Abstract

Simple “solar systems” are generated with planetary orbital radiirdistributed uniformly random in logrbetween 0.2 and 50 AU, with masses and order identical to our own Solar System. A conservative stability criterion is imposed by requiring that adjacent planets are separated by a minimum distance ofktimes the sum of their Hill radii for values ofkranging from 0 to 8. Least-squares fits of these systems to generalized Bode laws are performed and compared to the fit of our own Solar System. We find that this stability criterion and other “radius-exclusion” laws generally produce approximately geometrically spaced planets that fit a Titius-Bode law about as well as our own Solar System. We then allow the random systems the same exceptions that have historically been applied to our own Solar System. Namely, one gap may be inserted, similar to the gap between Mars and Jupiter, and up to 3 planets may be “ignored,” similar to how some forms of Bode's law ignore Mercury, Neptune, and Pluto. With these particular exceptions, we find that our Solar System fits significantly better than the random ones. However, we believe that this choice of exceptions, designed specifically to give our own Solar System a better fit, gives it an unfair advantage that would be lost if other exception rules were used. We compare our results to previous work that uses a “law of increasing differences” as a basis for judging the significance of Bode's law. We note that the law of increasing differences is not physically based and is probably too stringent a constraint for judging the significance of Bode's law. We conclude that the significance of Bode's law is simply that stable planetary systems tend to be regularly spaced and conjecture that this conclusion could be strengthened by the use of more rigorous methods of rejecting unstable planetary systems, such as long-term orbit integrations.