GEOPHYSICAL RESEARCH LETTERS, VOL. 33, L07201, doi:10.1029/2005GL025163, 2006
Self-gravity wakes in Saturn’s A ring measured by stellar occultations
from Cassini
J. E. Colwell,1 L. W. Esposito,1 and M. Sremčević1
Received 8 November 2005; revised 7 February 2006; accepted 17 February 2006; published 1 April 2006.
[1] An azimuthal brightness asymmetry in Saturn’s A ring
is caused by ephemeral agglomerations that continually
form under the mutual gravity of the ring particles only to
be torn apart by Keplerian shear. We calculate the shape and
spacing of the self-gravity wakes from Cassini stellar
occultations. The wakes are highly flatttened structures,
with height/width ratio of 0.15 to 0.37, increasing outward
across the A ring. The spacing between wakes increases
with their height from a low value in the inner A ring of less
than the wake width to >3 times the wake width in the outer
third of the ring. The opacity of gaps between wakes also
increases in the outer part of the ring where the wakes appear
to be less coherent than in the inner and middle A ring. We
calculate the vertical opacity of the A ring is 15– 35%
higher than previously reported. Citation: Colwell, J. E.,
L. W. Esposito, and M. Sremčević (2006), Self-gravity wakes in
Saturn’s A ring measured by stellar occultations from Cassini,
Geophys. Res. Lett., 33, L07201, doi:10.1029/2005GL025163.
1. Introduction
[2] The particles in Saturn’s main rings range in size from
centimeters to meters and collide at speeds <1 cm/s every
few hours. Tides from Saturn prevent them from accreting
into a moon. Ephemeral agglomerations of ring particles can
form near the Roche limit if the ring mass density is large
enough. Imaging observations of Saturn’s A ring have
revealed an azimuthal brightness asymmetry where the
reflectance of the rings has maxima separated by
180 degrees in ring longitude referenced to the Saturnobserver line and minima in between [e.g., Camichel, 1958;
Ferrin, 1975; Reitsema et al., 1976; Lumme and Irvine,
1976; Lumme et al., 1977; Thompson et al., 1981; Gehrels
and Esposito, 1981]. The amplitude of the asymmetry varies
across the A ring and reaches a peak near the middle of the
ring [Dones et al., 1993]. Colombo et al. [1976] proposed
transient agglomerations of ring particles similar to stellar
gravitational wakes in galaxies [Toomre, 1964], and Dones
and Proco [1989] showed that these spiral density wakes
reproduce the azimuthal brightness asymmetry in Voyager
images. Numerical simulations confirm the formation of
these density enhancements [Salo, 1992]. If the mass
density of the disk is high enough, gravitational instability
leads to the formation of clusters of particles which are
sheared apart, leading to a characteristic orientation that is
canted about 20 degrees from the local azimuthal direction
[Porco et al., 1999, 2001, 2003; Salo and Karjalainen,
1
Laboratory for Atmospheric and Space Physics, University of
Colorado, Boulder, Colorado, USA.
Copyright 2006 by the American Geophysical Union.
0094-8276/06/2005GL025163$05.00
2003; Salo et al., 2004]. While these structures are not
actually ‘‘wakes’’ (a ‘‘turbulent condition of the air or other
fluid left behind by a body moving through it’’, Webster’s
Third New International Dictionary; there is no body
moving through the ring to create these clumps), that
nomenclature has been used in the literature for three
decades. We therefore call them self-gravity wakes to
emphasize the role of interparticle gravity and to distinguish
them from satellite wakes which are also gravitational, but
which depend on the gravity of a distant moon and not the
self-gravity of the ring.
[3] The preferred orientation of the self-gravity wakes
produces the azimuthal brightness asymmetry which has
also been seen in transmitted microwave radiation [Dunn et
al., 2004] and radar echoes [Nicholson et al., 2005a]. Here
we model the azimuthal transparency asymmetry of the
A ring seen in Cassini ultraviolet observations of stellar
occultations of the rings. The same phenomenon has also
been observed in infrared observations of stellar occultations [Nicholson et al., 2005b] and radio occultations
[Marouf et al., 2005].
2. Observations
[ 4 ] The Cassini Ultraviolet Imaging Spectrograph
(UVIS) includes a High Speed Photometer (HSP) that
measures the intensity of starlight between 110 and
190 nm with a 2 – 8 ms sampling interval [Esposito et al.,
2004]. 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]. The paths of three of
these occultations were chords across the rings that made
two radial cuts across the A ring. Combined with the
Voyager PPS stellar occultation [Lane et al., 1982] and a
solar occultation observed by the Cassini UVIS, this gives
13 measurements of the A ring optical depth at ultraviolet
wavelengths. 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 (f, where f = p/2 when the line of
sight is tangent to the rings at the occultation point).
[5] If the ring particle positions are uncorrelated, then the
normal optical depth from a given occultation is given by
I0
tn ¼ m ln
I b
ð1Þ
where I0 is the unocculted intensity of the star, I is the
measured signal, b is the background signal (primarily
sunlight scattered by the rings), and m = sin(B) is the
projection factor to convert from line of sight opacity to
normal (vertical) optical depth. If particles or clumps of
particles in the rings have a preferred alignment, however,
then the opacity also varies with f. The effect on the
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on the value of I0. We bin the occultations to an effective
resolution of 4 km to increase the signal to noise. The
sampling size is thus large compared to the scale of the selfgravity wakes (<100 m, see below) and small compared to
the radial scale of optical depth variations in the A ring far
from density and bending waves. For the fainter stars some
regions of the A ring were opaque so that the number of
useful measurements varied with position in the rings due to
ring opacity and star brightness as well as data dropouts.
Figure 1. Geometric model of self-gravity wakes in
Saturn’s rings illustrating the model parameters that affect
the measured ring opacity. B is the angle of the line of sight
out of the ring plane, and f– fwake is the viewing angle f
with respect to the wake alignment, measured in the ring
plane.
measured opacity from aligned structures depends on the
shape, spacing, and relative opacity of these structures
(Figure 1). When the line of sight from Cassini to the star is
aligned with the self-gravity wakes (f– fwake 0, p), less
starlight is occulted than when the view is across the wakes
(f –fwake ±p/2). The change in opacity with viewing
angles f and B depends on the parameters identified in
Figure 1.
[6] We calculate the normal optical depth across the A
ring for each occultation. The UVIS occultations span a
useful range of values of B (9.5 to 32.2) and f (33.3 to
348.8 with a gap in coverage between 146 and 258 (the
planet blocks the view of the rings at low B and f 180))
to probe the self-gravity wakes. Two occultation profiles
from the same star together with a third profile roughly
aligned with the wakes illustrate the dramatic effect of f on
the measured opacity (Figure 2). Most prominent features in
the A ring optical depth are density waves launched at the
locations of orbital resonances with nearby moons, primarily Janus, Pandora, and Prometheus, orbiting just outside the
rings. Some of these waves are visible as spikes in the
optical depth profiles in Figure 2. In regions of high optical
depth the star signal can be completely occulted, depending
Figure 2. Three optical depth profiles measured by the
Cassini UVIS of Saturn’s A ring show variations in opacity
of up to a factor of two due to viewing geometry with
respect to aligned self-gravity wakes in the ring. The upper
two profiles are measurements of the same star (26 Tau,
B = 21) at f = 88 ± 2 (middle) and f = 327 ± 2 (top). The
lower profile is of the star Delta Aquarius (B = 29, f =
52 ± 1). All three profiles are shown at a radial resolution
of 10 km.
3. Model and Results
[7] We model the self-gravity wakes as regularly spaced,
aligned structures in the rings with a normal opacity twake,
an alignment relative to the local radial direction of fwake,
dimensions H, W, and L, and spacing, S, as shown in
Figure 1, and with normal opacity between the self-gravity
wakes tgap, this model is conceptually the same as that of
Dunn et al. [2004]. We calculate the attenuation of starlight,
I/I0, by averaging the attenuation of individual rays traversing
a ring with periodic self-gravity wakes with the parameters
shown in Figure 1. The attenuation of each ray is computed
from its path length through the wakes and the inter-wake
gaps and the values of twake and tgap. The path lengths
depend on the viewing angles f and B and the position of the
ray within the model field of view which spans W + S. The
average attenuated signal, hIi, is then converted to a model
optical depth, tm, via tm = m ln (I0/hIi).
[8] Our results are insensitive to the value of L, provided
L > W, due to the absence of measurements at very low
values of B, so we assume L is infinite and use a onedimensional set of N = 100 rays to get the best fit values for
the remaining self-gravity wake parameters. We compute
the wake parameters which provide the best fit to the
measured opacities, ti, by minimizing D:
D¼
13
X
ðti tm Þ2 :
ð2Þ
i¼1
The model reproduces the amplitude of the optical depth
variations with viewing angle for the range of B sampled in
this initial set of occultations. The variation of D with four
model parameters for a ring radius of 128,000 km is shown
in Figure 3. These show that our best-fit values for H and
Figure 3. Variation of D for model parameters holding all
other parameters fixed at the best-fit value at a radial
location of 128,000 km.
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tgap are essentially upper limits and our best-fit values for
twake are lower limits: D increases slowly with increasing
(decreasing) twake (H, tgap) beyond the minimum value. We
are not able to place constraints on the length, L (except
L > W), of the self-gravity wakes because the path length of
the occultation line of sight through the rings is shorter than
the wakes at the values of B sampled. Values of fwake are
determined to ±5 degrees. Our best-fit values of H/W and
tgap are shown in Figure 4 for locations where D < 0.01.
[9] We find that the self-gravity wakes are highly flattened structures, with H varying from 0.15W near the
inner edge of the A ring to H = 0.37W beyond the Encke
gap at 133,400 km (Figure 4). The high optical depths
between 122,000 km and 124,000 km prevent a determination of wake properties except in the moderate opacity
plateau at 123,200 km where the height/width ratio and
interwake spacing S/W are smallest. The separation, S, of the
wakes tracks the wake heights, H, and the value of S/H is
nearly constant across the ring with hS/Hi = 6.8 in the inner
and middle A ring and hS/Hi = 7.7 beyond the Encke gap.
Our model references these dimensions to the wake width,
W. The physical scale of the wakes (W + S) is predicted to
be 30 m based on the gravitational instability length scale
[Toomre, 1964]. Excess variance in the occultation data
provides direct measurements of the wake spatial scales that
are consistent with this predicted scale [Sremčević et al.,
2005].
[10] Two of the stellar occultation measurements did not
extend to the outer edge of the A ring. The wake properties
beyond the Encke gap are therefore computed from a fit to
11 or 12 measurements, depending on the exact orbit radius,
while for most of the rest of the ring there are 13 measurements. We have computed the wake properties throughout
the A ring excluding the measurements from these two
occultations for the 1481 locations in Figure 4. The radial
trends of increasing H/W, S/W and tgap are unchanged with
the reduced set of measurements, but the average value of D
increases by a factor of 1.9 and scatter in the wake
parameters increases in some of the ring. The increased
scatter in H/W at R > 132,000 km (Figure 4) is therefore due
in part to the smaller number of occultation measurements
in that part of the ring.
[11] Both the height and the separation increase rapidly
with distance from Saturn while the ring optical depth
decreases out to the strong density wave due to the Janus
4:3 resonance at 125,300 km. Both parameters increase
slowly and approximately linearly from that point to the
Encke gap. We find that the wake alignment varies across
the ring, with fwake = 64.4 ± 4.4 interior to the Janus 4:3
density wave, fwake = 66.6 ± 4.9 between the Janus 4:3 and
Janus 5:4 (R = 130,700 km) density waves, fwake = 62.7 ±
5.0 between the Janus 5:4 wave and the Encke gap, and
fwake = 57.2 ± 6.4 beyond the Encke gap. The student’s T
test shows that the decrease in fwake from the Janus 4:3
density wave to the outer edge of the ring is statistically
significant at the level p < 1 – 109 (probability p of
occurrence by chance). Smaller values of fwake correspond
to wakes that are canted at a larger angle to the local
azimuthal direction. The more radial alignment of the wakes
at larger distances from Saturn (smaller fwake) is consistent
with weaker Keplerian shear there. Recent radar observations place the orientation of the self-gravity wakes at
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Figure 4. Calculated height/width ratio of self-gravity
wakes (asterisks), optical depth of the gaps between the
wakes (diamonds), and the measured normal optical depth
of the A ring from the Cassini UVIS Sigma Sagitarius
occultation (line). Model values are only shown for
locations where D < 0.01, and have been median filtered
by 5 points for clarity.
fwake = 67 ± 4 and 247 ± 4 [Nicholson et al., 2005a],
consistent with our results and with earlier reflected light
imaging observations [Lumme and Irvine, 1976; Lumme et
al., 1977; Thompson et al., 1981; Dones et al., 1993],
theoretical expectations [Colombo et al., 1976; Franklin
and Colombo, 1978], and numerical simulations [Daisaka
and Ida, 1999; Salo et al., 2004]. Measurements by the
Cassini Composite Infrared Spectrometer (CIRS) instrument in azimuthal scans of the A ring cover virtually all
values of f allowing a determination of an average value of
fwake = 69 ± 1 and 249 ± 1 [Ferrari et al., 2005].
[12] We find tgap 0.12 across the inner two-thirds of
the A ring out to the strong Janus 5:4 density wave where
there is a local enhancement in the total optical depth and
gap optical depth (Figure 4). Between the Janus 5:4 resonance and the Encke gap the mean gap opacity is
htgapi 0.16, and beyond the Encke gap htgapi 0.24.
Thus the wakes are not as well organized in this part of the
ring as in the middle part of the ring. Our model finds lower
limits to the wake optical depths which are not tightly
constrained (Figure 3), though the mean values of twake
peak in the middle of the A ring where the self-gravity
wakes are most prominent and the azimuthal asymmetry is
largest. Stochastic variations in the wake structure due to
their weaker nature in the outer A ring likely contribute to
scatter in the derived wake properties (Figure 4).
[13] Despite the large number of density waves in the
outer A ring, the waves damp within 30 km, and most of
the ring, even beyond the Encke gap, is unperturbed by
density waves. The length scale of the self-gravity wakes is
<100 m, so the presence of density waves is not the cause of
the breakdown of the wakes. The dispersion of density
waves in the A ring shows a decrease in the surface mass
density from >40 g-cm2 in the inner and middle A ring to
<20 g-cm2 beyond the Encke gap [Spilker et al., 2004].
This drop is consistent with a decrease in the formation of
self-gravity wakes. The optical depth in this outer part of the
ring is higher than in the middle of the ring, so the lower
surface mass density is due to fewer large particles. Our
results show that self-gravity wakes beyond the Encke gap
account for a smaller fraction of the surface area of particles
in the ring than in the middle and inner part of the A ring,
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perhaps due to the lower self-gravity of the ring or higher
interparticle velocities from perturbations by nearby moons.
[14] The true vertical optical depth of the ring, that which
would be observed at B = 90, can be estimated using our
derived properties of the self-gravity wakes. From a pole-on
view of the rings the opacity has no f dependence, and the
average optical depth over a wake wavelength is the true
normal opacity that should be used to estimate the number
density in numerical simulations of the rings. We use the
calculated wake parameters to estimate the optical depth of
the ring that would be observed at large values of B. We
predict that the true normal optical depth of the A ring is
higher than that measured in the cross-wake occultations,
such as the Cassini Sigma Sagitarius occultation (Figure 4),
by 35 per cent interior to the Janus 5:4 density wave. The
predicted enhancement drops to 15 per cent beyond the
Encke gap. Cassini occultations at high sub-spacecraft
latitudes beginning in late 2006 will confirm or refute these
predictions.
[15] Acknowledgment. This work was supported by NASA through
the Cassini project.
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