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Resonance - Part 5 of 5

Several important conclusions result from this work

1. Chaotic orbits are common, contrary to the results of previous work:
collision probabilities with  the Earth and Mars appear adequate to
remove the asteroids with chaotic orbits within the age  of the solar
system.

2. Sudden large increases in the eccentricity of asteroids occur near
the 3/1 commensurability, and  likely others as well.

3. The outer boundary of the chaotic region corresponds precisely to the
boundary of the 3/1  Kirkwood gap, thus showing that the gravitational
hypothesis can adequately explain the gap.

This line of research has been explored further by Murray (1986), who
found extensive chaotic regions in the vicinity of both the 2/1 and 3/2
resonances. In the central region of the 2/1 resonance is an area where
asteroids will become Jupiter crossing, whereas no such central area
exists at 3/2. He suggests that this difference in the high-eccentricity
regions may explain why asteroids are found at the 3/2 resonance but
absent at 2/1.
Before leaving the subject, mention must be made of secular resonances
and libration of the argument of perihelion, w. Secular resonances arise
if the rate of the node or perihelion of an asteroid matches a frequency
of one of the fundamental, long-period oscillations of the planetary
system. Several such resonances are known, and all are greatly depleted
in asteroids. Secular resonances may have caused the present
distribution of eccentricity and inclinations of the asteroids (Scholl
and Froeschlé, 1985). They will be considered further in Chapter 13.
While not usually classed as a resonance, the argument of perihelion, w,
of an asteroid can librate about 90° or 270° when a combination of e and
i exceeds a critical value. Only 1373 Cincinnati and the Mars crosser
1974 UB exhibit such behavior, both librating about 90°. In the case of
Cincinnati, the period of libration is 11,000 years.


Best regards,

Bernd

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