## 1. IntroductionThe Titius-Bode law is the most famous of the remarkable
relationships among planetary and satellite parameters concerning the
solar system. This simple geometric progression describes with good
precision the distances of most, but not all, of the planets from the
sun. Known for over two centuries, this law still lacks an explanation
based upon physical laws. Although it has recently been shown that
simply respecting both scale and rotational invariance can yield an
endless collection of theoretical models predicting a Titius-Bode law,
irrespective to their physical content (Dubrulle& Graner, 1994),
early comparisons of the Titius-Bode law with the predictions of the
Bohr theory for atoms immediately raised the question of the
applicability of (some, at least) principles of Quantum Mechanics
( Once the so-called `inner solar system' (telluric planets) was quite reasonably given the increasing series of `quantum Bohr numbers' , which raises the problem of the existence of Vulcanus at , two questions remain unanswered: i) does anything correspond to in the solar system? ii) why is the `outer solar system' (i.e. J, S, U, N & P) described by Bohr numbers that are no more in sequence? The present paper provides a definite negative answer to the first question. Concerning the second question, one immediately notices that the mean quantum number interval between the planets of the outer solar system is 5. Therefore if a `renormalized' quantum number and a `renormalized' Bohr radius are defined, it seems that there is a two-stage quantization process according to the Bohr law written as for . Hence Jupiter would now rank , and then Saturn (), Uranus (), Neptune () and Pluto at (note that, again, the planet does not exist). Nottale (1993, 1996a,b, 1997) has proposed such a two-stage quantization in order to suggest a convincing explanation of the mass distribution in the solar system by a cascade mechanism for the planetesimals and their final accretion into planets. But Nottale did more. He proposed an explanation for the amazingly
good Bohr account of the main solar system parameters, which is indeed
the great mystery of the problem (in fact, the situation concerning
the `macro-quantization' of the solar system right now is not far from
what existed at the onset of the quantum description of the atomic
world, where people had Balmer et al.'s very precise formulas about
the spectral lines of, say, hydrogen, but lacked any convincing
explanation for such good fits...). By applying his `Scale Relativity'
principle to the chaotic - and hence fractal - matter flow which is
believed to have dominated the later stage of the solar system
formation (Laskar, 1989) and by borrowing some technical tools from
Nelson's Stochastic Quantum Mechanics (Nelson, 1966; Kyprianidis,
1992), Nottale was able to derive a macroscopic Schroedinger equation
for the description of the Chaotic Proto-Solar-System, hereafter
refered as the Actually, a highly non-conventional approach to planetary dynamics
that yields an equation very much like Schroedinger's one for the
stationary states of © European Southern Observatory (ESO) 1998 Online publication: August 6, 1998 |