New Approaches to Statistical Dynamics of Gravitational Systems

Abstract

It is pointed out that equations of motion (i.e., the continuity equations and the Euler-Bernoulli equation for the motion of a system of particles in gravitational interaction) are best and most advantageously formulated in terms of a Schr\odinger-type equation in which the ${h}^{3}$ characterizing the uncertainty principle is replaced by a macroscopic phase-space volume which characterizes the coarse-grainedness of the mass distribution in phase space. This method is applied to the study of a gravitational system of the type of our solar system in its recent development, as the present system is assumed to be preceded by a system of a large number of smaller particles in gravitational interaction. The distribution of orbital elements of the planets in the solar system, and also of the elements of the satellites of the major planets, shows distinct regularities. One of them, the commensurability of the periods of revolution, has been repeatedly discussed. Another, the small eccentricities and inclinations, has found a simple explanation in the Kant-du Ligond\`es hypothesis of the development of the solar system. Still another one, concerning the distribution of the semimajor axes (or periods), has found a numerological formulation in the Titius-Bode law, and its explanation has been attempted in terms of a distribution of vortex rings in the early stage of the Kant-du Ligond\`es model. We are here proposing a simpler analysis of the regularities in this distribution of the semimajor axes. It is suggested that statistically there are changes toward preferential orbital elements due to the gravitational interaction in such a system. Instead of discussing the issue in terms of perturbation expansions of individual orbits, however, it may be more advantageous to pursue the analysis in terms of the statistical distribution of integrals of the equations of motion, in particular the Jacobi integral. These distributions of the integrals (orbital elements) are most elegantly formulated in terms of distributions of the stationary-state wave functions which correspond to those integrals. The calculations yield, for the planets and for the satellites, sequences of orbital elements which coincide strikingly well with the observed elements.