Astronomy and Astrophysics – Astrophysics
Scientific paper
Feb 1976
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1976cemec..13....3w&link_type=abstract
Celestial Mechanics, vol. 13, Feb. 1976, p. 3-37. Research supported by the Smithsonian Astrophysical Observatory and Harvard Co
Astronomy and Astrophysics
Astrophysics
9
Asteroids, Jupiter (Planet), Orbit Calculation, Poincare Problem, Resonance, Solar Orbits, Statistical Analysis, Three Body Problem, Canonical Forms, Dynamic Models, Eccentricity, Equations Of Motion, Integral Equations, Kepler Laws, Orbital Elements, Orbital Mechanics
Scientific paper
Application of the statistical method of phase mixing to the approximate Poincare solution to resonant motion is considered. The two Poincare integrals of motion for the restricted problem of three bodies are introduced to first order in eccentricity. The theory of phase mixing of an initial ad hoc distribution of particles is developed for this dynamical system, and the absence of significant evolution of the system far from resonance is verified. Some results for the 2:1, 3:1, and 5:2 resonances are given which show, in general, a peak on the low side of exact resonance and a gap on the high side. The amplitudes of both the peak and the gap decrease, and their relative separation increases, as the resonance order increases or as the initial distribution is shifted to higher eccentricities. Comparison with large numbers of numerically integrated orbits gives good agreement with the model, at least for small eccentricities. However, the model is unable to exhibit the clean gaps shown by the real asteroid belt. Hence, a purely statistical model of the Kirkwood gaps is ruled out, and an additional mechanism must be sought.
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