Encyclopedia of the Solar System 2nd ed

802 Encyclopedia of the Solar System

2 3 4 5

1000

2000

3000

4000

5000

Number of Asteroids

Semimajor Axis (AU)

3:1 5:2 2:1 3 :2 1:1

FIGURE 17 A histogram of the distribution of the numbered
asteroids with semimajor axis together with the locations of the
major jovian resonances. Most objects lie in the main belt
between 2.0 and 3.3 AU, where the outer edge is defined by the
location of the 2:1 resonance with Jupiter. As well as gaps (the
Kirkwood gaps) at the 3:1, 5:2, 2:1, and other resonances in the
main belt, there are small concentrations of asteroids at the 3:2
and 1:1 resonances (the Hilda and Trojan groups, respectively).

an infinite sequence of first-order resonances that lie closer
together as its semimajor axis is approached. For example,
the 2:1, 3:2, 4:3, and 5:4 resonances with Jupiter lie at 3.3,
4.0, 4.3, and 4.5 AU, respectively. Since each (p+1):pres-
onance (wherepis a positive integer) has a finite width in
semimajor axis that is almost independent ofp, adjacent
resonances will always overlap for some value ofpgreater
than a critical value,pcrit. This value is given by

pcrit≈ 0. 51

(

m

m+M

)− 2 / 7

(44)

where, in this case,mis the mass of Jupiter andMis the

mass of the Sun. This equation can be used to predict that

resonance overlap and chaotic motion should occur forp

values greater than 4; this corresponds to a semimajor axis

near 4.5 AU. Therefore chaos may have played a significant

role in the depletion of the outer asteroid belt.

The histogram in Fig. 17 also shows a number of regions

in the main belt where there are few asteroids. The gaps

at 2.5 and 3.3 AU were first detected in 1867 by Daniel

Kirkwood using a total sample of fewer than 100 asteroids;

these are now known as the Kirkwood gaps. Their locations

coincide with prominent Jovian resonances (indicated in

Fig. 17), and this led to the hypothesis that they were created

by the gravitational effect of Jupiter on asteroids that had

orbited at these semimajor axes. The exact removal mech-

anism was unclear until the 1980s, when several numerical

and analytical studies showed that the central regions of

these resonances contained large chaotic zones.

The Kirkwood gaps cannot be understood using the

model of the circular restricted three-body problem de-

scribed in Section 4.2. The eccentricity of Jupiter’s orbit,

although small (0.048), plays a crucial role in producing the

large chaotic zones that help to determine the orbital evo-

lution of asteroids. On timescales of several hundreds of

thousands of years, the mutual perturbations of the planets

act to change their orbital elements and Jupiter’s eccentric-

ity can vary from 0.025 to 0.061. An asteroid in the chaotic

zone at the 3:1 resonance would undergo large, essentially

unpredictable changes in its orbital elements. In partic-

ular, the eccentricity of the asteroid could become large

enough for it to cross the orbit of Mars. This is illustrated in

Fig. 18 for a fictitious asteroid with an initial eccentricity

of 0.15 moving in a chaotic region of the phase space at

the 3:1 resonance. Although the asteroid can have periods

of relatively low eccentricity, there are large deviations and

2000 4000 6000 8000 10000

0.1

0.2

0.3

0

Eccentricity

Time (Jupiter periods)

Mars crossing

FIGURE 18 The chaotic evolution of the

eccentricity of a fictitious asteroid at the 3:1

resonance with Jupiter. The orbit was integrated

using an algebraic mapping technique developed

by J. Wisdom. The line close toe=0.3 denotes the

value of the asteroid’s eccentricity, above which it

would cross the orbit of Mars. It is believed that

the 3:1 Kirkwood gap was created when asteroids

in chaotic zones at the 3:1 resonance reached high

eccentricities and were removed by direct

encounters with Mars, Earth, or Venus.