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Resonance - Part 4 of 5

Gravitational Hypothesis

The gravitational hypothesis, the most widely accepted explanation for
the origin of the Kirkwood gaps, says the gaps are formed by purely
gravitational interactions with Jupiter.
Both Stanley Dermott and Carl Murray in 1981 and 1983, and Jack Wisdom
in 1982 and 1983 have provided strong evidence to support the
gravitational hypothesis. In the first study, 757 main-belt, non-family
asteroids were used. To analyze the properties of asteroids and their
orbits, Dermott and Murray defined a parameter d as the measure of the
displacement of an asteroid from the nearest strong resonance. An exact
resonance corresponds to d = 0, while /d/ = 1 is the greatest
displacement from resonance that an asteroid can have.
A direct plot of absolute magnitude, B(1,0) versus d is shown in Fig.
5-9. Figure 5-10 shows a smoothed number density contour plot based on
the same data. Since the number density increases continuously as d
increases, this implies the effects of resonance pervade the whole
distribution of main-belt asteroids.
Dermott and Murray showed that observational selection produces an
excess of high magnitude, low inclination asteroids. Thus, they reduced
their data set to 144 asteroids, constituting a bias-free set, from
which they concluded there is no magnitude-frequency distribution change
near the Kirkwood gaps. This result was mentioned as tending to disprove
the collisional hypothesis.
More importantly, they found the tendency for both eccentricity and
inclination to increase with increasing /d/ is a fundamental property of
the Kirkwood gaps (Fig. 5-11). Since the magnitude of the gravitational
disturbing function acting on an asteroid increases with e and i, this
result strongly suggests that Jupiter's gravitational effect creates the
Kirkwood gaps.
They conclude that at least some of the Kirkwood gaps have been formed
since the time of formation of the solar system, and that they are not
simply regions of small asteroid number density since the effects of
resonance pervade the entire asteroid belt. They confirmed and extended
these conclusions in 1983. Fig. 5-12 shows quite dramatically the lack
of asteroids at resonant positions in the belt, particularly at 3.3 AU
and 4.0 AU where resonances overlap. The few asteroids that remain in a
libration region tend to have eccentricities much higher than the
average of their neighbors.
Near the 3/1 commensurability, for example, only two asteroids are known
with librating orbits: Alinda (e = 0.55) and Quetzalcoatl (e = 0.58).
According to the gravitational mechanism, Iarge eccentricities can be
expected for librating orbits whose rate of change of perihelion
longitude is near zero. As shown by Scherbaum & Kazantsev (1985), this
is precisely what is found. Wisdom (1982) used a set of 300 'test
asteroids' in the neighborhood of the 3/1 commensurability to determine
if gravitational influences over a period of 2 million years could
produce a gap. Such a gap formed, but was narrower than the actual one.
Using improved computational techniques, he repeated the test.
"Of the 300 test asteroids in the random distribution, 89 were found to
have chaotic trajectories and only 11 were quasiperiodic librators. All
but five of the chaotic trajectories became Mars crossing within 300,000
years and only one had not reached an eccentricity of 0.3 within 1
million years. The predicted gap is now in satisfactory agreement with
the full distribution of real asteroids." (Wisdom, 1983). As Fig. 5-13
clearly shows, the boundaries of the chaotic zone discovered by Wisdom
match the boundaries of the real 3/1 Kirkwood gap quite well.

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