Scale relativity, when combined with the laws of gravitation, provides us with a general theory of the structuring of gravitational systems (Nottale 1996a , 1997). In this new approach, we do not any longer follow individual trajectories, but we jump to a statistical description in terms of probability amplitudes. Indeed, we have demonstrated that, under only three simple hypotheses (large number of potential trajectories, fractal geometry of each trajectory and local irreversibility), Newton's equation of dynamics can be transformed and integrated in terms of a generalized Schrödinger equation. This result suggests, in accordance with recent similar conclusions (Ord 1996; Ord & Deakin 1996; El Naschie 1995), that the Schrödinger equation could be universal, i.e. that it may have a larger domain of application than previously thought, but with an interpretation different from that of standard quantum mechanics.
It has been shown (Nottale 1993 , 1994), that this approach accounts for several structures observed in the Solar System, including planet distances, eccentricities, and mass distribution (Nottale et al. 1997), obliquities and inclinations of planets and satellites (Nottale 1998a), giant planet satellite distances (Hermann et al. 1998), parabolic comet perihelions (Nottale & Schumacher in preparation). Moreover, it also allows one to predict and understand structures observed on a large range of scales, from binary stars (Nottale & Schumacher 1998), to binary galaxies (Nottale 1996a; Tricottet & Nottale in preparation) and the distribution of galaxies at the scale of the local supercluster (Nottale & Schumacher 1998). A similar kind of approach has been applied by Perdang (1995) to a statistical description of HR diagrams.
It has been also demonstrated that the first newly discovered extra-solar planetary systems come under the same structures, in terms of the same universal constant as in our own Solar System (Nottale 1996b). The system of three planets discovered around the pulsar PSR B1257+12 also agree with the theoretical prediction with a very high precision of some (Nottale 1996b , 1998b).
The number of exoplanets discovered around solar-like stars has now been multiplied by five since these first studies, so that it seems worthwhile to check again whether the new observational data continue to prove to be non-uniformly distributed.
We indeed find that the semi-major axes of the presently known exoplanet orbits, (measured in terms of the natural gravitational unit for each system, given by the mass of their parent star), are non-uniformly distributed. Namely, their distribution show peaks of probability density that are consistent with the law , where the constant is a priori fixed to the value 144 km/s as in our own inner Solar System and in extragalactic data. Moreover, most of these exoplanets (51 Peg-type objects) fall in the fundamental probability density peaks ( AU/ and in the second orbital ( AU/ predicted by the theory.
© European Southern Observatory (ESO) 2000
Online publication: September 5, 2000