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«... Already 100 years
passed by from the discovery of the Kirkwood “gaps” in the distribution of the asteroid
semi major axes and, still [1981], a theory able to explain their origin has not been
found. [...] (1) »
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i These are the most popular
hypotheses:
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Gaps are only a statistical occurrence
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Gaps are formed by gravitational forces alone
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Gaps form, because near to resonance asteroids tend to assume a
large eccentricity and due to collisions with other asteroids can be
ejected
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Asteroids never occupy the gaps
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The statistical assumption led to the conclusion that asteroids near
to resonance never cross it
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The gravitational hypothesis was studied analytically with the
planar model of the restricted three bodies problem. In such a
hypothesis asteroids located in the “gap”, never leave it
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The collision hypothesis was studied following a procedure related
with the plane-elliptical problem, to test asteroids very close to
resonance. Eccentricity showed no evidence it could increase much. Furthermore, the
study of the collision probabilities showed very small values
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The dynamic hypotheses, necessary to explore the possibility that
asteroids may never be found within the “gaps”, were not yet thoroughly
examined
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How to explain the formation of the “gap” concerning the 3:1 resonance of the asteroids
with Jupiter was analytically taken into consideration and developed with the
help of the computer of Jack Wisdom (1), a contemporary of Bepi
Colombo
on this topic.
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The
problem was gradually examined, starting from the most simple planar models.
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The
negative answers given by the simplified models led to take into
consideration a 3D model in place of a planar one; to consider the
inclination of Jupiter’s orbit; Jupiter’s orbit is not static, but is subject
to age-old perturbations and, from a strict mathematical point of view, in
the processing of the equations also the terms of higher order [up to least 4th
level] in the eccentricity were included; also included were the long period
resonance terms and the high frequency perturbations.
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Additionally,
it was observed that the time requested for a gap is longer than what was
estimated till now.
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So,
this more flexible model, after a suitable period of time, discovered that
the eccentricity of the orbit of the asteroid near to the 3:1 resonance, suddenly
sharply raised, from low values, [less than 0,10] to higher values [more than 0,3], sufficient to
cross the orbit of Mars.
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A gap
shape is obtained, quite similar to the one measured in the distribution of
the semi major axes, assuming that within time Mars may remove through a “slingshot effect”
["fly-by"] the asteroids within
the 3:1 resonance band, with an eccentricity more than 0.3.
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Lifetime
of a “Mars
crosser” [eccentricity more than 0,3] has been estimated in 200 million years.
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Gap 3:1 is completed in a
period of time of 18 million years.
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Results of the study are summarised as follows:
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