Negative Diffusion in Planetary Rings

Collisional simulations of planetary rings that have been perturbed by a nearby moon have a tendency for material to clump together, moving into regions of higher density. This page contains a number of images and animations exploring this behavior. It also has raw data and SwiftVis analysis files that can be used to explore this behavior more fully.

Surface Plot Images and Data

Below is the list of simulations that were performed and the values measured for them.

Tau

E (*10-5)

Widthmin/Widthinitial

taumax/tauinitial

r=13cm

r=39cm

r=65cm

r=130cm

r=13cm

r=39cm

r=65cm

r=130cm

0.001

1.07

0.950

0.948



1.03

1.31



0.003

1.07

0.927

0.927



1.10

1.25



0.005

1.07

0.919

0.921

0.920


1.14

1.18

1.26


0.010

1.07

0.913

0.914

0.915


1.11

1.09

1.12


0.030

1.07


0.912

0.912



1.10

1.07


0.050

1.07


0.912

0.914



1.10

1.08


0.100

1.07



0.918

0.924



1.09

1.03

0.300

1.07



0.852

0.853



1.19

1.24

0.500

1.07



0.828

0.831



1.21

1.30

0.001

1.61

0.909

0.902



1.03

1.16



0.003

1.61

0.852

0.853



1.19

1.24



0.005

1.61


0.828

0.831



1.21

1.30


0.010

1.61


0.804

0.807



1.11

1.16


0.030

1.61


0.791

0.792



1.22

1.10


0.050

1.61


0.790

0.792



1.17

1.10


0.100

1.61



0.798

0.809



1.13

1.13

0.300

1.61



0.822

0.847



1.13

1.62

0.500

1.61



0.836

0.869



1.45

1.19

0.001

2.15

0.874

0.867

0.853


1.09

1.31

1.44


0.003

2.15

0.784

0.785

0.779


1.36

1.47

1.46


0.005

2.15

0.724

0.728

0.727


1.55

1.49

1.46


0.010

2.15

0.653

0.655

0.653


1.73

1.60

1.61


0.030

2.15

0.591

0.598

0.596


1.87

1.81

1.68


0.050

2.15

0.580

0.585

0.591

0.603

1.70

1.66

1.53

1.56

0.100

2.15


0.593

0.602

0.627


2.39

2.18

1.73

0.300

2.15


0.630

0.653

0.697


3.20

2.56

2.02

0.500

2.15



0.683

0.739



2.07

1.63

0.001

3.24

0.860

0.858



1.09

1.17



0.003

3.24

0.786

0.787



1.34

1.34



0.005

3.24


0.722

0.713



1.53

1.43


0.010

3.24


0.602

0.588



1.87

1.90


0.030

3.24


0.085

0.097



16.80

10.90


0.050

3.24


0.073

0.093



14.80

10.50


0.100

3.24



0.112

0.165



8.07

5.08

0.300

3.24



0.184

0.283



4.81

3.16

0.500

3.24



0.244

0.385



3.68

2.41

0.001

4.35

0.887

0.856



1.02

1.16



0.003

4.35

0.838

0.840



1.21

1.20



0.005

4.35


0.797

0.806



1.26

1.40


0.010

4.35


0.721

0.727



1.36

1.40


0.030

4.35


0.063

0.078



20.10

13.40


0.050

4.35


0.053

0.069



17.50

12.90


0.100

4.35



0.091

0.131



9.71

6.27

0.300

4.35



0.150

0.223



5.94

3.92

0.500

4.35



0.190

0.284



4.74

3.17

 

Collective Results

The results of these simulations can be summarized in four different plots. These plots show the data points along with an interpolated surface from the data points in the tau vs. eccentricity plane. The data points are drawn as colored circles. There is a dark outline around groups of circles with the same tau and eccentricity values. The size of the circle reflects the size of the particles for the simulation. So each black circle will have at least two colored circles inside of it. Often they are so close in color that you won't be able to tell there is more than one. The first plot shows the minimum width of the ring in each of the simulations.

This image shows that the negative diffusion process does not change the width much when either the optical depth or the forced eccentricity are too low. The most significant contraction of the ring occurs around an optical depth of 0.03-0.05 with a forced eccentricity of 3e-5 or greater. What might surprise some about this plot is that when negative diffusion is most effective, the width of the ring can be cut by more than on order of magnitude after only a single pass by the moon and, as was shown above, the majority of that change happens very quickly. The location of maximum effect is also seen in the next plot which shows the maximum optical depth attained in the simulation divided by the maximum optical depth at the beginning of the simulation.

Here we see that the maximum optical depth can also be enhanced by more than an order of magnitude. This might lead one to wonder how high the optical depths can get. The next plot shows the maximum optical depths attained in each simulation.

Here we see that the maximum optical depth keeps increasing as the average optical depth and the forced eccentricity increase. The key point to note here is that this process can get the optical depth at the core of a narrow ring to be well above the unity. The points in the top right corner of the plot are almost all above an optical depths of 2 with a highest value of 3.8.

The last plots of collected data looks at the efficiency of the process. The migration by collisions can't move particles farther than the radial drift induced by their forced eccentricities. This plot shows the change in the width divided by the magnitude of these excursions from the semimajor axis. This is basically a measure of how efficient the process is.

Here we see an island in the middle of the plot with a high efficiency. Unfortunately, it isn't clear if the lower values at the top of the plot are real or not because the original ringlet was only 2.2e-5*R_0 in width. Eccentricities higher than that might require broader rings in order to properly measure the efficiency. Still, there are several remarkable things about this plot. At the peak, the ring width is decreasing by 60% of the forced eccentricity. That is incredibly efficient. Equally impressive is that the majority of the parameter space has an efficiency of 25% or higher.

 

Mechanism

The mechanism that drives negative diffusion is a systematic drift in guiding centers that occurs when the streamlines begin to shear through one another. At that point, the particles move through the wake peak three times. The first time is the wake peak overtaking them, then they move back through with the wake peak and finally the wake peak moves on past them. The figure below shows a column of particles with lines drawn to where their guiding centers would be. The A frame is the last wake peak before the streamlines begin to shear through. The B frame is after the shear through.

There is an animation showing this same simple model of the system as the particle propagate downstream. One of the particles in this has been highlighted to help the viewer see its motion. (Divx encoded MPlayer movie) We also have an animation showing the velocity components. (Divx encoded MPlayer movie)

Simulation Velocity Curves

This movie shows the optical depth and guiding center optical depth surfaces for the simulation with optical depth 0.3, particle radius of 65 cm, and initial forced eccentricity of 3.24e-5. When watching the movie, note how the velocity curves mirror the cartoon model, particularly right before and right after the streamlines intersect.

Animations

These are animations that show how negative diffusion occurs. Four different simulations are presented and each animation shows several wakes passing through the simulation cell. The aminations begin with wakes prior to Y=Y_crit, the location of kinematic streamline corssing, where the distribution of semimajor axes is not significantly modified then runs through the Y=Y_crit region where the process is seen to be effective. The frames of the animation show five different views of the data. The bottom region displays particle streamlines colored by the ratio Y/Y_crit. They are black when this value is less than 0.9 then turn red from 0.9 to 1.0. If the eccentricities do not damp enough to keep Y less than Y_crit, the color goes from red to yellow up to a value of 1.1 then from 1.1 to 2.0 it changes from yellow to blue and remains blue above 2.0.

The top panel shows four different plots of data binned along the radial axis using either Cartesian or guiding center coordinates, x or X. In each of these plots there is a black line showing the total distribution of particles as well as a red line and a green line. The red lines are counts of particles for which the value of e or X is decreased over the next 50 time steps while the green line is counts of particles for which the value has increased. In these simulations, e and X are constant unless the particles undergo collisions.

Some things to note about these animations. First, very few collisions occur outside of the wake peaks. Second, the wake peaks prior to Y=Y_crit have collisions and modify particles, but not in a systematic way. In those wake peaks, the number of particles moving inward is roughly equal to the number moving outward at all distances and the number losing eccentricity is roughly balanced by the number gaining. Once the streamlines intersect however, the collisions in the wake peaks now systematically lower eccentricity and do so in such a way that particles above the wake peak systematically move down and those below the wake peak move up. The way in which this distrubance propagates through the region leads to the negative diffusion.

Name Optical Depth Average Forced Eccentricity Particle Size Distribution GIF Divx AVI
  0.03 0.5 5e-8 Gaussian 17 MB 9 MB
  0.03 1 5e-8 Gaussian 20 MB 8 MB
  0.03 2 5e-8 Gaussian 22 MB 9 MB
  0.03 1 3e-8 Square 21 MB 11 MB