Commensurability in the Solar System

The mean orbits of all planets, including Ceres, the largest asteroid, are correctly predicted [13] by the relative distances from the spiral center. With the orbital radii expressed as rational fractions, a quantized distribution of major planets, as numbered, is revealed. On this scale the orbit of Ceres measures r and those of the inner planets are rational fractions of the golden ratio. The same pattern was shown to repeat itself for the orbital motion of planetary moons and rings. 

From this observation we infer self-similarity on a cosmic scale, from atoms to galaxies, which implies the same numerical basis for the atomic models of Nagaoka and Bohr, assumed to be self-similar with the rings of Saturn and the planets, respectively. In principle, the periodic accumulation of extranuclear electron density on an atom could also be optimized by specifying an appropriate convergence angle. 

---