Icarus 216 (2011) 280–291
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Icarus
journal homepage: www.elsevier.com/locate/icarus
Morphology and variability of the Titan ringlet and Huygens ringlet edges
Richard G. Jerousek a, Joshua E. Colwell a,⇑, Larry W. Esposito b
a
b
Department of Physics, University of Central Florida, Orlando, FL 32816, United States
Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80309, United States
a r t i c l e
i n f o
Article history:
Received 8 April 2011
Revised 1 September 2011
Accepted 3 September 2011
Available online 12 September 2011
Keywords:
Planetary rings
Saturn, Rings
Occultations
a b s t r a c t
We present a forward modeling approach for determining, in part, the ring particle spatial distribution in
the vicinity of sharp ring or ringlet edges. Synthetic edge occultation profiles are computed based on a
two-parameter particle spatial distribution model. One parameter, h, characterizes the vertical extent
of the ring and the other, d, characterizes the radial scale over which the ring optical depth transitions
from the background ring value to zero. We compare our synthetic occultation profiles to high resolution
stellar occultation light curves observed by the Cassini Ultraviolet Imaging Spectrograph (UVIS) High
Speed Photometer (HSP) for occultations by the Titan ringlet and Huygens ringlet edges.
More than 100 stellar occultations of the Huygens ringlet and Titan ringlet edges were studied, comprising 343 independent occultation cuts of the edges of these two ringlets. In 237 of these profiles
the measured light-curve was fit well with our two-parameter edge model. Of the remaining edge occultations, 69 contained structure that could only be fit with extremely large values of the ring-plane vertical thickness (h > 1 km) or by adopting a different model for the radial profile of the ring optical depth.
An additional 37 could not be fit by our two-parameter model.
Certain occultations at low ring-plane incidence angles as well as occultations nearly tangent to the
ring edge allow the direct measurement of the radial scale over which the particle packing varies at
the edge of the ringlet. In 24 occultations with these particular viewing geometries, we find a wide variation in the radial scale of the edge. We are able to constrain the vertical extent of the rings at the edge to
less than 300 m in the 70% of the occultations with appropriate viewing geometry, however tighter constraints could not be placed on h due to the weaker sensitivity of the occultation profile to vertical thickness compared to its sensitivity to d.
Many occultations of a single edge could not be fit to a single value of d, indicating large temporal or
azimuthal variability, although the azimuthal variation in d with respect to the longitudes of various
moons in the system did not show any discernible pattern.
Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction
Saturn’s rings contain a number of ringlets and gaps with dramatically sharp edges over which the measured optical depth
drops from s > 1 to s = 0 over a radial scale on the order of tens
of meters. In general, the natural tendency for an unperturbed
ring or ringlet edge is to spread due to collisions with neighboring
ring particles on the ring side and an absence (or decreased number) of balancing collisions. This produces an outward-directed
angular momentum flux caused by the dissipative effects of particle collisions (Lewis and Stewart, 2000). Numerical simulations
suggest that the majority of the spreading occurs in the outermost few tens of meters (Weiss, 2005). Numerical models of
edges that are affected by orbital resonances with moons also
show that the majority of the drop in surface density occurs over
⇑ Corresponding author. Fax: +1 407 823 2012.
E-mail address: colwell@ucf.edu (J.E. Colwell).
0019-1035/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.icarus.2011.09.001
only the outermost few tens of meters (Borderies et al., 1982).
Borderies et al. (1984) also found that the reversal of angular
momentum flux, which can occur at density peaks near a perturbed edge, can evolve a sharper edge even as the ring itself
spreads.
The morphology of these sharp edges is not solely governed by
viscous spreading, however. Edges may also exhibit large variations in surface density, vertical thickness, and azimuthal structure
due to the gravitational influence of distant perturbers. While the
work of Borderies et al. (1982) primarily focused on edges near
Lindblad resonances, Lewis and Stewart (2000) specifically targeted shepherded ring edges such as the Encke gap edges. In the
case of shepherded ring edges, the compression of particle streamlines downstream from a satellite will create kinematic wakes
which drastically alter the surface density of the ring-plane and
even cause vertical splashing of material at wake peaks (Lewis
and Stewart, 2000). By more accurately measuring quantities that
describe the morphology of these edges we hope to better under-
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450
(a)
HSP Counts
(a)
BET CEN (104) Ingress
300
150
0
77896.0 77896.1 77896.2 77896.3 77896.4 77896.5
Ring−Plane Radius (km)
500
(b)
(b) ZET ORI (47) Egress
HSP Counts
400
300
200
100
0
117785.0
Fig. 1. Differing radial resolution between occultations. The drop in light levels as
the Huygens ringlet inner edge occults one of the binary stars in a-Crucis (100). (a)
Covers 10 points and covers about three points in the occultation of a-Arae (32) (b).
stand the dynamics and evolution of the rings on the scale of the
largest common ring particles.
The sharpness of an edge is an indicator of the elasticity of
ring particle collisions. Elastic collisions would lead to a broader
edge region, while inelastic collisions would lead to a greatly
diminished radial scale over which the optical depth varies at
the ring edge. In the A Ring, numerical simulations combined
with photometric modeling of self-gravity wakes indicate velocity-dependent normal coefficients of restitution of en 0.1 (Porco
et al., 2008). Experimental studies of coefficients of restitution at
the low speeds of ring particle collisions (v < 1 cm/s) produce results with virtually any value, depending on the properties of the
surfaces of the colliding particles. Impacts into regolith result in
en 0.1, or even sticking (Colwell, 2003) while impacts between
hard ice particles at low speeds are highly elastic with en > 0.5
(Supulver et al., 1995). The morphology of edges, as with other
small-scale structures in the rings such as self-gravity wakes, offers a window into the structure and properties of individual ring
particles. This in turn will help constrain the mass of the rings
and their rate of collisional evolution, both key factors in understanding the origin and long-term evolution of the rings (Esposito,
2010; Cuzzi et al., 2010).
Stellar occultations can directly constrain some properties of a
ring at its edge. Under the assumptions that a ring has an absolutely sharp rectangular edge cross-section with particles small
compared to the ring thickness and a uniform particle packing density up to the edge, the vertical extent of the ring-plane, h, can be
determined from the apparent sharpness of the edge (Lane et al.,
1982):
117790.0
117795.0
117800.0
Ring−plane Radius (km)
Fig. 2. Binary star occultations showing two edge profiles in a single occultation
light curve due to the angular separation between the individual stars. (a) The
occultation of the stars b-Centauri A and B (104) by the Titan ringlet outer edge with
a projected radial separation of 150 m. (b) The occultation of the stars f-Orioinis A
and B (47) by the Huygens ringlet inner edge with a projected radial separation of
11 km.
ds
Ds
¼
dr h tan jh0 j
ð1Þ
where s is the normal optical depth, r is the ring-plane radius and h0
is the angle between the line-of-sight and the ring plane normal
vector projected into the plane formed by the radial vector and ring
plane normal vector. (Below we use a, which is the complement of
h0 .) Using this method with Voyager PPS data of the occultation of d
Scorpii, an upper limit of 200 m was placed on the vertical extent of
the rings (Lane et al., 1982). Temporal sampling of the Voyager Photopolarimeter Subsystem (PPS) limited the spatial resolution to
100 m. The Cassini Ultraviolet Imaging Spectrograph (UVIS) measures stellar occultations with millisecond temporal sampling with
its High Speed Photometer channel (HSP) translating to 10 m sampling resolution in ring plane radius (Esposito et al., 1998, 2004).
These occultations span a range of viewing geometries with ring
plane elevation angles, B, ranging from 2.7° to 70°. With these data
we resolve the structure and morphology of edges of optically thick
rings at scales <100 m for the first time. The high spatial resolution
and large number of occultations at different geometries allows us
to model the edge morphology in the vertical and radial dimensions
and look for patterns in the azimuthal dimension.
In this paper we present the various morphologies observed in
Cassini UVIS HSP occultation data of the Titan1 and Huygens ringlet
inner and outer edges classifying these morphologies into three ba1
The Titan ringlet is located in the Colombo gap and is also referred to as the
Colombo ringlet. Its resonant association with Titan has led it to be called the Titan
ringlet.
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(a)
(b)
(c)
(d)
Fig. 3. Examples of occultations with symmetric edges. These edge profiles show a clearly defined and well-resolved transition while maintaining relatively consistent signal
levels interior and exterior the boundary. Occultations by (a) the inner edge of the Titan ringlet, (b) the inner edge of the Titan ringlet, (c) the inner edge of the Huygens
ringlet, and (d) the outer edge of the Huygens ringlet.
sic classes (Section 2). These edges were selected for our initial analysis because the ringlets are optically thick (so the contrast across
the edge is large even for occultations of faint stars) and also because
they typically do not have as complicated structure as the outer
edges of the A and B rings, for example. We describe two twoparameter optical depth models for sharp ring edges, both of which
contain a logistic (sigmoid shape) radial distribution of ring particles,
but with different vertical profiles. We demonstrate the sensitivity of
these models to viewing geometry as well as the two independent
parameters, h and d, representing the vertical thickness of the ring
and the characteristic radial length scale of the edge, respectively
(Section 3). We use a simple ray-tracing method to generate synthetic occultation light curves to find the best-fit values of the independent parameters, h and d (Section 4). We are also able to make
direct (model-independent) measurements of the radial scale of
the edge in occultations where the occultation line-of-sight and
track in the ring plane made the light curve insensitive to ring thickness. Our results, discussed in Section 5, show that a single set of
parameters could not be fit to any of the four edges studied, suggesting complex azimuthal structure.
2. Observations
2.1. Occultation data
We include 103 of the 115 occultations recorded by the Cassini
UVIS HSP as of October 2010 (Colwell et al., 2010). The remaining
12 occultations had too low signal-to-noise for analysis of our se-
lected ring edges. The number of useful occultations of any particular edge varied due to whether or not the quality of the data in the
vicinity of the edge warranted fitting by our model. In addition, not
all occultations include both ringlets. In this paper, we label occultations by the star followed by the rev number in parenthesis, i.e.
a-Virginis (34). ‘‘Rev’’ refers to the number of the orbit or revolution of Cassini around Saturn.
We chose to focus our attention on edges of relatively opaque
ringlets. For this initial application of the model we examine the
inner and outer edges of the Huygens and Titan ringlets. The Huygens ringlet is a non-circular ringlet in the inner Cassini Division
(within the Huygens Gap) with a variable width of 20 km. The Titan ringlet (also referred to as the Colombo ringlet) resides in the
Colombo gap in the inner C ring and also has a variable width ranging from 16 km at periapse to 34 km at apoapse. The ringlet is
associated with the apsidal resonance with Titan where the apsidal
precession rate of ring particles matches Titan’s mean motion (Porco et al., 1984; Nicholson and Porco, 1988).
The raw occultation data are measured in photon counts per
integration period, with typical integration periods of 1 or 2 ms
(Esposito et al., 2004; Colwell et al., 2010). The data constitute a
one-dimensional trace of the transparency of the ring material
along the occultation path in the ring plane. Since the raw data
points are evenly spaced temporally, the varying radial velocity
along the occultation path produces differing radial spacing between data points. Over the span of a ring edge, this variation is
usually negligible. In some occultations the line-of-sight path to
the star may trace a chord across the ring plane, offering a period
of time when the radial velocity of the occultation track ap-
R.G. Jerousek et al. / Icarus 216 (2011) 280–291
283
(a)
(b)
(c)
(d)
Fig. 4. Examples of occultations with asymmetric edges. These edge profiles show a rapid transition from the unocculted signal in the gap followed by a more gradual
decrease in the count rate further into the ring. Occultations by (a) the inner edge of the Huygens ringlet, (b) the inner edge of the Huygens ringlet, (c) the outer edge of the
Titan ringlet, and (d) the outer edge of the Titan ringlet.
proaches zero and the radial resolution of the HSP data increases
to >104 points per kilometer (Colwell et al., 2010). An example of
the differences in radial resolution between occultations is shown
in Fig. 1. The occultation geometry was calculated as described in
Colwell et al. (2006, 2010).
Many of the occultations observed with the UVIS HSP were occultations of binary stars. In these cases, the same ring structure
was measured twice, offset by a small amount in the ring plane.
In cases where the apparent shift in ring plane radius of the edge
was sufficiently large (>250 m), the two observations were treated
as separate occultations and compared to our model independently. In cases where the edge profiles from the two components
of the binary star are less than 250 m, we model the full light curve
including the light from both stars. Examples of each of these binary star occultations are shown in Fig. 2.
Asymmetric edges are defined by a relatively abrupt transition in
signal at the gap-edge boundary but show an asymmetric,
slower decline in the rate of change of optical depth on the ring
side of the edge transition (Fig. 4). This may be indicative of the
radial structure of the ringlet near the edge or of a large ring
thickness (Section 3.3). Asymmetric edges accounted for
roughly 20% of the edge profiles which were fit by our twoparameter model.
Complex edges show structure that cannot be fit by our simple
continuous edge model (Fig. 5). Roughly 10% of the individual
stellar occultations fell into this category.
3. Edge model
3.1. Description of optical depth model
2.2. Edge morphology
After experimenting with fitting our model to the data (Section 3), we found that the radial profiles of the edges can be
grouped into three broad categories based on their shapes.
Symmetric edges are edges in which the unocculted signal interior and exterior to the half-signal level are relatively symmetric (Fig. 3). Occultations of these edges show an abrupt but
resolved transition between an easily defined maximum and
minimum signal not deviating significantly from the respective
mean signal on either side of the edge. Symmetric edges
account for roughly 70% of the analyzed edge profiles.
We use a forward-modeling approach to create synthetic ring
edge occultation light curves using a ray-tracing technique and a
two-parameter optical depth model. We then compare the synthetic light curves to HSP occultation data in order to determine
the best-fit values of the edge model parameters. This technique
allows us to model the ring edge in terms of these parameters from
a broad family of functions without needing to examine, a priori,
the physics governing the dynamics of particles on the scale of
the edge transition region.
We assume a rectangular cross-section geometry for the ring
edge and vary the particle number density within this slab as a
function of ring-plane radius and height above the mid-plane of
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R.G. Jerousek et al. / Icarus 216 (2011) 280–291
HSP Counts
300
(a) EPS CEN (65) Ingress
200
Fig. 7. The slab represents an optically thick ring or ringlet. The red line, l0 (t) is the
projection of the line-of-sight path from Cassini to the occulted star into the radial
direction. Within the rectangular geometry of the ring slab we vary the optical
depth as a function of r and z. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
100
0
77851.6 77851.8 77852.0 77852.2 77852.4 77852.6
Ring−Plane Radius (km)
50
(b) DEL PER (36) Egress
HSP Counts
40
30
20
10
0
77871.1
77871.3
77871.6
77871.8
Ring−Plane Radius (km)
77872.0
Fig. 5. Examples of occultations of complex edges. These edge profiles have
structure that is inconsistent with our model (Section 3). Occultations by (a) the
inner edge of the Titan ringlet and (b) the outer edge of the Titan ringlet.
intensity point of the occultation light curve. The parameter d is a
radial length scale, and sringlet is the mean optical depth away from
the ring edge transition region. In this simple model, the optical
depth does not vary with longitude on the scale of the edge transition region. The radial length scale over which the ring optical
depth drops from its maximum value to the gap value is several
times the e-folding-length d. Weiss (2005) modeled unconstrained
edges and found that a sharp edge would diffuse into a profile defined by the complementary error function. This has the same sigmoid shape as our logistic function, and the argument of his error
function includes the ring viscosity and time. We chose a simple
parametric steady-state description since we have snapshots of
each edge profile, and because the edges are presumably confined
by some mechanism.
We fit two models to the data: in the uniform model we assume
that there is no vertical variation in particle packing with a slab
thickness of h, and in the gaussian model we assume a truncated
gaussian for the vertical distribution of material, centered symmetrically within the ring-slab, with a full-width of h, representing the
region extending 2 r both above and below the ring mid-plane.
The amount of light transmitted through our model ring depends on the spatial model of the ring density described above,
as well as the orientation of the line-of-sight of the starlight to
the ring plane and to the radial direction. Two angles are relevant
in the computation: the occultation elevation angle, B, which is
measured from the ring plane to the line-of-sight to the star, and
the azimuthal angle, /, which is measured from the radius vector
to the projection of the line-of-sight onto the ring plane, dlr
(Fig. 6), with |/| < p. Thus, an occultation that crosses the edge with
a line-of-sight tangent to the edge has / = p/2. Such occultations
contain no information on the vertical thickness of the ringlet,
and the observed radial profile is the actual radial profile of ring
Fig. 6. The slab represents a marginally optically thick ring or ringlet. The red line is
the line-of-sight path from the Cassini UVIS HSP device to the occulted star. a is a
projection of the ring plane elevation angle, B, onto the plane defined by the radial
vector and a vector normal to the ring plane. (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
the ring. We model the radial optical depth variation in the vicinity
of the edge with a logistic function:
h
sðrÞ ¼ sringlet 1 þ exp
r r i1
edge
d
ð2Þ
where r redge is the radial distance of the occultation data point at
ring plane radius r from redge, defined as the location of the half-
Fig. 8. Z = R(I(h = 1m) I(h))2 versus h for the s(r) and s(r, z) models with a = 30°,
d = 30 m and h spanning from 1 to 100 m. By plotting the sum of the squares of the
differences between light curves, we can see smaller variations in the light curves of
varying h. While both models produce essentially identical results over the range of
h, the similarities diverge slightly for larger h values.
R.G. Jerousek et al. / Icarus 216 (2011) 280–291
285
tan jBj
cos j/j
ð6Þ
tanðaÞ ¼
as shown in Fig. 6.
The particle size distribution is assumed to be uniform and
small on the scale of the vertical and radial extent of the ring
boundary. That is, we are modeling the ring as a continuous medium defined by s(r, z) where the optical depth of each infinitesimal
unit volume of the ring is independent of viewing geometry. We
also assume for simplicity that there is no radial variation of the
thickness of the ring (h) over the radial extent of the edge transition that we model. With these constraints, the slant-path optical
depth of an infinitesimal volume element of a cross section of
the ring with total normal optical depth sn and height h is a function of the distance from the ringlet edge, r redge and the height, z,
from the bottom of the ring:
Fig. 9. 20 light curves overplotted for a relatively low viewing angle of a = 15°. The
10 dashed (red) curves represent various values of h, spanning from h = 5 m to
h = 100 m while d was fixed at d = 10 m. The 10 dotted (blue) curves correspond to
various values of d, spanning from d = 5 m to 100 m while h was fixed at h = 1 m.
The greater contribution of increasing d than h to the radial softening of the light
curves can clearly be seen. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
ds ¼
sn
h
h 2 =2 2
eðz2Þ
r
rredge 1
0
1þe d
dl
06z6h
ð7Þ
The path length dl0 across infinitesimal ring elements confined to
the plane defined by the radial vector and ring plane normal is then
given by
0
dl ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
1 þ tan2 ðaÞ dr
ð8Þ
The resulting optical depth for a particular viewing geometry is
computed by integrating along the radial projection of the line-ofsight through the ring plane:
sðrÞ ¼
Z
1
1 þ exp½ðr 0 r edge Þ=d
#
2 ,
h
0
2r2 dr
exp jr r 0 j tan a 2
r
Fig. 10. D(h, d) for the a-Arae (79) ingress occultation by the Huygens ringlet outer
edge. The valley in D singles out a best-fit value for d, but there is no significant
variation in D within the range of h. The surface, D(h, d), varies significantly between
occultations, sometimes only allowing for either lower or upper limits to be placed
on the parameter, d.
optical depth. For values of / closer to 0 or p, the vertical extent of
the ring affects the observed radial profile. In an effort to simplify
the ray-tracing integration, we define the rate of change in elevation of the line-of-sight with respect to the radial vector in terms
of a new angle, a:
tanðaÞ ¼
dz
0
dr
ð3Þ
where
dz ¼ dl sin jBj
ð4Þ
and
0
dr ¼ dl cos jBj cos j/j
ð5Þ
The infinitesimal path length dl is along the line-of-sight to the star.
The dependence of a on B and / is given by
h cot a
sn h cos a
" ð9Þ
where z has been replaced by r0 tan(a), and r is the ring-plane radius
of the particular occultation point on the side of the ring nearest to
the spacecraft, and redge is the location of the half-signal point defining the edge. In our two models, r = h/8 for the gaussian model and
r is infinite for the uniform model corresponding to no variation of
s with z.
pffiffiffiffiffiffiffiffi
Over the range of occultations, the Fresnel zone (F ¼ 2kd) for
ultraviolet light with a wavelength k 150 nm, at a distance, d,
from the HSP to the ring plane occultation point was typically
20 m, consistent with our treatment of the ring slab as a continuous medium since particles are at most a few meters in size.
Noticeable diffraction spikes were observed in several occultations
at ringlet edges including the egress occultation of the Huygens
ringlet inner edge by a-Crucis (104) with a Fresnel zone of
17.5 m (Fig. 1a) and the egress occultation of the Titan ringlet inner
edge by b-Centauri (104) with a Fresnel zone of 17.4 m (Fig. 2a).
However, in most of our edge profiles there is no obvious diffraction peak. While this does not rule out a contribution from diffraction to the observed light curve, diffraction would not explain the
complex or asymmetric light curves. In this first paper we are characterizing the edge morphology and azimuthal and temporal variability in the ringlet edges. The inclusion of diffraction in the model
might affect the best-fit values for d, but the magnitude and width
of the diffraction peak when observed indicates that this would be
a small effect.
3.2. Model occultation profiles
We calculate the model line-of-sight optical depth by numerically integrating Eq. (9) along the radial projection of the line-ofsight for each value of the ring plane radius of the occultation
point, r, in the neighborhood of redge as the ring moves in front of
the star. From this we calculate synthetic light curves, I(r), for di-
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R.G. Jerousek et al. / Icarus 216 (2011) 280–291
Fig. 11. Four plots of symmetric edge profiles with their best-fit model light curves (red) overplotted. Model light curves corresponding to d = 1=4 , ½, 2, and 4 times the best-fit
value are overplotted in dashed (blue) lines to give an indication of the uncertainty in our determination of d. In these four plots, the s(r) and the s(r, z) models returned
identical results for the best-fit h and best-fit d values. (a) an occultation by the outer edge of the Huygens ringlet, (b) an occultation by the outer edge of the Huygens ringlet,
(c) an occultation by the outer edge of the Titan ringlet, and (d) an occultation by the outer edge of the Huygens ringlet. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
Table 1
Mean values of best-fit model parameters.
Edge
Number of occultations
Range of |a| (deg.)
Mean h (m)
Mean d (m)
sringlet (normal)
Titan ringlet (I)
Titan ringlet (O)
Huygens ringlet (I)
Huygens ringlet (O)
45
45
87
87
32.64–89.28
32.56–89.29
13.75–89.57
13.75–89.57
93
81
81
72
9
10
10
13
1.51–4.88
1.79–4.81
0.56–4.17
0.54–4.15
(I): Inner edge. (O): Outer edge.
rect comparison with the data. Fig. 7 shows the apparent motion of
the line-of-sight path into a cross-section of the ring slab as a function of time. We calculate I(r) = exp(s(r)) for values of a between
0° and 90° in one degree increments with values of h and d between 1 and 100 m in 1 m increments. These synthetic light curves
span 1 km in ring plane radius and are calculated at 5-m radial
resolution.
We generated two libraries of these light curves: one with our
two parameter optical depth model, s(r, z), with r = h/8 (such that
99.9% of the particles were distributed within h/2 of the center of
the ring slab), and one with a uniform vertical particle spatial distribution, depending only on the logistic radial variation in optical
depth, s(r) (Eq. (2)).
3.3. Sensitivity of light curves to viewing geometry
For values of h < 100 m, the s(r, z) and s(r) models were virtually
indistinguishable for a > 15° (Fig. 8). We focus on the s(r, z) family
of curves for fitting the parameter h between 0 and 100 m. Fig. 8
compares the sensitivity of both models to h for constant values
of a and d. By plotting Z, the sum of the squares of the differences
between the points along the h = 1 m curve, I(h = 1 m) and I(h) for a
range of h values
Z¼
X
ðIn ðh ¼ 1mÞ In ðhÞÞ2
ð10Þ
n
We can see small variations between the s(r, z) and s(r) models
which grow with h, however the models produce nearly identical
light curves.
The sensitivity of the light curves to viewing angle, a, is greatly
diminished in synthetic light curves in which vertical variation in
optical depth was excluded, indicating the significance of the
gaussian portion of the optical depth function, however the sensitivity of either model to the variation of the independent parameters was greater at viewing angles less than 15°, of which we
have only one occultation, the f-Orionis (47) egress occultations
R.G. Jerousek et al. / Icarus 216 (2011) 280–291
287
actual radial extent of the edge transition, and h only affects the
observed transition by projection. However, even in geometries
that maximize the effect of h on the light curve profile, it is still
only weakly constrained by the data. The possible exception is
the case of the asymmetric edges, discussed further in Section 5.
The relative insensitivity of our model to h does, however, mean
that essentially all the symmetric edge profiles provide a direct
determination of d. The decreased sensitivity to the vertical thickness parameter, h, compared to that of d, even at the relatively low
viewing angle of a = 15°, is shown by the range of model light
curves with varying values of the two parameters in Fig. 9.
4. Results
4.1. Comparison to HSP data
For each occultation we rebin the model light-curve vectors for
the value of a that matches the occultation geometry and for all
values of h and d to the radial resolution of the Cassini UVIS HSP
data in the vicinity of the edge. Since the typical scale between
the optically thick ring and the transparent gap at the ringlet edge
is much less than 1 km, the radial resolution of the data is essentially constant over this subset of the data. We scale the model
light curves to the data by matching the ringlet and gap values
to the mean values of the HSP signal inside and outside of the ringlet, Iringlet and Igap respectively.
IðrÞ ¼ hIgap i hIringlet i esðrÞ þ hIringlet i
Fig. 12. Binary star occultations of the Titan ringlet inner edge and Huygens ringlet
outer edge with best-fit model light curves (red) overplotted. Synthetic curves
corresponding to d = 1=4 , ½, 2, and 4 times the best-fit value are overplotted in
dashed (blue) lines. (a) The best-fit parameters for the b-Centauri (102) occultation
of the Titan ringlet inner edge were h = 250 m and d = 0.7 m. (b) The best-fit
parameters for the k-Scorpii (44) occultation of the Huygens ringlet outer edge were
h = 75 m and d = 0.1 m. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
ð11Þ
We align the model curves to the HSP data by matching the
half-intensity point of the smoothed data to the model light-curve.
We find the best-fit values of h and d by minimizing the quantity,
D, where
D¼
N
X
ðIn;i Im;i Þ2
i
I2n;i
ð12Þ
In,i is the measured signal, and Im,i is the model signal, a function of h
and d, at point i.
For symmetric edge profiles, surface plots of D as a function of h
and d (Fig. 10) show a well-defined minimum for d, but negligible
sensitivity to h as discussed in the previous section. This is true for
all viewing geometries. Examples of fits to symmetric edges are
shown in Fig. 11. In the cases of symmetric edges the two models
produced identical results. We summarize the results of the s(r, z)
model in Table 1. The results from binary star occultations separated by more than 250 m in ring-plane radius are also included
in Table 1.
4.2. Uncertainties in the determination of h and d
Fig. 13. Individual best-fit d values for the s(r, z) model. Occultations at small a
result in a smaller maximum measurable optical depth so that the light-curve
saturates closer to the edge and gives smaller values of d. (I): Inner edge. (O): Outer
edge.
by the Huygens ringlet edges, which is discussed in detail in
Section 4.5.
The radial scale of the observed transition of the stellar signal at
an edge from unocculted to fully occulted depends much more on d
than on h. This is perhaps not surprising since d characterizes the
The extent of our certainty in best-fit values of h and d is best
represented by the variation in the values of D(h, d) (Eq. (12)) for
the various model light curves. The shape of the D(h, d) surface
plots between occultations vary from simple troughs (Fig. 10) to
more asymetric surfaces. In the cases where the surfaces are simple troughs, we define upper and lower limits on the determination
of the best-fit value of the parameter, d by the values d takes on
where D is approximately twice the value of the minimum of D.
Our certainty in best-fit values of d typically ranges from approximately one half to twice the best-fit value in occultations which allow us to determine both upper and lower limits (see for example
the curves in Figs. 11 and 12). Error bars in Figs. 13–15 were also
computed using this method.
Our accuracy in the determination of best-fit parameters is also
dependent on the brightness of the occulted star and occultation
viewing angle since the measurement of the signal from bright
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R.G. Jerousek et al. / Icarus 216 (2011) 280–291
Fig. 14. The d, g versus edge radius for the Titan ringlet inner edge (a), the Titan ringlet outer edge (b), the Huygens ringlet inner edge (c), and the Huygens ringlet outer edge
(d).
stars will still be distinguishable from the background signal further into the ringlet. In other words, occultations of faint stars or
at small viewing angles, a, result in a smaller maximum measurable optical depth so that the light-curve saturates closer to the
edge and gives smaller values of d, as illustrated in Fig. 13.
We found good fits to binary occultations of the ringlet edges
with h 100–200 m and d 0.1–5 m. Due to the uncertainty in
the ratio of signal strengths and the projected radial separation
of the two stars, best-fit models were computed by manually estimating the best-fit parameters and computing minima in D(h, d) in
the neighborhood of the estimations.
4.3. Fitting binary occultations with small radial separations
4.4. Direct measurements of radial extent of ringlet edges
Binary star occultations that are closely spaced in ring-plane radius, such as in Fig. 12, offer a unique opportunity to model the
edge at two locations that are also closely spaced in azimuth at
essentially the same time. (Binary stars whose components were
separated by more than 250 m in the ring plane could be treated
as isolated occultations following the procedure described in Section 4.1.) For closely spaced binaries, we calculate model light
curves using the same ray-tracing integration technique mentioned above, but with the star signal integrated along two parallel
line-of-sight paths to the instrument, separated in ring-plane radius by the apparent radial separation of the two stars. We assume
that the value of d is the same at the longitudes of both components of the binary star. Based on this assumption, we fit certain
binary occultations with d < 1 m in occultations with significant
variation in D at these low values. We find satisfactory fits to the
binary star light curves with this assumption, indicating that the
radial structure of the edge does not change significantly over
the azimuthal separation of the two stars. Azimuthal motion of
the ring particles during the time interval between the two stars
crossing the edge is typically 400 m. The exact separation in
the particle frame depends on the individual occultation geometry
and the projection of the two stars on the ring plane, but is less
than 1 km. The results of fitting occultations of closely spaced
binaries by the Titan and Huygens ringlet edges are summarized
in Table 2.
Several occultations resulted in viewing geometries nearly tangent to the ring edge. In these special cases we measure the change
in radial particle packing while we are completely insensitive to
the vertical thickness parameter, h. The measured radial extent
over which the signal drops to 95% of its value in the adjacent
gap, g, is given in Table 3 for each ringlet edge and occultation with
appropriate viewing geometry. This parameter is related to, but
larger than, d (Eq. (2)). The morphology of the edge occultation profiles at these unique viewing geometries included symmetric,
asymmetric and complex edges, pointing toward the possibility
that the asymmetric edge profiles do contain structure that is not
described by our two-parameter model. Although there was only
a small number of occultations for which g was measurable, values
of g also did not show any perceivable pattern with longitude from
any of the relevant moons (Fig. 15).
4.5. Asymmetric edges
While many of the edge occultations are fit well with our model
curves, a few of the occultations showed an asymmetry between
the variation in signal exterior to and interior to the half-signal
point that was not fit with any combination of h and d spanning
0–100 m, but can be fit by extending the range of h to much larger
values (1 km). We find a better fit to the asymmetric edges with
289
R.G. Jerousek et al. / Icarus 216 (2011) 280–291
Table 3
Direct measurements of edge sharpness.
Occultation
Morphology
|a|
Radial separations Mean d sringlet
(m)
(m)
(normal)
Titan ringlet (I)
Titan ringlet (O)
Huygens ringlet (I)
Huygens ringlet (O)
49.23–85.41
49.23–85.44
13.75–89.57
13.75–89.57
10–250
10–250
5–300
30–275
2.3
3.1
1.9
2.4
0.76–5.93
0.41–4.93
0.087–4.71
0.23–4.50
Mean best-fit values of d for each ringlet edge and radial ring plane separations of
the two stars. (I): Inner edges. (O): Outer edges. Normal optical depths immediately
inside the ringlet edges are also given.
the s(r) model than with the s(r, z) model, suggesting that if the
edge is, in fact, vertically extended, that the vertical distribution
of particles near the edge is nearly uniform. An alternative to these
large edge thicknesses is that the radial distribution of material is
itself less symmetric than the radial distribution assumed in our logistic model (Eq. (2)). Some examples of asymmetric edges were
best-fit by values of h > 1 km although other observations of asymmetric edges at large a angles could not be fit by our model. As
mentioned above (Section 4.3) the observation of the asymmetric
profiles in occultations where the geometry is completely insensitive to h indicates that at these locations we are seeing radial structure in the ring and not a projection effect of a very thick ring edge.
The combination of the absence of a single value of d which fits
all of the edges and the large variation of g values for the directly
sringlet
(normal)
38
99
46
3.5
5.0
4.7
1.74
3.80
3.01
Titan ringlet outer edge
S
d-Persei (37) (I)
S
d-Persei (39) (I)
A
75.2
88.9
89.3
174
57
91
3.5
5.0
4.7
1.81
3.21
3.22
Huygens ringlet inner edge
S
S
S
b-Centauri (77) (E) A
b-Centauri (81) (I) S
b-Centauri (85) (I) S
b-Centauri (89) (I) A
b-Centauri (96) (I) S
k-Ceti (28) (I)
C
89.1
89.1
89.9
89.4
89.2
87.9
87.9
89.5
76.8
200
257
14
91
248
54
95
265
30
14.0
7.1
7.3
7.6
6.7
13.4
6.8
6.8
18.6
1.23
3.19
3.86
2.11
2.13
2.23
2.70
3.05
0.63
Huygens ringlet outer edge
A
S
S
b-Centauri (77) (E) A
b-Centauri (81) (I) S
b-Centauri (85) (I) S
b-Centauri (89) (I) S
b-Centauri (96) (I) S
k-Ceti (28) (I)
C
89.1
89.1
89.9
89.4
88.2
87.9
87.8
89.5
76.8
142
50
29
107
40
40
33
47
19
14.2
7.1
7.3
7.7
6.7
13.4
6.8
6.8
18.7
2.86
3.94
3.18
3.31
4.50
4.43
4.46
4.04
0.65
a-Arae (85) (I)
a-Arae (86) (I)
a-Arae (90) (I)
Occultation
Radial
resolution (m)
75.2
89.9
89.2
a-Arae (85) (I)
a-Arae (86) (I)
a-Arae (90) (I)
Table 2
Binary occultations with small radial separations.
g (m)
Titan ringlet inner edge
a-Virginis (34) (E) S
d-Persei (37) (I)
C
d-Persei (39) (I)
S
a-Virginis (34) (E)
Fig. 15. (a) d versus edge longitude relative to Titan for the Titan ringlet inner and
outer edges. (b) d versus longitude relative to Mimas for the Huygens ringlet outer
edge. d values show no perceivable pattern associated with perturbing satellites for
any of the ringlet edges studied. In the case of the Huygens ringlet, Mimas is shown
as an illustrative example due to its dynamical connection with the nearby B ring
edge (Spitale and Porco, 2010). We found no correlation between d and any of
Saturn’s moons.
|a|
Radial extents of edge transitions, g, for edge occultations with |a| > 75°. Edge
morphology is categorized by ‘‘S’’ for symmetric, ‘‘A’’ for asymmetric, and ‘‘C’’ for
complex (see text). (I): Ingress occultation. (E): Egress occultation. Normal optical
depths immediately inside the ringlet edges are also given.
measured edges indicates that the edges are strongly perturbed,
whether by external perturbers or the ringlet’s own self-gravity
or embedded moonlets. Particle crowding and traffic jams due to
finite-particle-size effects may be producing the large variations
we observe in the vicinity of the edges, analogous to the structures
seen in numerical simulations of edges near moonlets (Lewis and
Stewart, 2000). The best-fit values of d also show no apparent pattern as a function of the radial distance of the occultation from Saturn’s center, or as a function of longitude from known moons.
These results are shown in Figs. 14 and 15 respectively. The range
of d values determined from fitting our model as well as the directly measured g values at occultations with appropriate viewing
geometries is consistent with radial length scales of unconstrained
edges evolving through viscous diffusion in numerical simulations
over a time interval of <5 orbits (Weiss, 2005), suggesting an upper
limit on the time scale over which confining perturbations occur.
4.6. f-Orionis (Rev 47)
As mentioned in Section 4.1, the egress occultation of f-Orionis
(47) was the only occultation with a viewing geometry which allowed us to probe the differences between the s(r) and s(r, z) models, however several other factors make this occultation by the
Huygens ringlet edges worth a more detailed discussion. f-Orionis
is a bright, binary star with an occultation elevation angle of only
2.66° and a radial separation between the two stars of 10 km
in the ring plane. With all other occultations, we have only had
the opportunity to focus our study of these ringlet edges on small
scale structure (<1 km) because of the limiting factors of large temporal and azimuthal variability between each occultation. With the
large radial separation between the two stars in this binary occul-
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R.G. Jerousek et al. / Icarus 216 (2011) 280–291
Table 4
f-Orionis (47) egress occultation by the Huygens ringlet.
Star/edge
f-Orionis
f-Orionis
f-Orionis
f-Orionis
A (47) (I)
B (47) (I)
A (47) (O)
B (47) (O)
Ds(r,z)/Ds(r)
h for s(r) (m)
h for s(r, z) (m)
d for s(r) (m)
d for s(r, z) (m)
9.6
6.4
11.9
10.0
40
4
10
6
1
1
1
70
1
7
16
10
1
1
23
7
(I): Huygens ringlet inner edge. (O): Huygens ringlet outer edge.
tation, we determine the variation in the best-fit parameter d between two points separated by 2 km in longitude in the frame of
the ring particles.
For f-Orionis (47), best-fit values of D for the s(r, z) model were
about ten times larger than those of the s(r) model, indicating a
better fit of the s(r) model light curves to the HSP data over the
two occultations and providing four fits which appear to point toward a more uniform vertical optical depth distribution. At both
the inner and outer edges of the Huygens ringlet, a 13.75° for
both stars. The results from both models are summarized in Table 4. The radial resolution of the data for all four ring edges was
30 m. While we do find different values for d at the inner and outer edges between the two components of this star, the differences
are not large enough to conclude that there are real structural differences over the 2 km in longitude between the two profiles.
5. Conclusions
In roughly 70% of the usable edge occultations, deemed ‘‘symmetric’’ edges, the HSP data in the vicinity of an edge was fit well
with our two-parameter model, allowing us to estimate the radial
scale over which particle density varied at any particular viewing
geometry. In the remaining occultations, the edges showed significant structure within the adjacent 300 m that was not fit well by
our continuous edge model unless we used very large values of the
vertical thickness parameter (>1 km). Several of these occultations
had lines of sight nearly tangent to the ring edge, in which the
light-curve is insensitive to the thickness of the ring, and are better
fit by an asymmetric (i.e., not following the sigmoid function of our
model) radial optical depth profile. The small-scale radial structure
of the edge is qualitatively different in these asymmetric edge locations than in the majority of the sampled locations, highlighting
the azimuthal variation not only in the quantitative measures of
the edge gradient, but also its overall shape.
Certain occultations at elevation angles B > 60° with respect to
the ring-plane, as well as occultations nearly tangent to the ring
edge, allow direct measurement of g, the radial scale over which
particle density varies in the vicinity of the edge. We find a mean
value of g = 94 m over 24 such occultations and no values of g
greater than 257 m indicating that these ringlets edges are sharp
on this scale. This radial length scale is consistent with those found
in numerical simulations of the evolution of unconstrained edges
through viscous diffusion (Weiss, 2005) after no more than five
particle orbits, providing a possible upper limit on the time scale
over which confining perturbations occur.
We find values of d, the parameter that characterizes the radial
extent of the edge within the logistic radial optical depth model
(Eq. (2)), that are less than 80 m with uncertainties typically
ranging from ½d to 2d. This limits our ability to determine the vertical extent of the ring-plane even from the occultations with small
B. While we are unable to constrain the vertical thickness of the
rings at sharp edges due to this weak sensitivity of the occultation
profile to h compared with its sensitivity to d, we are able to calculate an upper limit on h from the cases where d is directly measured. We determine an upper bound of h in the vicinity of the
edge of 300 m in occultations with appropriate viewing geometry
and where we can determine d to be smaller than this scale. This is
consistent with Lane et al. (1982), however the limiting factor in
their work was the radial resolution of the Voyager PPS occultation
data. The previous limit was calculated under the assumption that
the radial extent of the edge is zero (d = 0). Here we are able to resolve and measure d for the first time, but we are unable to improve on the upper limit on ring thickness at edges set by Lane
et al.
Occultations of a single edge could not be fit to a single value of
d over multiple occultations. Furthermore, in occultations where d
was directly measured (proportional to g in Table 3), a single value
of d for any edge was inconsistent with the observations. We conclude that there is significant temporal and/or azimuthal variability, which may be due to increased random velocities in the
vicinity of embedded moonlets or particle clumps, streamline
crossing, or other perturbations due to internal or external forcings. The azimuthal variability of d at the Titan ringlet edges does
not appear to correlate to the longitude relative to Titan or any
of Saturn’s known moons, nor does the variability of d at the Huygens ringlet edges, however the relatively large uncertainties in the
best-fit values of d for certain occultations severely limit our ability
to draw any conclusions about the effects of satellite forcing or particle streamline crowding on the sharpness of the ringlet edges. In
cases of closely-spaced binary stars as well as the f-Orionis (47)
occultation with a larger azimuthal separation of 2 km, we are
able to fit the light curve with similar values of d suggesting that
the scale of azimuthal variability is greater than 2 km.
In this initial study we focused on the edges of only two ringlets.
Many other sharp edges, including the B ring outer edge and the
Encke and Keeler gap edges, can be modeled with this approach.
These edges are known to be strongly perturbed by Mimas, Pan,
and Daphnis, respectively and to exhibit large ring thicknesses at
some satellite-relative longitudes (Weiss et al., 2009). In future
work we will study these edges and look for signs of large values
of h due to satellite perturbations. In addition, diffraction can be included to constrain the particle size distribution and number density at the edge.
Acknowledgments
This material is based upon work supported by the National
Aeronautics and Space Administration under Grant No.
NNX10AF20G issued through the Cassini Data Analysis Program,
and by the Cassini project and the UVIS team. We thank two referees for thorough reviews that improved the quality of this paper.
We are indebted to the Cassini Rings Target Working Team and
Rings Working Groups led by Phil Nicholson, Brad Wallis, Kelly
Perry, and Jeff Cuzzi whose support for, and integration of, these
observations has made this research possible.
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