Linear Stability of Ring Systems
Egemen Kolemen
MAE Dept, Princeton University
joint with
Robert J. Vanderbei
ORFE Dept, Princeton University
New Trends in Astrodynamics and Applications
Dept. of Astrophysical Sciences, Princeton University
Princeton, NJ August 17th, 2006
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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.
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• In 1859, Maxwell’s Adams
Prize winning essay showed
that the rings have to be
composed of small particles.
• Modeled the ring as n coorbital particles of mass m.
3D Rendering of Saturn and his rings
• The ring system is stable “if”
Simplified model of the ring system
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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.
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• Equation of motion,
where
• Equilibrium point,
where
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• 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
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• Solving for the derivative term of the eigenvalue
• Setting
, 2n x 2n eigensystem reduces to:
• Block Circulant Matrix property gives the eigenvector:
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• 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 nth 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.
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• 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.
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Finding the m/M ratio for n¸7
• At bifurcation point
• Solving,
• Substituting  in f, m/M ratio is the root of
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• Expanding in power of n. Leading term gives Maxwell’s result.
• Computing the higher order terms, m/M normalized by n3 versus n
Unstable
n3m/M
Stable
n
• An approximate bound on the density of an icy boulder ring is
which matches with the observed optical density of Saturn rings
(0.05-0.25)
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References
• P. Hut, J. Makino, and S. McMillan, Building a better leapfrog, The Astrophysical
Journal Letters, 443:93–96, 1995.
• J.C. Maxwell. On the Stability of Motions of Saturn’s Rings. Macmillan and
Company, Cambridge, 1859.
• P. Saha and S. Tremaine, Astronomical Journal, 108:1962, 1994.
• 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, 145-176, March, 1935.
• Goldreich, P. and Tremaine, S., The dynamics of planetary rings, Ann. Rev. Astron.
Astrophys.,249-283, 20, 1982
• Scheeres, D. J. and Vinh, N. X., Linear stability of a self-gravitating ring, Celestial
Mechanics and Dynamical Astronomy, 83-103, 51, 1991
• Salo, H. and Yoder, C. F., The dynamics of coorbital satellite systems, AAP, 309327, October, 1988
• Willerding, E., Theory of density waves in narrow planetary rings, AAP, 403-407 ,
161, June, 1986.
• Saturn 3D Rendering: http://www.mmedia.is/bjj/satsys_rend.html
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