ON THE DERIVATION OF THE TITIUS-BODE LAW
Abstract
In Chapter One a history of the Titius-Bode law is presented. Many different theories have been proposed to explain this law; however, this paper concentrates on the gravitational theories. Among the different formulations of the law, the Richardson formulation as outlined in Appendix D is the only one considered. The astronomer Michael Ovenden explained the tendency for a system of planets or satellites to relax into a Bode's-law type of configuration through his Principle of Least Interaction Action. The principle states that this relaxation occurs when the time-mean of the action associated with the mutual interaction of the satellites is an overall minimum. Ovenden tested this principle by determining the configuration of minimum interaction action available to any system of satellites, and his algorithm for doing this is presented in Chapter Two. However, by applying the ergodic hypothesis that the time mean of the disturbing function is equal to the space mean, a new algorithm allowing more realistic assumptions is presented in Chapter Two. 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. Justification of the ergodic hypothesis is presented in Appendix B for circular motion of the satellites and in Appendix C for quasiperiodic motion. 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 from the satellites to the primary in a given system are very close to the minimum interaction-action configuration for that system. The possibility of a planet in our solar system lying beyond Pluto is investigated and it is found that, by assuming the initial distance of the planet is determined from Richardson's distribution and by varying the planet's mass and inclination, the orbit and inclination of such a planet would be similar to Pluto's. Finally some of my results are compared with those of Ovenden's 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 giant asteroid lying between Mars and Jupiter.
Degree
Ph.D.
Subject Area
Mathematics
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