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

## TL;DR

This paper provides a physical interpretation of the Titius-Bode rule for planetary orbits, linking it to gravitational work and revealing new properties of closed orbits consistent with Bertrand's theorem.

## Contribution

It introduces a new physical perspective on the Titius-Bode rule and uncovers novel properties of Bertrand's closed orbits related to planetary and exoplanetary systems.

## Key findings

- The Titius-Bode rule can be interpreted through gravitational work on perturbed particles.
- Exoplanets also follow the Titius-Bode relation, supporting its physical basis.
- New properties of Bertrand's closed orbits are revealed through this interpretation.

## Abstract

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. But 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 and known since 1873.