A Physical Interpretation of the Titius-Bode Rule and Its Connection to the Closed Orbits of Bertrand's Theorem

We consider the geometric Titius-Bode rule for the semimajor axes of planetary orbits. We derive an equivalent rule for the midpoints of the segments between consecutive orbits along the radial direction and we interpret it physically in terms of the work done in the gravitational field of the Sun by particles whose orbits are perturbed around each planetary orbit. On such energetic grounds, it is not surprising that some exoplanets in multiple-planet extrasolar systems obey the same relation. However, it is surprising that this simple interpretation of the Titius-Bode rule also reveals new properties of the bound closed orbits predicted by Bertrand's theorem, which has been known since 1873.