Solar System Dynamics: Regular and Chaotic Motion 795
a test massless particle moves under the gravitational effect
of both masses but does not perturb their orbits. From this
work, Poincar ́e realized that despite the simplicity of the
equations of motion, some solutions to the problem exhibit
complicated behavior.
Poincar ́e’s work in celestial mechanics provided the
framework for the modern theory of nonlinear dynamics
and ultimately led to a deeper understanding of the phe-
nomenon of chaos, whereby dynamical systems described
by simple equations can give rise to unpredictable behav-
ior. The whole question of whether or not a given system
is stable to sufficiently small perturbations is the basis of
the Kolmogorov-Arnol’d-Moser (KAM) theory, which has
its origins in the work of Poincar ́e.
One characteristic of chaotic motion is that small changes
in the starting conditions can produce vastly different final
outcomes. Since all measurements of positions and veloci-
ties of objects in the solar system have finite accuracy, rela-
tively small uncertainties in the initial state of the system can
lead to large errors in the final state, for initial conditions
that lie in chaotic regions inphase space.
This is an example of what has become known as the
“butterfly effect,” first mentioned in the context of chaotic
weather systems. It has been suggested that under the right
conditions, a small atmospheric disturbance (such as the
flapping of a butterfly’s wings) in one part of the world could
ultimately lead to a hurricane in another part of the world.
The changes in an orbit that reveal it to be chaotic may
occur very rapidly, for example during a close approach to
the planet, or may take place over millions or even billions
of years. Although there have been a number of significant
mathematical advances in the study of nonlinear dynamics
since Poincar ́e’s time, the digital computer has proven to be
the most important tool in investigating chaotic motion in
the solar system. This is particularly true in studies of the
gravitational interaction of all the planets, where there are
few analytical results.
4.2 The Three-body Problem as a Paradigm
The characteristics of chaotic motion are common to a wide
variety of dynamical systems. In the context of the solar
system, the general properties are best described by con-
sidering the planar circular restricted three-body problem,
consisting of a massless test particle and two bodies of
massesm 1 andm 2 moving in circular orbits about their
common center of mass at constant separation with all bod-
ies moving in the same plane. The test particle is attracted
to each mass under the influence of the inverse square law
of force given in Eq. (5). In Eq. (16),ais the constant sep-
aration of the two masses, andn= 2 π/pis their constant
angular velocity about the center of mass. Usingxandy
as components of the position vector of the test particle
referred to the center of mass of the system (Fig. 5), the
equations of motion of the particle in a reference frame
m (^1) m
2
r 1 r 2
r
O
y P
x
FIGURE 5 The rotating coordinate system used in the circular
restricted three-body problem. The masses are at a fixed distance
from one another and this is taken to be the unit of length. The
position and velocity vectors of the test particle (at pointP) are
referred to the center of mass of the system atO.
rotating at angular velocitynare
x ̈− 2 ny ̇−n^2 x=−G
(
m 1
x+μ 2
r 13
−m 2
x−μ 1
r 23
)
, (37)
y ̈+ 2 nx ̇−n^2 y=−G
(
m 1
r 13
- m 2
r 23
)
y, (38)
whereμ 1 =m 1 a/(m 1 +m 2 ), andμ 2 =m 2 a/(m 1 +m 2 ) are
constants and
r^21 =(x+μ 2 )^2 +y^2 , (39)
r^22 =(x−μ 1 )^2 +y^2 , (40)
wherer 1 andr 2 are the distances of the test particle from
the massesm 1 andm 2 , respectively.
These two second-order, coupled, nonlinear differential
equations can be solved numerically provided the initial
position (x 0 , y 0 ) and velocity (x ̇ 0 ,y ̇ 0 ) of the particle are
known. Therefore the system is deterministic and at any
given time the orbital elements of the particle (such as its
semimajor axis and eccentricity) can be calculated from its
initial position and velocity.
The test particle is constrained by the existence of a con-
stant of the motion called the Jacobi constant,C, given by
C=n^2 (x^2 +y^2 )+ 2 G
(
m 1
r 1
m 2
r 2
)
−x ̇^2 − ̇y^2. (41)
The values of (x 0 ,y 0 ) and (x ̇ 0 ,y ̇ 0 ) fix the value ofCfor
the system, and this value is preserved for all subsequent
motion. At any instant the particle is at some position on
the two-dimensional (x,y) plane. However, since the actual
orbit is also determined by the components of the velocity
(x ̇,y ̇), the particle can also be thought of as being at a partic-
ular position in a four-dimensional (x,y,x ̇,y ̇) phase space.
Note that the use of four dimensions rather than the cus-
tomary two is simply a means of representing the position
andthe velocity of the particle at a particular instant in time;