Encyclopedia of the Solar System 2nd ed

Solar System Dynamics: Regular and Chaotic Motion 803

Sun Sun

orbit of

Mars

orbit of asteroid

e = 0.15

orbit of

Mars

orbit of asteroid

e = 0.33

FIGURE 19 The effect of an increase in the

orbital eccentricity of an asteroid at the 3:1 Jovian

resonance on the closest approach between the

asteroid and Mars. Fore=0.15, the orbits do not

cross. However, fore=0.33, a typical maximum

value for asteroids in chaotic orbits, there is a clear

intersection of the orbits, and the asteroid could

have a close encounter with Mars.

ecan reach values in excess of 0.3. Allowing for the fact
that the eccentricity of Mars’s orbit can reach 0.14, this im-
plies that there will be times when the orbits could intersect
(Fig. 19). In this case, the asteroid orbit would be unstable,
since it is likely to either impact the surface of Mars or suffer
a close approach that would drastically alter its semimajor
axis. Although Jupiter provides the perturbations, it is Mars,
Earth or Venus that ultimately removes the asteroids from
the 3:1 resonance. Figure 20 shows the excellent correspon-
dence between the distribution of asteroids close to the 3:1
resonance and the maximum extent of the chaotic region
determined from numerical experiments.
The situation is less clear for other resonances, although
there is good evidence for large chaotic zones at the 2:1
and 5:2 resonances. In the outer part of the main belt, large
changes in eccentricity will cause the asteroid to cross the
orbit of Jupiter before it gets close to Mars. There may also

be perturbing effects from other planets. In fact, it is now

known that secular resonances have an important role to

play in the clearing of the Kirkwood gaps, including the

one at the 3:1 resonance. Once again, chaos is involved.

Studies of asteroid motion at the 3:2 Jovian resonance in-

dicate that the motion is regular, at least for low values of

the eccentricity. This may help to explain why there is a

local concentration of asteroids (the Hilda group) at this

resonance, whereas others are associated with an absence

of material.

Since the dynamical structure of the asteroid belt has

been determined by the perturbative effects of nearby plan-

ets, it seems likely that the original population was much

larger and more widely dispersed. Therefore, the current

distribution of asteroids may represent objects that are ei-

ther recent collision products or that have survived in rela-

tively stable orbits over the age of the solar system.

0.4

0.3

0.2

0.1

0.0

Eccentricity

Semimajor Axis (AU)

2.45 2.50 2.55

FIGURE 20 The eccentricity and semimajor axes of asteroids in the

vicinity of the 3:1 jovian resonance; the Kirkwood gap is centered

close to 2.5 AU. The two curves denote the maximum extent of the

chaotic zone determined from numerical experiments, and there is

excellent agreement between these lines and the edges of the 3:1

gap.