With this applet, you can investigate mass ratios and eccentricities that lead to stable ring systems vs. unstable ones.
The parameter ecc controls the eccentricity. A value of zero produces circular orbits. Positive values produce an eccentric ring.
For a circular ring, if the mass of each ring body is no more than 2.3 times the mass of Saturn divided by the cube of the number of ring particles, then the system can be expected to be stable; otherwise not. See Linear Stability of Ring Systems for the derivation of this inequality.
In the applet below, M is the mass of Saturn (in Earth-masses), m is the mass of an individual ring body, and n is the number of ring bodies.
The textfield labeled gamma is the ratio m*n^3/M. If this value is smaller than 2.3, the system will be stable. Large values will be unstable.
You will find that you don't need to increase m very much to make the system unstable. If you set "warp" to 100, the integrator will show the instability very quickly. Give it a whirl.
Notes: (1) The warp parameter only controls how often the screen is updated---large
values mean that many time steps of the integrator are performed between each
screen update. This makes the simulation run much faster as updating the
screen image is more time consuming that a step of the integrator.
(2) Due to unresolved technicalities, the time-step parameter dt can only be changed
if 'ecc' is set to zero (i.e., circular orbits).
(3) For WebGL version of applet,
click here.