Linear Stability of Ring Systems
Abstract
We give a selfcontained modern linear stability analysis of a system of n equalmass bodies in circular orbit about a single more massive body. Starting with the mathematical description of the dynamics of the system, we form the linear approximation, compute all of the eigenvalues of the linear stability matrix, and finally derive inequalities that guarantee that none of these eigenvalues have a positive real part. In the end we rederive the result that Maxwell found for large n in his seminal paper on the nature and stability of Saturn's rings, which was published 150 years ago. In addition, we identify the exact matrix that defines the linearized system even when n is not large. This matrix is then investigated numerically (by computer) to find stability inequalities. Furthermore, using properties of circulant matrices, the eigenvalues of the large 4n×4n matrix are computed by solving n quartic equations, which further facilitates the investigation of stability. Finally, we implement an nbody simulator, and we verify that the threshold mass ratios that we derive mathematically or numerically do indeed identify the threshold between stability and instability. Throughout the paper we consider only the planar nbody problem so that the analysis can be carried out purely in complex notation, which makes the equations and derivations more compact, more elegant, and therefore, we hope, more transparent. The result is a fresh analysis that shows that these systems are always unstable for 2<=n<=6, and for n>6 they are stable provided that the central mass is massive enough. We give an explicit formula for this massratio threshold.
 Publication:

The Astronomical Journal
 Pub Date:
 February 2007
 DOI:
 10.1086/510457
 arXiv:
 arXiv:astroph/0606510
 Bibcode:
 2007AJ....133..656V
 Keywords:

 planets: rings;
 Astrophysics
 EPrint:
 20 pages, associated website is http://www.princeton.edu/~rvdb/JAVA/astro/galaxy/StableRings.html