Dept. of Astrophysical Sciences, Princeton University Princeton, NJ August 17 th, 2006 Linear Stability of Ring Systems 1/12 Egemen Kolemen MAE Dept, Princeton University joint with joint with Robert J. Vanderbei ORFE Dept, Princeton University New Trends in Astrodynamics and Applications

2/12 ABSTRACT Linear Analysis of Circular Ring Formations in a modern, concise, efficient manner is performed. Stability criterion is obtained. Via numerical simulations transition from stable to unstable formations is shown.

3/12 In 1859, Maxwell’s Adams Prize winning essay showed that the rings have to be composed of small particles. Modeled the ring as n co- orbital particles of mass m. The ring system is stable “if” 3D Rendering of Saturn and his rings Simplified model of the ring system

4/12 Why are we looking at an old problem? There were a few mistakes and many hand waving arguments in the original paper. Subsequent papers provided full mathematical rigor. But they kept the old formulation which led to obscure derivations. And the full analysis is spread across different papers. Our aim is to provide a unified, concise and modern analysis. Hopefully, others will use this model as a fundamental formulation of the particle ring systems as opposed to the currently popular fluid models.

5/12 Equation of motion, where Equilibrium point, where

6/12 Linearizing the equation of motion around the equilibrium point, To find stability, find the eigenvalues of the 4n x 4n system: First 2n equations give

7/12 Solving for the derivative term of the eigenvalue Setting, 2n x 2n eigensystem reduces to: Block Circulant Matrix property gives the eigenvector:

8/12 All the equations reduce to one and the same. That is, the 2n x 2n system reduces to a 2 x 2 system. where the  j ’s are the n th roots of 1 Characteristic equation (with replaced by i Find when this equation has 4 real values. For n<7 the system is always unstable. For n ¸ 7 the stability is controlled by j = n/2.

9/12 For n<7, has the following shape with only 2 possible real solutions. Thus, the system is always unstable. For n ¸ 7, f( ) has the following shape and have the possibility of stability.

10/12 Finding the m/M ratio for n ¸ 7 At bifurcation point Solving, Substituting in f, m/M ratio is the root of

11/12 Expanding in power of n. Leading term gives Maxwell’s result. Computing the higher order terms, m/M normalized by n 3 versus n An approximate bound on the density of an icy boulder ring is which matches with the observed optical density of Saturn rings ( ) Unstable Stable n 3 m/M n

12/12 References P. Hut, J. Makino, and S. McMillan, Building a better leapfrog, The Astrophysical Journal Letters, 443:93–96, J.C. Maxwell. On the Stability of Motions of Saturn’s Rings. Macmillan and Company, Cambridge, P. Saha and S. Tremaine, Astronomical Journal, 108:1962, F. Tisserand, Traité de Méchanique Céleste, Gauthier-Villars, Paris, 1889 C. G. Pendse, The Theory of Saturn's Rings, Royal Society of London Philosophical Transactions Series A, 234, , March, Goldreich, P. and Tremaine, S., The dynamics of planetary rings, Ann. Rev. Astron. Astrophys., , 20, 1982 Scheeres, D. J. and Vinh, N. X., Linear stability of a self-gravitating ring, Celestial Mechanics and Dynamical Astronomy, , 51, 1991 Salo, H. and Yoder, C. F., The dynamics of coorbital satellite systems, AAP, , October, 1988 Willerding, E., Theory of density waves in narrow planetary rings, AAP, , 161, June, Saturn 3D Rendering: