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a/M (AU / M )
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Figure 141 Histogram of the observed distribution of variable n = 483 (a/M )1/2 where a indicates the semi-major axis and M star mass (in solar system units, ie, astronomical unit AU and solar mass M ), for the recently discovered exoplanets and the planets of our inner solar system We theoretically expect probability peaks for integer values of this variable The probability of obtaining such a statistical agreement by chance is lower than 4 10 5
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Figure 142 Histogram of the distribution of k = ne, where m is the principal quantum number (which characterizes the semi-major axes) and e eccentricity, for the exoplanets and the planets of the inner solar system The theory predicts probability peaks for integer values of this variable The probability of obtaining such an agreement by chance is lower than 10 4 Combined probability to obtain by chance the two distributions (semi-major axes and eccentricities) is 3 10 7 , ie, a level of statistical signi cance reaching 5
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motion of a solid, for the equation of motion with a dissipation function, for Euler and Navier-Stokes equations, or even for scalar eld equations [NOT 97a, NOT 08] Among the possible generalizations of the theory, we can also mention abandoning the differentiability, not only in the usual space (which leads, as we saw, to the introduction of a scale-space governed by differential equations acting on scale variables, in particular on the spatio-temporal resolutions), but also in the space of scales itself All the previous construction can again apply to this deeper level of description This leads to the introduction of a scale-quantum mechanics [NOT 08] In this framework, which is equivalent to a third quantization, fractal objects of a new type can be de ned: rather than having structures at well-de ned scales (the case of the ordinary fractal objects), or than having variable scale structures described by traditional laws (the case of scale-relativistic fractals considered in this chapter), they are now characterized by an amplitude of probability for scale ratios ( quantum fractals) With regard to the applications of this approach, we gave only two of their examples, concerning electron mass and planetary systems Let us recall, nevertheless, that it could be applied successfully to a large number of problems of physics and astrophysics which were unresolved with the usual methods, and that it also allowed theoretical prediction of structures and of new relations [NOT 96a, NOT 08] Thus, the transformation of fundamental dynamic equation into a Schr dinger equation under very general conditions (loss of information on the individual paths and irreversibility) leads to a renewed comprehension of the formation and evolution of gravitational structures This method, besides semi-major axes and eccentricities of planets discovered around solar-type stars, brie y considered earlier, was also applied successfully to the three planets observed around pulsar PSR 1257+12 [NOT 96b], to obliquities and inclinations of planets and satellites of the solar system [NOT 98a], satellites of giant planets [HER 98], double stars, double galaxies, distribution of the galaxies on a very large scale and other gravitational structures [NOT 96a, NOT 98b, DAR 03] 148 Bibliography
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[AIT 82] A ITCHISON I, An Informal Introduction to Gauge Field Theories, Cambridge University Press, Cambridge, 1982 [BEN 00] B EN A DDA F, C RESSON J, Divergence d chelle et diff rentiabilit , Comptes rendus de l Acad mie des sciences de Paris, s rie I, vol 330, p 261 264, 2000 [BER 96] B ERRY MV, Quantum fractals in boxes , J Phys A: Math Gen, vol 29, p 6617 6629, 1996 [CAS 02] C ASH R, C HALINE J, N OTTALE L, G ROU P, Human development and log-periodic laws , CR Biologies, vol 325, p 585, 2002
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