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Resonance - Part 3 of 5

Collisional Hypothesis

The Kirkwood gaps are created by collisions between asteroids according
to this hypothesis. These collisions result either in destruction of the
asteroid in the gaps, or an orbital change that sends them out of the
gaps.
In 1975, T.A. Heppenheimer investigated this problem by a Monte Carlo
simulation of collisions at the 2/1 commensurability. The evolution of
50 fictitious asteroids was followed, and all remained in the gap even
after numerous collisions. He also concluded that the fragmentation of
gap asteroids by collision is only slightly faster than for nonresonant
asteroids outside of the gaps, again militating against the collisional
hypothesis.
Stanley Dermott and Carl Murray provided further evidence militating
against this hypothesis in 1981. They found there is no significant
tendency for low-magnitude objects to be further away from the
resonances than high-magnitude objects. Such a tendency was predicted by
the the collisional hypothesis. Binzel (1986) has also concluded that
inter-asteroid collisions are not an important process in clearing the
Kirkwood gaps.

Cosmogonic Hypothesis

This hypothesis suggests the gaps represent regions where asteroids
failed to form during the early history of the solar system. This highly
speculative hypothesis was examined by Heppenheimer in 1978. On the
assumption that planetesimals break up at velocities exceeding 100
m/sec, his model predicted that Jupiter would increase the
eccentricities, and therefore velocities, of the planetesimals at the
resonant positions. The surrounding nebula would decrease the
eccentricities outside these areas, thus allowing the planetesimals to
remain in stable orbits without breaking up.
This model suffers from severe initial constraints, however. It requires
an appropriate density, mass and temperature for the nebula, and implies
that all asteroids are formed in almost circular orbits in a thin disk.
It also requires the Hilda asteroids, which are in resonant motion, to
be formed at a different place. And since low eccentric orbits behave
like nonresonant orbits, asteroids with small eccentricities should be
visible in the Kirkwood gaps, a prediction not supported by observation.

Dermott & Murray (1983) conclusively eliminated the cosmogonic
hypothesis. They showed that the resonant locations in the asteroid belt
must have formed after the asteroids dispersed from the disk in which
they accreted.

Statistical Hypothesis

Just as a pendulum spends the majority of its time away from the bottom
of its swing, or equilibrium point, so asteroids near commensurabilities
undergo large variations in their semi-major axes, spending most of
their time outside the gaps.
While this hypothesis was considered by Brendel in 1924 and Brown in
1928, it was not put to a numerical test until 1969, when François
Schweizer used a modern computer. He calculated the orbits of 185
asteroids around the Hecuba gap at 2/1, 20 orbits at 5/2, and 13 orbits
around the Hestia gap at 3/1, as well as four orbits inside of gaps.
In each case, he took the osculating mean motion n0, which performs
nearly sinusoidal oscillations around a mean value x with a period on
the order of several hundred years. While quite a short time scale, it
is thought his results are valid up to at least 10^5 years. When
Schweizer plotted the distribution function for n, the result still
showed Kirkwood gaps. If asteroids really crossed the gaps, this time
averaged plot would have filled them in.
Dermott & Murray (1983) showed that the Kirkwood gaps "cannot possibly
be a statistical phenomenon. " For it to be tenable, most asteroids near
a gap must be librators, but observations show the regions where they
should exist are almost devoid of asteroids.

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