Solar System Dynamics: Regular and Chaotic Motion 801
0.4 0.6 0.8
-1
-0.5
0
0.5
1
0.2
x
x
.
C = 3.13
5:2
5:2
5:2
2:1
9:4
9:4
9:4
FIGURE 15 Representative surface of section
plots forx 0 =0.262, 0.3, 0.34, 0.35, 0.38, 0.42,
0.52, 0.54, 0.7, and 0.78 withx ̇ 0 =0, y 0 =0, and
Jacobi constantC=3.13. Each trajectory was
followed for a minimum of 500 crossing points. It
is clear from a comparison with Fig. 14 that the
phase space is more regular; chaotic orbits still
exist for this value ofC,but they are more difficult
to find. The major resonances are identified.
that bring them close to Earth. One of the most striking
examples of the butterfly effect (see Section 4.1) in the
context of orbital evolution is the orbit of asteroid 2060
Chiron, which has a perihelion inside Saturn’s orbit and
an aphelion close to Uranus’s orbit. Numerical integrations
based on the best available orbital elements show that it is
impossible to determine Chiron’s past or future orbit with
any degree of certainty since it frequently suffers close ap-
proaches to Saturn and Uranus. In such circumstances, the
outcome is strongly dependent on the initial conditions as
well as the accuracy of the numerical method. These are
the characteristic signs of a chaotic orbit. By integrating
several orbits with initial conditions close to the nominal
values, it is possible to carry out a statistical analysis of
the orbital evolution. Studies suggest that there isa1in
8 chance that Saturn will eject Chiron from the solar sys-
tem on a hyperbolic orbit, while there is a 7 in 8 chance
that it will evolve toward the inner solar system and come
under strong perturbations from Jupiter. Telescopic obser-
vations of a faint coma surrounding Chiron imply that it
is a comet rather than an asteroid; perhaps its future orbit
will resemble that of a short-period comet of the Jupiter
family.
Numerical studies of the orbital evolution of planet-
crossing asteroids under the effects of perturbations from all
the planets have shown a remarkable complexity of motion
for some objects. For example, the Earth-crossing asteroid
1620 Geographos gets trapped temporarily in a number of
resonances with Earth in the course of its chaotic evolution
(Fig. 16).
A histogram of the number distribution of asteroid or-
bits in semimajor axis (Fig. 17) shows that apart from
a clustering of asteroids near Jupiter’s semimajor axis at
5.2 AU, there is an absence of objects within 0.75 AU of
the orbit of Jupiter. The objects in the orbit of Jupiter are
the Trojan asteroids (Section 3.2), which are located∼ 60 ◦
ahead of and behind Jupiter.
The cleared region near Jupiter’s orbit can be under-
stood in terms of chaotic motion due to the overlap of adja-
cent resonances. In the context of the Sun–Jupiter–asteroid
restricted three-body problem, the perturber (Jupiter) has
Time (years)
-100,000 0 100,000
1.20
1.22
1.24
1.26
Semi-major axis (AU)
14:19
11:15
13:18
8:11
11:15
FIGURE 16 A plot of the semimajor axis of the near-Earth
asteroid 1620 Geographos over a backward and forward
integration of 100,000 years starting in 1986. Under
perturbations from the planets, Geographos moves in a chaotic
orbit and gets temporarily trapped in a number of high-order,
orbit–orbit resonances (indicated in the diagram) with Earth.
The data are taken from a numerical study of planet-crossing
asteroids undertaken by A. Milani and coworkers. (Courtesy of
Academic Press.)