Edges of Saturn’s rings are fractal
© Li and Ostoja-Starzewski; licensee Springer. 2015
Received: 8 January 2015
Accepted: 15 March 2015
Published: 1 April 2015
The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn’s rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 “Mapping Clumps in Saturn’s Rings”, slide #54 “Scattered Sunshine”, slide #66 taken two weeks before the planet’s Augus’t 200’9 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2’ mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63 ~ 1.78. This clarifies in what sense Saturn’s rings are fractal.
Results and discussion
All the images analyzed in this paper yield fractal dimensions in the range 1.63 to 1.78. This is a consistent estimate of the fractal dimension of the rings’ edges, regardless of the various image sources we utilized. Indeed, the fact that the rings’ edges are fractal provides one more hint to developing models of the intricate mechanics and physics governing these structures of granular matter. Interestingly, somewhat related studies (Feitzinger and Galinski 1987; de la Fuente and de la Fuente 2006a, b) found average fractal dimension ~1.7 for the projected fractal dimension of the distribution of star-forming sites (HII regions) in a sample of 19 spiral galaxies.
Modified box counting using boxes with shape being self-similar to the global image. This method is well suited for generally rectangular images (Xu and Lacidogna 2011), where the boxes are rectangles self-similar to the whole image. The selection of the ratio of image size to box size is in powers of 2 for optimal log(n)-log(r) regression. When the ratio does not give an integer box size, the box size was chosen to be the closest integer at that ratio.
Power 2 box counting using boxes with sizes as powers of 2, possessing optimal log(n)-log(r) regression. Here the partial boarder effects are evident generally when the image size was not powers of 2. In this case the image was embedded in an empty image with size being powers of 2 closest to the original image size. The box counting was then performed on the ‘enlarged’ image.
Divider box counting using boxes with sizes being the dividers of the image size. Subsequent box size may be too close for log(n)-log(r) regression, while the border effects can be eliminated.
This work was made possible by the NSF support under the grant CMMI-1030940.
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