Mark C. Lewis (Trinity University)
Glen R. Stewart (Laboratory for Atmospheric and Space Physics)
Figure 1 – This figure shows a slightly edited section of the “Wakemaker Moon” image from ciclops.org next to an animated gif of one orbital period from one of our simulations. This simulation has a moon with a mass of 1.5e-13 Saturn masses and an eccentricity of 2e-5. It includes 27 million particles that have interparticle collisions with a velocity dependent coefficient of restitution. Particle self-gravity is ignored. The oscillations in the simulation are due to the fact that the moon is on an eccentric orbit. See below for a moon on a circular orbit.
This page presents some graphics from our simulations. Most of these are animated gifs which allow you to see the evolution of the system over an orbital period. We are working on a short document that will describe in more detail what is seen in these short "movies". It should be posted here within a few days.
The simulations posted here are of a system similar to the outer edge of the Keeler gap. They all use a large, fixed, semiperiodic cell method, have an optical depth of 0.1 using particle 13.6 m in radius. Due to the size of the cell, this means all of these simulations include slightly over 21 million particles. The simulations include inter particle collisions, but due to the large number of particles, they do not include particle self-gravity. In the various simulations presented, the mass of the moon and the eccentricity of its orbit are varied to see how those values impact the behavior of the system. Each of the figures below show geometric optical depth, eccentricity, and guiding center "optical depth" for a region of the simulation. The guiding center location is basically the semimajor axis and average longitude of the particle. These images use a color spectrum instead of gray scale to help the viewer to see more details. All figures on this page were produced with SwiftVis (convert was used to turn PNG images into animated GIFs).
The first simulation we present uses an eccentricity of 2e-5 and a moon mass of 1.5e-13. These are the largest values that we use for either of those parameters. Notice how the orbits of the particles are modified in bulk as they pass through each wake peak. This behavior is seen in both the eccentricity and guiding center optical depth plots. Click on the still image to see an animated GIF that shows the evolution of the system. The role of collisions is very significant in this simulation as it results in migration of the semimajor axes of particles, and the damping of eccentricities. Those two behaviors are actually closely linked.
To give you some idea of what the eccentricity of the moon is doing in this simulation, we have run a simulation where the moon is on a circular orbit. If you click on the image to see the animation, you will notice that the system is basically in a steady state as any particle that passes the moon at the same distance always gets the same perturbation and any particle at any location has the same history in a statistical sense.
For this next simulation, we left the mass of the moon at 1.5e-13 Saturn masses, but cut the eccentricity of the moon's orbit in half, down to 1e-5. The primary impact of this, as would be expected, is to reduce the variability across a single orbital period. Click on the still image to see an animated GIF that shows the evolution of the system.
This simulation shows what happens if we reduce the mass of the moon slightly to 1.25e-13 Saturn masses. This is closer to mass estimates based on Cassini observations. In this simulation the data was also binned a bit further out so that the peaks of the wavy edges would not be cut off so much. Most of the features in this simulation have not changed significantly from the simulation with the larger mass. It is interesting to note however that if you catch this region at the time when the most strongly perturbed particles are at periapse, the edge of the gap is almost perfectly flat beyond the 4th peak. We picked that frame for the still image below. This situation occurs in the other simulations, but is a bit more obvious in this one. Click the still image to see the animation.
This next image comes from a simulation that did not include the effects of collisions. In the collisionless case, the system behaves quite differently. The top plot shows the reason for this. In this plot we show some of the streamlines for the particles in the system. Notice how as the streamlines shear out they pass directly through one another. This process leads to characteristic "double peaks" features that are seen in the bottom figure if one looks at optical depth on a given radial slice. Those features are absent in the collisional simulations shown above. As above, you can click the single image shown here to see an animation of this simulation over one orbital period. This simulation used a moon mass of 1.25e-13 Saturn masses and the moon had an eccentricity of 2e-5.
This this next figure, we use those same parameters again and turn collisions back on. This time we plot the streamlines for that simulation along with both the optical depth and guiding center optical depth. The former two plots have plotted on top of them lines that show the separation between the particle guiding centers and the positions of the physical particles. In order to explain what exactly is shown here we should also describe the method of making the streamline plots in a bit more detail. At the same time that the optical depths of the guiding centers are calculated, we also calculate the average epicyclic phase angles and average eccentricities of the particles in those bins. We then select some of those bins and plot where a particle would be given those averaged values.
The extra information shown in the bottom two plots puts little dots over some of those binned guiding center locations and draws in the offsets for the given eccentricity and epicyclic phase. Because we pick certain bins, you don't see the guiding centers move in this figure. The reason for showing this information is that it helps to show that when guiding centers are moving around in the bottom plot, the particles those guiding centers belong to are actually in or near the wake peaks where the highest optical depths are.
The reader should also compare this plot to the collisionless one and notice how the streamlines do not shear through one another in any significant manner when collisions are considered. If you look back at the plot two higher on this page, it used the same parameters as this simulation, but also plots the eccentricities. What you notice is that the eccentricities drop steaply in the wake peaks. This is how the steamlines are prevented from shearing through one another.
One of the things that should jump out in this plot is that the top streamline that is drawn has regions and period of what looks like high noise levels. The particle counts in these regions are not actually low enough to produce that much noise. Instead, these are regions where the epicyclic phases have been fairly randomized by collisions, but the eccentricities remain large.