# Edges of Saturn’s rings are fractal

- Jun Li
^{1}and - Martin Ostoja-Starzewski
^{2}Email author

**4**:158

https://doi.org/10.1186/s40064-015-0926-6

© Li and Ostoja-Starzewski; licensee Springer. 2015

**Received: **8 January 2015

**Accepted: **15 March 2015

**Published: **1 April 2015

## Abstract

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.

## Keywords

## Background

## Results and discussion

*D*comes from estimation of the slope of log(

*n*)-log(

*r*) in

*n*∝

*r*

^{− D }, where

*n*is the number of boxes with size

*r*needed to cover the region of interest. The local slopes of log(

*n*)-log(

*r*) are also acquired to determine optimal cut-offs of box sizes. The cut-offs are specified where the local slope varies strongly. The log(

*n*)-log(

*r*) plots of the three box counting methods for images of Figure 1 (a), (d), and (e) are shown in Figures 2, 3 and 4, respectively. Since the plots for Figures 1 (b) and (c) are very similar to the others, they are not shown here in order to save space. Note that, for modified box counting,

*r*denotes the ratio of image size to box size, unlike power 2 or divider box counting, where

*r*is the box size.

## Conclusions

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.

## Methods

- 1.
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. - 2.
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. - 3.
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.

*n*)-log(

*r*). Figure 6 shows an example of the local slope of log(

*n*)-log(

*r*) for power 2 box counting applied to Figure 1 (a) with

*r*= 2 to

*r*=

*b*/2, where

*b*denotes the image size (after extended to powers of 2). The fine box size

*r*= 2 tends to be below the average spacing of ring particles, whereas the very coarse box count (

*r*=

*b*/2) usually fails to capture structural details. The lower and upper cut-offs of box sizes are then 4 and b/4.

## Declarations

### Acknowledgment

This work was made possible by the NSF support under the grant CMMI-1030940.

## Authors’ Affiliations

## References

- Avron JE, Simon B (1981) Almost periodic Hill’s equation and the rings of Saturn. Phys Rev Lett 46(17):1166–8View ArticleGoogle Scholar
- de la Fuente MR, de la Fuente MC (2006a) The fractal dimensions of the spatial distribution of young open clusters in the solar neighbourhood. Astron Astrophys 452:163–8, Doi:10.1051/0004-6361:20054552View ArticleGoogle Scholar
- de la Fuente MR, de la Fuente MC (2006b) Multifractality in a ring of of star formation: the case of Arp 220. Astron Astrophys 454:473–80, Doi:10.1051/0004-6361:20054776View ArticleGoogle Scholar
- Feitzinger JV, Galinski T (1987) The fractal dimension of star-forming sites in galaxies. Astron Astrophys 179:249–54Google Scholar
- Fridman AM, Gorkavyi NN (1994) Physics of Planetary Rings. Springer, BerlinGoogle Scholar
- Maggi F (2008) Projection of compact fractal sets: application to diffusion-limited and cluster-cluster aggregates. Nonlin Proc Geophys 15:695–9View ArticleGoogle Scholar
- Mandelbrot BB (1982) The Fractal Geometry of Nature. W.H. Freeman & Co, New YorkGoogle Scholar
- Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge, EnglandGoogle Scholar
- Xu J, Lacidogna G (2011) A modified box-counting method to estimate the fractal dimensions. Appl Mech Mater 58:1756–61View ArticleGoogle Scholar

## Copyright

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.