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Astron. Astrophys. 361, 379-387 (2000)

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7. Discussion

7.1. Theory

Concerning the theory, we shall briefly discuss three points. A more complete discussion will be published elsewhere (Nottale in preparation).

(i) Let us first compare our approach with the standard theory of gravitational structure formation and evolution. We write instead of the Euler-Newton equation and of the continuity equation a unique, complex, generalized Euler-Newton equation that can be integrated in terms of a Schrödinger equation, completed by the Poisson equation. The square of the modulus of the Schrödinger equation then gives the probability density [FORMULA]. Now, when the `orbitals', which are solutions of the motion equation, can be considered as filled with the particles (e.g., planetesimals in the case of planetary systems formation), their mass density is proportional to the probability density. By separating the real and imaginary parts of the Schrödinger equation we get respectively a generalized Euler-Newton equation and the continuity equation (which is therefore now part of the dynamics), so that our system becomes:




Then this system of equations is equivalent to the classical one, except for the introduction of an extra potential term Q that writes:


In this case the new potential term (the "Bohm potential") is a function of the density of matter, as the usual Newton potential.

We recover the standard theory in the limit [FORMULA]. As a consequence, from the viewpoint of the application of the present approach to the formation and evolution of gravitational systems, the main question becomes to demonstrate from observational data that the parameter [FORMULA] does exist and is non vanishing, then to determine its form and its values in the various physical configurations. This is one of the goal of the present paper and of forthcoming works.

Conversely, the existence of this additional term demonstrates that there is indeed new physics here, i.e. that one cannot obtain such a system of equations from a simple extrapolation of the standard approach.

(ii) This remark leads us to briefly discuss other related approaches which also suggested the use of a quantum-like description for planetary systems and for other gravitational structures.

The suggestion to use the formalism of quantum mechanics for the treatment of macroscopic problems, in particular for understanding structures in the solar system, dates back to the beginnings of the quantum theory (see e.g. Jehle 1938; Liebowitz 1944; and Reinisch 1998for additional references). However these early attempts suffered from the lack of a convincing justification of the use of a quantum-like formalism and were hardly generalizable (e.g., the Schrödinger-like equation derived by Liebowitz applied only to stationary states of n particles and allowed only for a real (non-complex) wave function). Anyway these works clearly anticipated the recent understanding of the universality of the Schrödinger equation.

More recently, several works attempted to develop a macroscopic quantum theory under the motivation of describing the various Tifft effects of redshift quantization (Greenberger 1983; Dersarkissian 1984; Carvalho 1985; Cocke 1985). The problem with these attempts is that they did not allow an understanding of the origin of the observed quantization and of its meaning. Recall indeed that our interpretation of these effects is different from theirs. They were interpreted by Tifft as an occurrence of anomalous, precisely quantized non-Doppler redshifts, while in the present approach we interpret them as peaks of the probability distribution of standard velocity redshifts. The present result that the solar system planets and exoplanets (for which we are certain that we deal with true velocities) show the same structures supports our interpretation.

A third, better theoretically motivated, approach, has been the suggestion to use Nelson's stochastic mechanics (Nelson 1966) as a description of the diffusion process in the protosolar nebula (Albeverio et al. 1983; Blanchard 1984). The problem with such a suggestion is that Nelson's twin diffusion process is not a standard diffusion process. Indeed, in standard diffusion one can establish a forward Kolmogorov equation (also called the Fokker-Planck equation), then a backward Kolmogorov equation in which the average velocity is the same as in the forward one (see e.g. Welsh 1970). On the contrary, Nelson's theory introduces, in addition to the forward Kolmogorov equation (which he calls forward Fokker-Planck equation), a "backward Fokker-Planck equation" in which the average backward velocity is different from the forward one. This backward Fokker-Planck equation is therefore incompatible with the backward Kolmogorov equation of standard diffusion. Moreover, the form of the mean acceleration in stochastics mechanics must be arbitrarily postulated. Fianally, a justification is lacking for the application of such a non standard diffusion process to macroscopic systems.

Recall that in our own present attempt, these problems are overcome since (i) the two-valuedness of the mean velocity vector is explained as a consequence of the [FORMULA] symmetry breaking (local irreversibility); (ii) we use neither Fokker-Planck equations nor a diffusion description; (iii) our basic equation, written in terms of the new "covariant" time derivative operator, keeps the standard form of the equation of dynamics, and it now includes the continuity equation.

(iii) A last point that we shall briefly discuss here is Reinisch (1998)'s proposal according to which the [FORMULA] mode should correspond to purely radial motion and then be singular and therefore forbidden. He suggests to use non linear modes that give a distance law [FORMULA]. These solutions, when fitted to the inner solar system, yield [FORMULA] AU, and then give probability peaks at 0.14, 0.38, 0.71, 1.14 and 1.67 AU. The agreement is bad for the Earth (1 UA) and Mars (1.52 UA), but, more importantly, the theoretical prediction of this model for extrasolar planets at intramercurial distances is one unique peak at [FORMULA]. This is clearly rejected by the observed distribution of exoplanets close to their stars (see Table 1 and Fig. 1 and Fig. 4), in particular by the well-defined peak at [FORMULA]. The problem raised by Reinisch is that the fundamental mode is characterized by a secondary quantum number [FORMULA] for [FORMULA], and should therefore correspond to zero angular momentum. However, one should not forget that this behaviour is obtained from a highly simplified treatment of planetary formation, namely, pure Kepler two-body problem with spherical symmetry. This is clearly an oversimplication, since a more complete approach should also account for flattening, self-gravitation of the disk and other effects. The work aiming at developing such a more realistic treatment is now in progress. Anyway, the above question is already simply solved by jumping to a 2-D Schrödinger equation, in order to describe the high flattening of the initial dust disk: we indeed find in this case that the radial modes are suppressed, since the secondary quantum number now varies from 1/2 to [FORMULA] (Nottale in preparation). Moreover, in all cases studied up to now, the [FORMULA] orbital happens to be sub-structured.

7.2. Error analysis

The number of exoplanets ([FORMULA]) now known allows one to perform an analysis of the distribution of errors. As already remarked, the main source of error is about the parent star mass. The relative error is [FORMULA], so that the uncertainty on [FORMULA], which is proportional to [FORMULA], is [FORMULA].

Then we can compute the expected number of exoplanets for which [FORMULA] (the difference between [FORMULA] and the nearest integer) fall outside the interval [-0.25, +025] due to the error on mass alone. By using a Student law corresponding to the observed number of exoplanets in each "orbital", we find a total number of 4.1 exoplanets expected to be discrepant, to be compared with the observed number of 9 among 35 exoplanets.

This indicates that, as expected from our model, there is an intrinsic dispersion around the "quantized" values [FORMULA] of the semi-major axes. This dispersion is expected to be non zero, but smaller than that of the calculated orbitals, given by generalized Laguerre polynomials. Indeed, we interpret these orbitals as describing the density distribution of planetesimals in the initial disk. The [FORMULA] values give their density peaks, so that, after the accretion process, the planet is expected to lie with higher probability at about this distance from its star. A theoretical estimate of the expected dispersion of the planet positions around the peak values remains to be done. It should depend on the various conditions and perturbations that intervene during the accretion process, as indicated, e.g., by the fact that the observed dispersion for the three planets around the pulsar PSR B1257+12 (Nottale 1996b) is far smaller that for the planets of our Solar System.

The inner Solar System, for which the planet positions are known with precision, and the exoplanet [FORMULA] orbital, for which the contribution of the error on the star mass remains small, allow us to obtain an observational estimate of the intrinsic dispersion on [FORMULA]. We get for the inner Solar System [FORMULA] (7 objects), a value which is confirmed by the [FORMULA] exoplanets (7 objects), for which [FORMULA]. We can now make again the preceding analysis by combining the intrinsic dispersion, assumed to be [FORMULA], and the error from the mass star (see Table 2). We find a total expected number of [FORMULA] discrepant exoplanets (outside the interval [-0.25,0.25]), which compares well with the observed number of 9.


Table 2. Error analysis of the semi-major axis distribution of exoplanets. The table gives: the quantum number n; the observed number of exoplanets in each orbital; the expected total dispersion on [FORMULA] obtained from a combination of the estimated intrinsic dispersion [FORMULA] and of the dispersion coming from the error on the star mass, [FORMULA]; the value of the Student t variable which corresponds to the observed number of exoplanets, for a difference [FORMULA]; the corresponding expected rate of discrepant exoplanets; the final number of expected discrepant exoplanets in each orbital.

7.3. Consequences for exoplanets

We have shown that the new exoplanets discovered since three years agree in a statistically significant way with the theoretical prediction of the scale-relativity approach. This is a confirmation of the result which was established with the firstly discovered ones (51 Peg, ups And, tau Boo, 55 Cnc, HD114762, 47 UMa, see Nottale 1996b). The existence of a candidate planet around Prox Cen, which was taken into account in (Nottale 1996b), has not been confirmed since and therefore it was excluded from the present analysis. The candidate planet around HR 7875 also remains to be confirmed.

One of the possible consequences of our result is that the quantization of planet interdistances to their stars may be used as a filter to help discovering secondary planets. For example, in the case of HD 114762, the power spectrum published by Mazeh et al. (1996) shows, in addition to the main peak at 84 days, a secondary smaller but isolated peak corresponding to a period of 22.2 days. Such a peak in a power spectrum, though statistically unsignificant when there is no theoretical prediction, may become significant provided its value was a priori predicted. In the case of HD 114762, the secondary small peak is one of the possible periods for this star predicted from the [FORMULA] law. Indeed, we find [FORMULA] AU/[FORMULA] and [FORMULA]. If confirmed with future improved data, this would yield a new object on the [FORMULA] orbital (see Fig. 1), in addition to the already discovered [FORMULA] planet around this star.

Note also that our theoretical methods may help solving some of the problems encountered in the standard models of planetary formation. In the description of the protoplanetary nebula based on the present approach, the 51 Peg-type planets could be formed in situ, precisely at the distances close to their stars where they are observed. But our generalized Schrödinger equation could also be applied as a statistical description of the chaotic motion of a planet formed at a Jupiter distance, and which would spiral in the disk toward the inner planetary system. In both cases, the same final position is predicted, in the first case as a peak of planetesimal density and in the second as a peak of probability of presence for the planet.

Anyway, it has already been remarked (e.g. Marcy et al. 1999) that in the standard paradigm of planetary formation, the distribution of semi-major axes and eccentricities of giant planets presents an unsolved puzzle. Indeed, in the inward migration scenario, circular orbits are expected while most exoplanets have non-circular orbits, and, moreover, no mechanism is known to halt the migration. Our present result according to which, when plotted in terms of [FORMULA] instead of a, this migration should be stopped in most cases at [FORMULA] AU/[FORMULA], and [FORMULA] AU/[FORMULA] is therefore still harder to understand in the standard scheme, while these values are the theoretically predicted ones in our own model.

We conclude this discussion by noting that, in a recent paper, Laskar (2000) developed a simplified model of planetary accretion that also yields a [FORMULA]-like distance law (of the form [FORMULA]) for a particular choice of the initial mass distribution. This is an interesting convergence of the standard approach with the generalized one considered here; however, the standard methods will be confronted to the difficult problems of explaining why, as well in the Solar system as for extrasolar systems, the constant [FORMULA] is zero, the ratio [FORMULA] is a universal constant, and why this [FORMULA] law holds even in the case of one single planet.

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© European Southern Observatory (ESO) 2000

Online publication: September 5, 2000