A Subjective Evaluation of Bode's Law and an ‘Objective’ Test for Approximate Numerical Rationality

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Abstract
Bode's law has usually been evaluated subjectively by means of overall judgment and opinions differ regarding its significance. Here we analyze it in some detail, with use of the axioms of probability, but still subjectively. The subjective Bayes factor in favor of Bode's law's being not a mere accident is given as between about 300 and 700. More tentatively, the existence of Bode's law seems to provide a factor of about 20 in favor of a non-cataclysmic origin of the solar system and hence in favor of there being an extremely large number of planetary systems in the galaxy. These factors would I think be much increased by taking into account “Dabbler's Law,” which is concerned with the mean distances of the saturnine satellites. It is pointed out that Dabbler's law has had predictive value and can be used for a further prediction. An “objective” test is given for whether a number is “nearly rational”. The ratios of the sidereal periods of pairs of solar bodies, for a given primary, are found not to be significantly close to rational numbers except that there is a significant clustering near 1/2. (Tail area about 0.01). All ratios close to 1/2 are related to known or new Laplacian relations.

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