The theory of relativity of scale results in representing the orbits like fractal lines on very small scale; there is then an infinity of possible trajectories, and a statistical equation governs their probabilities. These probabilities are not uniform, and one can predict how planets will be distributed around a given star, according to its mass. Laurent Nottale and his collaborators calculated in particular predictions on the semi major-axes and the eccentricities of the orbits.

The theoretical prediction resulting from this new approach is that average speeds of planets must show peaks of probability for discrete values given by: v n = w 0 / n where n is an integer and where w 0 is a fundamental and universal speed. Independent observations on extragalactic scales showed that this constant was about 144 km/s. The ratio ag = w0/c plays in fact the role of a gravitational coupling constant. Because of the third law of Kepler, v varies like (M/a) 1/2 or like (M/P) 1/3 , where M is the mass of star, a the semi major-axis of the planetary orbit and P its period. One thus expects peaks of probability for integer discrete values of the quantity 4.83(P/M) 1/3 = 4.83(a/M) 1/2 . The constant 4.83 is the value of the fundamental speed (144 km/s) in units of the Earth speed; the period (which is the quantity directly observed for the new exoplanets, through the movement of the star they induce) is in years; the semi major-axis (a) is in astronomical unit (1 UA = 149.7 million of kilometers, defined by the distance Earth-Sun); finally the mass of the star (M) is in solar mass. The observed values for exoplanets discovered for five years (see figures 1 and 2 ) are in excellent agreement with these predictions: for example one finds for the first two peaks of probability a/M = 0.0438 + / - 0.0017 UA/Msol and 0.168 + / - 0.014 UA/Msol, to compare with the predicted values 0.0430 and 0.171 UA/Msol. It is the same for the three planets system discovered around the B1257+12 pulsar, for which the relative agreement between theory and observations is of some ten-thousandth.

Figure 1. Distribution observed for the semi major-axes (a) of the orbits of the exoplanets and the solar system intern planets (in natural length unit for these systems, derived from the mass M of the star). The reported numerical values are those of a/M and average speed v (connected via the third law of Kepler); the scale of the layout is of square root of (a/M). The regions predicted with high probability are plotted in yellow, and in blue those of low probability. Let us recall that the existence of these peaks (thus the possibility of great proximity between the exoplanets and their star, which astonished at the time of their discovery) was predicted theoretically (L Nottale, 1993, Fractal Space-Time and Microphysics, World Scientific) before this discovery, which goes back to 1995 as far as stars of the solar type are concerned.

Figure 2. Histogram of the distribution of the semi major-axes (normalised to the mass of their star), for the extrasolar and internal solar planets. The masses of stars used to build this histogram are those given by the authors (uncertainties of about 10%, which thus becomes 3% for (P/M) 1/3 ). The distribution observed is not uniform: peaks of density of probability appear very precisely at the predicted positions, without any adjustment. The probability of obtaining such a result by chance is lower than 2 10 -5 .

The distribution of exoplanets eccentricities (e) is predicted to show peaks for e = k/n, where k is an integer and varies from 0 to n-1. The distribution observed is statistically in significant agreement with the theoretical prediction (see figure 3 ). One thus expects the existence of possible circular orbits whatever the principal quantum number, but also of elliptic orbits. One thus accounts for the existence of large observed values for eccentricity, which constituted, with the small values of the semi major-axes of certain exoplanet orbits, one of the major enigmas within the framework of standard theories of formation of planetary systems.

Figure 3. Histogram of the distribution of the eccentricities (multiplied by " the principal quantum " number), for the extrasolar planets and planets of the internal solar system. One predicts peaks for integer values of this product, which is confirmed by observation. The probability of obtaining such a result by chance is lower than 10 -4 . The probability combined for the two effects of structuring (semi major-axes and eccentricities) is 3 10 -7 . These results go in the direction of a universality of structure of the various planetary systems, including our own solar system.

### References:

L. Nottale, G. Schumacher & E.T. Lefèvre, 2000,[Astron. Astrophys. 361, 379: "Scale relativity and quantization of exoplanet orbital semi-major axes" Nottale, N. Tran Minh, 2001, Astron. Astrophys. (soumis pour publication): "Scale relativity and quantization of exoplanet eccentricities"

### Contact:

- L. Nottale

(DAEC, Observatoire de Paris) - N. Tran Minh

(DAMAP, Observatoire de Paris)

Last update on 25 October 2013