6. Schroedinger equation on the scale of the solar system
where defines the gravitational potential given by Eq. (46), is the mass of the sun and is the gravitational constant. Eq. (60) yields the following mass-independent Bohr radius for the solar system:
Equivalently, one may consider in the above theory of the hydrogen atom a formal mass-dependent CPSS "quantum of action", namely:
in accordance with Eqs. (53) and (60). As a consequence, for , the radial-action quantization given by Eq. (58) yields:
regardless of the energy level N. Therefore the radial-velocity planetary solitons have all the same norm :
6.1. Radial-velocity solitions
When considering the Kepler system described by Eqs. (46-51) for , we have seen in Sect. 5.2 that the abscissa that defines the initial conditions given by Eqs. (40-43) is defined by Eq. (57). Since Eqs. (37), (47-49) and (54) yield, together with , the following classical wavefunction (which is singular, as expected, at the classical turning points ):
the initial amplitude of both the regular Schroedinger eigenfunction and the nonlinear mode at the initial abcissa is:
Now we note an interesting symmetry of both Eqs. (21) and (55), namely multiplying the solution by amounts to dividing the Wronskian A by . In particular, choosing reduces the ODE problem defined by Eqs. (40), (55) with , (57) for the choice of the initial abscissa and (66) to the following one:
The radial-velocity soliton field that is defined by Eqs. (22), (30), (56) and (60) and that corresponds to the nonlinear mode defined by Eqs. (67-68) reads:
where is the Kepler-Bohr velocity:
simply resulting from the combination of Kepler's third law for the revolution period at the energy level (56) and Bohr's quantization formula given by Eq. (57) (Malisoff, 1929; Nottale, 1997). Indeed:
The radial velocity defined by Eq. (69) must be complemented by the orbital velocity field that is defined by Eq. (32). By use of Eqs. (57) and (62), this latter field becomes at :
for and . Eqs. (69) and (72) completely define the quantized velocity field in terms of its radial and orbital components. The case is singular since it corresponds to a purely radial motion. We conclude that it does not yield any orbiting planetary motion in the CPSS.
6.2. Quantized patterns
Fig. 1 shows the radial nonlinear mode that is solution to Eq. (67) and corresponds to (Mercury), and to the initial conditions defined by Eqs. (68) at (cross). The classical counterpart defined by Eq. (65) (with ) is displayed in bold line. As expected, it diverges at the classical turning points . Note that the corresponding eccentricity agrees to an order of magnitude with Eq. (59). Clearly, the nonlinear mode is both quite flat and quite close to the classical state within the classically allowed region.
Fig. 2 shows, in units of the Kepler-Bohr velocity , the corresponding radial-velocity static soliton field that is defined by Eq. (69), together with its classical counterpart defined by Eqs. (38-39) and (65) and displayed in bold line. Figs. 3 and 4 display the same plots for (Jupiter). There is clearly a `quantum dressing' about the classical radial velocity field that is localized between the classical turning points . It mostly accounts for the tunnel effect which extends fairly beyond the classical turning points and which is related to the divergence of the radial nonlinear mode about the boundaries and of the system.
6.3. Conjecture about planetary orbit quantization
Because the present macro-quantum theory is, like the stationary Schroedinger equation itself, conservative and reversible, we believe that it is not able to account for the final stage of planet accretion in the CPSS. Indeed, the process by which the static radial-velocity solitons would concentre into singularity-planets seems to be highly irreversible. Therefore, in the frame of the present theory (i.e. without making use of the standard Born postulate related to ), the Bohr formula (57) for the planet accretion could tentatively be explained by assuming that the radial CPSS matter flow, which is by the definition (40) fairly uniform about (see Figs. 2 and 4), ultimately concentres through irreversible processes about the center-part of the classically allowed region, namely . Fig. 5 displays the locations of the nine planets of the solar system divided by (crosses = semimajor axis: McGraw-Hill, 1983), compared with their respective corresponding prediction, namely (points). The average of the absolute error that this formula yields with respect to the nine planet semimajor-axis locations is 3 and its standard deviation is 11, while the sequence of relative errors between and the nine planet locations is: , , , , , , , and .
Let us recall as a final result that the case is singular (cf. Eq. (72)). Therefore the orbital quantization of the CPSS begins at , possibly corresponding to the hypothetical intramercurial planet `Vulcanus'. No orbiting pattern is expected at .
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998