Planetary Systems and the Hidden Symmetries of the Kepler Problem

József Cseh

doi:10.3390/sym12122109

Abstract

The question of whether the solar distances of the planetary system follow a regular sequence was raised by Kepler more than 400 years ago. He could not prove his expectation, inasmuch as the planetary orbits are not transformed into each other by the regular polyhedra. In 1989, Barut proposed another relation, which was inspired by the hidden symmetry of the Kepler problem. It was found to be approximately valid for our Solar System. Here, we investigate if exoplanet systems follow this rule. We find that the symmetry-governed sequence is valid in several systems. It is very unlikely that the observed regularity is by chance; therefore, our findings give support to Kepler’s guess, although with a different transformation rule.

Highlights

Kepler’s laws of planetary motion [1,2] played an essential role in the celestial mechanics, and in the foundation of physics
There was still an expectation by Kepler which was not fulfilled: he was looking for the proportions of perfection in the sequence of planets [2,3]. He expected to find Divine Harmony in the distances of the planets. He thought that the orbits of the six planets are related to each other by inserting the five regular polyhedra between them
Afterwards, we investigate some exoplanet systems in order to test if their observed characteristics follow the symmetry-inspired regularity

Summary

Introduction

Kepler’s laws of planetary motion [1,2] played an essential role in the celestial mechanics, and in the foundation of physics. There was still an expectation by Kepler which was not fulfilled: he was looking for the proportions of perfection in the sequence of planets [2,3] He expected to find Divine Harmony in the distances of the planets. He thought that the orbits of the six planets (known at his time) are related to each other by inserting the five regular polyhedra between them. In other words, he was looking for a transformation, which determines the outer planetary orbitals from that of the first one

Methods

Results

Conclusion