A Limit Case of the “Ring Problem”

E. Barrabés1, J.M. Cors2 and G.R.Hall31 Universitat de Girona 2 Universitat Politècnica de Catalunya 3 Boston University

Music: The Planets. Op.32 Saturn, Gustav Holst

“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 m0 around a central mass m.

“string of pearls”

Maxwell found that such a ring is linearly stable provided that =m/m0 is sufficiently large.

In 1994 Moeckel, added the assumption n≥7.

“string of pearls”

In 1998 Roberts, carried the analysis a step further. How large must be such that the configuration is linearly stable?

> kn3, k=0.435

Science fictionSuppose 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 kn3 grows faster than 81n. The Earth's mass is approximately 81 times that of the moon's.

Multitude of small particlesAn intermediate step between arealistic model and the cartoon“string of pearls” model is thestudy of a swarm of infinitesimalbodies moving under the influence of a large planet and n smallequal bodies in a regular n-gonmoving in a circular orbit. Assumingthe infinitesimal particles do notinteract, it suffices to study themotion of a single infinitesimal bodyat a time.

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 theplane of the n-gon, a two degree of

freedom system is obtained.

Limit CaseSince 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 n3.

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.

−∞

The limit ring system

3

3

n0

8lim0

n

k

k

kyx

kyxy

kyx

xxyx

2/3220

2/3220

))((

1'2''

))((

13'2''

where

2/3220

2/3220

)('2''

)(3'2''

sfirst term

yx

xxy

yx

xxyx

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

The phase space

Equilibrium points for big values of 0

– L1 at (±x1,0)

• x1 ~ 1/(30)1/3

• center x saddle

– L2 at (0,1/2)

• complex saddle 5.905 < 0 < 431.643

• center x center 0 > 431.643

• zvc dissapear

Hill’s regions

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

Linear SystemEquations:

xy

xyx

2''

3'2''

)(')(

)(')(

2/3)cos(2)sin(2)(

)sin()cos()(

tytv

txtu

dcttBtAty

ctBtAtx

Solution

Poincaré section

0constant Jacobi ,0',0 Cuu

2π-periodic function

we can define the Poincaré return map on any point on

∑ following the flow 2π units of time

]5.0,5.0[ mod )2(6

),,(),,(

:

000

00000

0

vxyy

vyxvyx

P

Linear System

Fixed points of the Poincaré return map on the cylinder 2/1y

Lemma

The return map P0 satisfies the monotone twist condition

03360

x

y

x = const are invariant curves

For big values of x, the ring problem is

well approximated by the linear system

Limit ring problem

TheoremFor each choice of the Jacobi constant C0, if (x0,y0,u0,v0)

is an initial point with C (x0,y0,u0,v0) =C0 and x0 is

sufficiently large, then there is a constant b=b(C0,x0) 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.

Numerics 1100

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 Wu of a Lyapunov orbit by symmetry we obtain the branches of Ws

Their intersection gives rise to homoclinic connections

Connections between different peripherals its possible to connect paths visiting the bodies of the ring

Numerics 1100 •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

Numerics 1100

As the Jacobi constant decreases, the chaotic zones spread towards the right

C=0.05

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

244.0ring mass

planet massn

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.