A statistical theory of resonance motion in the Sun-Jupiter system |
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| Authors: | W E Wiesel |
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| Institution: | (1) Center for Astrophysics, Harvard College Observatory and Smithsonian Astrophysical Observatory, 02138 Cambridge, Mass, USA;(2) Present address: 2610 Waldean St., 80909 Colorado Springs, Col, USA |
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| Abstract: | We consider the application of the statistical method of phase mixing to the approximate Poincaré solution to resonant motion. The two Poincaré integrals of the motion for the restricted problem of three bodies are introduced to first order in the eccentricity. The theory of the phase mixing of an initialad hoc distribution of particles is then developed for this dynamical system, and the absence of significant evolution of the system far from resonance is verified.A selection of results is given for the 2 1, 31, and 52 resonances, 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 we must search for an additional mechanism. Some speculation on possible additional mechanisms is offered. |
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