Linear Stability of Ring Systems
Robert J. Vanderbei
Operations Research and Financial Engineering, Princeton University
rvdb@princeton.edu
Egemen Kolemen
Mechanical and Aerospace Engineering, Princeton University
ekolemen@princeton.edu
ABSTRACT
We give a self-contained modern linear stability analysis of a system of n
equal mass 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
positive real part. In the end, we rederive the result that J.C. 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. Further-
more, using properties of circulant matrices, the eigenvalues of the large 4n ×4n
matrix can be computed by solving n quartic equations, which further facilitates
the investigation of stability. Finally, we have implemented an n-body simulator
and we verify that the threshold mass ratios that we derived mathematically or
numerically do indeed identify the threshold between stability and instability.
Throughout the paper we consider only the planar n-body problem so that the
analysis can be carried out purely in complex notation, which makes the equa-
tions 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 mass-ratio
threshold.
Subject headings: planets: rings
– 2 –
1. Introduction
One hundred and fifty years ago, Maxwell (1859) was awarded the prestigious Adam’s
prize for a seminal paper on the stability of Saturn’s rings. At that time, neither the
structure nor the composition of the rings was known. Hence, Maxwell considered various
scenarios such as the possibility that the rings were solid or liquid annuli or a myriad of small
boulders. As a key part of this last possibility, M axwell studied the case of n equal-mass
bodies orbiting Saturn at a common radius and uniformly distributed about a circle of this
radius. He concluded that, for large n, the ring ought to be stable provided that the following
inequality is satisfied:
mass(Rings) ≤ 2.298mass(Saturn)/n
2
.
The mathematical analysis that leads to this result has been scrutenized, validated, and
generalized by a number of mathematicians over the years.
We summarize briefly some of the key historical developments. Tisserand (1889) derived
the same stability criterion using an analysis where he assumed that the ring has no effect on
Saturn and that the highest vibration mode of the system controls stability. More recently,
Willerding (1986) used the theory of density waves to show that Maxwell’s results are correct
in the limit as n goes to infinity. Pendse (1935) reformulated the stability problem so
that it takes into account the effect of the rings on the central body. He proved that,
for n ≤ 6, the system is unconditionally unstable. Inspired by this work, Salo and Yoder
(1988) studied coorbital formations of n satellites for small values of n where the satellites
are not distributed uniformly around the central body. They showed that there are some
stable asymmetric formations (such as the well-known case of a pair of ring bodies in L4/L5
position relative to e ach other—i.e., one leading the other by 60 deg). Scheeres and Vinh
(1991) extended the analysis of Pendse to find the stability criterion as a function of the
number of satellites when n is small. The resulting threshold depends on n but for n ≥ 7, it
deviates only a small amount from the asymptotically derived value. More recently, Moeckel
(1994) studied the linear stability of n-body systems for which the motion is given as a
simple rotation about the center of mass and under the condition that all masses except one
become vanishingly small. This latter condition greatly simplifies the analysis. Under these
assumptions, Moeckel shows, using the invariant subspace of the linearized Hamiltonian, that
symmetric ring systems are stable if and only if n ≥ 7. For n ≤ 6, he gives some examples
of the stable configurations discussed in Salo and Yoder (1988) where the ring bodies are
not uniformly distributed. Finally, Roberts (2000) expanding on Moeckel’s work obtained
stability criteria that match those given by Scheeres and Vinh (1991).
In this paper, we give a self-contained modern linear stability analysis of a system of
equal mass bodies in circular orbit about a single more massive body. We start with the
– 3 –
mathematical description of the dynamics of the system. We then form the linear approxi-
mation, compute all of the eigenvalues of the matrix defining the linear approximation, and
finally we derive inequalities that guarantee that none of these eigenvalues have p ositive real
part. In the end, we get exactly the same result that Maxwell found for large n. But, in addi-
tion, we identify the exact matrix that defines the linearized system even when n is not large.
This matrix can then be investigated numerically to find stability inequalities even in cases
where n is not large. Furthermore, using properties of circulant matrices, the eigenvalues
of the large 4n × 4n matrix can be computed by solving n quartic equations, which further
facilitates the investigation. Finally, we have implemented an n-body simulator based on a
leap-frog integrator (see Saha and Tremaine (1994); Hut et al. (1995)) and we verify that
the threshold mass ratios that we derived mathematically or numerically do indeed identify
the threshold between stability and instability.
Throughout the paper we consider only the planar n-body problem. That is, we ignore
any instabilities that might arise due to out-of-plane perturbations. Maxwell claimed, and
others have confirmed, that these out-of-plane perturbations are less destabilizing than in-
plane ones and hence our analysis, while not fully general, does get to the right answer.
Our main reason for wishing to restrict to the planar case is that we can then work in the
complex plane and our entire 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.
Finally, we should point out that the relevance of this work to observed planetary rings
is perhaps marginal, since real ring s ystems appear to be highly collisional and the energy
dissipation associated with such collisions affects their stability in a fundamental way (see,
for example, Salo (1995)).
2. Equally-Spaced, Equal-Mass Bodies in a Circular Ring About a Massive
Body
Consider the multibody problem consisting of one large central body, say Saturn, having
mass M and n small bodies, such as boulders, each of mass m orbiting the large body in
circular orbits uniformly spaced in a ring of radius r. Indices 0 to n−1 will be used to denote
the ring masses and index n will be used for Saturn. Throughout the paper we assume that
n ≥ 2. For the case n = 1, Lagrange proved that the system is stable for all mass ratios
m/M.
The purpose of this section is to show that such a ring exists as a solution to Newton’s
– 4 –
law of gravitation. In particular, we derive the relationship between the angular velocity
ω of the ring particles and their radius r from the central mass. We assume all bodies lie
in a plane and therefore complex-variable notation is convenient. So, with i =
√
−1 and
z = x + iy, we can write the equilibrium solution for j = 0, 1, . . . , n − 1, as
z
j
= re
i(ωt+2πj/n)
(1)
and
z
n
= 0. (2)
By symmetry (and exploiting our assumption that n ≥ 2), force is balanced on Saturn itself.
Now consider the ring bodies. Differentiating (1), we see that
¨z
j
= −ω
2
z
j
. (3)
From Newton’s law of gravity we have that
¨z
j
= −GM
z
j
− z
n
|z
j
− z
n
|
3
+
X
k6=j,n
Gm
z
k
− z
j
|z
k
− z
j
|
3
. (4)
Equations (3) and (4) allow us to determine ω, which is our first order of business. By
symmetry it suffices to consider j = 0. It is easy to check that
z
k
− z
0
= re
iωt
e
πik/n
2i sin(πk/n) (5)
and hence that
|z
k
− z
0
| = 2r sin(πk/n ). (6)
Substituting (5) and (6) into (4) and equating this with (3), we see that
−ω
2
= −
GM
r
3
+
n−1
X
k=1
Gm
4r
3
ie
πik/n
sin
2
(πk/n)
(7)
= −
GM
r
3
−
Gm
4r
3
n−1
X
k=1
1
sin(πk/n)
+ i
Gm
4r
3
n−1
X
k=1
cos(πk/n)
sin
2
(πk/n)
. (8)
It is easy to check that the summation in the imaginary part on the right vanishes. Hence,
ω
2
=
GM
r
3
+
Gm
r
3
I
n
(9)
where
I
n
=
1
4
n−1
X
k=1
1
sin(πk/n)
. (10)
With this choice of ω, the trajectories given by (1) and (2) satisfy Newton’s law of gravitation.
– 5 –
3. First-Order Stability
In order to carry out a stability analysis, we need to counter-rotate the system so that
all bodies remain at rest. We then perturb the system slightly and analyze the result.
A counter-rotated system would be given by
e
−iωt
z
j
(t) = re
2πij/n
= z
j
(0).
In such a rotating frame of reference, each body remains fixed at its initial point. It turns
out to be better to rotate the different bodies different amounts so that every ring body is
repositioned to lie on the x-axis. In other words, for j = 0, . . . , n − 1, n, we define
w
j
= u
j
+ iv
j
= e
−i(ωt+2πj/n)
z
j
. (11)
The advantage of repositioning every ring body to the positive real axis is that perturbations
in the real part for any ring body represent radial perturbations w hereas perturbations in
the imaginary part represent azimuthal perturbations. A simple counter-rotation does not
provide such a clear distinction between the two types of perturbations (and the associated
stability matrix fails to have the circulant property that is crucial to all later analysis).
Differentiating (11) twice, we get
¨w
j
= ω
2
w
j
− 2iω ˙w
j
+ e
−i(ωt+2πj/n)
¨z
j
. (12)
From Newton’s law of gravity, we see that
¨w
j
= ω
2
w
j
− 2iω ˙w
j
+
X
k6=j
Gm
k
ξ
k,j
|ξ
k,j
|
3
, (13)
where
m
k
=
m, for k = 0, 1, . . . , n − 1,
M, for k = n,
(14)
ξ
k,j
= e
iθ
k−j
w
k
− w
j
(15)
and
θ
k
= 2πk/n. (16)
Let δw
j
(t) denote variations about the fixed point given by
w
j
≡
r, for j = 0, 1, . . . , n − 1,
0, for j = n.
(17)
