1. A Limit Case of the “Ring Problem” E. Barrabés 1, J.M. Cors 2 and G.R.Hall 3 1 Universitat de Girona 2 Universitat Politècnica de Catalunya 3 Boston University Music: The Planets. Op.32 Saturn, Gustav Holst

2. “string of pearls” In 1859 Maxwell published his study of the rings of Saturn. As a first approximation, he treats the ring as a rigidly rotating regular polygon of n equal masses m 0 around a central mass m.

3. “string of pearls” Maxwell found that such a ring is linearly stable provided that  =m/m 0 is sufficiently large. In 1994 Moeckel, added the assumption n≥7.

4. “string of pearls” In 1998 Roberts, carried the analysis a step further. How large  must be such that the configuration is linearly stable?  > kn 3, k=0.435

5. Science fiction Suppose that the Moon (in a nearly circular rotation about the Earth) is suddenly split into n equal pieces, with each new moon landing close to a vertex of a regular n-gon with the Earth at its center. Is the Earth heavy enough to maintain this configuration? No, unless 7 ≤ n ≤ 13, as kn 3 grows faster than 81n. The Earth's mass is approximately 81 times that of the moon's.

6. Multitude of small particles An intermediate step between a realistic model and the cartoon “string of pearls” model is the study of a swarm of infinitesimal bodies moving under the influence of a large planet and n small equal bodies in a regular n-gon moving in a circular orbit. Assuming the infinitesimal particles do not interact, it suffices to study the motion of a single infinitesimal body at a time.

7. The Model Consider the N+1-body problem, with N=1+n. That is, the motion of an infinitesimal point mass under the gravitational forces of a central planet and n small equal masses in a regular n-gon in a circular orbit about the planet. Restricting the infinitesimal body to the plane of the n-gon, a two degree of freedom system is obtained.

8. Limit Case Since there is no natural choice for the number n of bodies in the regular n-gon, we consider a limiting problem as n tends to infinity. For this process to give an interesting limit, the central mass must be of order n 3.

9. The limit ring problem The limit problem consists of forces from infinitely many point mass bodies arranged along the integer points of the y-axis plus the force corresponding to the (infinitely distant) planet. −∞

10. The limit ring system where

11. The phase space Symmetries with respect y=±k and x=0 Configuration space: {(x,y); -1/2 ≤ y ≤ 1/2 }

12. The phase space Equilibrium points for big values of  0 –L 1 at (±x 1,0) x 1 ~ 1/(3  0 ) 1/3 center x saddle –L 2 at (0,1/2) complex saddle 5.905 <  0 < 431.643 center x center  0 > 431.643 zvc dissapear Hill’s regions

13. Aims Show that the KAM Theorem can be used to prove the boundedness of orbits for a fixed energy level –The basic idea is to compare the flow of the limit ring problem to an approximating linear system for x large. –Applying the KAM Theorem requires to verify a geometric “monotone twist” condition an analytical circle intersection property. Determine the width of a ring using only gravitational effects

14. Linear System Equations: Solution Poincaré section 2π-periodic function we can define the Poincaré return map on any point on ∑ following the flow 2π units of time

15. Linear System Fixed points of the Poincaré return map on the cylinder Lemma The return map P 0 satisfies the monotone twist condition x = const are invariant curves For big values of x, the ring problem is well approximated by the linear system

16. Limit ring problem Theorem For each choice of the Jacobi constant C 0, if (x 0,y 0,u 0,v 0 ) is an initial point with C (x 0,y 0,u 0,v 0 ) =C 0 and x 0 is sufficiently large, then there is a constant b=b(C 0,x 0 ) such that |x(t)| > b for all t. The first KAM invariant tori separates the regular motion (far from the ring) from the chaotic region (close to the bodies of the ring). From its location we can obtain a mesure of the width of the ring.

17. Numerics Close the ring: complicated dynamics due to the presence of the invariant manifolds associated to the hyperbolic equilibrium points and the family of Lyapunov orbits Branches of W u of a Lyapunov orbit  by symmetry we obtain the branches of W s Their intersection gives rise to homoclinic connections Connections between different peripherals  its possible to connect paths visiting the bodies of the ring

18. Numerics Far from the peripherals: the system is well approximated by the linear system Fixed an energy level, the KAM invariant torus separates the regular and the chaotic zone C = 0.18

19. Numerics As the Jacobi constant decreases, the chaotic zones spread towards the right C=0.05

20. Conclusions The limit ring problem is a cartoon model of a planetary ring of dust particles under the influence of a central planet and a large number n of ring bodies. It exhibits the complicated dynamics that can be expected in a Hamiltonian system with many unstable equilibria and periodic orbits It only applies to a specific range of mass ratios between the ring and the planet

21. Conclusions The model could be only applicable to rings with very small mass Very narrow rings are obtained if only the gravitational forces are taken into account While it is possible for rings to persist under their own gravitational interactions in isolation around a planet, those rings we see about Saturn and the other major planets are either bands of particles in unrelated orbits or coherent rings constrained by resonances with larger moons, shepherding by smaller moons or other factors.