Icarus 206 (2010) 431–445 Contents lists available at ScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarus Estimating the masses of Saturn’s A and B rings from high-optical depth N-body simulations and stellar occultations Stuart J. Robbins a,*, Glen R. Stewart a, Mark C. Lewis b, Joshua E. Colwell c, Miodrag Sremčević a a Laboratory for Atmospheric and Space Physics, University of Colorado at Boulder, UCB 392, Boulder, CO 80309, United States Department of Computer Science, Trinity University, Department of Computer Science, San Antonio, TX 78212, United States c Department of Physics, University of Central Florida, 4000 Central Florida Blvd., Orlando, FL 32816, United States b a r t i c l e i n f o Article history: Received 22 November 2008 Revised 3 September 2009 Accepted 12 September 2009 Available online 26 September 2009 Keywords: Saturn Saturn, Rings Occultations a b s t r a c t We have completed a series of local N-body simulations of Saturn’s B and A rings in order to identify systematic differences in the degree of particle clumping into self-gravity wakes as a function of orbital distance from Saturn and dynamical optical depth (a function of surface density). These simulations revealed that the normal optical depth of the ﬁnal conﬁguration can be substantially lower than one would infer from a uniform distribution of particles. Adding more particles to the simulation simply piles more particles onto the self-gravity wakes while leaving relatively clear gaps between the wakes. Estimating the mass from the observed optical depth is therefore a non-linear problem. These simulations may explain why the Cassini UVIS instrument has detected starlight at low incidence angles through regions of the B ring that have average normal optical depths substantially greater than unity at some observation geometries [Colwell, J.E., Esposito, L.W., Sremčević, M., Stewart, G.R., McClintock, W.E., 2007. Icarus 190, 127– 144]. We provide a plausible internal density of the particles in the A and B rings based upon ﬁtting the results of our simulations with Cassini UVIS stellar occultation data. We simulated Cassini-like occultations through our simulation cells, calculated optical depths, and attempted to extrapolate to the values that Cassini observes. We needed to extrapolate because even initial optical depths of >4 (r > 240 g cm2) only yielded ﬁnal optical depths no greater than 2.8, smaller than the largest measured B ring optical depths. This extrapolation introduces a signiﬁcant amount of uncertainty, and we chose to be conservative in our overall mass estimates. From our simulations, we infer the surface density of the A ring to be r ¼ 42—54 g cm2 , which corresponds to a mass of 0:5 1019 kg—0:7 1019 kg. We infer a minimum surface density of r ¼ 240—480 g cm2 for Saturn’s B ring, which corresponds to a minimum mass estimate of 4 1019 kg—7 1019 kg. The A ring mass estimate agrees well with previous analyses, while the B ring is at least 40% larger. In sum, our lower limit estimate is that the total mass of Saturn’s ring system is 120–200% the mass of the moon Mimas, but signiﬁcantly larger values would be plausible given the limitations of our simulations. A signiﬁcantly larger mass for Saturn’s rings favors a primordial origin for the rings because the disruption of a former satellite of the required mass would be unlikely after the decay of the late heavy bombardment of planetary surfaces. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction It is surprisingly difﬁcult to make an accurate measurement of the total mass of Saturn’s rings. When we observe a local smallscale patch in the rings, it is natural to assume that the ring particles are uniformly distributed across the plane of the rings within that patch as long as it is small compared to the large-scale (>10 km) radial structures that are observed in the rings. This assumption implies that the ratio of the normal optical depth, s\, to the surface density, r, of the rings should be roughly constant. By measuring the wavelength of density waves in the rings, we can infer the sur* Corresponding author. Fax: +1 303 492 6946. E-mail address: stuart.robbins@colorado.edu (S.J. Robbins). 0019-1035/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2009.09.012 face density which can be combined with the optical depth in the regions that contain waves to obtain a value for the opacity, j = s\/r. The value of the opacity obtained from density waves in the A ring is about 102 cm2 g1 (Esposito et al., 1983; Tiscareno et al., 2007). In the Cassini Division, the opacity is 5 times larger, indicating a smaller maximum particle size than in the A ring (Colwell et al., 2009). Using measured values for the optical depth of the entire ring system and the opacity inferred from A ring density waves, Esposito et al. (1983) calculated a total mass of 2:8 1019 kg for Saturn’s rings. However, this value is likely to only be a lower bound on the ring mass because it neglects the possibility of particles clumping into large gravitational aggregates. One of the most surprising results of the Cassini mission to Saturn is that the transparency of the rings measured during stellar 432 S.J. Robbins et al. / Icarus 206 (2010) 431–445 occultations is not accurately modeled by the path length of the starlight through the rings (Colwell et al., 2006, 2007; Hedman et al., 2007). Stellar occultations of the more massive B ring observed by the Cassini UVIS instrument ﬁnd that the line-of-sight optical depth is roughly independent of the viewing angle out of the ring plane, which implies that the B ring is primarily composed of opaque clumps of particles separated by nearly transparent gaps (Colwell et al., 2007). Dynamically, this is thought to be the result of self-gravity wakes, ﬁrst predicted in stellar systems by Julian and Toomre (1966) and shown to occur in Saturn’s rings by Salo (1992, 1995). As a result, the observed transparency of the B ring is really only measuring the fractional area of the rings that is not covered by opaque clumps of particles; the measurement does not yield the mass of ring particles contained in the opaque clumps. Hence, to obtain a more accurate estimate of the mass of the B ring, we must model the structure of the unresolved particle clumps in the ring. Our approach to the problem is to measure the transparency as a function of viewing geometry of simulations of local patches in the rings. By comparing the transparency of our simulations with the Cassini UVIS measurements of the rings, we can constrain the internal mass density of the ring particles that best match the observations. We can then use our simulations to estimate the mass of the densest regions in Saturn’s rings where density waves are not available to measure the surface mass density directly. Previously, Salo et al. (2004), French et al. (2007), and Porco et al. (2008) have used N-body simulations to model the well-known azimuthal brightness asymmetry of Saturn’s rings. A good independent check of our procedure is that we obtain essentially the same internal particle density as was found in these previous studies. We extend these previous studies by making a much more extensive exploration of parameter space and by simulating the highest optical depth rings that have been reported in the literature so far. In Section 2, we explain our numerical methods and parameter space. In Section 3, we brieﬂy discuss the qualitative aspects of the meso-scale structure of the rings in our simulations (deca to hectometers). In Section 4, we discuss our process and results of simulating occultations through our simulated ring cells and the constraints on parameters from Cassini results. In Section 5, we discuss the implications of our new estimate for the mass of the rings for the feasibility of various models for the origin of Saturn’s rings. 2. N-body code 2.1. Numerical methods The simulations performed in this work are all N-body simulations that assume smooth, inelastic, hard spheres. The particle motions are modeled using a rotating pseudo-Cartesian coordinate system in the usual Hill approximation (see Stewart, 1991, for example): 2 d x 2 2X dy @V SG 3X2 x ¼ dt @x 2 þ 2X dx @V SG ¼ dt @y 2 þ X2 z ¼ dt 2 d y dt 2 d z dt ð1Þ @V SG @z where VSG is the gravitational potential due to the self-gravity of all the other particles in the simulation. The (x, y, z) coordinates are deﬁned in terms of cylindrical coordinates as x = r r0, y = r0(h Xt), z = z, where X is the angular velocity of a reference circular orbit at a distance r0 from the center of Saturn. The numerical solution to these equations was broken up into two parts during each time step in the simulation. In the ﬁrst part of the time step, the simulation ignored the right hand sides of Eq. (1), and moved the particles forward using the following analytic solution of the unperturbed Hill’s equations (Stewart, 1991): x=r0 ¼ X e cos u y=r 0 ¼ Y þ 2e sin u ð2Þ z=r 0 ¼ i cos f where X, e, and i are constants of the unperturbed motion and Y, u, and f are linear functions of time: YðtÞ ¼ Y 0 ð3=2ÞX Xt uðtÞ ¼ u0 þ Xt ð3Þ fðtÞ ¼ f0 þ Xt In analogy with common usage in plasma physics, this collection of variables (X, Y, u, e, f, i), will be referred to in this paper as ‘‘guiding center variables” since ( X, Y) gives the coordinates of the orbit-averaged motion on an unperturbed eccentric and inclined orbit. In other words, X is essentially the scaled semimajor axis and Y is the mean anomaly measured in the local rotating reference frame. Thus, every time step requires a transformation of every particle’s Cartesian coordinates and velocities to its guiding center variables. This approximation ignores the oblateness of Saturn, which would produce small fractional changes in the timerate-of-change of Y, u, and f. Although these frequency shifts are important for the dynamics of density waves, they are not expected to have a signiﬁcant effect on the local formation of selfgravity wakes. In the second part of the time step, the simulation calculates the acceleration of every particle due to the gravitational accelerations of the other particles. The self-gravity is calculated using a tree method similar to that of Barnes and Hut (1986) with the octree replaced by a k-D tree (Bentley, 1975). In this method, the simulation region is divided into a nested series of cells until the smallest cell of each branch of the k-D tree only contains a small number of particles. Calculation of the gravitational acceleration on a particle begins at the largest cell size in the tree. First the angle that the cell subtends as seen from the accelerated particle is computed. If this ‘‘opening angle” is less than a critical value, b, the acceleration on the particle from the entire cell is calculated from the cell’s gravitational monopole. If the opening angle is larger than the critical value, we descend down the tree to the cell’s children and the same test is applied to each daughter cell. If the opening angle is still too large, we descend further and repeat the process. This process repeats until either a small enough cell is found or the descent terminates in a cell that is a leaf (single particle) in the tree (Lewis and Stewart, 2008). In this latter case, the acceleration is computed from each particle separately. The simulations presented in this paper use b = 0.3 in effective units of radians. Using this method, the time required for gravity calculations scales as N log N rather than N2. After the gravitational accelerations for the time step have been calculated and the trajectories of the particles have been modiﬁed, the collisions are processed in a discrete-event style for the duration of time in a single step. Collisions between particles are treated as smooth, inelastic, hard spheres with a velocity-dependent coefﬁcient of restitution of the form (Bridges et al., 1984): eðv ? Þ ¼ min 0:34v ?0:234 ; 1 ð4Þ According to this rule, only the component of the relative velocity that is perpendicular to the plane of contact is altered by the collision. We adopted the link-cell method to search for collisions between ring particles (Hockney et al., 1974). In this method, the S.J. Robbins et al. / Icarus 206 (2010) 431–445 simulation region is broken up into a two-dimensional spatial grid. Each cell in this grid keeps a list of the particles whose centers are located in that cell. This eliminates the need to ever perform an exhaustive search among all pairs of particles. Instead, the search can be done just between pairs of particles in adjacent cells. As a consequence, the choice for the size of the cells in the grid and the size of the time step are linked in order to preserve the assumption that only adjacent cells must be searched. In particular, we choose the cell size to be the size of the particles plus several times the product of the time step and the average relative velocity between the particles. This last product tells us how far any two particles are expected to move relative to one another over the course of one time step. The average relative speed is determined by sampling pairs of nearby particles at regular intervals during the simulation. In general, the time step is a small fraction of an orbit period, so the motion of a particle during a time step can be approximated from the equations of motion quite easily. The search for collisions is done by running through the grid, and for each cell, searching against its own particles as well as those in the eight surrounding adjacent cells. Each pair of particles is checked to see if they overlap during the course of the time step. When a pair of particles is found that undergoes a collision during the current time step, that pair is added to a list that is sorted by the collision time. After the entire grid has been processed in this way, the collisions are processed. This is done by advancing the particles to the point of the collision and then updating their velocities appropriately. The particles are then marked with a time stamp saying that their current positions are for that point in time. All subsequent collisions for those two particles are then removed from the list of collisions and the grid is used to check for new collisions with the particles with new trajectories. This process is repeated until the end of the time step is reached. Like Wisdom and Tremaine (1988), we use periodic, sliding box boundary conditions. In this method, the central computational box is surrounded by eight ‘‘ghost” boxes that are exact images of the central box. The three ghost boxes located farther from Saturn slide uniformly in the negative y-direction, while the three ghost boxes closer to Saturn slide uniformly in the positive y-direction so as to maintain the uniform shear rate predicted for circular orbits in the Hill’s approximation. Any particles that leave the central box are replaced by particles entering the central box from one of the surrounding ghost boxes. The error in the average vertical component of the self-gravity of the ring particles introduced by using only one level of mirror cells for our simulations was on the order of 1%. For the initial conditions of the simulations, particle guiding centers were placed randomly through the cell with equal probabilities for all radial and azimuthal positions. The eccentricities, inclinations, and epicyclic phase angles were also selected at random from uniform probability distributions. The eccentricities were distributed between 0 and 109 to give the effect of a cold distribution with small free eccentricities. The inclinations were distributed between 0 and 2 108 radian. The larger range in inclinations was used to effectively reduce the chances of particles originally overlapping with one another for numerical reasons. The epicyclic phases were selected randomly between p and +p. The results presented here for all simulations were taken at a time when the choice of initial conditions had been erased by the evolution of the system, which occurs by orbit number 5. This was determined by visual inspection, monitoring the Toomre Q stability value, and calculating the velocity dispersion (Toomre, 1964). 2.2. Parameter space One hundred sixty-two simulations were run to 10 full orbital periods for this study. We ran two main categories of simulations 433 – single-sized particles and simulations with a particle size distribution governed by the power-law dN/dR / R3, where R is the particle radius (this is near the value found by French and Nicholson (2000)). Single-sized particle simulations had particles with a radius of 1 m. Simulations with a particle size distribution had sizes constrained to provide the same surface density as their singlesized simulation counterparts for a given optical depth; for simulations with particle radii spanning a factor of 10, radii ranged between 0:256 m 6 R 6 2:56 m. Simulation cell sizes were square with length L in the x- and ydirections. Most simulations were run with internal particle densities q of either 0.85 g cm3 or 0.45 g cm3; we also ran a few simulations with q = 0.65 g cm3. We chose these densities in order to bracket the assumed range of possible densities of ring particles based on spectroscopic results and inferred moonlet densities. The former have shown that the rings are mostly water ice, which has a density 0.92 g cm3, though actual ring particles are likely to have some degree of porosity. Our lower limit is based on the inferred densities of small moonlets in the saturnian system (e.g., Renner et al., 2005; Porco et al., 2007) as well as to match simulations performed by other groups (e.g., French et al., 2007). B ring simulations were set at a distance of 100 Mm from Saturn (near the inner edge of the B ring core), and A ring simulations were 130 Mm from Saturn (mid-A ring). We initially laid out the simulations to run with N 100,000 particles. However, we expect gravitational instabilities to form at the Toomre critical wavelength, kcr ¼ pGr=X, where G is the gravitational constant, r is the surface density, and X is the orbital frequency (Toomre, 1964). For low optical depths, kcr is very small (2 to 10 m) and it is not necessary to conduct these simulations with 100,000 particles. We established an upper-limit on the size of the simulation cell L at 100kcr , such that if 100,000 particles would result in L > 100kcr , we reduced the number of particles until the cell size was appropriately small. This is still signiﬁcantly larger than other researchers (e.g., French et al., 2007), even though at the low end we included only 3800 particles. On the other end of the size range, we did not conduct any simulations with cell sizes L < 6kcr ; we increased L and hence the particle number to ensure this lower bound. This caused our most populated simulation to have 1.3 million particles. We provide a complete listing and parameter table on the website http://saturn.sjrdesign.net that includes N for each simulation. Previously completed simulations by Heikki Salo (e.g., French et al., 2007) typically occupy L ¼ 4kcr , while simulations presented by Porco et al. (2008) use 4kcr 10kcr . Our simulations would need to be compared with identically chosen parameters to determine if our L choice is preferable over Porco et al.’s since the area is comparable, but larger cell sizes than used by French et al. (2007) allow us to better constrain the properties of each simulation for two main reasons. First, a larger cell size allows us to average over a larger area and more structures when deriving properties such as tendency for clump formation, the effective height of the structures, or optical depth. Second, numerical artifacts tend to occur when cell sizes are too small due to the mirror cell boundary conditions. A larger physical cell suppresses these artifacts and allows larger structures to develop than would otherwise be possible. The main parameter we varied was the dynamical optical depth, sdyn, which is equal to the sum of the cross-section area of the particles divided by the area of the cell. Though we discuss this further in Section 4.1, from this point on in the paper we will use s to refer to sdyn unless otherwise stated. We varied s by starting with s = 2.0 and multiplying it successively by factors of 21/4: 0.0884, 0.105, 0.125, 0.149, 0.177, 0.210, 0.250, 0.297, 0.354, 0.420, 0.500, 0.595, 0.707, 0.841, 1.000 for the A and B rings, in addition to 1.189, 1.414, 1.682, and 2.000 for the B ring. All of these were run with parameters q = 0.45 g cm3 and R ¼ 1 m; for the 434 S.J. Robbins et al. / Icarus 206 (2010) 431–445 size distribution, we did not simulate s = 2.000 in the B ring nor s = 1.000 in the A ring. In addition, we completed s = 2.378, 2.828, 3.364, and 4.000 simulations for the B ring, where R ¼ 1 m. For q ¼ 0:85 g cm3 and R ¼ 1 m, s = 2.000 was not simulated for the B ring, and s = 1.000 was not simulated in the A ring. For a size distribution at this high internal density, we only simulated up to s = 1.000 and s = 0.500 for the B and A rings, respectively. We did not do these simulations due to the computer time required. We realize that it may be difﬁcult to follow all the various combinations, and so we have created several tables on the website http://saturn.sjrdesign.net to provide a complete parameter-space listing of our simulations. Surface density was calculated by r = 4sRq/3. For q ¼ 0:45 g cm3 simulations, it ranged between r ¼ 5:30 g cm2 for s = 0.0884 up to r ¼ 240 g cm2 for s = 4.000. For q ¼ 0:85 g cm3 simulations, it ranged between r ¼ 10:0 g cm2 for s = 0.0884 up to r ¼ 191 g cm2 for s = 1.682. The simulations with the highest internal particle density and those simulations with a particle size distribution were more limited in s because of the L P 6kcr constraint and the subsequent large amounts of computer time required. All simulations were performed on an Apple MacPro computer with dual quad-cores operating at 2.8 GHz with 6 GB of RAM. Most were run with the version 4.2 g++ compiler, though the highest-s simulations were run with an Intel C++ compiler. Simulations took anywhere from 0.5 h of CPU time for the smallest R ¼ 1 m, q ¼ 0:45 g cm3 , s = 0.0884 simulation in the B ring with N 3800 particles up to 3200 CPU hours for the largest simulation with our particle size distribution, q ¼ 0:45 g cm3 , and s = 2.000 with N 524,000 particles. We note that 27,000 CPU hours went into the simulations presented here. 3. Qualitative end-state image analysis At the end of each simulation, an image was created with the particles to-scale to show the ﬁnal state of the cell. Each end-state was placed into a montage, though for readability we have selected only a few representative ones to illustrate the general trends. We show s P 1:000 for the B ring in Fig. 1, and s P 0:500 for the A ring in Fig. 2, in multiplicative intervals of 2N/2. We make several observations that have implications for the ring masses. (1) The A and B rings are signiﬁcantly different in their mesoscale structure. In the B ring, the self-gravity effects create long, wake-like structures that form a thick net of clumps with more transparent gaps in between. In the A ring, the netting gives way to a less organized structure where we observe some wake-like behavior that tends to change to more randomized clumping. These clumps are sometimes isolated, but they are more often connected by a slightly over-dense trail of material to a larger clump or wake. (2) Increasing the internal particle density signiﬁcantly affects clumping in the rings, regardless of where the simulation is in the ring system and regardless of the particle sizes. In general, besides increasing kcr , it makes the clumps themselves denser. In the B ring, the higher density causes the clumps to be more efﬁcient at clearing the gaps of particles, resulting in an overall greater transparency through the cell, as shown in the two bottom rows of Fig. 1. (3) Using a size distribution can signiﬁcantly affect the transparency of the gaps between self-gravity wakes, decreasing it by adding more small particles. This is most obvious in the high-q simulations. It has an interesting effect in the lowq A ring simulations shown in the second row of Fig. 2, making the gaps much less transparent but also making the clumps in the s = 0.707 simulation have a more diffuse edge. 4. Evolution of optical depths 4.1. Deﬁnitions There were three different ‘‘optical depths” that we examined: dynamical optical depth, projected optical depth, and calculated optical depth. Dynamical optical depth, s, is determined by summing the area of all particles within the simulation and dividing by the total area of the cell. This quantity is bound by 0 at the lower limit of no particles within the cell, and it is unbound at the high end except by computational resources, [0, 1). Note that it is only unbound in theory; in practice, the rings must have a ﬁnite mass within a ﬁnite volume, and hence this quantity is not unbound in the real world. This quantity does not change throughout a simulation, and it is used as the horizontal axis in Fig. 3. It is also sometimes referred to as ‘‘classical” or ‘‘geometric optical depth,” and though it is normally introduced as sdyn, most drop the subscript and use s as we do. Transparency, T, is measured by compressing the simulation cell along the z-axis into a 2D plane and calculating the fraction of light from a hypothetical star at normal incidence to the ring that would not be blocked by ring particles. This has values in the range [0, 1], and it changes throughout the simulation. Note that 1 sproj = transparency, where sproj is a ‘‘projected optical depth,” and it is sometimes referred to instead. We use transparency to directly compare our results with Cassini observations in Section 4.4 in order to constrain ring parameters from our simulations. Calculated (sometimes referred to below as ‘‘instrument”) optical depth, scalc, is directly related to T by scalc ¼ sinðBÞ ln I0 T b ð5Þ where I0 is the unattenuated stellar intensity and b is the background intensity. These were set to 1 and 0, respectively, in our calculations, but they vary in actual Cassini UVIS data. This optical depth can be directly compared to the ‘‘apparent normal optical depth” from stellar occultations (Colwell et al., 2006, 2007). The sin(B) factor is a line-of-sight path-length correction, where B is the observation angle relative to the ring plane (90° is normal to the plane). The factor l = sin(B), which converts the calculated optical depth from ‘‘slant” to ‘‘normal,” is not ideally suited for a scenario that includes particle clumping since it assumes a uniform distribution of particles. However, this is what is generally used in the literature (e.g., Colwell et al., 2006), and so it was used here for the sake of comparison. The values of scalc are bound as s is, [0, 1), though as a practical matter the largest optical depths that can be measured by UVIS are scalc 5–7. This value is used on the vertical axis in Fig. 3, which we use to extrapolate the masses of Saturn’s rings. 4.2. Technique From the simulations, optical depth was determined by ﬁrst calculating T for a given set of angles B and /. B is as deﬁned in Section 4.1. / is also referred to as the ‘‘clock angle,” ‘‘azimuth angle,” or ‘‘cant angle,” where / = 0 is radially away from Saturn. Calculating T was done by rotating the simulation cell by B and /. A circular aperture could then be projected through the simulated ring section and, if needed, multiple surrounding mirror cells (we used a radius of 100 m as this is larger than the spot size of an individual Cassini UVIS measurement). The area within the aperture was divided into a ﬁnite grid based upon an integer subdivision of the radius of the smallest particle. All particles’ positions in the simulation (and any mirrors) were measured relative to S.J. Robbins et al. / Icarus 206 (2010) 431–445 435 Fig. 1. Montage of simulation cells from the B ring (100 Mm). The top two rows were run with q ¼ 0:45 g cm3 , the bottom two q ¼ 0:85 g cm3 . Odd rows are R ¼ 1 m and even rows are 0:256 m 6 R 6 2:56 m. Scale is 500 m on each side of each cell, and Saturn is to the right and orbital direction is down. the aperture in the center of the cell; if they were within the aperture, then the corresponding grid cells they covered were ﬁlled if they had not yet been covered by another particle. The number of cells still empty at the end was divided by the total number in the aperture to yield T. After this was determined, scalc was calculated from Eq. (5). Our method is limited by three factors. First, it is limited by the resolution of the grid and overestimates T because of the square discretization of circular particles in a circular aperture. Experiments were conducted with grid resolutions of 1/4, 1/8, and 1/16 the radius of the smallest particle, and the ﬁnal scalc only differed by 0.1%. This amount was deemed acceptable because it is less than the difference of using a different timestep of the simulation (different timesteps are addressed in Sections 4.3 and 4.4). The second limitation is computer speed. Simulated observation angles close to normal to the ring plane were fairly fast because no mirrors need to be searched – these took on the order of 100 ms. However, more complicated simulations with 100,000 particles at a very low B angle (such as 2°) along the diagonal of the cell (/ = 45°) could take several minutes to complete. On average, for a single simulation timestep analyzed for a 2° 2° grid of B and / angles, this code required 6–36 h to run with a median time of 12 h. Lastly, as alluded to in the second limitation, we cannot calculate the optical depth at very small B angles because the grazing 436 S.J. Robbins et al. / Icarus 206 (2010) 431–445 Fig. 2. Montage of simulation cells from the A ring (130 Mm). The top two rows were run with q ¼ 0:45 g cm3 , the bottom two q ¼ 0:85 g cm3 . Odd rows are R ¼ 1 m and even rows are 0:256 m 6 R 6 2:56 m. Scale is 500 m on each side of each cell, and Saturn is to the right and orbital direction is down. incidence angle to the ring plane requires a very large number of mirror cells. The simulations with high surface densities would also sometimes completely ﬁll all grid cells, resulting in a ‘‘saturated” grid and scalc = 1. Since the Cassini UVIS measurement of the ring’s optical depth are generally limited to B > 2.5°, this limitation did not affect our comparison with observations. Despite these limitations, our method is sufﬁcient for our purposes because we are mainly interested in comparing our simulations with stellar occultations observed by Cassini, where the star is sufﬁciently bright and the ring particles sufﬁciently dark in the UV that the observed signal is mostly due to direct transmission through the rings as opposed to scattering and/or reﬂection (Colwell et al., 2006). 4.3. Results observing normal to the ring plane Once self-gravity wake structures form in the simulations, scalc could change by up to a factor of 2 depending upon the intra-wake particle density and the size of the wakes relative to our chosen aperture (100 m in radius). In order to calculate a scalc normal to the ring plane (scalc\) that is representative of the simulations, we averaged the scalc\ results for 200 different time steps, spaced 10 steps apart (each step being 1/1000th orbit), covering the last two simulated orbits. This averaging captures the effect of the spatial averaging in the UVIS stellar occultation data used to reduce the effect of stochastic variations (noise) in the observations. The observed and S.J. Robbins et al. / Icarus 206 (2010) 431–445 437 Fig. 3. scalc\ for the last two full orbits of each simulation, averaged over 200 time steps with the error bars indicating the variance. Top frame is the B ring, bottom frame is the A ring. The dotted diagonal line in both images is a 1:1 reference correlation. The q ¼ 0:65 g cm3 data fell directly between the other two particle densities, so it has been removed from the graphs to aid readability. simulated optical depths are thus a function of viewing geometry and not on the particular conﬁguration of self-gravity wakes within a single UVIS footprint nor single simulation time step. The resulting calc? are shown for all simulations, grouped by rings (B and values of s calc? over A), in Fig. 3. We also calculated the standard deviation of s the measured timesteps, which are shown as error bars. We make the following observations from these data: (1) Once scalc\ and s are no longer linearly related – a point which is tightly correlated with the onset of self-gravity wake/clump formation – our simulations show monotonic behavior in gencalc? levels off to a near-constant eral in the B ring, where s value. However, this is not the case for the q ¼ 0:45 g cm2 , calc? starts to level off but then R ¼ 1 m simulations, where s begins to increase again, at least as far in s as we have been able to simulate. We expect the other parameter sets to calc? again, but we have no way of estimating at increase in s what s that will occur without actually conducting the simulations. This has signiﬁcant implications on the overall mass of the B ring because Cassini UVIS has observed scalc\ > 7 for portions of that ring (see Fig. 4). Previous estimates for the mass of the B ring (e.g., Esposito et al., 1983) have assumed a linear relationship between s and scalc\, which our results show may signiﬁcantly under-estimate the mass of the B ring. (2) A linear relation between the calculated optical depth and the dynamical optical depth in the B ring exists over a much larger range of s for the lowest particle density simulations. The effects of introducing a particle size distribution do not correspond to a predictable behavior of where, as a function of s, the linear behavior will end. For q ¼ 0:45 g cm2 , the size-distribution simulation deviates at s 0.9 ðr 55gcm2 ), while the R¼1m simulations do not deviate until s 1.1 (r 65 g cm2 ). For q ¼0:85gcm2 , the behavior is the opposite, where linear behavior persists up to s 0.5 (r 55 g cm2 ) for the size distribution, but only up to s 0.4 (r 45gcm2 ) for the single-sized particles. For q ¼0:65gcm2 , the behavior is similar as for the higher density, but the deviation occurs at a slightly larger s. A possible explanation of the switch in behavior could be due to the relative importance of gravitational scatter versus collisions and the ease with which gravitational aggregates form as a result. In the higher density particle simulations, the higher masses of the wakes may increase the velocity dispersion of the smaller particles by gravitational scattering and the higher mass small particles may scatter each other more effectively. (3) Linear behavior in the A ring follows the same general pattern observed in the B ring in that the higher the internal calc? remains similar particle density, the range over which s to s is shorter. However, in the A ring, more predictable behavior is observed in terms of how introducing a particle size distribution affects this turn-off. Our simulations show that the linear behavior persists to larger values of s in simulations with a size distribution. (4) In the A ring, the behavior is more difﬁcult to predict due to the apparent scatter in Fig. 3, although the simulations calc? once the linear do appear to show a leveling off of s 438 S.J. Robbins et al. / Icarus 206 (2010) 431–445 Fig. 4. scalc\ for Saturn’s B ring (top panel) and A ring (bottom panel), from a Cassini UVIS stellar occultation. Note that the scales are different. To estimate the true normal optical depth we used the data from the ingress occultation of the star b Centauri on Rev 077 (bCen077(1), 2008-202–2008-203). We binned the data to regular 10-km intervals in ring plane radius and computed the slant-path-corrected optical depth. For b Centauri, l = sin(B) = 0.92, so the effects of self-gravity wakes on the observed optical depth is small. In the B ring, at measured optical depths of 5 we estimate the uncertainty in s to be ±0.4 where the main sources of the uncertainty are systematic errors in background and instrument response. The uncertainty in scalc\ increases rapidly with scalc\ and values greater than 7 are indistinguishable from completely opaque in this occultation. relationship ceases. All we can say at this point is that, for a s calc? < s, the observed deviathat is large enough such that s tions are mirrored for both simulations with and without a particle size distribution. However, this behavior appears to be dependent upon q and is not predictable from these simulations. 4.4. Results at other angles and comparison with Cassini UVIS occultations In the previous section, the results were only for observations normal to the ring plane. This section removes that constraint and allows us to make direct comparisons with Cassini UVIS stellar occultations. To-date, these observations have spanned 2.7° < B < 68.2°. For our simulated scalc, we had six dimensions of parameter space: (1) distance from Saturn, (2) internal particle density, (3) particle radius, (4) s, (5) B, and (6) /. In order to cope with the large space (and the requisite 162 graphs to analyze them, one s(B, /) per simulation), we plotted the results independent of / and s, creating the 12 graphs shown in Fig. 5 (B ring) and Fig. 6 (A ring). The data are averaged over the last full orbit of each simulation, with a resolution of 10° in both B and /; when plotting, the data are placed regardless of / as well as s, though we colored the results based on s. In these ﬁgures, the horizontal axis is the projection factor l = |sin(B)|. The vertical axis is transparency, exp (scalc/l), though it can also be calculated as shown in Section 4.1. Over-plotted are Cassini UVIS data. The utility of graphing in this manner is that the vertical range of values varies depending upon the parameters used; it is most dependent upon distance from Saturn, and within that sub-set, the particle density and size distribution of particle radii interact to affect the shape as well. Our transparency plots can therefore provide reasonable constraints upon q and R based on the overplotted Cassini data. From Fig. 5, we conclude that, of the six sets of parameters within the ranges that we studied, the Cassini data best ﬁt the q ¼ 0:45 g cm3 particle density and the R ¼ 1 m particle radii as the parameters for the B ring. We acknowledge a monodisperse particle population is likely an unphysical result and discuss it further in Section 5.4, but it is the best match with our simulations. From Fig. 6, we conclude that the Cassini data best ﬁt the q ¼ 0:45 g cm3 particle density with the particle size distribution as the parameters for the A ring. This value for the particle density was also found to be the best ﬁt to observations of the azimuthal brightness asymmetry of the rings (Salo et al., 2004; French et al., 2007; Porco et al., 2008). We use these to estimate the total mass of Saturn’s rings. There are two more constraints that we can apply to the parameters of Saturn’s A ring. First, Tiscareno et al. (2007) show from 439 S.J. Robbins et al. / Icarus 206 (2010) 431–445 calc results for the B ring averaged over 100 time steps (1 full orbit) in 10° by 10° resolution in observation angle B and azimuthal viewing angle /. Data are graphed Fig. 5. s independent of / and s (though s is indicated by color). Over-plotted grey symbols are Cassini UVIS data. The top row is q ¼ 0:45 g cm2 , the middle row q ¼ 0:65 g cm2 , and the bottom row q ¼ 0:85 g cm2 . The left column is for single-sized particles / (R = 1 m) and the right column is for a size distribution (0:256 m 6 R 6 2:56 m). wavelet analysis of density waves that the surface density of the A ring at 130 Mm is r 44 g cm2 (weighted average over the four closest locations to our 130 Mm position). Our A ring simulation that most closely matches this, given the Cassini UVIS stellar occultation constraint above, is s = 0.707 for q ¼ 0:45 g cm3 (r 42:4 g cm2 ). calc? 0:46 0:04. Using the rectangular For this s, we calculate s slab model for self-gravity wakes (Colwell et al., 2006), we can calculate the optical depth that would be observed by an occultation at normal incidence to the rings from the self-gravity wake properties that they determined. That gives scalc\ 0.61 for 130 Mm from Saturn. The measured apparent normal optical depth at this location is scalc\ = 0.58 in the b Cen (Rev 077 ingress) occultation for which l = 0.92, and therefore the effects of wakes on measured optical depths are small. Since our simulated size distribution did not calc? include particles smaller than 0.256 m, we would expect our s to be smaller than the value observed in Saturn’s A ring at the fewpercent level (discussed further in Section 5.4). 5. Estimating the mass of the B and A rings 5.1. Methods Based on the correlations between our simulations and the Cassini science results, we can provide an estimate for the mass of Saturn’s rings by using the simulation parameters that are closest matches to the data. We provide separate estimates for the A and B rings. The total ring mass may be measured directly by tracking the Cassini spacecraft for the end-of-mission maneuvers pro- posed, which will maneuver Cassini between the rings and Saturn; the mass-resolution of these determinations should be approximately that of the moon Mimas (3:8 1019 kg) or better (Seal and Bufﬁngton, in preparation). The sum of our values below (A ring + B ring masses) are our prediction for what Cassini will measure. We provide a range of estimates for the masses of the A and B rings, as well as the B ring core. They are based on the following equation: MRing ¼ p a2outer edge a2inner edge L2 ¼ p a2outer edge a2inner edge r N X 4 pq R3i 3 i¼1 ! ð6Þ as a disk with an area deﬁned by We treat each ring p a2outer edge a2inner edge . We divide this disk by the area L2 of our simulation cell (for the simulation that best matches the observed properties) to calculate the effective number of cells that ﬁt into the area of the ring. We then multiply this by the total mass of that simulation cell to get the mass of the ring. Note that this is equivalent to multiplying our derived surface densities by the surface area of the rings. We provide two different mass estimates for each ring, and we provide an additional one for the B ring. Our minimum mass estimate for each ring is based on the closest-match simulation that we have completed based upon Figs. 5 and 6. This is expected to grossly under-estimate the mass of the B ring because we have 440 S.J. Robbins et al. / Icarus 206 (2010) 431–445 calc results for the B ring averaged over 100 time steps (1 full orbit) in 10° by 10° resolution in observation angle B and azimuthal viewing angle /. Data are graphed Fig. 6. s independent of and s (though s is indicated by color). Over-plotted grey x symbols are Cassini UVIS data. The top row is q ¼ 0:45 g cm2 , the middle row q ¼ 0:65 g cm2 , and the bottom row q ¼ 0:85 g cm2 . The left column is for single-sized particles (R ¼ 1 m) and the right column is for a size distribution (0:256 m 6 R 6 2:56 m). not been able to perform simulations that yet match the optical depths observed there. The second mass estimate assumes that there is a 1:1 linear relationship between s and scalc\ after our largest completed simulation (for the parameter set that best matches per Figs. 5 and 6). We use this linear relationship to extrapolate our scalc\ values to those observed by Cassini (Fig. 4). We assume this linear relationship with a slope of 1 based upon that correlation at low-s and because we actually expect the slope to be less, but we do not know by how much; with a true slope of less than 1, we will again necessarily under-estimate the mass from this technique, lending support to our purpose of setting a new, conservative lower limit to the mass of the saturnian ring system. Our third estimate is based on a parameterization of how clumping affects scalc\ throughout the B ring and it is explained in Section 5.3 where we discuss our B ring mass estimates. We only use this third estimate for the B ring because the A ring is sufﬁciently constrained by the simulations we were able to do as well as by spiral density waves. Clumping also varies much less throughout the A ring than the B ring. We compare our values with optical depth instead of transparency even though transparency is the actual value measured by both Cassini and our code. The reason is that transparency varies linearly with the light transmitted through the rings (and with s until s = 1), but the mass does not – mass varies linearly with optical depth. As we have shown in Fig. 3, scalc\ does not vary linearly with s. Therefore, while a transparency decrease from 0.95 to 0.98 would result in very little change in our derived mass, the corresponding optical depth increase from 3.0 to 3.9 increases our derived mass signiﬁcantly. 5.2. The A ring mass estimates Saturn’s A ring extends over approximately a0 = 122.05– 136.78 Mm (see Fig. 4). Again using the self-gravity wake slab calc? 0:69 for the A model of Colwell et al. (2006), we calculate s ring using the average self-gravity wake parameters (values averaged from Fig. 4). Our simulation that best ﬁts with the Cassini constraints (Fig. 5) is for the parameters s ¼ 0:707; q ¼ 0:45 g cm3 , and the size range of particles (0:256 m 6 R 6 2:56 m). This corresponds to an opacity j ¼ 1:1 102 cm2 g1 and a surface density r ¼ 42 g cm2 . From this simulation, our ﬁrst estimate – which is the lowest – calc? ¼ 0:46 is 5:1 1018 kg. Our second estimate extrapolates our s calc? 0:69, which would for this simulation to Cassini’s observed s require s 0.94 (r ¼ 54 g cm2 ). This value of s results in a mass of 6:8 1018 kg. Due to the constraints from Tiscareno et al. (2007), we do not extrapolate to larger optical depths because the implied surface density becomes several factors larger than has been derived. Consequently, we estimate the mass of the A ring to be 14–18% the mass of Mimas. We note that this is fairly close to previous estimates by other researchers using average surface density estimates to extrapolate the mass (e.g., Horn and Russell, 1993; Spilker et al., 2004). 5.3. The B ring mass estimates Saturn’s B ring extends over approximately a0 = 92.00– 117.59 Mm, while the core of the B ring extends over a0 = 98.82– 117.16 Mm (Fig. 4). Over the entire ring, the optical depth observed 441 S.J. Robbins et al. / Icarus 206 (2010) 431–445 calc? 3:1, and the core is at least s calc? 3:9 (values is at least s averaged from Fig. 4). Note that these are minimums because parts of the ring are so opaque that their optical depth cannot yet be measured, and as a consequence all of our B ring mass estimates should be considered as minimum masses. The optically thick regions of the B ring core are currently estimated to have scalc\ > 7.5 (Colwell et al., 2007). Of the q and R parameters we examined, those that best ﬁt the transparency observed have q ¼ 0:45 g cm3 and R ¼ 1 m (Fig. 5). Although it is unlikely that there is a monodisperse population of particles, that is the set of simulations in this study that best ﬁt the observational data; we discuss this issue in the next section. As an absolute lower limit, we can use our s = 4.000 simulation as a template for the rings, even though it only reached scalc? ¼ 2:1 0:7 (which has a corresponding opacity of j ¼ 0:42 102 cm2 g1 and r ¼ 240 g cm2 Þ. Using this procedure for the entire ring, we calculate a minimum mass of 4:0 1019 kg ð110% MMimas Þ. For just the core of the B ring, this procedure yields 3:0 1019 kg (80% MMimas). Our second estimate calc? 2:1 (at s = 4.0) and linearly scales it to 3.1 and 3.9, takes s requiring s ¼ 5:0 ðr ¼ 300 g cm2 ) and 5.8 ðr ¼ 350 g cm2 ). This scales our minimum mass estimate to 5:1 1019 kg (140% MMimas) for the entire ring, and 4:3 1019 kg (120% MMimas) for the B ring core alone. Our third mass estimate of the B ring is still a conservative lower estimate, but it is based upon a more complex extrapolation than the previous two. It is based on the idea of taking into account the effects of gravitational clumping as one moves away from Saturn through the extent of the B ring, while simultaneously calc? observations by Cassini that we cannot yet accounting for s reach in our simulations. To do this, we need two main equations. First, our B ring simulations at 100 Mm from Saturn only went calc? ¼ 2:1 0:7. In order to estimate up to s = 4.0, which yielded a s calc? will reach the values actually observed a likely s at which s (Fig. 4), we performed a linear ﬁt through 2:378 6 s 6 4:000 (four data points): scalc? ða ¼ 100; sÞ ¼ 0:11 þ 0:48s ð7Þ The uncertainties on the ﬁt parameters are ±0.84 and ±0.30, respectively. The ﬁt, along with an example power-law ﬁt overlaid, are shown in Fig. 7 (we did not use the power-law ﬁt due calc? to insufﬁcient data to justify it). When we apply this ﬁt, s calc? = 3.9 at reaches 3.1 for s ¼ 6:3 ðr ¼ 380 g cm2 Þ, and s s ¼ 8:0 ðr ¼ 480 g cm2 ). Note that in the rest of the derivation calc? will be referred to as what Cas(until Eq. (10)), this value of s obs . We cannot simulate these at present and estisini observes, s mate they would require a minimum of 40,000 and 130,000 CPU hours, respectively. The second equation we need is one that will allow us to paramcalc? relates to s as a function of a (distance from Sateterize how s urn). To do this, we performed an additional 20 simulations for the parameters q ¼ 0:45 g cm3 and R ¼ 1 m at 95, 105, 110, and 115 Mm from Saturn for s = 0.707, 1.000, 1.414, and 2.000. A plot similar to Fig. 3 is shown for these simulations in Fig. 8, illustrating calc? as the effects of distance from Saturn on clumping that affect s a function of s. We ﬁt a power-law slope to the s = 2.0 results: scalc? ða; s ¼ 2Þ ¼ 0:30 þ 9:3 108 a4:4 ð8Þ where a is in Mm. The formal uncertainties in the ﬁt parameters are ±3.3, ±4.4 1010, and ±11. While we note that all of these uncertainties are greater than the actual values, we believe the ﬁt is still reasonable, as shown in Fig. 9. A linear ﬁt has better formal uncertainties in the ﬁt parameters, but it does not follow the data as well and is less supported by theory. At this stage, we went through the following steps: calc? ða; s ¼ 2Þ that will allow us to (1) We have an expression s calculate the observed optical depth for s = 2 at any location a within the B ring. calc? ða ¼ 100; s ¼ 2Þ is from our simula(2) We know what s calc? ða ¼ calc? ða; s ¼ 2Þ s tions. Taking the difference s 100; s ¼ 2Þ will allow us to calculate how much greater or calc? will be at our arbitrary a for s = 2 than this smaller s known simulation. calc? ða ¼ 100; sÞ that will allow us (3) We have the expression s calc? from an input s for the location to calculate any value s a = 100 in the rings. We can use this to determine what value s is needed when given an input scalc? . At this point, we also obs from Cassini. calc? is equivalent to a s note that an input s (4) Putting steps (2) and (3) together, we have the expression scalc? ða; s ¼ 2Þ scalc? ða ¼ 100; s ¼ 2Þ þ scalc? ða ¼ 100; sÞ, which, when we assume the rings can be parameterized by well-behaved functions such as these, will be equal to scalc? ða; sÞ at both an arbitrary a and s. (5) Finally, we can solve the above equation for s, using arbitrary Greek letters for the ﬁt parameters at this stage: scalc? ða; s ¼ 2Þ scalc? ða ¼ 100; s ¼ 2Þ þ scalc? ða ¼ 100; sÞ calc? ða; sÞ ¼s calc? ða ¼ 100; s ¼ 2Þ þ ðf þ nsÞ ¼ s calc? calc? ða; s ¼ 2Þ s )s scalc?=obs scalc? ða; s ¼ 2Þ þ scalc? ða ¼ 100; s ¼ 2Þ f )s¼ n ð9Þ We integrated Eq. (6) for our monodisperse particle size R ¼ 1 m after substituting in Eq. (9) for s to derive our third plausible lower bound mass estimate: Z 4qR amax obs Þda a sða; s 3 amin Z amax 8pqR sobs scalc? ða; s ¼ 2Þ þ scalc? ða ¼ 100; s ¼ 2Þ f a M¼ da 3 n amin Z obs ð0:30 þ 9:3 108 a4:4 Þ þ 1:3 0:11 8pqR amax s a da M¼ 3 n amin Z obs 9:3 108 a4:4 þ 1:5 8pqR amax s M¼ a da 3 0:48 amin M ¼ 2p Fig. 7. This shows the same data as in the top panel of Fig. 3 for the B ring simulations with parameters q ¼ 0:45 g cm2 and R ¼ 1 m for s > 2.0. Over-plotted are two best ﬁts – a linear and a power-law. The latter was not used due both to its near-quantitative similarity with the former, and because there are not enough data points to justify a power-law ﬁt at this time. Using a linear ﬁt will not affect our goal of providing a new minimum mass estimate for the ring system, however, because a linear ﬁt will necessarily under-estimate a power-law ﬁt at large values, as can be calc? ¼ ð0:11 0:84Þþ seen towards the right edge of this ﬁgure. The ﬁts are s calc? ¼ ð0:9 2:1Þ þ ð0:06 0:68Þ a2:27:2 . ð0:48 0:30Þ a0 and s 0 ð10Þ 442 S.J. Robbins et al. / Icarus 206 (2010) 431–445 calc? for the last two full orbits of various simulations averaged over 200 time steps with the error bars indicating the variance. The dashed diagonal line is a 1:1 linear Fig. 8. s calc? and s throughout the range of the B ring. Simulations were reference correlation. This shows our attempt to parameterize how clumping affects the difference between s conducted in 5 Mm intervals for q ¼ 0:45 g cm2 and R ¼ 1 m. A slight horizontal offset is included in the ﬁgure for 95, 105, and 115 Mm symbols to aid readability. A powerlaw was ﬁt through the s = 2 simulations for use in our third minimum mass estimate of the B ring. calc? versus a0. Fig. 9. This shows the same data as in Fig. 8, though it is displayed as s Over-plotted are two best ﬁts – a linear and a power-law. The former was not used, despite the better formal certainties, while the latter was used in the text due to it calc? ða0 ; better representing the data as well as support from theory. The ﬁts are s s ¼ 2Þ ¼ ð7:1 1:3Þ þ ð0:056 0:013Þ a0 and scalc? ða0 ; s ¼ 2Þ ¼ ð0:3 3:3Þ þ 8 10 . ð9:3 10 4:4 10 Þ a411 0 Table 1 Summary of different ring mass estimates. Mass is in 1019 kg. Ring Estimate 1 Estimate 2 Estimate 3 A B, core B, total 0.51 3.0 4.0 0.68 4.3 5.1 – 6.6 7.0 calc? ¼ 3:1; 3:9, we calculate a mass of Integrating for s 7:0 1019 kg (190% M Mimas ) for the B ring and 6:6 1019 kg (180% MMimas ) for the B ring core. While these are both larger than the previous two mass estimates – and larger than has been previously derived – we still think that this still likely under-estimates the mass. We address why in Section 5.4 and note that all of the mass estimates are summarized in Table 1. 5.4. Caveats There are ﬁve primary sources of uncertainty in the above estimates for the mass of the rings. First, while the particle density that best ﬁts with the Cassini results is q ¼ 0:45 g cm3 , we only conducted simulations for q ¼ 0:45; 0:65; 0:85 g cm3 . This is a fairly coarse resolution in density, and we did not go lower than that value. A better match for the rings could be as high as q ¼ 0:55 g cm3 , but it could also be lower (though it would be difﬁcult to conceive a realistic material with a density of less than half that of ice – they would need to be very loosely packed ‘‘snowballs” despite being subjected to compacting collisional processes over their long history). Second, we somewhat arbitrarily chose R ¼ 1 m for the size of our particles, and then we chose our size range over a factor of 10 in R to yield the same surface density. Altering the range of sizes to yield a different surface density for a given optical depth could result in increasing or decreasing the requisite s for a given scalc? , altering the derived mass. Also, the actual size-frequency of ring particle radii may not follow a b = 3 power-law. Note, however, that increasing the range of R (while maintaining the previous surface density) will likely not change the results signiﬁcantly more than what we have shown between the single-sized particles and the 10 size range; it will decrease the transparency slightly as the gaps become ‘‘dirtier,” but we estimate this effect to be at calc? from the few-percent level. This is based on the difference in s a simulation with or without a particle size distribution (see Fig. 3). It is also based on constraints of sgap < 0.3 (Nicholson et al., 2008) (the optical depth of the gaps between the gravitational aggregates), which our simulations have already reached. Third, we assumed a Bridges et al. (1984) velocity-dependent coefﬁcient of restitution, e (see Section 2). Other researchers (Salo et al., 2004; French et al., 2007) often use a single value, such as e = 0.5. Recently, Porco et al. (2008) argue that the particles are less elastic than Bridges et al. (1984) inferred, based on ﬁtting their measurements to the A ring. If e is smaller, then we would expect the particles to clump more than in our simulations. This additional clumping would act to decrease the scalc\ for a given s, resulting in an increase in the mass of the rings. Fourth, we treated the rings as uniform slabs at constant optical depth. As a consequence of this, any differences in clumping effects on the true optical depth with distance from Saturn were not taken into account. This is unlikely to signiﬁcantly affect the mass estimate in the A ring because this ring has a fairly uniform optical depth that slowly decreases with increasing distance from Saturn. Our simulations show that clumping increases with distance from Saturn and lowers scalc\ for a given s, which is consistent with what is observed (Fig. 4). In the B ring, this is not the case, and the densest parts are 100–102 Mm and 104– 108 Mm, both of which are beyond where we did our primary simulations in the B ring. Based on increased clumping at larger S.J. Robbins et al. / Icarus 206 (2010) 431–445 443 a, this fourth limitation is also likely to only increase our mass estimate for the B ring. Fifth – and this is more applicable to the B than the A ring – we do not know if a linear extrapolation from our available scalc? ða ¼ 100; sÞ data (Fig. 3) is realistic, especially considering that it is based on only four data points. There are both reasons to think it is and is not. On the con side, Fig. 3 fairly conclusively shows that there is not a simple relationship between scalc? and s once self-gravity wake or clump structures form. Without simulating larger optical depths, we do not know if the trends we see at lower s will continue. On the pro side, we are in a physical situation where we know that, for some value of s, scalc\ must reach what we observe in the actual rings. In addition, the largest few simulations do seem to follow a reasonably linear relationship in Fig. 3, and we may have reached a regime where the simulations are now self-similar and will continue to follow that trend. This is supported by the end-stage simulation cells shown in Fig. 10 where, as the cell size and optical depth increases, the structure remains the same except in scale. With these limitations in mind, it is difﬁcult to place an uncertainty on our mass of 0:51 1019 kg—0:68 1019 kg for the mass of the A ring, 3:0 1019 kg—6:6 1019 kg for the minimum mass of the B ring’s core, and 4:0 1019 kg 7:0 1019 kg for the minimum mass of the entire B ring. We roughly estimate that the ﬁrst and second issues noted above will likely play only minor roles in shifting these to larger or smaller masses. Based on sgap constraints that are no more than 20% larger than our simulation results, adding more smaller particles will not increase scalc? enough to account for Cassini observations at currently simulated values of s. The third point will likely increase the masses of both rings. The fourth is likely to not change the A ring mass estimate by any signiﬁcant amount. It should, however, increase the B ring estimate but we do not know by how much without doing an actual point-by-point integration with our extrapolated ﬁts compared to the Cassini results; going through that process is unwarranted due to the large uncertainties in our ﬁts at this time. The ﬁfth issue could signiﬁcantly alter the B ring’s mass, but we do not know if it will alter it to be larger or smaller. Finally, we have neglected the mass of the Cassini Division and the C ring here. Colwell et al. (2009) estimate the Cassini Division mass to be 1:4 1017 kg. Using their measured Cassini Division opacity of 0:1 cm2 g1 and applying it to the C ring gives a C ring mass of 1017 kg. Even if the opacity of the C ring is as high as it is in the A ring, both of these ring regions are likely negligible in terms of the total ring mass, and the factors we note above likely play a much larger role than this. 6. Conclusions and discussion We have explored a large simulation parameter space of possible particle densities, size distributions, and optical depths for Saturn’s rings and constrained them to likely values based on comparison with Cassini UVIS stellar occultations. By extrapolating our simulation results to the larger optical depths measured in the core of the B ring, we have demonstrated that the mass of Saturn’s rings is likely to be larger than has been previously estimated by at least a factor of 2. While this initial parameter space provides an important constraint, there is still a large amount of unexplored parameter space, speciﬁcally narrowing down the internal particle density, trying different size distribution ranges, and varying the coefﬁcient of restitution (this last one we did not vary at all). Internal particle densities substantially smaller than 0:45 g cm3 appear to be unlikely, however, since such low particle densities would be difﬁcult to match both the surfaces densities measured by density waves and the observed optical depths and photometric properties Fig. 10. Montage of the four s > 2.0 simulations in the B ring (q = 0.45 g cm3, R ¼ 1 m). Scale is 600 m on each side of each cell shown. Saturn is to the right, and orbital motion is down. This montage illustrates the developing self-similar gravitational aggregate structure at large optical depths. While the optical depth increases, the scale of the structures increase, but the types of structure generally stays the same, which is why we think it is plausible to extrapolate to higher optical depths. 444 S.J. Robbins et al. / Icarus 206 (2010) 431–445 in the A ring (Porco et al., 2008). It is also necessary to conduct simulations throughout the B ring at s > 4:0 ðr > 240 g cm2 Þ until the scalc? from the simulations reach what is observed by Cassini; unfortunately, these are very CPU-intensive due to the requisite particle number and were not done for this study. Extending the particle size distribution down to centimetersized particles could potentially yield larger optical depths in the B ring with a smaller total mass, but our simulations using a truncated size distribution show that most of the smaller particles reside on opaque clumps in the B ring rather than ﬁll the gaps between these clumps. Cassini UVIS and VIMS observations of stellar occulations for a range of ring opening angles are best explained by gap optical depths of less than 0.3 throughout the A and B rings, which also implies that most of the smaller particles reside on opaque clumps (Colwell et al., 2007; Nicholson et al., 2008). The optical depths measured in the dense B ring core are therefore likely to be primarily due to the fractional area covered by opaque clumps of particles and adding more small particles will not likely alter this fractional area by more than a few tens of percent. While this may seem like a large amount, the actual area covered by these gaps is fairly small at s > 4.0 (Fig. 10) and so we do not expect this to be a signiﬁcant effect, especially when compared with the caveats mentioned above that would act to increase our minimum mass estimate. Porco et al. (2008) argue that the particle collisions are more dissipative than we have assumed in our simulations, but more dissipative collisions will most likely cause the smaller particles to have smaller random velocities and to spend more time stuck on larger clumps of particles where they reside inside the Hill sphere of the larger particles or clumps of particles. This latter possibility may even explain why our B ring simulation using a single particle size was a better ﬁt to the UVIS stellar occultation data than was our simulation using a particle size distribution with relatively elastic particle collisions: if small particles really spend most of their time in opaque clumps, they would not contribute to the observed transparency of the B ring. Since the tendency for particles to clump increases as we move farther from Saturn, and since all of our s > 2 B ring simulations were done at an orbital radius of 100 Mm near the inner edge of the B ring core, we have most likely underestimated the degree of particle clumping in most of the B ring core and therefore have systematically underestimated its mass. We have attempted to account for this variation in clumping across the B ring core by including a dependence from Saturn extrapolated from our s = 2 simulations. If all these considerations are additive, they have the potential to increase our mass estimate by at least a factor of 2. A larger ring mass could make it less probable for the rings to be a young system (i.e., less than 109 years), partially due to the unlikelihood of such a massive moon breaking up in recent Solar System history. If the rings are debris from the disruption of a progenitor satellite, then a mass several times that of Mimas may be consistent with the disruption of the moon occurring during the late heavy bombardment (Charnoz et al., 2009). This scenario requires the progenitor satellite to have migrated from the protosatellite disk inward to an orbital radius just outside of the orbit that is synchronous with Saturn’s rotation rate at 1.86 Saturn radii so that subsequent tidal evolution of the satellite’s orbit would not cause the satellite to spiral into the planet before the late heavy bombardment event some 700 Myr after the formation of Saturn. An attractive feature of this scenario is that the massive core of the B ring straddles the synchronous orbital radius just as we would expect if it spread inward and outward from a collisionally disrupted satellite. Perhaps the greatest single observational difﬁculty with the age of Saturn’s rings comes from the apparent lack of meteoroidal contaminants to their predominantly water–ice composition (Cuzzi and Estrada, 1998; Poulet et al., 2003; Nicholson et al., 2005). It is also possible that the less massive A ring was added later by the tidal disruption of an ice-rich centaur (Dones et al., 2007). Such tidal disruption events preferentially capture the ice-rich mantle of a differentiated object, so a younger, less contaminated A ring may be easier to explain than an older, uncontaminated B ring core. In principle, increasing the mass of the B ring provides a greater volume of icy ring material in which dark contaminants can be hidden, preserving the bright surfaces of the ring particles (Esposito et al., 2008). Esposito et al. (2008) show that a ring 10 more massive would have proportionally less meteoritic pollution, and thus have no discernable spectral features from carbon in its UV spectrum. This suggests that dark contaminants can be hidden in a more massive ring, and so arguments for a young age from a lack of dark contaminants are less tenable. Acknowledgments This research has greatly beneﬁted from numerous discussions with Larry Esposito. We would also like to thank our reviewers D. Richardson and M. Tiscareno whose comments increased the quality of this paper. S.J. Robbins and M. Sremčević were partially supported by NASA through the Cassini project at JPL. J.E. Colwell was supported by the NASA Cassini Data Analysis Program Grant Number NNX08AQ73G. G.R. Stewart and M.C. Lewis were supported by the NASA Planetary Geology and Geophysics program Grant Number NNG06GG46G. References Barnes, J., Hut, P., 1986. A hierarchical OðN logðNÞÞ force-calculation algorithm. Nature 324, 446–450. Bentley, J.L., 1975. Multidimensional bilinear search trees used for associative searching. Commun. ACM 18, 509–517. Bridges, F.G., Hatzes, A., Lin, D.N.C., 1984. Structure, stability, and evolution of Saturn’s rings. Nature 309, 333–335. Charnoz, S., Morbidelli, A., Dones, L., Salmon, J., 2009. Did Saturn’s rings form during the late heavy bombardment? Icarus 199, 413–428. Colwell, J.E., Esposito, LW., Sremčević, M., 2006. Self-gravity wakes in Saturn’s A ring measured by stellar occultations from Cassini. GRL 33, L07201. doi:10.1029/ 2005GL025163. Colwell, J.E., Esposito, L.W., Sremčević, M., Stewart, G.R., McClintock, W.E., 2007. Self-gravity wakes and radial structure of Saturn’s B ring. Icarus 190, 127–144. doi:10.1016/j.icarus.2007/03/018. Colwell, J.E., Cooney, J.H., Esposito, L.W., Sremčević, M., 2009. Density waves in Cassini UVIS stellar occultations. 1. The Cassini Division. Icarus 200, 574–580. doi:10.1016/j.icarus.2008.12.031. Cuzzi, J.N., Estrada, P.R., 1998. Compositional evolution of Saturn’s rings due to meteoroid bombardment. Icarus 132, 1–35. Dones, L., Agnor, C.B., Asphaug, E., 2007. Formation of Saturn’s rings by tidal disruption of a centaur. DPS Meeting 39, #7.07. Esposito, L.W., Ocallaghan, M., West, R.A., 1983. The structure of Saturn’s rings – Implications from the Voyager stellar occultations. Icarus 56, 439–452. Esposito, L.W., Elliot, J.P., Albers, N., 2008. Regolith growth and darkening of Saturn’s ring particles. AGU Meeting, A1295. French, R.G., Nicholson, P.D., 2000. Saturn’s ring’s. II. Particle sizes inferred from stellar occultation data. Icarus 145, 502–523. French, R.G., Salo, H., McGhee, C.A., Dones, L., 2007. HST observations of azimuthal asymmetry in Saturn’s rings. Icarus 189, 493–522. Hedman, M.M., Nicholson, P.D., Salo, H., Wallis, B.D., Buratti, B.J., Bains, K.H., Brown, R.H., Clark, R.N., 2007. Self-gravity wake structures in Saturn’s A ring revealed by Cassini-VIMS. Astron. J. 133 (6), 2624–2629. Hockney, R.W., Goel, S.P., Eastwood, J.W., 1974. Quiet high-resolution computer models of a plasma. J. Comput. Phys. 14, 148–158. Horn, L.J., Russell, C.T., 1993. Mass of Saturn’s A Ring. LPSC Proceedings XXIV, 671– 672. Julian, W.H., Toomre, A., 1966. Non-axisymmetric responses of differentially rotating disks of stars. Astrophys. J. 146, 810–830. Lewis, M.C., Stewart, G.R., 2008. Features around embedded moonlets in Saturn’s rings: The role of self-gravity and particle size distributions. Icarus 199, 387– 412. Nicholson, P.D., French, R.G., Campbell, D.B., Margot, J.-L., Nolan, M.C., Black, G.J., Salo, H.J., 2005. Radar imaging of Saturn’s rings. Icarus 177, 32–62. Nicholson, P.D., Hedman, M.M., Salo, H.J., and the Cassini VIMS Team, 2008. CassiniVIMS observations of self-gravity wakes in Saturn’s rings – II. AAS DDA Meeting #39, #18.01. S.J. Robbins et al. / Icarus 206 (2010) 431–445 Porco, C.C., Thomas, P.C., Weiss, J.W., Richardson, D.C., 2007. Saturn’s small inner satellites: Clues to their origins. Science 318, 1602–1607. 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. Poulet, F., Cruikshank, D.P., Cuzzi, J.N., Roush, T.L., French, R.G., 2003. Composition of Saturn’s rings A, B, and C from high resolution near-infrared spectroscopic observations. Astron. Astrophys. 412, 305–316. Renner, S., Sicardy, B., French, R.G., 2005. Prometheus and Pandora: Masses and orbital positions during the Cassini tour. Icarus 174, 230–240. Salo, H., 1992. Gravitational wakes in Saturn’s rings. Nature 359, 619–621. Salo, H., 1995. Simulations of dense planetary rings. III. Self-gravitating identical particles. Icarus 117, 287–312. 445 Salo, H., Karjalainen, R., French, R.G., 2004. Photometric modeling of Saturn’s rings. II. Azimuthal asymmetry in reﬂected and transmitted light. Icarus 170, 70–90. Spilker, L.J., Pilorz, S., Lane, A.L., Nelson, R.M., Pollard, B., Russell, C.T., 2004. Saturn A ring surface mass densities from spiral density wave dispersion behavior. Icarus 171, 372–390. Stewart, G.R., 1991. Nonlinear satellite wakes in planetary rings. Icarus 94, 436– 450. Tiscareno, M.S., Burns, J.A., Nicholson, P.D., Hedman, M.M., Porco, C.C., 2007. Cassini imaging of Saturn’s rings. II. A wavelet technique for analysis of density waves and other radial structure in the rings. Icarus 189, 14–34. Toomre, A., 1964. On the gravitational stability of a disk of stars. Astrophys. J. 139, 1217–1238. Wisdom, J., Tremaine, S., 1988. Local simulations of planetary rings. Astron. J. 95, 925–940.