Collisional Damping of Wakes in Perturbed Planetary Rings. M. C. Lewis. Department of Computer Science. Trinity University.

Transcription

1 1 Collisional Damping of Wakes in Perturbed Planetary Rings M. C. Lewis Department of Computer Science Trinity University One Trinity Place San Antonio, TX and G. R. Stewart Laboratory for Atmospheric and Space Physics Campus Box 392 Boulder, CO pages 24 figures 5 tables

2 2 We continue the work from our earlier paper (Lewis, M. C. & Stewart, G. R., AJ, 120,1395, 2000) looking at the dynamics of wakes near the edge of the Encke gap induced by the moonlet Pan. Explicitly we look at how wake damping depends on particle size, optical depth, and coefficient of restitution. Correspondence should go to Mark C. Lewis Department of Computer Science Trinity University One Trinity Place San Antonio, TX

3 3 Abstract The collisional damping of satellite wakes at the edge of the Encke gap of Saturn's rings is studied using N-body simulations. The self-gravity of the ring particles is neglected. The wake damping rate is determined as a function of ring optical depth, particle size, and the coefficient of restitution of colliding particles. The amplitude decay of the wakes is nonuniform and is not a simple function of the azimuthal distance from the satellite. Surprisingly, the wake decay rate is found to be a non-monotonic function of the ring optical depth. The evolution of the wakes is quite sensitive to changes in the assumed functional form of the coefficient of restitution. Secondary wakes that evolve over multiple synodic periods are observed at larger radial separations from the perturbing satellite. The radial migration of particle guiding centers that was noted in our previous paper is explained and more fully explored. Keywords: Planetary Rings, Saturn, Collisional Physics

4 4 1. Introduction The Voyager flybys of Saturn changed our view of the solar system s most visible ring system in significant ways. Far from the homogenous disk many expected, it was shown to contain a variety of structures on both large and small scales. Many of those features were linked to the effects of Saturn s satellites. Near the outer edge of the A- ring the details surrounding the Encke gap indicated that it was due to the presence of an unseen moonlet orbiting in the gap itself (Cuzzi & Scargle 1985 and Showalter et al. 1986). Closer analysis of the Voyager images later revealed the small body now known as Pan (Showalter 1991). In addition to clearing out the particles in the gap region, Pan gravitationally perturbs the ring particles around the gap. These perturbations induce the formation of wake structures in the ring. Near the edge of the gap, these wake structures form quite quickly and begin damping due to collisions between the particles. Farther from the edge the wakes form and are damped more slowly. In this paper we look at the larger scale behaviors of the wakes, and in particular, the rate at which the wake structures damp out downstream from the perturber. We also explore a number of tangent paths where interesting phenomena were observed in the simulations. One of these tangents was to explore the behavior of the wake further from the edge of the Encke gap. Evidence for the slower formation and damping of the wakes in this region was found by Horn, et al. (1996) when they performed a Burg analysis on the Voyager occultation data sets. They found that in certain regions there were oscillations in the data with wavelengths characteristic of what would be expected for a second-order Pan wake, or a wake that was present in its second synodic period. In addition, the Burg spectra for much of the region around the Encke gap showed many fairly strong signals

5 5 that could not be associated with either Pan or resonant interactions with more distant moons. A potential explanation for some of those features is presented in section 7 of this paper. To explore the collisional evolution of the Encke gap satellite wakes we use local N-body simulations with particles sizes near the upper end of the size range measured for the outer A ring (Nicholson & French 1998). An analysis of the resolved dynamics of the region near the edge of the Encke gap was presented in the first paper in this series (Lewis & Stewart 2000, hereafter paper I). The primary conclusion of that paper were that despite the fact that previous analytic models by Borderies, Goldreich, and Tremaine (1982, 1983, 1985, 1986, 1989) had some significant limitations, they still appeared to do a good job of explaining much of the physics the Encke gap system. We were also able to show that in our simulations, the angular momentum luminosity became negative at roughly the location of the observed edge of the Encke gap. The process of wake damping has significance largely because it is observable using information from occultations. For that reason, we explore the dependence this damping rate has on various parameters such as particle size and coefficient of restitution. We also present results on how it varies with optical depth. Because of the small number of azimuthal locations sampled by Voyager, most of these results can not be reliably compared to the current data that is available. However, we expect that Cassini will provide enough occultation measurements that damping rates can be measured and compared to these results. To date, the only other work we know of that has looked at wake damping in a system like the Encke gap was that performed by Hertzsch, et al. (1997). They used a

6 6 global simulation method, which resulted in poor resolution. It was a good start to looking at the problem, but lacked the detail to properly investigate the system in any detail. Work has also been done on a slightly different type of system that shares some commonalties. That is the type of system where ring material is perturbed by a moon at a significant distance (Haenninen and Salo 1992, 1994). This type of perturbation is only significant when looking at the area around low m-number first order Lindblad resonances. In particular the 2:1 was the focus of the work by Haenninen and Salo. By contrast the system explored here sits in the area around the 549:550 resonance and spans multiple resonances. In this system the significant forcing comes in the form of a kick in a single pass by the moon. This removes many of the periodicities from the system. It is also worth a short note to mention that there are two common uses for the term wake in ring discussions. In this paper we will always be referring to the large-scale compression and rarefaction of material that results from close passage by a perturbing body. The term is also commonly used to describe the clustering of particles that occurs when there is sufficient surface density and the particles are self-gravitating. The two phenomena are extremely different, as is the body of work that has been collected on them in recent years. This paper is organized into eight main sections. Following this introduction section, we will look at the physics of the wakes at the Encke gap and various descriptions of how they behave. The third section will present the behavior of a collisionless system. Section 4 deals with the process of fitting individual wake peaks while section 5 explores the way in which those peaks decay. Section 6 presents the measurements of the damping rates as functions of various simulation parameters. In

7 7 section 7 we present the results of simulations of particles located several hundred kilometers from the edge of the gap where the wakes can survive beyond a synodic period. The last section presents conclusions and potentially significant Cassini observations that could be made in the Encke gap region and compared with this work. 2. Description of the Physics The formation of the wakes at the Encke gap can be modeled as the compression of streamlines that occurs when the perturbed particles drift downstream from the point of closest approach to the moonlet. In paper I, we described a kinematic model of the wakes in which particle collisions are ignored. The kinematic solution shows the wakes growing until a point when the streamlines cross. Streamline crossing produces a singularity where the density goes to infinity. Beyond that point the streamlines pass completely through one another, the magnitude of the wakes decreases, and two distinct peaks are formed. Although the kinematic model could be used to accurately predict the location of Pan, it obviously fails beyond the point where the streamlines are predicted to cross (Showalter, et al., 1986 and Showalter 1991). Stewart (1991) developed a more advanced model of the kinematics of this region by looking at the motions of the particles in guiding center coordinates. The use of guiding center coordinates is continued in this paper both to simplify the process of integrating the particle paths and to visualize what is occurring with some of the behaviors that are observed. In addition to these kinematic models, a number of fluid descriptions of the dynamics have been developed. Paper I compared the results of the fluid models of

8 8 Borderies, Goldreich, and Tremaine (1982, 1983, 1985, 1986, 1989) to the results of our simulations for a particle size and optical depth resembling that which has been measured in the outer A-ring. In section 5 we will further those comparisons by looking at the behavior of the nonlinearity parameter, q, which is used extensively in the Borderies, Goldreich, and Tremaine papers. One process that was noted in paper I, which is not generally handled in standard fluid models, is the radial migration of particle guiding centers. The combination of dense packing at the wake peaks and the steep density gradients present at the wakes was seen to cause the streamlines themselves to shift radially in our simulations. This process was realized as a systematic movement of the locations of the guiding centers. Further exploration has enabled the construction of a more complete picture of why this process occurs in strongly perturbed systems. As we will show in section 6.4, this process has implications beyond what was initially observed in the simulations. To understand the guiding center migration, one must understand the behavior of particles as described in guiding center coordinates as well as the way in which wake structures alter the general behavior of the ring. Guiding center coordinates are useful because they give a simple way to express the motion of particles on low eccentricity orbits. The conversion between Cartesian coordinates, (x, y, z) and their derivatives, and guiding center coordinates, (X, Y, e, i, ϕ, ζ), is given by the following x r 0 y r 0 z r 0 = X e cosϕ = Y + βesin ϕ = i cosζ (1) where the motion is described by

9 9 ϕ( t) = κt + ϕ ζ ( t) = µ t + ζ Y ( t) = 2A Xt + Y (2) and X, e, and i, are constant. In this paper, we ignore the oblateness of Saturn and assume 3 Keplerian motion, so κ = µ = Ω 0, A = 0, and β = 2. Note that the motion of a particle 0 Ω 4 in this system is described by elliptical motion about a central point, which is drifting Fig 1 downstream at a constant rate. Figure 1 shows schematically how this looks. One of the most significant conclusions from paper I was that perturbed rings do not behave as uniform fluids. This unusual behavior occurs in more than one way. The most obvious difference between the perturbed ring and a normal ring is that the wake peaks can have significantly higher optical depths than the rest of the ring, and as a result, the fluid properties are definitely not uniform. In a perturbed ring, with optical depth well below unity, a particle is far more likely to collide with another particle in the wake peak than at any other time. This is manifest in the evolution of the forced eccentricities, which drop steeply at the wake peaks yet remain virtually unchanged elsewhere (see Figure 6 in paper I). The second major way in which the perturbation changes the behavior of the ring is that it roughly aligns the epicyclic phases of the particles. This becomes important when the wakes become strong, because it means that the particles systematically experience more collisions at certain epicyclic phase angles. To see what epicyclic phase angles the collisions tend to occur at one need only look at a plot of the streamlines of the Fig 2 particles in a simulation. Such a plot is shown in figure 2. From it one can see that the particles are in the wakes when ϕ is between roughly π/4 and 3π/4 radians. Note that the particle is directly below its guiding center when ϕ=0 as is shown in the leftmost set of

10 10 particles in figure 1. So when the particles enter the wake they are located below their guiding centers, and by the time they leave the wakes they are above their guiding centers. This is significant in light of the fact that the eccentricities drop primarily during the particles time in the wake peak. One must keep in mind that in any individual collision, the position of the particle does not change ( x= y=0). However, the particle s velocity does change, which moves the location of the guiding center. If a collision reduces the eccentricity of a particle, the guiding center of the particle effectively moves toward the particle as a result of the collision. How much the guiding centers move can be found explicitly by solving for the relationship between the change in velocity and the location of the guiding center using equations (1) and (2). This produces x κ = Y = r0 β y = r Ω Y 1 ( βκ A ) X = Ω X (3) The net result is that when any given particle enters the wake, its guiding center will move inward to a smaller radius when the eccentricity declines because the particle is located below its guiding center. As the particles leave the wake region they have advanced in their phase so that the physical particle is now located at a larger radial distance than the guiding center, and as a result, collisions that occur then will systematically move the particle guiding centers outward. It should be noted here that conservation of angular momentum requires that in any given collision the values X 1m1 = X 2m2. This implies that for the eccentricities to behave as observed some form of collective behavior must be exhibited in the wake peaks. Because of the pitch angle of the wakes, particles farther out in the x-direction Fig 3 reach the wakes later than those that are close in do. Figure 3 shows both the behavior of

11 11 the guiding center densities at a wake peak as well as an illustration of the motion of the guiding centers that produces this result. In the figure we have drawn the individual guiding centers having neither inward nor outward net movement. This is not necessarily the case. However, understanding the radial motion of the guiding centers of the particles is not exactly straightforward. Given just the assumption that the forced eccentricities damp significantly in the wake peaks but not elsewhere, one can create models of how the particles could perform large-scale radial motions that vary with the particular configuration of the ring particles. If all of the forced eccentricities were identical for particles at different radial distances, it is reasonable to assume that the majority of the eccentricity damping could be done when the particles first enter the wake peaks, especially if tight packing occurs in the wake. However, the forced eccentricities are smaller at larger radial distances so the particles spend more time in the wakes while their phase angles are greater than π/2 than they do prior to that point. This implies that if the radial gradient in the forced eccentricities is large, the particles will move outward. If the eccentricity gradient is too small, the particles are more likely to drift inward, especially if the optical depth of the system is fairly high. Since the satellite excites an eccentricity on the order of 1/X 2, the eccentricity gradient is of the order 1/X 3, which decreases rapidly as the radial separation from the satellite increases. These arguments are based upon potential behaviors of single particles though, and do not inherently conserve angular momentum. While it is clear that for any given collision, the movements of the guiding centers of the two particles must offset one another, the fact that the collision occur in a very systematic way allows them to produce complex behaviors. For the longterm evolution of the system, the fact that the ring can experience a torque from the

12 12 satellite also becomes important though that is beyond the scope of this paper. The motion of guiding centers will be discussed in a number of specific cases in section 6.4. Looking again at figure 2 one can see how altering the streamline plots would result in lower wake peaks. In particular there are four values that determine the height of the wake peaks at a given location: radial gradient of the epicyclic phase angles, magnitude of the forced eccentricity, distribution of the eccentricities, and distribution of the epicyclic phase angles. The first of these is determined by the position downstream. Our results show that collisions in the wakes do slow the advance of the epicyclic phases with respect to what would occur without collisions, however, it does so in a generally uniform way across different radial distances so the gradients are not that strongly altered. In paper I, we briefly explored the effect of the magnitude of the forced eccentricities and the distribution of the epicyclic phase angles. In the streamline formalism, the forced eccentricities simply modify the amplitude of the streamlines radial motion, while the distribution in epicyclic phases smears out the streamlines along the azimuthal direction. The distribution in the eccentricities effectively smears out the streamlines in the radial direction. The role of collisions is quite straightforward in causing both the drops in the eccentricity and the randomization of epicyclic phase angles. The question is how parameters such as particle size, optical depth, and coefficient of restitution affect the rates at which collisions alter these values. There are two significant properties of the impacts that will determine these rates. Those are the frequency of impacts and the impact velocities. While it is obvious that more impacts will alter the distribution more if the nature of the impacts doesn t change, the role of impact velocities is not so clear. The

13 13 most significant change in the role of collisions due to impact velocities is that in most of our simulations the coefficient of restitution of the particles is velocity dependent. The role of these parameters will be explored in section 6 and compared to some simple analytic models based on energy dissipation. Before looking at the results of the simulations, there are a few points that need to be discussed about the model used for the simulations that have changed from paper I. We use a local cell method where collisions are searched for using a spatial hash. Some optimizations were made to the collision handling routines, particularly in the handling of the collision lists, but the basic implementation remains the same as was presented in the previous paper. A more detailed explanation of the methods used in our simulation code can be found in two papers on simulation methodology (Lewis & Stewart 2002, Lewis & Stewart 2003). The most significant change that has been made was to the boundary conditions. As noted in paper I, the boundary conditions that we used resulted in particle loss over the course of the simulation. To prevent this loss of particles the boundary conditions were revised. In the new boundary conditions, when a particle passed beyond one of the azimuthal boundaries of the cell, its guiding center is offset so as to put the particle at the other end of the cell, and the epicyclic phase angle is updated. The amount that the epicyclic phase angle is updated by is calculated from equation (2). Given the radial location of the guiding center, the amount of time it takes a particle to move from one edge of the cell to the other is used to calculate the amount by which the epicyclic phase angle would change. This amount is given by the expression κ Y ϕ = 2A X 0 (4)

14 14 where Y is the value by which the azimuthal position of the guiding center is changed. This choice of boundary conditions was decided upon based on the result from the previous work that the average epicyclic phase of the particles is not significantly altered over the length of a simulation cell, which is on the order of 10-4 radians in the simulations presented here. We find, however, that collisions do slow the progression of the epicyclic phases somewhat over the course of an entire simulation with either of the two boundary conditions that have been applied. This delay could be used to explain the longer than expected radial wavelengths found by Horn et al. (1996) for wakes significantly downstream from Pan. These changes were validated by testing the code in two ways. First, the new code was tested to make sure that it could reproduce the results of Wisdom and Tremaine (1988) for the measurements of the components of the velocity ellipsoid for an unperturbed ring. Second, measurements of a number of the values that will be presented in the following section were done with both the previous code and the updated code, and the results were in agreement. The next five sections present the results from the simulations. The data is frequently analyzed by looking at azimuthal slices through the simulation region. In all sections, azimuthal slices are taken at 6.5 km from the outer edge of the Encke gap unless otherwise noted. We begin in a manner similar to that taken in paper I by looking at what occurs in a system where collisions between particles are ignored. 3. Collisionless Behavior

15 15 The simplest simulation that can be performed, is one in which the collisions are ignored. This case is interesting to explore as it provides a link between idealized analytic solutions and the collisional simulations, because while it ignores collisions, it has the finite particle counts of the simulations. A slice through a sample output surface Fig 4 density is shown in figure 4. Unlike in collisional simulations, the reduction in the wake peak heights in this simulation has nothing to do with altering the eccentricities or epicyclic phases of the particles. In this simulation the decrease is due solely to the fact that the streamlines begin passing through one another. When this happens, the wakes become double peaked, and their height drops dramatically. It continues to drop as the two peaks move farther apart. The jumps that are apparent in the simulation data further downstream are caused when the peaks have spread far enough apart that they meet the peaks that were originally associated with the wake peak that had been upstream or downstream from it. To see what the double peaks look like up close refer to figure 3 in paper I. In the next section we will look in depth at the process of trying to fit the wake structures to measure the rate at which the wake magnitudes decrease. Here I briefly discuss the more limited scope of trying to fit the way in which the wake peaks drop in the collisionless case, because it will be significant it the discussion in the next section. The term damping has been attached to the decrease in amplitude of the wake peaks, and exponential functions have been used to try to fit the envelope of the wake maxima (Hertzsch, et al., 1997). At first this seems to be a valid approach, and in global simulations that lacked resolution it fit the data as well as could be expected. However, Fig 5 trying to apply such a fit to the simulation without collisions fails miserably. Figure 5a

16 16 shows the first section of the data in figure 4 with a single exponential fit to illustrate this point. The data points that are used for this fit are the local maxima for each wake peak. When looking for an alternate function to use in fitting the behavior of the wake maxima in this region we look to the analytic expressions that have been used to describe the structure of the wakes. In particular Showalter et al. (1986) derived the following expression to describe the surface density in the wakes, which has been converted to guiding center coordinates, σ σ 0 ( X ) ( X, Y ) =, (5) 1+ Σ( X, Y ) cos( ϕ ) where σ 0 is the unperturbed surface density, Σ ( X, Y ) = Y Y ( X ), and Y crit (X) is the azimuthal position at which the streamlines cross for a particular radial distance. The absolute value of the denominator has been taken so that the formula stays reasonably well behaved beyond the point where Σ>1. At any particular epicyclic phase angle in successive wakes this simplifies to σ ( ) ( X ) σ X, Y 0, (6) = 1+ C Y where C is some constant. Figure 5b shows the results of performing such a fit to the crit simulation data. It is clear that this form fits the collisionless case better than an exponential. 4. Fitting Wakes Equation (5) provides us with a valid expression to fit the large-scale behavior of the wake maxima and worked extremely well for predicting the location of Pan (Showalter, et al., 1986). However, it does not do so well for predicting the structure of

17 17 Fig 6 the individual wakes in either collisional or collisionless simulations. Figure 6 shows an azimuthal slice from the beginning of a collisionless and a collisional simulation to just beyond the highest peak. Superimposed on these plots are the best fits for a function of the same form as equation (5). What is most obvious in this figure is that the best fit is actually quite poor. The reason for this is that the simulated wake peaks are extremely sharp. One of the properties of equation (5) is that when ϕ is between π/2 and 3π/2 the surface density will always be greater than σ 0. This behavior is not seen in the simulation results for most longitudes. Instead, the peaks are extremely sharp when Y is similar to Y crit, and only in a very narrow region around ϕ=π are the surface density values greater than the unperturbed surface density. The origin of this discrepancy is a subtle approximation made by Showalter et al. (1986) in their evaluation of the Jacobian of the transformation from local coordinates to orbital coordinates. In particular, Showalter et al. do not transform the Jacobian into local coordinates, but rather evaluate the local surface density by setting the local coordinates equal to the orbital coordinates. This turns out to be a poor approximation when the wakes are strongly nonlinear, i.e. when Σ(X,Y) is close to unity. A more accurate expression for wake profiles which reproduces the narrow peaks seen in our N- body simulations can be derived from the formalism described by Stewart (1991) and will be described at length in a future publication. The implications of this adjustment to the theory of wakes could be significant. The models developed by Borderies, Goldreich, and Tremaine are based on the same approximation used by Showalter et al. (1986) and their results could be significantly skewed by this change. The reason this change is so significant is that in the simulations

18 18 one sees extremely high particle densities in very localized regions at the wake peaks. As a result, most of the modifications to particle orbits occur in a very regular way in very specific locations. This is clearly illustrated by the motion of the guiding centers as was discussed in section 2 and which will be explored further in section 6.4. If the wakes behaved as described in equation (5), the higher optical depths would be present over a significantly larger range of epicyclic phases. As a result, the collisions would occur in a more isotropic manner, which would reduce the impact of the anisotropic collision distribution that is actually seen in the simulations. 5. Features of Decay Despite this current inability to explain the shape of a single wake structure, we have found that using the maximum in the simulated data for each wake works sufficiently well for analyzing their long-term behavior. This is due to that fact that we can perform simulations with large numbers of particles that afford us a high signal to noise ratio. The next step is to look at the wake maxima and see how they evolve Fig 7 downstream from the perturbation. Figure 7 shows a slice through a large azimuthal range of a collisional simulation. It is equivalent to what is shown in figure 4 for a simulation without collisions. In this simulation 100,000 particles 13m in radius were used in a cell with an optical depth of The height of the wakes builds for the first radians, then begins to decrease. This is similar to the collisionless case except that the highest peak occurs one orbital period later. The decrease in this simulation displays a fairly consistent decay until the wakes are gone. The original growth very closely

19 19 models what is seen in the kinematic description as well as in the collisionless simulations. One can make the features of the decay more apparent by taking only the maximum points of each of the wakes. These points are connected in figure 7 to show the envelope bounding the full slice through the data set. Using this subset of the data, one can analyze the rate at which the wakes decay. In this case a single exponential can be nicely fit to the wake peaks. Such a fit is also shown in figure 7. In this particular case the wake maxima behave in a manner that one might have expected. Before radians the wakes grow in magnitude in a way that is reasonably well fit by equation (5). Beyond radians the magnitude of the wakes drops off in a manner that is well fit by an exponential. In this case is seems fairly clear that the magnitude of the wakes behaves in such a way that the term decay is appropriate for describing it. In many ways though, this simulation is an ideal case because two basic types of Fig 8 behavior fit it quite well. In most of the simulations this is not the case. Figure 8 shows a plot equivalent to figure 7 only for a simulation with an optical depth of using 50,000 13m particles. This particular case is more characteristic of what is found in simulations with lower optical depths or smaller particle sizes. In this situation the drop in the height of the wakes is very non-uniform. It begins with a very steep decent that is followed by a slow decay. It should be quite obvious that this structure is not going to be well fit by a single exponential. Instead, we have chosen to fit it with the broken function that begins with a function of the form of equation (6) and then changes to an exponential beyond a certain point. The best fit for this data is shown in figure 8. This is the functional form that is used for the fits presented in section 6 unless otherwise noted.

20 20 The reason for using the function that we did for the initial region of rapid decay is not straightforward when one considers the fact that the streamlines do not shear through one another in most of the simulations. To be more specific, the particleaveraged streamlines do not shear through one another. Those are the streamlines defined by averaging the eccentricities and epicyclic phases of a group of particles in a particular region. It was the shear through process that resulted in the inversely proportional nature of the drop in the wake magnitudes in the collisionless simulation. In the collisional case, the rapid decay after the larger wake peak can instead be attributed to a spreading of the averaged streamlines as was discussed in section 2. Those first few very dense wake peaks where the streamlines would begin to shear through one another instead cause the dispersion in the forced eccentricities and epicyclic phases to increase Fig 9 rapidly. This effect is shown in Figure 9 where the dispersions in the forced eccentricity and epicyclic phases of the particles are plotted along with the optical depths. This figure clearly shows how the dispersion in both the forced eccentricity and the epicyclic phase angles grows quite rapidly in the region where the wake peaks drop sharply. Beyond that region, the dispersion in the epicyclic phase angles grows very slowly while the dispersion in the eccentricity actually decreases significantly. One might at first attribute this drop in the eccentricity dispersion to the fact that the eccentricities themselves are dropping, however we find that even the quotient of the dispersion over the forced eccentricity decreases significantly here. Of the three parameters that control the magnitudes of the wakes, the forced eccentricity is the most straightforward to understand as well as the most well behaved. As was observed in paper I, the forced eccentricities are generally flat prior to the highest

21 21 wake peak and begin a process of stepping down when the streamlines begin to touch. Late in the simulations, what had been sharp drops at the wake peaks become less distinct and are replaced by a steady decline that appears to be well fit by an exponential. However, this exponential does not also fit the initial regions. To determine what type of function can be used to fit the initial behavior of the forced eccentricity one can use the fact that the averaged streamlines do not intersect. In other words, at the wake peaks x X = 0. Using the expressions for x in equations (1) and (2) this is equivalent to x X = κy ( X e cos( ϕ ) X 0 2 A0 X eκy = 1+ 2A X 0 2 κy sin( ϕ ) = A0 X (7) For this expression we have replaced the ϕ with an expression that does not explicitly depend on time and instead requires only the guiding center location, (X, Y). For a description of this refer to equations (30) and (40) of Stewart (1991). The wake peaks occur when the term with the sine function is minimized which depends on the sign of Y. Because this paper looks only at wakes exterior to the moonlet the value of Y is always negative and the wake occurs when the argument of the sine function has a value of π + 2πn so the sine function has a value of unity. The fact that the wake maxima occur at 2 this epicyclic phase angle can also be arrived at by inspecting figure 2. Solving (7) for e produces the following 2 2A0 X e =. (8) κy In practice, the behavior is not smooth because the compression of the streamlines happens at azimuthal locations as prescribed by the assumption about the epicyclic phase angle. Nevertheless, equation (8) does a fairly good job of fitting the behavior of the

22 22 forced eccentricities early in the simulations. By using a broken function in the same way that was done for the wake peaks, a good fit can be found for the entire simulation. One very important fact to notice about the system when the eccentricity is damped is that the value of Y crit changes. The relationship between the two is that Y crit 1/e (Showalter, et al. 1986). We find that this alters the behavior of the wake maxima in a number of simulations. Specifically it can cause the location of the highest wake peak to move downstream from where it would normally be found. Unfortunately, the impact this has on the wake height is less obvious because the assumption of how compressed the streamlines can get actually determines the height of the peak. For (7) we assumed that they could compress until touching but not pass through one another so the far right side has a value of zero. The streamline plots show that the wake peaks go down with the forced eccentricities. The exact relation between the two is not clear, however. In previous analytic work on the topic of perturbed rings, the state of the system is often characterized by the nonlinearity parameter, q. In BGT83 this value is defined as q = 2 de dϕ ( 1+ X ) + ( 1+ X ) e dx dx 2 (9) This definition differs slightly from that given in the BGT83 paper because the value we use for ϕ appears in that paper as mϕ, where m is the length of the synodic period at that location measured in orbital periods. This value, q, is a measure of how compressed the streamlines are and is effectively equivalent to Σ(X,Y) in equation (5). While it has already been demonstrated that in many ways the detailed behavior of this system does not agree with the model used by Borderies, Goldreich, and Tremaine, it is instructive to

23 23 compare how q behaves in the simulations to the prediction of its behavior given in BGT89. This will give a somewhat better indication of how great an impact these discrepancies in the details have on the global evolution of the system. Fig 10 Figure 10 shows the value of q for a simulation with an optical depth of using particles that are 13 m in radius. Because the slice through the data is taken extremely close to the location of the edge of the gap the q value should be comparable to Fig 11 figure 1a in BGT89 which is reproduced as figure 11 here. Our q value does not compare all that closely to what Borderies, et al. found. In both cases, the initial region is a linear growth in q that is caused by the shearing of the epicyclic phase angles. In the BGT89 plot the maximum value is followed by a plateau that bends into a drop off before smoothly going to zero. In the simulations, after q reaches its maximum value, it immediately begins to drop off in a nearly linear manner. Of the plots given in BGT89, the structure of the behavior seen in our simulations is closest to that seen in their figure 1c. However, one must also take into account the azimuthal scales of the plots. Our simulation only ran for one radian downstream or just over 57 degrees. The amount by which q drops during that time is more than in any of the plots calculations in BGT89. It also appears that the maximum value of q attained in the simulations is greater than what was found in BGT89. In addition to the q value, figure 10 shows the curve produced by joining the wake peaks of the optical depth along the same slice and a predicted optical depth that is found by putting q into (5) in place of Σ. What is quite remarkable about this is that the predicted optical depth mirrors the actual optical depth peaks quite closely through most of the simulation. The deviation is only significant in the region around the highest wake

24 24 peaks. The implication here is that the spreading of the streamlines that was discussed in section 2 actually has little impact on the height of the wake peaks except when they are strongest. It also implies that the reduction in the forced eccentricity seen in equation (8) is responsible for some of the sharp drop seen in the wake peaks in many simulations instead of the spreading of the streamlines. As it happens, the difference between the actual peak heights and the prediction in this data set is well correlated to the dispersion in the eccentricities. There are exceptions where this prediction does not well match to actual wake peak heights. When we look at extremely low optical depth simulations the two differ significantly. It is not clear if this is because of an effect of the streamline spreading or if the small amount of shear through that causes the prediction based on the q value to be off. To get a plot that should be more comparable to BGT89 figure 1c we used the data sets from the simulations discussed in section 7 of wakes further from the gap edge. Fig 12 This is shown in figure 12, which has the same three values that were plotted in figure 10. The behavior exhibited is very similar to the prediction in BGT89. This simulation was actually carried out for multiple synodic periods so one can see what happens when the system is re-perturbed. Once again the q value used with equation 5 does an excellent job of predicting the envelope of the wake maxima. One point that it interesting to note here is that for the q value shown in figure 10, the first term under the square root in equation (9) is insignificant. Where it plays a role is in figure 12 when the system passes back by the perturber a second and third time. The new perturbation is strong enough to realign the epicyclic phases so the second term in equation (9) drops steeply and the first term compensates that.

25 25 One last point to note about the plots in BGT89 is that the forced eccentricity plots shown in figure 2 are quite similar to what is seen in most of the simulations, at least early on. This is because they seem to be finding a drop in the q value of the form 1/Y, almost exactly like what was found in equation (8). In many of the simulations we see a clear transition from this type of behavior to an exponential decay that is missing in the BGT89 results. 6. Damping Rates Using this basic understanding of how the wake maxima behave, we can proceed to examine how the wake damping rate varies with different parameters of the simulations. For the most part we focus on the rate of damping at the downstream end of the maxima envelope where it is well fit by an exponential. However, results of other points of interest are also presented. First, we look at how the rate of damping is altered by changes in optical depth, particle size, and coefficient of restitution. These dependencies are examined not only to help us better understand the dynamics of the system, but also because knowing how these values alter the wake optical depth could prove to be useful in determining the properties of the ring particles in Saturn s rings from measurements of the rates of damping. In particular, the damping rate could help to determine the coefficient of restitution because the particle size and optical depth can be constrained through other observations. To look at these dependencies, we only vary one parameter at a time relative to the standard simulation using particles 13m in radius with an optical depth of and the Bridges, et al. (1984) coefficient of restitution. After

26 26 the sections discussing those dependencies there is a section that look at special low optical depth simulations which produced interesting results Optical Depth Dependence As was discussed in section 5, the way in which the magnitude of the wake peaks Fig 13 decrease in many of the simulations is far from uniform. Figure 13 shows the data and fits for two of the optical depth runs to illustrate how different they are. The fits were produced using a split function as described in section 5. These are also directly Tbl 1 comparable to figures 7 and 8. Table 1 shows significant information from the various simulations where optical depth was varied while particle size and coefficient of restitution were kept unchanged. For all of these simulations, particles 13m in radius were used with the velocity dependent coefficient of restitution found by Bridges, et al. (1984). The decay rate that is listed is for the exponential region farther downstream as described in the previous section. The fourth column gives the azimuthal location where the fit function changed from an inverse form to an exponential. This value gives us an idea of where the dynamics change from a spreading out of the streamlines and fast drop in eccentricity to prevent shear through to a slower exponential decay of the eccentricities. The fifth column is the azimuthal location of the highest peak measured in radians downstream from the perturber. In most of the simulations this occurs at the same point as where the streamlines would normally cross in the collisionless simulation. However, that is not always the case, as one can see that in the higher optical depth simulations it occurs farther downstream. The second to last column gives the height of the highest wake peak as a multiple of the unperturbed optical depth used for the

27 27 simulation and the last column gives the exponential decay rate of the forced eccentricities. Fig 14 The dependence of the damping rate on optical depth is shown in figure 14. The one data point that does not fit the trend and has a value significantly different from its neighbors was difficult to measure accurately because the particle guiding centers migrated an appreciable amount during the simulation. As such the guiding center density in the center of the cell was significantly altered. When that one data point is ignored, the most significant feature of this plot is that it is nonmonotonic. In particular, there is a minima located around an optical depth of To understand why the damping rate increases at larger and smaller optical depths we can look at an analytic model for energy dissipation in the rings. In BGT85 they give the rate at which energy is dissipated in collisions to be 3 de v, (10) dt s where v is the velocity dispersion and s is the average distance traveled between collisions. To approximate the dependence of the velocity dispersion on optical depth we can use the following expression 2 ε = 1 c 2 c 1 2 ( 1+ τ ), (11) where c 1 and c 2 are constants of order unity (Goldreich and Tremaine 1978), and ε denotes the perpendicular coefficient of restitution. While this expression is not valid when finite sized particles are close-packed as occurs in the wake peaks, it will suffice for this analysis. As was noted in paper I, the velocity dispersion rises quite quickly after the wake peaks back to the level they were at before the particles entered the peak. This

28 28 gives some indication that the low-density regions have just as much of an influence on the velocity dispersion as the high-density regions do. Combining equation (11) with the velocity dependent coefficient of restitution (Bridges, et al. 1984), we find that the velocity dispersion varies with the optical depth as τ v 2. (12) τ Equations (10) and (12) together make is clear why there is this non-monotonic behavior in the damping rate. At optical depths significantly below unity the velocity dispersion becomes quite large, resulting in a high rate of energy dissipation. At optical depths closer to unity, the velocity dispersion varies less with optical depth, but the distance between collisions becomes quite small which again increases the rate at which energy is dissipated. While developing such a theory based on standard fluid arguments is instructive and can help us to understand the behavior, it is very important to keep in mind that the system is not well described by a uniform fluid model when the wake peaks are large. The significance of this nonmonotonic nature is not clear at this point because these simulations used large particles. A more realistic scenario would be like that discussed in section 6.4 where the low optical depths are achieved by having small particles. The lower optical depth simulations all follow basically the same structure that was discussed in section 5. The simulations at and above an optical depth of have different structures. The simulation with an optical depth of produced a single decay region. A break point is listed in the table simply because a fit can be made that produces that result. The highest peak in this simulation occurs one wake peak later than where it does in the collisionless case. This trend to late maximum peaks is even more

29 29 pronounced at higher optical depths. In both the and optical depth simulations the wake peak magnitudes rise significantly after the critical azimuthal position. Beyond the highest point they also fall into a simple decay pattern. The reason the wakes continue to grow in these simulations beyond the point where the kinematic theory says that they should stop is not immediately obvious. The data does not allow an explanation involving the thickness of the streamlines as was used for the steep drop in the other simulations. A second hypothesis might be that in these simulations the particles could not become densely packed early on because the particle distribution is too hot. This hypothesis is also not supported by the data as the dispersion in the particle velocities rises in the region in question where this explanation would require it to fall. Indeed, the reason the wakes behave in this way is due to the fact that Y crit, as used in equation (5), is a function of eccentricity. To be more specific, Showalter, et al. (1986) found it to be inversely proportional to the forced eccentricity. This is significant because the forced eccentricities in these simulations decrease before reaching Y crit. The early eccentricity damping moves the expected location of the maximum wake peak downstream and, as such, the wake peaks continue to rise for a while. It turns out that these higher optical depth simulations behave very differently than the low optical depth ones in other ways as well. In particular, the behavior of the boundary layers in the highest optical depth simulations that were presented here is very different from that in other simulations. They do not show a significant buildup of particles at the edges. This likely is a result of the collisional diffusion rate winning out over the process that causes these buildups. The nature of that process is something we discuss more completely elsewhere.

30 30 Focussing on the behavior of the location where the fit transforms from an inverse structure to an exponential structure, one sees that the location for this transition generally moves downstream at lower optical depths. This seems somewhat natural, as one would expect that in the lower optical depth simulations the collisions take longer to significantly alter the structure of the wakes. What is interesting though is that around the optical depth where the minimum damping rate occurs, this transition point reaches 0.5 radians down stream and does not move significantly farther downstream. It is not clear if this is simply the result of the method of doing the fitting or if the higher collision velocities in these low optical depth simulations actually do prevent the transition region from moving farther downstream. The second to the last column in table 1 shows how much larger the optical depth in the highest peak is from the unperturbed optical depth in the simulation. Here again the value rises with lower optical depths until it reaches a maximum value of just above 10. Again, it is not clear if the drop seen in the lowest optical depth simulations is real and results from the same processes that cause the higher damping rates, or if it is merely due to having more noise in the simulation with fewer particles. Figure 5 in paper I showed how the forced eccentricity and dispersion in epicyclic phases varied for a single simulation. In that particular simulation the forced eccentricities became negligible about the same time that the epicyclic phase angles were completely randomized. While this behavior is generally seen across different optical depths, there are differences in the details of how it occurs. The last column in table 1 shows the decay rates of the forced eccentricities late in the simulations. The trends for these numbers generally mirror what is seen in those for the damping rate of the wake

31 31 maxima in that they have a minimum value around an optical depth of The decay rates for the forced eccentricities are generally larger than those for the wake peaks with the exception of very high and very low optical depths. As one might expect from the analysis of the behavior of the forced eccentricities given in section 5, the forced eccentricities are nearly identical through the early parts of most of the simulations. The two major exceptions to this occur when the optical depths are high enough that there is significant damping before the highest wake peak and when there is some shear through which occurs in the optical depth case. In cases where there is neither shear through, nor early damping of the eccentricities, the values of the forced eccentricities are nearly identical until between 0.2 and 0.25 radians downstream. At that point they diverge with the different exponential rates given in table 1. As was mentioned earlier, in the two highest optical depth simulations, the collision rates are high enough that the forced eccentricities begin to drop before the highest wake peak. However, this does not seem to alter the relative importance of the forced eccentricities and the epicyclic phase angle dispersions. The higher collision rates merely seem to cause the eccentricities to damp faster while the dispersions grow more quickly. The more interesting case is when the optical depth is very low and some shear through can occur. The general behavior of low optical depth systems is examined in section 6.4 with smaller particles. With the large particles the data is quite noisy, but it is still very clear that the forced eccentricities do not drop as quickly as they do in the other simulations. However, in the optical depth simulation the dispersion in the

32 32 epicyclic phase angles rises fairly quickly. As a result, it is the dispersion that dominates the observed damping of the wakes in this case Particle Size Dependence Tbl 2 In this section the effects of varying particle size on the damping of the wakes is explored. Table 2 has the significant results from the simulations where particle size was varied while the optical depth remained fixed at a value of and the coefficient of restitution was that found by Bridges, et al. (1984). The columns in the table are the same as for table 1, which were described in the previous section. In these simulations the location where the fit changes from inverse to exponential appears to be completely unaffected by the particle size. Looking at the location of the highest magnitude peak the situation is much less clear. What is not obvious from this table is that many of the simulations in this set have multiple peaks that are nearly the same height surrounding the highest peak. For this reason, small fluctuation can easily alter exactly which one is the Fig 15 highest. Figure 15 shows the data for two of these simulations as well as the best fits for them. Fig 16 Looking specifically at the damping rate at the end of the simulation there is a clear trend in the effects of particle size. Figure 16 shows both the measured damping rates as well as a polynomial fit to the data. This clearly shows that the damping rate goes as roughly the square root of the particle radius in the range that we have considered. This trend will almost certainly break down if the particle size becomes comparable to the size of the wake features, though it is unlikely that one could truly measure the wakes in that regime. What is less clear is what would happen at the other

33 33 end when the particles are very small. In a real system particles below a certain size would begin to feel the effects of other forces like plasma drag enough that the perturbation of the moonlet might become less significant. However, because most of the optical depth in many rings is provided by very small particles, understanding how they behave could be significant. This question will be addressed to some extent for lower optical depth systems in section 6.4. An explanation of why the dependence goes roughly as the square root of radius can be found using arguments similar to those presented in section 6.1. This explanation would have the damping rate depend on the way in which coefficient of restitution varies with collision velocity. To determine if this was the case, I have performed three simulations using a coefficient of restitution with a different velocity dependence. The coefficient used for these alternate simulations had the following form: ( v 0. 5 ) ε. = 0.01cm / s This form varies from that found by Bridges, et al. (1984) only in that the exponent has been multiplied by a factor of slightly greater than two. This was chosen because using the analysis that gave a square root dependence with the Bridges, et al. (1984) model it should have produced a damping rate that is independent of the particle size, a criteria that can be very easily checked with a small number of simulations. The results of these Tbl 3 simulations are given in table 3. They do show the damping rate having a dependence on the particle size. When fit with a power-law, a slope of 0.66 is found. We can only conclude that whatever is causing the dependencies that are seen is more complex than what can be found with a simple analyses based on previous fluid models. When comparing to what dominates the damping of the wakes for the original particle size simulations there are two things that stand out. At smaller particle sizes, the

34 34 randomization of epicyclic phase angles seems to occur more slowly relative to the damping of the forced eccentricities. This is most likely due to the tighter packing of particles that can occur in the wake peaks when particles are small. When the particles are packed together more tightly they are aligned in their epicyclic phases. The packing however, does nothing to help preserve the forced eccentricities. There is one other characteristic of the forced eccentricities that stands out when comparing simulations using particles of different sizes. The behaviors of the forced eccentricities early on in these simulations is not nearly as identical as it was in the simulations that used only different optical depths. While the curves exhibit very similar shapes, the steep drop off predicted by equation (8) occurs earlier in the simulations with larger particles. This would seem to imply that the larger particles prevent the streamlines from being compressed as far and as such, push upstream the location at which the forced eccentricities have to begin their rapid decent. However, this happens independently of the location of the highest peak. Because the highest peak occurs roughly where the streamlines are most compressed, there must be some type of relaxation process that prevents the streamlines from fully compressing early on. This type of behavior would support the observation that many of the simulations have several peaks at roughly the same height Coefficient of Restitution Dependence Let us now turn to one of the least well constrained features of Saturn s rings: particle coefficient of restitution. As has been shown in the previous subsection, it can have a dramatic impact on the behavior of the system. This section explores the effect on

35 35 damping rate caused by simple variations in two basic models of particle elasticity. The simpler of the two is to use a constant coefficient of restitution. This method is significantly simpler in analytic treatments and is often used in numerical treatments to be able to compare to previous analytic results. The other approach taken was to start with the standard velocity dependent coefficient of restitution from Bridges, et al. (1984) Tbl 4 and multiply it by different factors. Table 4 presents the results. All of these simulations used 100,000 particles 13m in size with an optical depth of The first 5 simulations listed used the velocity dependent model multiplied by the value in the first column. The last three used the specified constant coefficients of restitution. The first thing that jumps out when looking at this table is that the decay rate trends are non-monotonic for both forms of coefficient of restitution. In both cases, higher decay rates are found for both more elastic and less elastic collisions. In the former case this is because the collisions more effectively randomize the orbits of the particles and produce larger dispersions in the epicyclic phase angles and eccentricities. In the latter situation the collisions are more dissipative and cause the forced eccentricities to be reduced quickly. Fig 17 Figure 17 shows four of the data sets and best fits for different coefficients of restitution. In this figure the velocity dependent curves have been put on the top and the velocity independent curves are on the bottom. One point that should be immediately obvious is that the constant coefficient of restitution simulations behave in a different manner from what has been seen previously when the coefficient of restitution is velocity dependent. This is most obvious for the simulation where ε=0.6. The first difference is how early the maximum peak occurs. The second difference is that there is a region in

36 36 this simulation well downstream of the highest wake peak where the optical depths at the wake peaks rises substantially. This rise appears to be attributable to the behavior of the forced eccentricity and the dispersion in the epicyclic phase angles with the later being the more significant. During the region where the wake peaks are rising, the forced eccentricity drops only slightly so the Y value is advancing here faster than the Y crit value is being pushed downstream. More importantly though, the dispersion in the epicyclic phase angles decreases quite dramatically in this region. Many of the simulations present evidence of how the high densities in the wakes can organize the motion of the particles, but as was discussed in paper I, this process generally appears to have only short range implications. In the ε=0.6 simulation that is definitely not the case. The difference is probably attributable to the fact that as the wakes build, the impact velocities increase. In a velocity-dependent-ε model this would result in more energy dissipation, allowing the particles to pack together more tightly. With the fixed, high elasticity of the collisions, they instead produce a significant randomizing effect. It is not until later in the simulation, after significant energy has been removed, that the wakes can become densely packed enough for the packing to begin organizing the particle motion. While the difference seen here is probably not a problem for most of the studies that have been performed using a constant coefficient of restitution, this should serve as a warning that doing so can drastically alter the behavior of the system in certain situations. This also suggests that more work needs to be done in determining the collisional behavior of the ring particles. It is interesting that the results imply that the Bridges, et al. (1984) model nearly minimizes the damping rate compared to similar models with the

37 37 same basic functional dependence. Unfortunately, the non-monotonic nature of the results would prove a difficulty in any attempt to use a damping rate determined from observations to constrain the nature of the collisions. However, there is still hope that if more detail than just a damping rate can be determined from observations, then some constraints could be placed on the collisional model used to describe it Low Optical Depth Rings In the last three sections, we explored how changing one parameter of the simulation while leaving the others fixed alters the rate of decrease in the wake magnitude. In this section, we look at the effects of reducing particle size while leaving the number of particles constant to produce low optical depth rings. This also allows us to collect less noisy data, as noise was a problem in the low optical depth simulations where the particle size was kept constant. The initial motivation for performing these simulations was to find the point at which collisions no longer prevent the streamlines from passing through one another. While they served this purpose, they also give an indication of how single satellite shepherding might be accomplished through wake-like phenomena. Tbl 5 Table 5 presents the characteristics of simulations that were done with small particles and low optical depths. All of the simulations used 100,000 particles and the standard velocity-dependent coefficient of restitution. Only 4 simulations were performed because that was sufficient to give us an idea of roughly when the streamlines could begin to shear through one another. In the simulations with an optical depth of and , the streamlines did shear through. However, this fact is not

38 38 apparent from the optical depth data. In the collisionless simulation, the shear through was obvious because it produced double peaks in the optical depth data. In the optical depth simulation there are no double peaks. Instead, one has to actually plot the Fig 18 averaged streamlines in order to see that they have sheared through. Figure 18a shows both the streamlines and the particle densities from a region of this simulation. The lack of double peaks can be attributed to the fact that collisions have produced a dispersion in the epicyclic phase angles and eccentricities. This spreading out of the streamlines prevents them from creating the sharp structures that are required for double peaks. In the lower optical depth case a similar behavior is seen though there is some indication of double peaks at certain azimuthal and radial locations. In spite of this, the optical depth data does have a characteristic that marks it as different from the simulations where shear through does not occur. In the lower optical depth case, the optical depth beyond the location of shear through looks much like a square wave, with one flat peak that stretches between the locations where the double peaks would be expected. In this simulation, we also see the type of jump in the optical depths that was seen in the collisionless case when the peaks spread far enough apart to run into their neighbors. The particles used in these simulations are not all that small, nor are the optical depths extremely small in comparison to what is seen in many dusty rings. Inevitably, in a strongly perturbed dusty ring system with micron sized particles and considerably lower optical depths, the behavior would be expected to mirror that seen in the collisionless simulations assuming that other forces like plasma drag did not dominate the behavior. The first simulation that was performed in this category used an optical depth of with particle 2.6m in radius. While this simulation did not exhibit streamlines

39 39 shearing through one another, it did display a potentially more interesting phenomenon. In the radian downstream that was simulated, the radial width of the particle distribution, which was originally 13.3km across, became roughly 2km or 15% narrower. Because the particles at the radial edges of the simulation cell retain a significant forced eccentricity, the measurement of the width is done on the location of the guiding centers. In effect this is to say that at the end of the simulation the particles cover a smaller range of semimajor axis values than they did originally. A similar, though smaller, effect is seen in the simulation using 1.3m particles with an optical depth of It is also present in the two lower optical depth simulations, but to a much smaller extent. Still, we can not rule out the possibility that it would be significant in such a system over many synodic periods. Fig 19 Before discussing the cause of this behavior it is helpful to see exactly what is happening in these simulations. Figure 19 shows surface plots of the optical depth of the physical particles (red) along with the density of the guiding centers (green) for the simulation using 2.6m particles. When looking at these plots keep in mind that the particles oscillate about their guiding centers as described by equation (1) and (2). Surface plot (a) shows their behavior in the region around the location of the highest peak for the inner edge of the simulation region 1. The smooth distribution of guiding centers and the building wake structures are clearly visible on the left-hand side of the plot. Once the highest peak is reached, the eccentricities begin to be reduced and the distribution of guiding centers is altered on the right-hand side. Surface plot (b) shows a region just before 1.0 radian downstream where the effects of the guiding center motion 1 Recall that Y max is a function of eccentricity and that there is a radial eccentricity gradient caused by varying proximity to Pan.

40 40 are clearly visible. One of these effects is the formation of a ringlet on the inner edge. A similar ringlet forms on the outer edge but is less pronounced due to the smaller forced eccentricities and the fact that the wakes take a longer time to build at larger radial distances. A blowup on this ringlet is shown in surface plot (c). In this region the highest optical depths found in the ringlet are just over 0.1. At this point, the forced eccentricities in the ringlet are still fairly large so it exhibits its own wakes. The density of guiding centers at the core of this ringlet is growing in the region shown in this figure and the epicyclic phases of the particles are becoming more aligned at each wake peak. However, this comes at the cost of further damping the forced eccentricities of the particles. It is not clear from this simulation if the eccentricities in the ringlet will be fully damped before a synodic period has passed. If it does, the question then becomes, how much will the ringlet diffuse before being reperturbed. The presence of this ringlet is quite significant because Lane, et al. (1982) observed a similar feature in the Voyager images. Unfortunately, the details of this ringlet are not reported there though their figure 9 implies that the ringlet is separated from the main edge of the ring by roughly a kilometer and that it has a lower optical depth than what is present in the main ring. While this seems to fit the results of our simulation, it should be reiterated that this entire simulation used a lower optical depth than is present in the main ring. Similar, though less drastic phenomena occur in our other simulations as was addressed in the previous paper. What is not clear at this point is how these structures might evolve over multiple synodic periods or with a distribution of particle sizes. It is quite possible that a combination of those two factors could produce what is seen at the Encke gap. We plan to explore this possibility in future work.

41 41 As is clear from Figure 19, the compression of the ring does not occur uniformly. Instead, the center of the ring remains relatively unchanged while the particles at the edges move inward to create regions of high density at both radial extremes of the simulation cell. This motion of particles begins roughly at the highest wake peak. The reason it occurs is a simple extension of the guiding center migration that was discussed in section 2. Recall that at the wake peaks the guiding centers move radially due to the reduction of the forced eccentricity and the fact that collisions there only occur at specific epicyclic phases. That discussion dealt with a particle that was embedded in the middle of the ring. In this example, we begin with particles that are at the edge. In the case of particles at the inner edge of the cell, they do not enter the wake until after their epicyclic phase angle has advanced beyond π/2 radians because there were no particles below them to form a wake at smaller epicyclic phases. As a result, damping of the forced eccentricities causes the guiding centers of the particle at the inner edge to all move outward. Of course, to conserve angular momentum, the guiding centers of other particles must move inward at the same time. As a result, the particles just inside the edge get pulled inward radially as they enter the wake then are pulled back outward as they are leaving the wake. This motion propagates outward with the wakes until it reaches the outer edge where the particle guiding centers move inward only. This process clearly causes the ring to become narrower. It is disrupted by density variations as will be discussed later. This process creates an initial build up of guiding centers on the edges. The arguments applied here though can equally well be used in any location where there is a radial gradient in the guiding center densities. In particular, the perturbation of the

42 42 moonlet appears to cause an instability where the density at higher density regions grows with time, effectively a negative diffusion. This effect can be seen in the radial slice Fig 20 through part of the region shown in figure 19b as is shown in figure 20. The high-density regions that form at the edges produce a gradient in the guiding center densities between the edge region and the main part of the ring adjacent to it. As a result, particles just inside the high-density region move toward the high-density region. This produces a region of low density and subsequently the particles just beyond the low-density region tend to move away from it. This alternating high and low density structure propagates away from the edge until the instability is no longer effective. This effect can be seen as a wave-like structure across the guiding center densities shown in figure 20. In order for this instability to work, the region of the ring in question must retain the primary characteristic of a perturbed ring: it must have forced eccentricities significantly larger than the free eccentricities and to be most efficient, the epicyclic phase angles of the particles must have a small dispersion. Because collisions throughout the ring cause the eccentricities to damp as particles move farther downstream from the perturbation, the magnitude of these density variations decreases further from the edges of the ring. In addition to the forced eccentricity and epicyclic phase requirements, the density of the ring must not be too high; otherwise the diffusion rate of the ring particles will occur on a timescale comparable to that over which these structures form. One very interesting feature of the formation of this inner ringlet is that the region between the ringlet and the main ring, where the particle density is reduced, has an extremely low forced eccentricity. This reduction is significantly stronger than the reduction in the particle densities. This is a major contributor to the fact that the ringlet

43 43 and the main body of the ring are dynamically separated. The only communication between the two must happen on a diffusion timescale instead of the much faster timescale that we find are caused by the wake phenomenon with large eccentricities. This large reduction in the forced eccentricity can not be explained by damping alone. Especially with the lower optical depth in this region, the damping rate for the eccentricity would be expected to be quite small as the optical depth has not gone so low that the damping rate rises. Instead, what is happening here is that all of the particles with large eccentricities are leaving the region because at their radial extrema they can collide with particles in either the main ring or the ringlet. Due to the fact that the migration mechanism only works for particles that have significant eccentricities, the particles that were in the lower end of the eccentricity distribution get left behind. It is not clear if this radial migration process has any visible effects at the Encke gap other than possibly to create a high-density edge and perhaps an imbedded ringlet. Unfortunately the periodic pattern it produces in the low optical depth simulation would be extremely faint in the Voyager data and would only propagate a few kilometers into the ring at most because by the time it would have advanced that far the forced eccentricities will be damped out. What is potentially most significant about the guiding center migration is the fact that it occurs in many of the situations and can be explained by a simple, direct mechanism. If this phenomenon occurs in a broad enough range of circumstances, it could successfully produce and maintain narrow rings. Another paper that focuses on this process exclusively in such systems is currently in preparation. 7. Distant Wakes

44 44 Up until now we have been looking only at wakes very near the edge of the Encke gap. This is interesting because it can be compared to Voyager images of the wavy edges (Cuzzi & Scargle, 1985) as well as some of the original analysis of the occultation data (Showalter, et al., 1986). It is also important to look at simulations of regions farther from the gap edge, where the magnitude of the forced eccentricity is smaller, to compare them with the results of Horn, et al. (1996). In that paper, a more sophisticated analysis of the PPS and RSS occultation data is performed to look for the signature of perturbations due not only to Pan, but also due to resonant interactions with more distant moons. One of the significant results of that work was the conclusion that at distances between 431km and 646km from the orbit of Pan, the wake structures survive for more than a synodic period. In particular, they found three of these long-lived wakes, which they refer to as second-order wakes, in the Voyager data. Two of the second-order wakes were found interior to Pan and the third exterior. At first this result might seem surprising, since it contradicts the simulation results that have been discussed so far, where the wakes damp out almost completely in less than a quarter of a synodic period. Before looking at the simulations of regions further from the gap, some discussion of why this result should not be quite so unexpected is in order. The first effect of being farther from the gap edge, and farther from Pan, is that the synodic period is significantly shorter. Because the wake peak occurs once each orbital period, this change results in fewer wake peaks per synodic period. In the case of the second-order wakes identified by Horn, et al. (1996), the particles had completed between 199 and 254 orbits at the locations were they were detected. Compare this to the roughly 550 orbital periods in a single synodic period at the gap edge. This is quite

45 45 significant because most of the collisions occur in the wake peaks. In paper I it was shown that the forced eccentricities drop significantly at the wake peaks and are nearly constant between the wakes. The shorter synodic period is also significant, because the wakes do not start damping immediately. Recall that in the edge simulations, the wakes generally grow until Y crit is reached. The second effect is that Y crit 1/e and that the magnitude of the forced eccentricity varies as 1/X 2 (Julian and Toomre, 1966). Combining these we see that the location at which one expects the wakes to begin to damp out moves downstream as X 2. Before looking at the results of simulations done far from the gap edge, there is one more topic that needs to be discussed. In paper I, we described the simulation method and the fact that it did not calculate the gravitational interactions between the ring particles and the moonlet. Instead, the particles were initialized using an analytically derived expression for the distribution of the particles after an impulsive interaction with the satellite. While this simplified the body of the code, it also made it impossible to simulate what would happen after more than one synodic period. For the simulations presented in this section, the code was altered to perform the gravitational interactions with the moonlet instead of using the analytically derived initial conditions from Stewart (1991). For comparison with the Horn, et al. (1996) results, we preformed a simulation at a radial distance ranging from km to km from Pan (134,130.5 km to 134,143.8 km from Saturn). This lies within the range covered by the second-order outer wake detected in the Voyager RSS data. For this simulation we used particles 13m in radius and an unperturbed optical depth of so that it is comparable to our earlier

46 46 simulations as well as being reasonably close to what is observed in the outer A-ring. Fig 21 Figure 21 shows a surface plot of the optical depth that covers the region where the Voyager outer RSS scan was taken. In the body of the ring there are three regions where the wake structures are strong, separated by regions where the wakes seem to be largely missing. Taking a radial slice through this reveals two periodicities. The shorter wavelength one is that of the second-order wake, which has a wavelength that generally agrees with that found by Horn, et al. (1996). The longer wavelength periodicity has a very different origin. The origin of this second periodicity lies in the dynamics of the original wake structure passing by the perturber a second time. At the location when it passes by, the particles are in different phases in their orbits, depending on their radial distance from the perturber. Those particles that are heading toward Pan at that time have their motion reinforced by its gravitational pull. This causes their forced eccentricities to increase and the dispersion in their epicyclic phases to decrease. On the other hand, the particles that are moving away from Pan when they pass have their motion cancelled by the gravitational pull, which has exactly the opposite effects. The result of this can be seen in Fig 22 figure 22 where the average streamlines of the particles before and after the encounter with Pan are shown. An alternate view of this effect is that the forced eccentricities are increased by the perturbations at the locations of first order Lindblad resonances and damped between them. This particular system provides a very graphic display of the width and spacing of these resonaces, and how they alter the behavior of a ring system. The way in which this structure forms causes it to have the same periodicity at all azimuthal positions and its frequency is equal to the frequency of the wake structures at

47 47 2π radians. Alternately this is the same as the radial frequency of the Lindblad resonances of the perturbing satellite. In this case the measured frequency of the eccentricity variations is cycles/km. Comparing this to the Burg contours in the Fig 23 outer RSS scan in figure 9b of Horn, et al. (1996), which is reproduced as figure 23 here, one sees that the Voyager data shows a strong signal at this frequency in the radial region that has been simulated here. Because the spacing of the Lindblad resonances is the same as the spacing of the wakes at the end of the synodic period, the frequency of this structure has the same radial dependence as the wake wavelengths. However, it occurs in the region between the first order and second-order wakes. Exactly where it should lie between those depends on the azimuthal location of the scan. For scans taken only slightly beyond 2π radians the wavelength should be closer to that of the second-order wake while just before 4π radians it would be very near the wavelength of the first order wakes. Structures that fit this expectation can be seen in both the inner and outer scans in figure 9 of Horn, et al. (1996). One interesting feature of the dynamics of these alternating high and low eccentricity regions is that they migrate radially outward over the course of a synodic period. During this migration, the amplitude of the variations in the eccentricity gets smaller. Not only do the maximum eccentricity regions damp down, the eccentricities of the lowest regions increase as well. The explanation for these behaviors lies in the fact that the majority of the collisions occur in the wake peaks where the particles are at very specific epicyclic phase angles. In particular, the wakes peak in the region where the particles are moving radially outward. In a region where the eccentricity gradient is negative, the collisions occur in a location where the higher eccentricity particles are

48 48 moving up into the lower eccentricity particles. These collisions on average move energy, and therefore eccentricities, outward. When the eccentricity gradient is positive, the region where the high eccentricity particles are moving down and could transfer their eccentricity occurs at a local minima in the optical depth. As a result very little happens. The result is that in regions where the radial eccentricity gradient is negative, the eccentricities at larger radial distances are increased. Where the gradient is positive the eccentricities only undergo the normal damping caused by collisions. So the periodic structure as a whole moves outward away from Pan over the course of a synodic period Fig 24 before being reset when the particles pass back by Pan. Figure 24 illustrates this process. Interestingly, this effect occurs with no migration of the physical particles. Unfortunately, this effect does not have any immediately obvious observational consequences though it will alter the way in which the Lindblad resonances are manifest in this system over many synodic periods. The significance of these resonances comes from the fact that the perturbations are additive over successive synodic periods. The migration of the eccentricity peaks reduces the efficiency of this process. 8. Conclusions By exploring the behaviors of wakes in simulations with a wide variety of parameters, we have been able to gain a more complete understanding of the dynamics of strongly perturbed ring systems. This includes understanding how the rate at which wakes damp depends on the optical depth, particle size, and coefficient of restitution. In particular, the results show that the exact nature of the coefficient of restitution can have a dramatic impact upon the behavior of the system. Unfortunately, this is probably the

49 49 least well-constrained property of the ring particle system. Variations in optical depth produced non-monotonic behavior in the damping rates with the highest damping rates occurring at the extremes of high and low optical depths. The most well behaved variations were those with particle size where the data was well fit by a power-law. Our results reiterate the fact that these strongly perturbed systems do not behave as described by ideal gas or fluid models. This is quite significant because these models are commonly used in explaining ring systems in analytic treatments. For example, in the fluid models produced by Borderies, Goldreich and Tremaine, the behavior of the system was generally averaged over an orbital period. In our simulations we see such strong variations in optical depth and other values over each orbital period that a more accurate description might be achieved if the evolution of the system was instead modeled by a localized kick at each wake peak. Perhaps the most intriguing feature of the simulations is the way in which the particles guiding centers migrate during the course of the simulation. We clearly see evidence that the strong perturbations can produce a negative diffusion in the system. This behavior is the result not only of having large forced eccentricities, but also of the fact that the epicyclic phase angles of the particles are well aligned in the wake peaks. This alignment of the epicyclic phases and strong inhomogeneities caused by the wake structures, cause collisions to occur preferentially at certain phases instead of being isotropically distributed. The phase-aligned collisions lead to rapid spatial evolution of the system in many of the simulations. When a lower optical depth is used, we see the negative diffusion instability play a prominent role and a narrow ringlet forms that is somewhat disjoint from the main ring. While this is related to the shear reversal

50 50 predicted by Borderies, Goldreich, and Tremaine, we found in paper I that shear reversal only occurs over a very narrow range of azimuthal locations while the negative diffusion is found to occur more generally in the simulations. The most difficult challenge may be connecting these results to what people should be looking for in Cassini observations. There should be sufficient coverage from the Cassini observations to calculate a damping rate for the wakes. In addition it is likely that measurements of the particle size distributions will be possible for most areas of the rings. The question there will be radial resolution. Unfortunately, it is unlikely that any strong conclusions will be attainable using just these results in conjunction with measurements of the damping rates and particle sizes. There are two main reasons for this. First, these results have not included particle size distributions and self-gravity, which will inevitably complicate the behavior of the system. Second, the variable that these results are most likely to help us understand is the coefficient of restitution of the particles. The true functional form of the coefficient of restitution will be difficult to constrain with observed wake damping rates. One possible way to improve the situation would be to look at wake behaviors and damping rates as a function of radial distance from the perturber. Cassini observations should be able to provide some measure of wake profiles at different distances from the perturber. This work focussed almost completely on the damping at the edge of the gap but it is quite possible that the smaller forced eccentricities further from Pan could produce different damping rates and behaviors in otherwise identical systems. One prediction that can be made from this work is that if the inner ringlet at the Encke gap does prove to exist, it is likely to be populated by smaller particles than are

51 51 found in the main body of the ring. This prediction is based on the fact that the only simulations in which we observed the formation of such a ringlet were those performed with smaller particles. This would not be too surprising as Voyager data showed that the edges of gap like the Encke gap appear to have enhanced number of small particles. Our simulations here would indicate that if a region of smaller particles did built up at the edge of the Encke gap, the negative diffusion caused by the wakes would tend to make said material collapse down into a narrow ringlet. Acknowledgements We would like to acknowledge financial support from???. References Borderies, N., Goldreich, P., & Tremaine, S. 1982, Nature, 299, , Icarus, 55, , Icarus, 63, , Icarus, 68, , Icarus, 80, 344 Bridges, F., Hatzes, A., & Lin, D. 1984, Nature, 309,333 Cuzzi, J. & Scagle, J. 1985, ApJ, 292, 276 Haenninen, J. & Salo, H. 1992, Icarus, 97, , Icarus, 108, 325 Hertzsch, J., School, H., Spahn, F., & Katzorke, I. 1997, A&A, 320,319 Horn, L., Showalter, M., & Russal, C. 1996, Icarus, 124, 663

52 52 Julian, W. H., & Toomre, A. 1966, ApJ, 146, 810 Lane, A. L., Hord, C. W., West, R. A., Esposito, L. W., Coffeen, D. L., Sato, M., Simmons, K. E., Pomphery, R. B., & Morris, R. B. 1982, Science, 215, 537 Lewis, M. & Stewart, G. 2000, AJ, 120, 3295 Lewis & Stewart 2002, IASTED Lewis & Stewart 2003, IASTED Nicholson, P. D., & French, R. G. 1998, BAAS, 30, 1043 Showalter, M., Cuzzi, J., Marouf, E., & Esposito, L. 1986, Icarus, 66, 297 Showalter, M. 1991, Nature, 351, 709 Stewart, G. 1991, Icarus, 94, 436 Wisdom, J., & Tremaine, S. 1988, AJ, 95, 925 Optical Particle Decay Break Point Peak Peak Eccentricity Depth Count Rate Location Height Decay Rate (radians) , ±0.28 N/A ± , ±0.04 N/A ± , ± ± ± , ± ± ± , ± ± ± , ± ± ± , ± ± ±0.11

53 , ± ± ± , ± ± ± , ± ± ± , ± ± ± , ± ± ±0.37 Table 1 - Significant values from simulations using different optical depths. All rates are given per radian and all positions are given in radians downstream from the perturber. Particle Particle Decay Break Peak Peak Eccentricity Radius Count Rate Point Location Height Decay Rate (meters) 104 1, ± ± ± , ± ± ± , ± ± ± , ± ± ± , ± ± ± , ± ± ± , ± ± ± , ± ± ± ,600, ± ± ±0.02 Table 2 - Significant values from simulations using different particle sizes. All rates are given per radian and all positions are given in radians downstream from the perturber.

54 54 Particle Particle Decay Break Peak Peak Eccentricity Radius Count Rate Point Location Height Decay Rate (meters) 26 25, ± ± ± , ± ± ± , ± ± ±0.03 Table 3 - Significant values from simulations using different particle sizes with a modified coefficient of restitution that had a different dependence on the collision velocity. All rates are given per radian and all positions are given in radians downstream from the perturber. Coefficient Decay Rate Break Point Peak Peak Height Eccentricity of Location Decay Rate Restitution ± ± ± ± ± ± ± ± ± ± ±0.05 * ± ± ± ± ±0.10 N/A ±0.32 constant

55 ±0.12 N/A ±0.11 constant ± ± ±0.11 constant Table 4 Significant values from simulations using different coefficients of restitution. The * denotes a simulation where a fit could be performed and a transition point measured, but the decay could be equally well fit by a single exponential curve. All rates are given per radian and all positions are given in radians downstream from the perturber. Particle Optical Decay Break Point Peak Peak Eccentricity Radius Depth Rate Location Height Decay Rate (meters) ± ± ± ± ± ± ± ± ± ± ± ±0.03 Table 5 Significant values from simulations with small particles and low optical depths. All rates are given per radian and all positions are given in radians downstream from the perturber. Figure 1 The motion of particles as described by guiding center coordinates can be easily visualized as motion in an ellipse around the guiding center which is moving

56 56 downstream at a rate proportional to the radial distance of the guiding center from the moonlet. This figure shows three particles at three different times as they move downstream. They begin at ϕ=0. Note that this illustration is very out of proportion for the edge of the Encke gap. In that system, the starting forced eccentricities are two orders of magnitude larger than the particle radii. Figure 2 A plot of the streamlines for a simulation shows clearly the fact that the wake peaks occur at particular phase angles. From this plot and equation (3), one can see that the density of streamlines, and therefore the density of particles, is highest between phase angles of roughly π/4 and 3π/4 radians. Note that from equation (1) periapse is at a phase angle of zero and the phase angle advanced to the right in this figure. Figure 3 A slice through the guiding center density of one of our simulations, (a), shows a quick drop followed by a steep rise before settling down to very near its initial value. This result can be produced by the guiding centers moving first inward, then outward as a result of particle phase alignment in the wakes and the fact that the farther out a particle is in the ring, the later it passes through the wake peak (b). Figure 4 An azimuthal slice through the middle of our simulation cell for a simulation without collisions shows how the wakes behave according to a purely kinematic description. In this model the wakes are reduced in height because the streamlines shear through one another. This reduction due to shear through is extremely fast and is best fit by a function of the form 1/x instead of by an exponential.

57 57 Figure 5 The first decay region of the collisionless simulation is plotted here with two different best fits. In (a) an exponential fit is used. It is quite apparent that this does not provide a good fit to the data. In (b) a function of the form A/(x-B) is used as might be predicted by equation (6). This alternate function does an extremely good job of fitting the simulated data in this case. Figure 6 Slices through the region from the beginning of the simulation to just past the highest peak are shown for a simulation without collisions, (a), as well as our nominal collisional simulation, (b). Plotted along with the slices through the simulation data are best fits for each data set using equation (5). The fits are poor near the maximum wake peaks because the wake peaks are much narrower in the simulation data than they are in the analytic expression. Also plotted are horizontal lines showing the unperturbed particle count and optical depth. The optical depth used is the geometric optical depth of the physical particles: τ=nπr 2 /A=πσr 2 /m. Figure 7 A plot of the optical depth in a slice through our baseline simulation. This simulation involved 100,000 particles each 13m in radius. They were positioned in a cell such that the average optical depth was The wake peaks have been connected to illustrate the behavior of the wake magnitudes. This simulation clearly shows two basic regions of wake behavior. The first region, from the point of the perturbation to 0.17 radians downstream, is where the wakes are increasing in strength because shear is compressing the particle streamlines. Beyond 0.17 radians the height of the wakes

58 58 decays out in a fairly smooth manner until the wakes are completely gone. An exponential function has been fit to the region of the wake decay. Figure 8 This plot is a slice through a collisional simulation with an optical depth of using 13m particles. The wake maxima have been connected to illustrate the bounding envelope and a two-part fit to the region where the wake heights are decreasing is shown. Once again the initial increase in the wake heights is well fit by equation (5). The drop in the wake magnitudes though displays two distinct behaviors. At first it drops very quickly before leveling off to a slow exponential decay. The fit that is shown used the same type of function that was used for the collisionless case to fit the first region and joined it to an exponential function that runs out through the rest of the data. Figure 9 This plot shows the dispersions in the eccentricity and phase angles of particles in the same simulation that was used to produce figure 8. These values effectively determine how diffuse the streamlines are. The spreading out of the streamlines produces roughly the same effect on the wake maxima as the streamlines shearing through one another. Figure 10 The nonlinearity parameter, q, is plotted along with the envelope of the wake peak optical depths and a predicted wake peak envelope calculated by substituting q for Σ in (5). This plot is for a slice close to the edge of the gap in a simulation with an optical depth of using 13 m particles. The structure of the q does not match what was found by BGT89 very well. It is interesting that the predicted envelope matches the

59 59 actual envelope very nicely. This implies that the spreading of the streamlines has little impact on the height of the wake peaks. Figure 11 This is a reproduction of figure 1 from BGT89. It shows the behavior of the nonlinearity parameter, q, at three different distances from the perturber. The parameter ξ is the ratio of the distance from Pan the distance of the observed edge from Pan. Our simulations use similar parameters to those for plot (a). Figure 12 This plot is the same as what is shown in figure 10 only for a slice through a simulation done far from the edge of the gap. This simulation also ran for multiple synodic periods. Here again we see that the predicted optical depth very closely matches the actual optical depth envelope. Figure 13 These plots show slices through the output for simulations with two different optical depths, both different from those used in figure 7 and 8. The slices have the optical depth data and the fit to the wake maxima. The most significant aspect of the plots is the variety of behaviors seen in the behaviors of the wake peaks. Figure 14 The above plot shows the variations in the final damping rates as a function of optical depth for simulations with particles 13m in radius and a velocity dependent coefficient of restitution. Notice that there is a minima that occurs around an optical depth of Above this optical depth the increase in the collision rate causes the viscosity to rise. Below this value the velocity dispersion increases, which not only

60 60 increases the viscosity, it also results in more dissipative collisions. Figure 15 Optical depth data and fits from the particle size simulations. Plot (a) shows data from a simulation that used particles 6.5 m in radius while plot (b) comes from a simulation with particles 52 m in radius. Figure 16 Figure showing the damping rates for the simulations where particle size was varied. A power-law fit is shown to the first 8 data points. The data set from the simulation with the largest particles was left out because the boundary layers in that simulation were thick enough that they touched in the center of the simulation cell, making it impossible to get a good measurement of the natural damping rate. Figure 17 Data from the coefficient of restitution simulations. The top two plots are for simulations with velocity dependent coefficients of restitution that had the general form given by Bridges, et al. (1984) but were multiplied by factors of 0.5 for (a) and 2.0 for (b). Plots (c) and (d) show data from simulations that had constant coefficients of restitution. The simulation that (c) was taken from used a value of 0.4 while the simulation that (d) came from used 0.6. Notice how small changes in the coefficient can drastically alter the behavior of the wake peaks. Figure 18 Plots of the streamlines and the optical depth from two collisional simulations with optical depths of , (a), and , (b). In both of these cases the streamline plots show that the average streamlines sheared through one another. The

61 61 shear through does not produce the double peaks seen in the collisionless simulations because collisions have effectively broadened the streamlines. However, the optical depth data does display a significantly broader peak than is found in the simulations where shear through does not occur. In the lower optical depth case the peaks are not only broader, but also very flat. Figure 19 Three surface plots from a low-density simulation show the formation of a narrow ringlet. In all three plots red represents the optical depth of the particles and green indicates the density of particle guiding centers. Plot (a) shows the region around Y crit for the inner edge. On the left side, the guiding centers are uniformly distributed and aren t moving significantly with time. On the right side, the higher optical depths in the wakes have started to cause the guiding centers to move. Plot (b) shows a region farther downstream where this guiding center migration has produced a ringlet on the inner edge. Plot (c) is a blowup on the inner edge that shows the structure of the inner ringlet more clearly. Figure 20 A radial slice through figure 19 (b) shows how a periodic pattern of higher and lower guiding center densities is set up in a simulation due to a reverse diffusion process that occurs as a result of the strong perturbation in a lower density ring. Figure 21 This surface plot of the optical depth in a region where second-order wakes are interacting with a first order perturbation shows the impact that Lindblad resonances have on the wakes. In the radial locations where we have m/(m+1) resonances, the

62 62 forced eccentricities are enhanced and the epicyclic phase angles strongly aligned. This results in stronger second-order wake structures. Between the resonance locations the eccentricities are reduced and the particles phases are more dispersed causing weaker wake structures. Figure 22 This plot of the streamlines at the end of the first synodic period shows how the phases of the particles entering the region where they pass back by the satellite produces the structures seen in the surface density plot. Figure 23 This plot is a reproduction of figure 9b from Horn, et al. (1996) showing the Burg spectra of the RSS outer scan from Voyager. Our simulation of the distant wakes covered the radial span from 134,130.5 km to 134,143.8 km from Saturn. Notice that in this region not only is a single with the expected frequency of a second order wake seen, but another feature with a wavelength of just under 0.3 km is present. This wavelength matches the spacing between the first order Lindblad resonances that were observed in our simulations. Figure 24 The left plot show the forced eccentricity at the beginning of the 3 rd synodic period as well as half way through it to show how the eccentricity structures migrate. Right of that is close up of the averaged streamlines from the simulation of secondary wakes to illustrate why this happens. On the top half of the figure is a region where the radial eccentricity gradient is negative. The compression of streamlines occurs in a region where streamlines with larger eccentricities are moving up into streamlines with

63 63 smaller eccentricities. On the lower half of the plot is a region where the gradient is positive. In this region, the larger eccentricity streamlines are moving toward the lower eccentricity streamlines in regions where the streamlines are most distant and fewer collisions are occurring.

64 64

65 0.120 radians radians 65

66 66 Guiding Center Optical Depth a) b) y (radians) Lower Normal density regions Higher

67 Particle Count E E E E E+00 y (radians)

68 68 (a) Particle Count E E E- 01 y (radians) 3.00E E E E E- 01 (b) Particle Count E E E E E E E E-01 y (radians) Data Wake Peaks Fit

69 69 a) Particle Count y (radians) b) Optical Depth y (radians) Unperturbed Data Fit

70 Optical Depth E E E E E E+00 y (radians) Data Wake Peaks Fit

71 Optical Depth E E E E E E+00 y (radians) Data Wake Peaks Fit

72 E E E E E E E E E E E E E+00 y (radians) Phase Angle Dispersion Eccentricity Dispersion

73 Optical Depth y (radians) Data Prediction q

74 74

75 y (radians) q Optical Depth Prediction

76 Optical Depth E E E E E E+00 y (radians) 0.14 Optical Depth E E E E E E+00 y (radians) Data Wake Peaks Fit

77 77 Damping Rate (1/radian) Optical Depth

78 78 a) Optical Depth E E E E E E+00 y (radians) b) Optical Depth E E E E E E+00 y (radius) Data Wake Peaks Fit

79 79 Damping Rate (1/radian) y = x Particle Radius (m)

80 80 a) 1.4 b) Optical Depth E E E+00 y (radians) Optical Depth E E E+00 y (radians) c) d) Optical Depth Optical Depth E E E+00 y (radians) E E E E- 01 y (radians) 8.00E E+0 0 Data Wake Peaks Fit

81 81 a) radians 0.35 radians 0.29 radians 0.35 radians b) radians 0.35 radians 0.29 radians 0.35 radians

82 (a) (b) (c)

83 Radial Distance from Pan (r Pan ) Optical Depth Guiding Center Density

84 radians 9.85 radians

85 radians 6.5 radians

86 86

87 87 Forced Eccentricity 3.00E E E E E E E Radial Position Negative eccentricity gradient Positive eccentricity gradient 19 radians 23 radians