Titius-bode's relation and distribution of exoplanets

Abstract

The distance distribution in our planetary system has been a controversial matter. Two kinds of important issues on Titius-Bode's relation have been discussed up to now: one is if there is a simple mathematical relation between distances of natural bodies orbiting a central body, and the other is if there is any physical basis for such a relation. We have examined, by applying it to exo-planetary systems, whether Titius-Bode's relation is exclusively applicable to our solar system. We study, with the χ2 test, the distribution of period ratios of two planets in multiple planet systems by comparing it with that derived from not only Titius-Bode's relation but also other forms of it. The χ2 value between the distribution of the orbital period derived from Titius-Bode's relation and that observed in our Solar system is 12.28 (dof = 18) with high probability, i.e., 83.3 %. The value of χ2 and probability resulted from Titius-Bode's relation and observed exo-planetary systems are 21.38 (dof = 26) and 72.2 %, respectively. Modified forms we adopted seem also to agree with the planetary system as favorably as Titius-Bode's relation does. As a result, one cannot rule out the possibility that the distribution of the ratio of orbiting periods in multiple planet systems is consistent with that derived from Titius-Bode's relation. Having speculated Titius-Bode's relation could be valid in exo-planetary systems, we tentatively conclude it is unlikely that Titius-Bode's relation explains the distance distribution in our planetary system due to chance. Finally, we point out implications of our finding.