A Close Correlation between Third Kepler Law and Titius-Bode Rule

TL;DR

This paper reveals a mathematical connection between Kepler's third law and the Titius-Bode rule, showing that planetary distances follow an exponential pattern derived from orbital momentum relations.

Contribution

It demonstrates that the Titius-Bode rule can be derived from Kepler's third law without assuming exponential distance, using a Taylor expansion of planetary distances.

Findings

Planet orbital momentum is effectively a function of planetary distance.

Planet distances approximate an exponential form in the linear regime.

Quantized orbital momenta follow a geometric progression.

Abstract

In this work we present a close correlation between third Kepler law and Titius-Bode empirical rule. Concretely, we demonstrate that third Kepler law, or, corresponding equilibrium condition between centrifugal and Newtonian gravitational force, implies that planet orbital momentum becomes effectively a function of the planet distance as unique variable and vice versa. Then, approximation of the planet distance by its first order Taylor expansion over planet orbital momentum holds an exponential form corresponding to Titius-Bode rule. In this way it is not necessary postulate exponential form of the planet distance (as it has been done by Scardigli) but only discrete values of its argument. Physically, it simply means that, in the linear approximation, "quantized" planets orbital momentums do a geometrical progression.