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Astron. Astrophys. 337, 299-310 (1998)

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1. Introduction

The 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 (QM ) to orbital systems in astronomy (Corliss, 1986). About fifty years ago, or even more, the following ranking of the planets of the solar system according to Bohr's quantum number N and its related law [FORMULA] became known with a weak percentage deviation for the observed mean distances [FORMULA] from sun if adopting the value [FORMULA] for the Bohr radius [FORMULA] (Caswell, 1929; Penniston, 1930; Barnothy, 1946): Mercury ([FORMULA]), Venus ([FORMULA]), Earth ([FORMULA]), Mars ([FORMULA]), Planetoids ([FORMULA]), Jupiter ([FORMULA]), Saturn ([FORMULA]), Uranus ([FORMULA]), Neptune ([FORMULA]). Bagby (1979) added the planet Pluto at [FORMULA]. Then from Kepler's third law and the distance relation, the planet velocities were also expected to be quantized in [FORMULA], and indeed it was found that they were in inverse proportion to simple integral numbers, while the periods of the planets were proportional to the cubes of the same integers (Malisoff, 1929). Moreover it was suggested that the innermost hypothetical planet Vulcanus expected to orbit at the distance [FORMULA] from the sun, between the sun and Mercury, should rank [FORMULA] (Barnothy, 1946). Bagby (1979), in his extensive comparison of the Titius-Bode law with the Bohr atomic orbitals, even suggested that the Earth's moon might have been this intramercurial planet, since the capture possibility for the Moon was envisaged (Gerstenkorn, 1970).

Once the so-called `inner solar system' (telluric planets) was quite reasonably given the increasing series of `quantum Bohr numbers' [FORMULA], which raises the problem of the existence of Vulcanus at [FORMULA], two questions remain unanswered: i) does anything correspond to [FORMULA] 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 [FORMULA] and a `renormalized' Bohr radius [FORMULA] are defined, it seems that there is a two-stage quantization process according to the Bohr law written as [FORMULA] for [FORMULA]. Hence Jupiter would now rank [FORMULA], and then Saturn ([FORMULA]), Uranus ([FORMULA]), Neptune ([FORMULA]) and Pluto at [FORMULA] (note that, again, the [FORMULA] 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 CPSS (Nottale 1993, 1996a,b, 1997). Therefore a convincing link was proposed for the first time between the CPSS and the Schroedinger equation, although the main scaling parameters of this latter (actually the Bohr radius [FORMULA]) had of course to be adapted to its macroscopic dimensions (hence the terminology macro-quantization used in the present paper).

Actually, a highly non-conventional approach to planetary dynamics that yields an equation very much like Schroedinger's one for the stationary states of n particles, with the same basic interpretation, was proposed as early as 1944 by Liebowitz. The corresponding theory drastically changes certain standard quantum mechanical ideas by constructing a "probability of presence" from first mechanical principles (Liebowitz, 1944). It is therefore amazing that one seems to get a much larger domain of relevance of the Schroedinger equation with respect to its reference atomic world as soon as one envisages new foundation principles for the corresponding quantum theory that remains otherwise (i.e. in all its full further technical developments) unchanged. Could it be a complete round-turn with respect to the early quantum theory where people tried to fit the classical laws of celestial mechanics at atomic scales? Could the Schroedinger equation actually yield some underlying physical principles to (part of) Classical Mechanics (CM) itself, in particular when chaos dominates and classical determinism disappears? We note with great interest the discovery, in the quite recent literature devoted to these questions, of deterministic chaos in the frame of the causal interpretation of QM (Parmenter & Valentine, 1995; Konkel & Makowski, 1998). Delivering definite answers to these fundamental questions is far too ambitious at the present state of the art, but the present paper intends to participate in this stimulating debate by pointing out indeed the existence of a macroscopic context for Schroedinger's equation.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998