Icarus 216 (2011) 280–291 Contents lists available at SciVerse ScienceDirect 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- 281 R.G. Jerousek et al. / Icarus 216 (2011) 280–291 450 (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. 282 R.G. Jerousek et al. / Icarus 216 (2011) 280–291 (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 284 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- 286 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 288 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- 290 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. References Borderies, N., Goldreich, P., Tremaine, S., 1982. Sharp edges of planetary rings. Nature 299, 209–211. Borderies, N., Goldreich, P., Tremaine, S., 1984. Unsolved problems in planetary ring dynamics in planetary rings. In: Greenberg, R., Brahic, A. (Eds.), Tucson. Univ. of Arizon Press, Arizona, pp. 713–734. R.G. Jerousek et al. / Icarus 216 (2011) 280–291 Colwell, J.E., 2003. Low velocity impacts into dust: Results from the COLLIDE-2 microgravity experiment. Icarus 164, 188–196. Colwell, J.E., Esposito, L.W., Sremčević, M., 2006. Self-gravity wakes in Saturn’s a ring measured by stellar occultations from Cassini. Geophys. Res. Lett. 33, L07291. Colwell, J.E., Esposito, L.W., Jerousek, R.G., Sremčević, M., Pettis, D., Bradley, E.T., 2010. Cassini UVIS stellar occultation observations of Saturn’s rings. Astron. J. 140, 1578. Cuzzi, J.N. et al., 2010. An evolving view of Saturn’s dynamic rings. Science 327, 1470–1475. Esposito, L.W., 2010. Composition, structure, dynamics, and evolution of Saturn’s rings. Ann. Rev. Earth Planet. Sci. 38, 383–410. Esposito, L.W., Colwell, J.E., McClintock, W.E., 1998. Cassini UVIS observations of Saturn’s rings. Planet. Space Sci. 46, 1221–1235. Esposito, L.W. et al., 2004. The cassini ultraviolet imaging spectrograph investigation. Space Sci. Rev. 115, 294–361. Lane, A.L. et al., 1982. Photopolarimetry from Voyager 2; preliminary results on Saturn, Titan, and the rings. Science 215 (4532), 543. Lewis, M.C., Stewart, G.R., 2000. Collisional dynamics of perturbed planetary rings I. Astron. J. 120, 3293–3310. 291 Nicholson, P.D., Porco, C.C., 1988. A new constraint on Saturn’s zonal gravity harmonics from Voyager observations of an eccentric ringlet. J. Geophys. Res. 93, 10209–10224. Porco, C.C., Nicholson, P.D., Borderies, N., Danielson, G.E., Goldreich, P., Holberg, P.B., Lane, A.L., 1984. The Eccentric Saturnian Ringlets at 1.29 R(S) and 1.45 R(S). Icarus 60, 17–24. Porco, C.C., Weiss, J.W., Richardson, D.C., Dones, L., Quinn, T., Throop, H., 2008. Simulations of the dynamical and light-scattering behavior of Saturn’s rings and the derivation of ring particle and disk properties. Astron. J. 136, 2172–2200. Spitale, J.N., Porco, C.C., 2010. Free unstable modes and massive bodies in Saturn’s outer B ring. Astron. J. 140, 1747–1750. Supulver, K.D., Bridges, F.G., Lin, D.N.C., 1995. The coefﬁcient of restitution of ice particles in glancing collisions: Experimental results for unfrosted surfaces. Icarus 113, 188–199. Weiss, J.W., 2005. The Physics of Unconstrained Edges in Planetary Rings. Thesis submission. University of Colorado, Dept. of Astrophysical and Planetary Sciences. Weiss, J.W., Porco, C.C., Tiscareno, M.S., 2009. Ring edge waves and the masses of nearby satellites. Astron. J. 138, 272–286.