Encyclopedia of the Solar System 2nd ed

800 Encyclopedia of the Solar System

method for the regular and chaotic orbits described here.
This leads to an estimate ofγ= 10 −^0.^77 (Jupiter periods)−^1
for the maximum Lyapunov characteristic exponent of the
chaotic orbit. The corresponding Lyapunov time is given by
1/γ, or in this case∼6 Jupiter periods. This indicates that
for this starting condition the chaotic nature of the orbit
quickly becomes apparent.
It is important to realize that a chaotic orbit is not neces-
sarily unbounded. The maximum Lyapunov characteristic
exponent concerns local divergence and provides no in-
formation about the global stability of the trajectory. The
phrase “wandering on a leash” is an apt description of ob-
jects on bounded chaotic orbits—the motion is contained
but yet chaotic at the same time. Another consideration is
that numerical explorations of chaotic systems have many
pitfalls both in how the physical system is modeled and
whether or not the model provides an accurate portrayal of
the real system.

4.2.3 LOCATION OF REGULAR AND CHAOTIC REGIONS

The extent of the chaotic regions of the phase space of a dy-
namical system can depend on a number of factors. In the
case of the circular restricted three-body problem, the crit-
ical quantities are the values of the Jacobi constant and the
mass ratioμ 2. In Figs. 14 and 15, ten trajectories are shown
for each of two different values of the Jacobi constant. In
the first case (Fig. 14), the value isC=3.07 (the same as
the value used in Figs. 8 and 11), whereas in Fig. 15 it is
C=3.13. It is clear that the extent of the chaos is reduced
in Fig. 15. The value ofCin the circular restricted problem
determines how close the asteroid can get to Jupiter. Larger
values ofCcorrespond to orbits with greater minimum dis-

tances from Jupiter. For the caseμ 2 =0.001 andC> 3 .04,

it is impossible for their orbits to intersect, although the

perturbations can still be significant.

Close inspection of the separatrices in Figs. 14 and 15

reveals that they consist of chaotic regions with regular re-

gions on either side. As the value of the Jacobi constant de-

creases, the extent of the chaotic separatrices increases until

the regular curves separating adjacent resonances are bro-

ken down and neighboring chaotic regions begin to merge.

This can be thought of as the overlap of adjacent resonances

giving rise to chaotic motion. It is this process that permits

chaotic orbits to explore regions of the phase space that

are inaccessible to the regular orbits. In the context of the

Sun–Jupiter–asteroid problem, this observation implies that

asteroids in certain orbits are capable of large excursions in

their orbital elements.

5. Orbital Evolution of Minor Bodies

5.1 Asteroids

With more than 130,000 accurately determined orbits and

one major perturber (the planet Jupiter), the asteroids pro-

vide a natural laboratory in which to study the consequences

of regular and chaotic motion. Using suitable approxima-

tions, asteroid motion can be studied analytically in some

special cases. However, it is frequently necessary to resort

to numerical integration. [SeeMain-BeltAsteroids.]

Investigations have shown that a number of asteroids

have orbits that result in close approaches to planets. Of

particular interest are asteroids such as 433 Eros, 1033

Ganymed, and 4179 Toutatis, because they are on orbits

0.4 0.6 0.8

-1

-0.5

0

0.5

1

0.2

x

x

. 2:1 3:2

5:3

5:3

5:2

5:2

5:2

C = 3.07

FIGURE 14 Representative surface of section

plots forx 0 =0.25, 0.29, 0.3, 0.45, 0.475, 0.5, 0.55,

0.56, 0.6, and 0.8 with=0,y 0 =0, and Jacobi

constantC=3.07. Each trajectory was followed

for a minimum of 500 crossing points. The plot

uses the points shown in Figs. 8 and 11 (although

the scales are different), as well as points from

other regular and chaotic orbits. The major

resonances are identified.