Planetary Systems and the Hidden Symmetries of the Kepler Problem

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.

Orbital Regularity of Exoplanets and the Symmetries of the Kepler Problem

“…He found that the orbital parameters of the planets in our solar system follow this rule to a good approximation. In a recent work, one of us investigated if the observed data of some selected exoplanetary systems follow the symmetry-governed transformations [7]. A remarkable agreement was found.…”

“…In [7], only a few quintuple-planet or bigger systems were taken into consideration. Here, our exploration covers as many cases as possible by looking at all exoplanetary systems having four or more planets.…”

“…There only six exoplanetary systems were considered, whereas here, we carry out a systematic investigation by studying all the 100 known systems with four or more planets. The connection between the (generalized) Titius-Bode rule and the transformations of the dynamical group of the Kepler problem has not been discussed in [7], nor in any other work to the best of our knowledge. Here we point out this connection, which might be worth for further attention.…”

“…Some preliminary aspects of this investigations were published in [7], as mentioned above. There only six exoplanetary systems were considered, whereas here, we carry out a systematic investigation by studying all the 100 known systems with four or more planets.…”

“…Here we point out this connection, which might be worth for further attention. Since the symmetries of the Kepler problem are not very widely known, and were mentioned only briefly in [7], we present it here with some details.…”