100-C 25-C 50-C 75-C C+M 50-C+M C+Y 50-C+Y M+Y 50-M+Y 100-M 25-M 50-M 75-M 100-Y 25-Y 50-Y 75-Y 100-K 25-K 25-19-19 50-K50-40-40 75-K 75-64-64CHAPTER 42
Solar System Dynamics:
Regular and
Chaotic Motion
Jack J. Lissauer
NASA Ames Research Center
Moffett Field, CaliforniaCarl D. Murray
Queen Mary, University of London
London, U.K.- Introduction: Keplerian Motion 6. Long-Term Stability of Planetary Orbits
- The Two-Body Problem 7. Dissipative Forces and the Orbits of Small Bodies
- Planetary Perturbations and the Orbits of Small Bodies 8. Chaotic Rotation
- Chaotic Motion 9. Epilog
- Orbital Evolution of Minor Bodies Bibliography
1. Introduction: Keplerian Motion
The study of the motion of celestial bodies within our solar
system has played a key role in the broader development
of classical mechanics. In 1687, Isaac Newton published his
Principia, in which he presented a unified theory of the mo-
tion of bodies in the heavens and on the Earth. Newtonian
physics has proven to provide a remarkably good descrip-
tion of a multitude of phenomena on a wide range of length
scales. Many of the mathematical tools developed over the
centuries to analyze planetary motions in the Newtonian
framework have found applications for terrestrial phenom-
ena. The concept ofdeterministicchaos, now known to
play a major role in weather patterns on the Earth, was
first conceived in connection with planetary motions (by
Poincar ́e, in the late 19thcentury). Deviations of the or-
bit of Uranus from that predicted byNewton’s Lawsled
to the discovery of the planet Neptune. In contrast, the
first major success of Einstein’s general theory of relativity
was to explain deviations of Mercury’s orbit that could not
be accounted for by Newtonian physics. But general rela-
tivistic corrections to planetary motions are quite small, so
this article concentrates on the rich and varied effects of
Newtonian gravitation, together with briefer descriptions
of non-gravitational forces that affect the motions of some
objects in the solar system.
Newton showed that the motion of two spherically sym-
metric bodies resulting from their mutual gravitational at-
traction is described by simple conic sections (see Section
2.4). However, the introduction of additional gravitating
bodies produces a rich variety of dynamical phenomena,
even though the basic interactions between pairs of objects
can be straightforwardly described. Even few-body systems
governed by apparently simple nonlinear interactions can
display remarkably complex behavior, which has come to be
known collectively as chaos. On sufficiently long timescales,
the apparently regular orbital motion of many bodies in
the solar system can exhibit symptoms of this chaotic
behavior.
An object in the solar system exhibits chaotic behavior
in its orbit or rotation if the motion is sensitively depen-
dent on the starting conditions, such that small changes
in its initial state produce different final states. Examples
ofchaotic motionin the solar system include the rota-
tion of the Saturnian satellite Hyperion, the orbital evolu-
tion of numerous asteroids and comets, and the orbit of
Pluto. Numerical investigations suggest that the motion of
the planetary system as a whole is chaotic, although there
are no signs of any gross instability in the orbits of the
planets. Chaotic motion has probably played an important
role in determining the dynamical structure of the solar
system.