### The Difference Between 6 and 7

A ring system consists of $n$  “light” point masses orbiting around a “heavy” central mass. Assume that the orbit is circular and the light masses are equally spaced around the circle. The relevant physical force is Newtonian gravity.

Here is a ring system with $n = 6$:

And here is a ring system with $n = 7$:

The ring system with $n = 6$ is always linearly unstable, while the ring system with $n = 7$ can be stable (if the central mass satisfies a certain condition).

See the detailed analysis here: https://arxiv.org/abs/astro-ph/0606510, by Robert J. Vanderbei and Egemen Kolemen.

This is closely related to work done by Maxwell (yes, the electromagnetism Maxwell) on the stability of the rings of Saturn. Speaking of which, some amazing pictures have been taken of Saturn’s rings:

The above is a high-resolution image in the Solar System sense – each pixel corresponds to about 3 square kilometers! The image was taken by the Cassini Orbiter. (More info here: https://photojournal.jpl.nasa.gov/catalog/PIA21628).