Solar System Dynamics: Regular and Chaotic Motion 805
and the 320-km-wide Encke gap at 133,600 km. The pre-
dicted radii of the icy satellites required to produce these
gaps are∼2.5 and∼24 km, respectively. In 1991, an analysis
ofVoyagerimages by M. Showalter revealed a small satel-
lite, Pan, with a radius of∼10 km orbiting in the Encke gap.
In 2005, the moon Daphnis of radius∼3–4 km was discov-
ered in the Keeler gap by theCassinispacecraft.Voyager 2
images of the dust rings of Uranus show pronounced gaps at
certain locations. Although most of the proposed shepherd-
ing satellites needed to maintain the narrow rings have yet
to be discovered, these gaps may provide indirect evidence
of their orbital locations.
6. Long Term Stability of Planetary Orbits
6.1 TheN-Body Problem
The entire solar system can be approximated by a system
of nine planets orbiting the Sun. (Tiny Pluto has been in-
cluded in most studies of this problem to date, because it
was classified as a planet until 2006. But Pluto does not sub-
stantially perturb the motions of the eight larger planets.)
In a center of mass frame, the vector equation of motion for
planetimoving under the Newtonian gravitational effect
of the Sun and the remaining 8 planets is given by
r ̈=G∑^9j= 0mjrj−ri
rij^3(j =i), (46)whereriandmiare the position vector and mass of planet
i(i=1, 2,... , 9), respectively,rij≡rj−ri, and the sub-
script 0 refers to the Sun. These are the equations of the
N-body problem for the case whereN=10, and although
they have a surprisingly simple form, they have no general,
analytical solution. However, as in the case of the three-body
problem, it is possible to tackle this problem mathematically
by making some simplifying assumptions.
Provided the eccentricities and inclinations of theNbod-
ies are small and there are no resonant interactions between
the planets, it is possible to derive an analytical solution
that describes the evolution of all the eccentricities, incli-
nations, perihelia, and nodes of the planets. This solution,
called Laplace–Lagrange secular perturbation theory, gives
no positional information about the planets, yet it demon-
strates that there are long-period variations in the planetary
orbital elements that arise from mutual perturbations. The
secular periods involved are typically tens or hundreds of
thousands of years, and the evolving system always exhibits
a regular behavior. In the case of Earth’s orbit, such periods
may be correlated with climatic change, and large varia-
tions in the eccentricity of Mars are thought to have had
important consequences for its climate.
In the early nineteenth century, Pierre Simon de Laplace
claimed that he had demonstrated the long-term stability
of the solar system using the results of his secular perturba-
tion theory. Although the actual planetary system violates
some of the assumed conditions (e.g., Jupiter and Saturn
are close to a 5:2 resonance), the Laplace–Lagrange theory
can be modified to account for some of these effects. How-
ever, such analytical approaches always involve the neglect
of potentially important interactions between planets. The
problem becomes even more difficult when the possibility
of near-resonances between some of the secular periods of
the system is considered. However, nowadays it is always
possible to carry out numerical investigations of long-term
stability.6.2 Stability of the Solar System
Numerical integrations show that the orbits of the plan-
ets are chaotic, although there is no indication of gross
instability in their motion provided that the integrations
are restricted to durations of 5 billion years (the age of
the solar system). The eight planets as well as dwarf planet
Pluto remain more or less in their current orbits with small,
nearly periodic variations in their eccentricities and inclina-
tions; close approaches never seem to occur. Pluto’s orbit is
chaotic, partly as a result of its 3:2 resonance with the planet
Neptune, although the perturbing effects of other planets
are also important. Despite the fact that the timescale for
exponential divergence of nearby trajectories (the inverse
of the Lyapunov exponent) is about 20 million years, no
study has shown evidence for Pluto leaving the resonance.
Chaos has also been observed in the motion of the eight
planets, and it appears that the solar system as a whole is
chaotic with a timescale for exponential divergence of 4 or
5 million years, although different integrations give differ-
ent results. However, the effect is most apparent in the
orbits of the inner planets. Though there appear to be no
dramatic consequences of this chaos, it does mean that the
use of the deterministic equations of celestial mechanics to
predict the future positions of the planets will always be
limited by the accuracy with which their orbits can be mea-
sured. For example, some results suggest that if the position
of Earth along its orbit is uncertain by 1 cm today, then the
exponential propagation of errors that is characteristic of
chaotic motion implies that knowledge of Earth’s orbital
position 200 million years in the future is not possible.
The solar system appears to be “stable” in the sense
that all numerical integrations show that the planets re-
main close to their current orbits for timescales of billions of
years. Therefore the planetary system appears to be another
example of bounded chaos, where the motion is chaotic but
always takes place within certain limits. Although an analyti-
cal proof of this numerical result and a detailed understand-
ing of how the chaos has arisen have yet to be achieved, the