          Astron. Astrophys. 335, 281-286 (1998)

## 1. Introduction

Scale Relativity theory is an extension of Einstein's principle of relativity, applied to scale laws. By giving up the differentiability of space-time coordinates at very large time-scale, one can describe the solar system in terms of fractal trajectories governed by a Schrödinger-like equation. The theory is due to L. Nottale (see for example Nottale, 1993, Nottale, 1996a, and Nottale, 1997a). In a previous paper (Nottale et al., 1997) we have shown how the theory can be applied to the solar system, and how the orbits of the planets are quantized. In the present paper we consider that not only the orbits of the planets are quantized, but also the orbits of their main satellites and maybe the rings.

Lets consider a gravitational system with a central mass (Kepler problem) and an orbiting body, and consider the case of almost circular orbits. This is the case of most planets and satellites in the solar system. We have shown that the radius of the orbits are quantized, and that their distribution is given by: where a is the semi-major axis of the orbit, M the mass of the central object, G the gravitational constant, w a constant having the dimension of a velocity, and n is the rank of the orbit. This relationship is valid in the framework of a theory of formation of a planetary system. The matter fills the orbitals with time, and then the planet, or the satellite, is formed by accretion at the mean distance given by 1 (see Nottale et al., 1997).

In the first part of this paper, we expose the method to determine the rank of the satellites, and we study the statistical significance of the results. In a second part, the results are given for every planetary system. In a third part we show the classification by rank of the main satellites of the whole solar system, suggesting that some orbits are unoccupied, or not yet discovered.    © European Southern Observatory (ESO) 1998

Online publication: June 12, 1998 