 Tau profiles (black and green curves) have different optical depths due to a
variation in φ indicating aligned structures in the ring. A smaller φ dependence
is seen in the α Leo profiles. A data gap in the 126 Tau (10) data is responsible
for the straight segment from 115,500 to 115,800 and near 116,400 km.
at virtually all locations in the B ring the line of sight optical depth, τ = τn /μ = − ln(T ) is roughly independent of B.
The ring cannot be entirely opaque, however, or no light would
be seen through it at any viewing angle. The observations can
be explained by alternating regions of totally opaque clumps
and intervening gaps of low optical depth. The observed transparency in a given viewing geometry then depends primarily
on the fraction of the ring plane that is obstructed by opaque
clumps. From this simple argument one can anticipate the result of our more detailed modeling presented below, namely
that the separation of self-gravity wakes in the B ring is in-
where τn,i is the measured normal optical depth, τm,i is the
model normal optical depth, and N is the number of usable
observations. Using the uncertainties in τn described in Section 2 we can also calculate a reduced χ 2 statistic. However,
because our model is an idealization (perfectly parallel and periodic structures with rectangular cross-section) there are nonstatistical errors in our fit to the data due to the assumptions
of the model that complicate interpretation of the value of χ 2 .
We find comparable results for the model parameters whether
we minimize D or χ 2 . Similarly, changing the optical depth to
transparency in Eq. (5) produces no significant changes. The
parameter D is equal to χ 2 for the case where the measurement errors are equal. That is, we treat each measurement with
equal weight, even though some have larger formal measurement errors as outlined in Section 2. This procedure effectively
gives the error associated with the model simplifications outlined above greater importance than the measurement errors, by
giving each measurement the same weight. Figs. 12–14 illustrate the sensitivity of D on different parameters for the model
fit to the data at one location in the outer B ring. Like the results for the A ring (Colwell et al., 2006), our model essentially
places an upper limit on H /W and a lower limit on τwake . We
are able to more tightly constrain S/W and τgap , though there
are still relatively large uncertainties for those parameters and
for φwake (Section 4.2).
Although this model oversimplifies the ring structure, it does
capture the basic behavior of the ring transparency with viewing geometry allowing us to deduce properties of clumps of
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J.E. Colwell et al. / Icarus 190 (2007) 127–144
Fig. 11. Measured normal optical depths (asterisks) at 6 locations in the B ring at 50 km resolution with the best-fit self-gravity wake model (diamonds) using the
analytic expression of Eq. (4). The dominant trend is for an increase in apparent normal optical depth with elevation angle B. Spread in the data at similar values of
B are due to the different values of φ of the observations.
particles within the B ring. More sophisticated models will be
needed when additional observations are available that probe
the rings at new geometries and which are able to penetrate the
optically thick B ring with greater signal. The best opportunities
to see through the densest parts of the B ring with stellar occultation measurements begin in mid-2008 when Cassini will be
on highly inclined orbits around Saturn.
4.2. Self-gravity wake properties in the B ring
The number of useful measurements of τn varies across
the B ring as different occultations have τ > τmax at different locations and not all occultations have full radial coverage
(Table 1). We are able to fit the observations with the selfgravity wake model across the B ring with the exception of
R = 100,000–101,000 km, and R = 104,000–110,000 km due
to the high optical depths in those regions. We calculate the
best-fit values of τgap , S/W , H /W , and φwake using leastsquares minimization (Eq. (5)) for all points in the B ring for
which there are at least eight UV occultation measurements
of the optical depth with τn < τmax and for which D < 0.1.
This corresponds to a mean error in the optical depth of 0.1.
When using the ray-tracing method (Colwell et al., 2006) we
also get best-fit values of τwake , although for most of the ring
these values are indistinguishable from infinity (i.e., opaque
wakes). An exception is the relatively low optical depth regions in the inner B ring. Nevertheless, even in the valley at
R = 95,000 km where τn ∼ 1 (Fig. 21), the mean calculated
τwake = 4.9. Thus, assuming τwake = ∞ is a reasonable approximation across the B ring. We checked this by comparing the
results for self-gravity wake parameters using the analytic expression of Eq. (4) with the results from the ray-tracing method
at both 10 and 50 km resolution. We do not see systematic
differences between the various techniques in the B ring. For
example, the calculated values of τgap are shown in Fig. 15 for
the two techniques.
If the B ring structure were azimuthally symmetric ringlets
of large optical depth separated by nearly empty gaps, that
Self-gravity wakes in Saturn’s B ring
Fig. 12. Contours of D (Eq. (5)) for the self-gravity wake model fit to the 14
observations of the B ring optical depth at R = 113,225 km (Fig. 11, lower left)
as a function of S/W and τgap . The particular solution for S/W and τgap found
by the automated minimization of D is shown by the asterisk, and the values of
D for the two minimum contours are labeled.
Fig. 13. Contours of D (Eq. (5)) for the self-gravity wake model fit to the 14
observations of the B ring optical depth at R = 113,225 km (Fig. 11, lower
left) as a function of φwake and τgap . The particular solution for φwake and
τgap found by the automated minimization of D is shown by the asterisk. The
minimum (heavy line) contour has D = 0.012, and the next contour (dashed)
has D = 0.016. This broad minimum in D makes it difficult to tightly constrain
φwake .
would explain the observed trend of τn with sin(B) (Fig. 11).
Aligned structures in the rings introduce variations in τn with
the observing angle φ. This variation is seen in the scatter of
points at equal values of B in Fig. 11. The best fit values of
φwake across the B ring and the A ring using the ray-trace
method are shown in Fig. 16. Although there is considerable
scatter in the values, the data clearly show that structures in
the B ring are not azimuthal and are canted from the azimuthal
direction at about the angle expected for trailing self-gravity
wakes. There is a clear trend for the cant angle (π/2 − φwake ) to
decrease from the inner to outer B ring, while the trend reverses
in the A ring (Colwell et al., 2006). The outer B ring has higher
optical depth, however the correlation between φwake and τm
(the maximum value of τn measured at any given location) for
all locations in the B ring is not statistically significant. Nevertheless, the difference in values of φwake between the inner
139
Fig. 14. Contours of D (Eq. (5)) for the self-gravity wake model fit to the 14
observations of the B ring optical depth at R = 113,225 km (Fig. 11, lower
left) as a function of S/W and H /W . The particular solution for S/W and
H /W found by the automated minimization of D is shown by the asterisk. The
minimum (heavy line) contour has D = 0.012, and the next contour (dashed)
has D = 0.026.
Fig. 15. Calculated values of τgap using the analytic expression of Eq. (4),
which assumes τwake = ∞, and using the ray-trace method of Colwell et al.
(2006). The ray-trace results have been median filtered by 5 points to make the
comparison between the two sets of results more clear.
and outer B ring may reflect limitations of the model (infinitely
long wakes, for example) that are emphasized in the outer B
ring where the low-B α Leo occultation measured much lower
optical depths than those at even slightly larger values of B.
The finite line-of-sight optical depth at B = 9.5 degrees in the
α Leo occultation may be the result of light passing over the
ends of self-gravity wakes.
The dominant trend in S/W is a correlation with total optical depth (Fig. 17). Large values of optical depth can only be
achieved if the self-gravity wakes are tightly packed. In other
words the normal optical depth of the ring is in large part determined by the fraction of the ring surface area that is blocked by
the clumping of particles into self-gravity wakes. Regions with
lower optical depth have more open space between the selfgravity wakes while the more opaque regions have the clumps
more tightly spaced. If they have large spaces between them,
then in order to have large overall optical depths there would
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J.E. Colwell et al. / Icarus 190 (2007) 127–144
Because S/W appears to be primarily determined by τm ,
which we use as a proxy for the true normal optical depth (i.e.,
τn for B = π/2), we can fit the results shown in Fig. 17 with an
exponential to get a usable value of S/W for any region of the
B ring:
W
(6)
= 0.28e1.616τm .
S
Values of τgap show large scatter, like those of H , with a
mean value of 0.25, and 90% of calculated values of τgap are
less than 0.42 and greater than 0.08. There are no correlations
between τgap and either position in the ring, total optical depth,
or other self-gravity wake parameters.
4.3. Small scale structure in the α Leo occultation
Fig. 16. Best fit values of φwake across the B ring (diamonds) and A ring (asterisks) using the ray-trace method at 10 km resolution and median filtered by
5 points.
Fig. 17. Inverse correlation between S/W , the relative spacing between
self-gravity wakes, and the maximum measured optical depth. High optical
depths require tightly packed self-gravity wakes (W/S 1).
need to be a significant τgap . This would in turn make the
ring optical depth sensitive to path length and destroy the actual strong dependence of apparent τn on B. Values of H /W
are widely scattered between 10−3 and 0.2, consistent with the
weak dependence of D on H below some critical upper limit
(Fig. 14). The mean value of H /S for the B ring is 0.2, with
scatter that is consistent with the uncertainty in the determination of H at each location. That is, to the limits of this model,
one can assume H /S = 0.2 and use the values of S/W determined by the model fit and shown in Fig. 17. A similar trend in
H /W with total optical depth was seen in the A ring (Colwell et
al., 2006). Fig. 18 shows H /W and τgap for the A and B rings,
where A ring values have been updated from Colwell et al.
(2006) with the new occultations presented here. Gap optical
depths are higher in the B ring than in the A ring, but because
the wake spacing is much higher in the A ring the gaps contribute a larger fraction of the overall optical depth than in the
B ring.
The turnaround of the α Leo occultation in the outer B ring
provided the first opportunity to study one radial location in
the rings with a nearly azimuthal track along the ring. The
small change in ring plane radius from point to point results in
an effective increase in the resolution of radial structure, and
the ingress and egress portions of the occultation track near
the minimum radius provide two measurements under identical
observing conditions of the same ring plane radii at different
longitudes. The increased signal to noise at fine radial resolution allows us to directly observe ring structures at scales of
∼100 m.
The turnaround radius of this occultation is R = 114,149.32
km (Fig. 10). The accuracy of the geometry solution is determined by a number of factors including the spacecraft trajectory
reconstruction and the pole of Saturn and is estimated to be typically ∼1 km for the stellar occultation results presented in this
paper. The precision of the calculation is better than 1 m. Here
we provide the more precise values to give the relative positions
of features at the turnaround, but a more accurate geometric solution that adjusts the pole of Saturn, the spacecraft trajectory,
the position of the star, and the timing of the data is beyond
the scope of this paper. There are 3.20 s of data (1600 points)
taken just in the innermost 680 m of the occultation, between
R = 114,149.32 km and R = 114,150.00 km, and there are
6.36 s of data interior to R = 114,152.00 km. Near the turnaround, the relative radial positions are affected only by the
uncertainty in the timing of the data which is comparable to
the integration period (2 ms). We bin the α Leo data to 0.1 s
for analysis of azimuthal structure near the turnaround radius.
The inner 2.7 km are shown in Fig. 19. The turnaround radius
is within a local minimum in total optical depth (Fig. 10), and
shifts in the data in an attempt to make the small scale radial
structure correlate between ingress and egress destroy the large
scale correlation between ingress and egress seen in Fig. 10.
The timing uncertainty for the data shown in Fig. 19 is much
less than a single point, so the correlation between the inbound
(ingress) and outbound (egress) portions of the occultation is
accurate as plotted.
While some areas show a positive correlation, such as at
R = 114,150.1–114,150.2 km, there are also several regions
of width ∼0.1 km that are anti-correlated between the ingress
Self-gravity wakes in Saturn’s B ring
Fig. 18. Gap optical depths and H /W ratio for self-gravity wakes across the B
and A rings using the ray-trace method at 10 km resolution and median filtering
by 5 points. The A ring values can be compared to Colwell et al. (2006) where
H /W values show more scatter due to fewer occultations used to derive model
parameters. H /W , like S/W , are inversely correlated with optical depth.
Fig. 19. T from data binned to 0.1 s of the α Leo occultation near the minimum
ring plane radius in the outer B ring sampled by this occultation.
and egress profiles in the fluctuations in optical depth (Fig. 19).
For the inner 2.7 km shown in Fig. 19 the correlation coefficient is −0.02. When the data are binned into equally spaced
bins in ring plane radius, the correlation coefficient between the
ingress and egress profiles is −0.001. We searched for a phase
lag by shifting the data in time resulting in shifting the minimum radius point on the light curve. No significant correlation
was found for any possible phase lag; larger shifts of the minimum radius point destroy the ingress–egress symmetry of large
scale circular features in the rings and so were not explored.
The power spectra of both the ingress and egress branches, however, do show a strong peak at a wavelength of 160 m. This is
the only wavelength above the noise (Fig. 20). When the data
are binned in equally spaced bins in ring longitude the power
spectrum does not show any dominant wavelength.
The line of sight distance to the occultation point in the rings
at the minimum ring plane radius is 3.12 × 105 km. The spectrum of α Leo convolved with the HSP quantum efficiency
as a function of wavelength shows that 75% of the photons
detected by the HSP were in the wavelength range of 132–
141
167 nm. The radius of the Fresnel zone on the plane of the
sky is therefore λFr = 6.8 m using an effective wavelength of
150 nm. Projected onto the rings the Fresnel zone becomes an
ellipse with its long axis in the φ direction and λFr /μ = 41 m
long, and its short axis equal to λFr . At this point in the occultation φ = 112 degrees meaning that the Fresnel ellipse is
rotated 22 degrees clockwise from the local azimuthal direction as seen from north of the rings. The diffraction-limited
resolution is therefore sin(φ − π/2) × 41 = 15 m in the radial direction and 38 m in the longitudinal direction. Interior
to R = 114,200 km the change in radius from one 2 ms integration period to the next is less than 1.5 m. The track in the
azimuthal direction in one 2 ms integration period, however, is
50 m long for R < 114,200 km. In 2 ms the ring particles travel
38 m along their orbits around Saturn in the opposite direction
as the occultation path, increasing the azimuthal smear relative
to the ring particles to 88 m. The angular diameter of α Leo is
1.3 milli-arcsec (Radick, 1981), and at the line of sight distance
to the occultation in the B ring of ∼3.1 × 105 km the projected
size of the star in the sky plane at the rings is less than 2 m. So
the effective radial resolution near the minimum radius is diffraction limited to 15 m. Thus the radial wavelength of 160 m
is a resolved radial length scale at this location in the ring.
Because the radial velocity of the occultation track is so slow
for the data in Fig. 19, in the time it takes to span one 160 m
radial wavelength, the occultation has sampled several km in
the azimuthal direction. These structures must therefore be
nearly azimuthal themselves and have much longer azimuthal
extent than self-gravity wakes. Similar wavelength structure
has been observed in power spectral analysis of all UVIS stellar occultations with sufficient radial sampling (Sremčević et
al., in preparation) and Cassini Radio Science (RSS) occultations (Thomson et al., 2006). The sub-km structure seen in
the highest resolution images (Porco et al., 2005) might also
be connected with the 160 m waves observed here. These azimuthally symmetric undulations co-exist with the self-gravity
wakes. This periodic wave-like azimuthal structure is likely
due to viscous overstability in regions of relatively high optical depth (Schmidt et al., 2001; Salo et al., 2001). Because the
model described here does not include both self-gravity wakes
and axisymmetric waves, such as produced by viscous overstability, locations in the rings where the overstability waves are
prominent may have different wake properties than those described here.
5. Discussion
We find that clumping in Saturn’s B ring produces variations
in the apparent normal optical depth of the same magnitude that
is seen in the A ring. Although it has long been recognized that
the azimuthal brightness asymmetry of the A ring is most easily
explained by aligned structures, it is a surprise that these aligned
structures persist throughout much of the B ring.
Recent theoretical attempts to model the formation of largescale structures in the B ring have treated the rings as a viscous
fluid with transport coefficients such as the viscosity and thermal diffusivity that can be modeled as monotonically increasing
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J.E. Colwell et al. / Icarus 190 (2007) 127–144
Fig. 20. Fast Fourier Transform in ring plane radius space of ingress (blue)
and egress (red) data from the α Leo occultation for the inner 6 km of the
occultation (R < 114,115.3 km). The peaks in the ingress and egress data are
at a radial wavelength of 160 m.
Fig. 21. Predicted normal optical depths for measurements at B = 90 degrees
at 10 km resolution compared to the λ Sco occultation, which measured the
highest overall optical depths in the B ring at most locations.
functions of the local number density of particles (Salo et al.,
2001; Schmidt et al., 2001). These models predict a viscous
overstability that produces azimuthally symmetric radial density variations that slowly grow in wavelength. Salo et al.’s
(2001) direct N -body simulations have demonstrated the initial development of the viscous overstability at optical depths
appropriate for the A ring. A viscous fluid model is needed to
extend the small scale N -body results to larger scales. The presence of aligned self-gravity wakes in the B ring places limits
on the applicability of a viscous fluid model for the B ring. In
particular, a viscous fluid model can only be justified on radial
scales that are larger than the characteristic wavelength of the
gravity wakes. The viscous and thermal transport coefficients
of such a fluid must therefore represent the spatially averaged
transport caused by the wake structures.
N -body simulations of self-gravity wakes in the A ring
suggest that the effective viscosity is dominated by the gravitational interactions between the self-gravity wakes and that
particle collisions play a secondary role (Salo, 1995; Daisaka
et al., 2001; Griv et al., 2003). These authors argue that the
scale of the self-gravity wakes is governed by Toomre’s critical wavelength for axisymmetric Jeans instabilities (Toomre,
1964), which scales linearly with the surface mass density of the
rings. Based on scaling arguments, Daisaka et al. (2001) argue
that the effective kinematic viscosity of the rings in the presence
of self-gravity wakes scales as the square of the Toomre wavelength and thus increases as the square of the surface mass density. This scaling has been used in the context of modeling the
spreading rate of the protolunar disk (Ward and Cameron, 1978;
Kokubo et al., 2000; Takeda and Ida., 2001). However, it may
not make sense to apply this scaling to Saturn’s B ring where
the optical depth within the self-gravity wakes is substantially
larger than unity.
The inferred structures in the B ring are highly flattened,
tightly packed, trailing spiral self-gravity wakes, consistent
with theoretical predictions and numerical simulations for
lower-optical depth rings. The variation of optical depth with
the azimuthal viewing angle, φ, is smaller than in the A ring,
and there is a stronger dependence on the elevation angle, B.
This paints a picture of broad, flat sheets of particles that are
nearly opaque with relatively empty space between them. The
sheets are loosely organized into the trailing spiral density
enhancements that give the azimuthal brightness asymmetry
observed in the A ring. The φ-dependent asymmetry is stronger
than that of the B ring. This may explain why a much weaker
asymmetry has been observed in the B ring in previous observations made at a limited range of the elevation angle, B (e.g.,
Nicholson et al., 2005).
These clumps persist across the B ring, although measurements in the dense core are limited and preclude a model fit
at all locations in the central B ring. While these structures are
qualitatively the same as the self-gravity wakes in the A ring,
they are apparently wider relative to the ring thickness than their
A ring neighbors. The height to width ratio of ∼0.1 in the B ring
implies that the B ring self-gravity wakes are monolayers of the
largest (r ∼ 5 m; Zebker et al., 1985) particles, with the smaller
particles filling in the spaces between the larger particles. The
resulting higher collisional viscosity of the B ring self-gravity
wakes likely produces structures that persist for many orbital
periods in contrast to the A ring where self-gravity wakes form
and break apart on the orbital timescale.
The viscous spreading timescale for the B ring would be
four times shorter than for the A ring if viscosity continues to
scale with the square of the surface mass density. We suggest
instead that the B ring self-gravity wakes are strongly nonlinear
structures with a characteristic wavelength that does not necessarily follow the scaling for marginal Jeans instabilities given
by Toomre’s critical wavelength. The azimuthal extent of individual self-gravity wakes in the B ring is unknown and is
difficult to estimate from N -body simulations due to the combination of large simulation cells needed and the high collision
rates. If the B ring self-gravity wakes do indeed persist for
many orbits and have a lower azimuthal frequency than their
A ring counterparts, they will contribute a smaller rate of angular momentum transport by gravitational interactions between
the wakes than would be implied by an extrapolation of the scal-
Self-gravity wakes in Saturn’s B ring
ing hypothesized by Daisaka et al. (2001) based on the Toomre
critical wavelength. In addition, the high density of particles
within the B ring self-gravity wakes raises the possibility that
the shear rate is substantially reduced below the Keplerian rate
due to finite yield stresses between tightly packed inelastic particles. This has been suggested in a recent model for the origin
of large-scale structures in the B ring (Tremaine, 2003). The
net result could be a shear-banded structure where the shear
rate exceeds the Keplerian value in the low-density regions
between the wakes. Exactly how the effective rate of angular
momentum transport varies with the average surface density on
scales much larger than the scale of the self-gravity wakes in
the B ring is currently unknown and deserves further theoretical study. A convincing explanation for the large-scale irregular
radial structure in the B ring must take into account the existence of these broad self-gravity wakes throughout the B ring.
Radial structures formed by viscous overstabilities are not excluded in this picture, but the radial scale on which they can
occur should be larger than the self-gravity wake scales. In the
example presented here in the α Leo occultation (Fig. 19) the
overstability radial scale is ∼160 m (Fig. 20) which is roughly a
factor of two larger than the Toomre length scale for self-gravity
wakes at that location. At regions in the B ring where the optical depth, and presumably the surface mass density, are higher,
the self-gravity wake scale could be significantly larger and any
overstability structures would be restricted to correspondingly
larger scales as well.
The self-gravity wake parameters allow us to predict the apparent normal optical depth (that is, the optical depth calculated
from Eq. (2)) of the B ring that would be measured at any value
of B and φ. The predicted normal optical depths (measured at
B = 90 degrees) are 1.18 times the largest values measured by
stellar occultation, which at most radii are the measurements
from the λ Sco occultation at B = 41.7 degrees. The predicted
true normal optical depths and the λ Sco optical depth profile
at 10 km resolution are shown in Figs. 21 and 22 for the inner
and outer B ring. We were only able to fit the wake model to a
few locations in the low-τ valleys in the middle B ring, so they
are not plotted. In contrast, the model fits virtually the entire
A ring where optical depths are generally lower. The predicted
true normal optical depth for the A ring is shown in Fig. 23
with the λ Sco profile and the maximum measured normal optical depth. The λ Sco optical depths are not necessarily the
maximum values because the wider spacing of the self-gravity
wakes in the A ring reduces the effect of increasing τn with B.
The predicted true normal optical depth in the inner half of the
A ring (R < 130,000 km), where self-gravity wakes are most
prominent, is 1.19 times the maximum optical depths measured
using the data presented in this paper, nearly identical to the
B ring ratio of 1.18. The ratio drops to 1.07 at R > 130,000 km
where self-gravity wakes are less well-organized (Colwell et al.,
2006).
There are no obvious trends between the self-gravity wake
parameters and location in the B ring other than the trend in φ.
Overall, B ring optical depth appears to be related to how
closely packed these dense self-gravity wakes are packed to
each other (Fig. 17). We cannot rule out the possibility that this
143
Fig. 22. Predicted normal optical depths for measurements at B = 90 degrees at
10 km resolution compared to the λ Sco occultation, which measured the highest overall optical depths in the B ring at most locations. Each location with a
self-gravity wake model solution is marked by a + on the red curve. Regions
of very high optical depth have no solution because there are not enough measurements. The region from R = 114,500 to 115,200 km is unusual in that the
optical depth is not extreme, but the self-gravity wake model failed to find solutions that met our criteria for a good fit.
Fig. 23. Predicted normal optical depths for measurements at B = 90 degrees
at 10 km resolution compared to the λ Sco occultation and the largest measured
normal optical depth at each location.
trend is an artifact of the model in that it may be that close
packing of the wakes is the only way for this regular geometric
model to produce higher optical depths. However, it is difficult
to reconcile the observations with an alternative explanation
(such as a locally homogeneous distribution of closely packed
particles). Light from faint stars is seen through the B ring at
very low incidence angles (B) where the inferred normal optical depth is much greater than unity. Virtually empty gaps in
the ring are the only explanation we have for this transparency
in low-B observations. The gaps between self-gravity wakes fit
the requirements of the observations.
Photometric models of the rings must include the wake
structure. The wakes make the optical depth for both incident
and emerging rays from the rings dependent on both B and φ.
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J.E. Colwell et al. / Icarus 190 (2007) 127–144
For example, the observer may be looking across the wakes and
see a relatively large ring cross-section, but if the Sun direction
is parallel to the wakes there will be a small illuminated crosssection and the ring would appear darker than an observation at
the same phase, incidence and emission angles but with the Sun
shining across the wakes. Geometry-dependent optical depths
can be estimated from the self-gravity wake model and parameters presented here for arbitrary viewing geometries. To a lesser
extent the wake structure will also affect the multiple scattered
light.
Further observations at a range of values of B and φ will
enable tests of models with fewer constraints than the one described here and will help refine the variations of the fine structure in the B ring with other global properties of the ring, such
as composition, surface mass density, and location.
Acknowledgments
This work was supported by NASA through the Cassini
project. G.R.S. was supported by NASA under grant NNG06GG46G issued through the Office of Space Science Planetary Geology and Geophysics Program. We thank Jürgen
Schmidt and an anonymous referee for detailed reports that
greatly improved this paper. We thank Brad Wallis, Jeff Cuzzi,
Phil Nicholson, and the Cassini Rings Working Group for
their invaluable assistance in making these observations a reality.
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