On the dynamical derivation of the Titius-Bode law

Abstract

The Principle of Least Action Interaction, developed by the dynamical astronomer Michael W. Ovenden, is tested using a new algorithm based on the ergodic hypothesis that the time mean of the disturbing function is equal to the space mean. This algorithm is an improvement over the one that Ovenden (1972) used in testing his principle, i.e. it can be applied to systems having more than three satellites without violating the conservation law of angular momentum and these satellites may have significant inclinations. This algorithm treats the problem of finding the configuration of least action interaction as a Lagrange multiplier problem. Renormalization group techniques and existing non-gradient optimization algorithms are incorporated into this new algorithm to reduce some of the numerical complexities.

This algorithm is tested on the planets and asteroids in our solar system and on the satellite systems of Jupiter, Saturn, and Uranus. In most cases the results show that the current distances of the satellites from their primary is very close to the minimum interaction-action configuration for that system. The possibility of a planet lying beyond Pluto is investigated using this algorithm.

Finally, some of my results are compared with those of Ovenden (1972) for our solar system. The results indicate that the interaction-action potential is lower using this new algorithm than the potential obtained from Ovenden's. Also, greater skepticism is raised concerning the one-time existence of a planet of 90 earth masses lying between Mars and Jupiter.