I am a Physics undergraduate at Trinity. My research this summer will focus on the structure and composition of the dense rings (A and B) of Saturn. In order to determine the maximum mass that could be contained in the regions with high optical depth, N-body simulations of A and B ring particles in environments with different surface and volume densities will be run. In addition, the amount of silicates in the rings will be examined by performing photometric analysis of N-body simulations of B-ring particles with different percentages of silicates. For reference, our observer theta will go from -PI to PI, our light theta will go to 2*PI, our observer phi will go from -1.4 to 1.4, and out light phi will go from 0.1 to 0.5.
Overview of Procedure
For this study, we produced sets of particles that had different surface densities, size ranges, and distances from Saturn via N-body simulations. Because we were studying the hiding of silicates, each simulation contained 0.5, 1, or 2% silicates by number of particles. The surface density we used was determined before the addition of silicates. Because of this, for a surface density of 30g/cm2, the effective surface density after silicate addition was 30.75 for 0.5%. 31.5 for 1%, and 33g/cm2 for 2%. Our distances from Saturn were those of the middle of the A and B ring, and our particle radius range was 9E-10—9E-9 that of Saturn. Because of time constraints, we only performed photometry on simulations with 30g/cm2. Once we had those files, we used the program SwiftVis to perform photometry and other processes. Photometry was done by tracking the path of around two million photons as they interacted with the particles.
Using a spherical surface with 400 bins, we recorded the total light intensity received by each bin and the amount of reflected light from silicates received. After repeating this process for 100 permutations of horizontal and vertical lighting angle, we made a plain text file that contained the lighting angles, the observation angles, and information about received light. In order to glean a truer sense of the properties of each system, we averaged the text files from three times in the N-body simulation. From there, we performed two procedures on the averaged files.
The first was to plot the ratio of reflected silicate light to total incoming light as a function of each of the four angles (ref RelIntensVAng). No clear relationship was evident on any of the plots we produced, with the exception of the vertical angle of the observer, where a marked difference between the light ratio above and below the plane of particles was present. To quantify this difference, we separated the plot into three sections, took the average of each, and found the ratios.
The second procedure was to make a plot of reflected silicate light as a function of total received light, or the relative intensity. and find the slope of the line of best fit (ref RelIntens). Once we had the best fit slope, we could plot the difference between the slope of the best fit line and the slope of a line from the origin to all points as a function of the phase angle. We did this for total, horizontal, and vertical phase angle (ref Phase). Again, a marked relationship was present in the slope difference as a function of the vertical phase angle, but not the horizontal. We found the slope and error of the line of best fit for this plot.
Additional operations we could perform with the light ratio data were to plot the residual and slope difference as functions of the angles(ref Residual & SlopeDiff. In order to display the four angles and the additional value, we used the main axes to represent two angles and colored by the additional value. Instead of a point at each value of the two major angles, we displayed a small plot whose axes were the two other angles. In this way, we displayed the values of the residual or slope difference across all applicable values of the four-dimensional parameter space of the angles. We then found interesting angles that were above and below the plane (for the observer) or in the plane and above it (for the incoming light). Our selection criteria was to acquire a diverse range of behaviors. We then performed photometry on the original N-body simulation data with these angles over a period of two orbits, which we made into movies in order to observe the temporal evolution of the system.
Hide Silicates
A photometric plot of a B-Ring N-body simulation with particle radii between 8e-10 and 8e-9 R0 with 2% silicates. The colored dots indicate visible silicates, while the cloudy areas represent other, icy material. Photometric plots such as this were produced for multiple combinations of different values of theta and phi, the angles of incoming light. In this case, the observer is directly over the particle plane.
This plot shows data from photometric analysis of a simulation of A-ring particles with 1% of the total particles as contaminants. Each pixel shows the deviation from the line of best fit for the observed silicate light - total incoming light relation, ranging from highly positive (red) to zero (green) to highly negative (black/purple). The plot axes depict the orientation of the observer with respect to the rings. The horizontal axis is theta and the vertical axis is phi. At each orientation, instead of a single point, there is a smaller plot showing the deviation from the line of best fit as a function of incoming light orientation. Thus, each pixel shows the deviation from the best fit line in a system with a certain incoming light and observer orientation. The brightness of each point indicates the amount of light received by the observer: faint points mean low light and dark points mean more light. From this, more light is received by the observer when the observer is at the highest or lowest possible phi. Also, when both the observer and light are below the ring, there is a positive deviation from the best fit slope. In order to determine other relationships, we will make other plots with different axes.
Photometric plots of B Ring particles with 1% contaminants with the observer below the particles and the light above them. The left plot depicts the temporal evolution of the system, while the right depicts the amount of light reflected by silicates (as shown by the size) and the z-coordinate (as shown by the color). The total time displayed is two orbits, and the total time elapsed before the first frame is 9 orbits.
A Ring, 0.5% Silicate
RelIntens - A graph of observed light reflected directly from silicates as a function of total incoming light intensity for many orientations of the incoming light and observer with respect to the ring material. A linear relationship is apparent, as the more light is put in the system, the more will be reflected from any contaminants. The green points near the origin are those with input light vertical orientations above the lowest point. Note that even though the points move away from the line of best fit as the total light increases, if a line were to be drawn from the origin through one of the green points far from the line, the slope of said line would differ from the slope of the best fit line more than the slope of a line drawn through a point of high intensity.
Residual - This shows the residual of each orientation, or the distance from the line of best fit of silicate reflected light intensity as a function of total light intensity. The horizontal axis is theta, with the observer theta determining the location of each subplot and the phi of the observer determining the horizontal location of points within each subplot. The vertical axis is for the incoming light and has similar large and small scale angles (theta and phi, respectively). The green means that the deviation is very small, while the red indicates a high positive deviation and the purple a high negative deviation. It is evident that the only high positive deviations occur at low incoming light phi and for certain values of the thetas. In order to explore this seemingly linear relationship, we also made plots of the difference between the slope of the line from the origin to each light intensity/silicate reflected light point and the slope of the line of best fit. SlopeDiff - This shows the difference in slopes. Oddly, the orientations that showed a high residual before now show a low difference in slope. This most likely means that they are points of high total light intensity, as points far from the origin can be far from the best fit line and still have slopes close to the best fit line compared to the points closer to the origin. This is also shown in that the points with low residual show high slope deviations. This plot still shows a linear relationship between the coordinates of points of high slope difference. Phase - Theses plots show the relative intensity as a function of the phase angle of the data, or the difference between the observer angle and the light angle. From left to right, the horizontal axes are total phase angle, difference in theta, and difference in phi for all points. The theta phase is not very interesting, but the phi phase shows a distinct difference between the difference above and below the plane of particles.When the total difference is found, an evident downward linear slope was found, as shown by the green line.
RelIntensVAng - This shows the relative intensity, or slope of silicate light reflected as a function of total light, of points as a function of each angle for A ring particles with 0.5% silicates. No obvious trends are present in any of the plots except for the observer phi, where the relative intensity is higher above the disk than below it. When the observer phi relative intensity was separated into three parts and each part was averaged, the ratio of the relative intensity of the particles when observed above the ring to that of the particles when observed below the ring was found. When compared to the same ratio for A ring particles with 1% silicates, they only differed by 0.3%, which encourages further verification with the other data sets.
When viewing the spherically binned photometry plots created at all permutations of observer and light vertical and horizontal angle, we found that the amount of silicate light received is correlated with the amount of total light received. This is trivial information, as the total light received acts as an upper bound to the amount of reflected silicate light received, so when the bound increases, so should the silicate light. In order to further examine this relationship, we looked at the slopes of our plots.
In order to quantify the behavior of the lines of best fit of the relative intensity and phase plots, we divided each by the percent silicate present in the set. As seen below, this is a fairly stable value over our data. Additionally, we found the ratio of the slope difference for observations on the unlit side to those on the lit side and plotted it as a function of the percent silicates. They are very close together, suggesting a constant ratio.
Here, the data is:
Surface Density
Silicate percentage
Ring
Slope of Fit Line
Slope of Phase Line
Intensity Ratio
30
0.5
A
1.072E-3
-9.782E-5
0.892
30
1.0
A
1.990E-3
-1.590E-4
0.885
30
2.0
A
3.890E-3
-2.829E-4
0.890
30
0.5
B
4.425E-4
-1.243E-4
0.594
30
1.0
B
7.602E-4
-1.740E-4
0.663
30
2.0
B
1.375E-3
-2.757E-4
0.646
Note that the best fit slope for the relative intensity plot is
$\frac{\text{Average Reflected Silicate Light}}{(\text{Average Total Light}*\text{Percent Silicates})}=\Gamma$
From these values, we found that the relative intensity divided by the percent silicates is equal to $\Gamma=$ (2.0±0.1)*10-1 for A ring particles and (0.8±0.1)*10-2 for B ring particles. For the ratios, which were equal to
$\frac{Average Reflected Silicate Light Below}{Average Total Light Below}/\frac{Average Reflected Silicate Light Above}{Average Total Light Above}=\beta$
we found values of $\beta=$(8.89±0.04)*10-1 for A and (6.3±0.4)*10-1 for B ring particles. Finally, from our phase slope, we found that
where $\lambda=$ (-1.7±0.3)*10-4 for A or (-1.9±0.6)*10-4 for B plus the average relative intensity divided by the percent silicates. This is something that could be tested against measurements.
Of course, these are only three points, so while they suggest a correlation, it is not totally trustworthy. To test these correlations further, additional silicate percentages must be tested. In addition, these data are only for the 30g/cm2 surface density and particles with radii between 9E-10 and 9E-9 that of Saturn, so other particle sets should be tested as well. Because of time restrictions, we were not able to complete all 100 permutations for all data sets, so these results are not quite trustworthy. When compared to a full 100 permutation spherical photometry set with only one million photons, most of our values fit, suggesting that, despite our lack of a complete set, these data are indicative of the real trend.