Astron. Astrophys. 322, 1018-1025 (1997)

Scale relativity and quantization of the solar system

L. Nottale ¹, G. Schumacher ² and J. Gay <sup>2

1</sup> C.N.R.S., D.A.E.C. Observatoire de Paris-Meudon, F-92195 Meudon Cedex, France
² Observatoire de la Côte d'Azur, Département Augustin Fresnel, URA 1361 du CNRS, av. Copernic, F-06130 Grasse, France

Received 2 April 1996 / Accepted 10 October 1996

Abstract

The scale relativity theory, by giving up the differentiability of space-time coordinates at very large time-scales, describes the solar system in terms of fractal trajectories governed by a Schrödinger-like equation. The predictions of the theory are expressed in terms of probability densities, that we interpret as a tendency for the system to make structures. Planets can no longer orbit at any distance from the Sun, but instead at preferential distances given at lowest order by: . In this formula, M is the mass of the Sun and  km/s is a fundamental constant which is observed from the planetary scales to the extragalactic scales. Our theoretical predictions agree very well with the observed values of the actual planetary orbital parameters, including those of the asteroid belts. In addition, since Mercury ranks in the above formula, there is good reason to anticipate a small planet or two between the Sun and Mercury. We propose to check the theory by searching for such an object, on the second orbit which has a semi major axis of  AU.

Key words: chaos diffusion gravitation planets and satellites: general solar system: general

Send offprint requests to: G. Schumacher

Contents

• 1. Introduction
• 2. Theoretical background
• 3. Application to the solar system
  • 3.1. Circularity of the orbits
  • 3.2. Distribution of the distances of planets
  • 3.3. Distribution of the masses
• 4. Possible existence of an intramercurial small planet
• 5. Conclusion
• Acknowledgements
• Appendix A: upper limit of the mass due to the uncertainty of Mercury perihelion's advance
• References

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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