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

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2. Theory: a short reminder

We have demonstrated (Nottale 1993 , 1996a , 1997) that Newton's fundamental equation of dynamics can be integrated in the form of a Schrödinger-like equation under the three following hypotheses:

(i) The test-particles can follow an infinity of potential trajectories: this leads us to use a fluid-like description, [FORMULA].

(ii) The geometry of each trajectory is fractal (of dimension 2). Each elementary displacement is then described in terms of the sum, [FORMULA], of a mean, classical displacement [FORMULA] and of a fractal fluctuation [FORMULA] whose behavior satisfies the principle of scale relativity (in its simplest "Galilean" version). It is such that [FORMULA] and [FORMULA]. The existence of this fluctuation implies introducing new second order terms in the differential equations of motion.

(iii) The motion is assumed to be locally irreversible, i.e., the ([FORMULA]) reflection invariance is broken, leading to a two-valuedness of the velocity vector that we represent in terms of a complex velocity, [FORMULA].

These three effects can be combined to construct a complex time-derivative operator which writes

[EQUATION]

where the mean velocity [FORMULA] is now complex and [FORMULA] is a parameter characterizing the fractal behavior of trajectories.

Since the mean velocity is complex, the same is true of the Lagrange function, then of the generalized action [FORMULA]. Setting [FORMULA], Newton's equation of dynamics becomes [FORMULA], and can be integrated in terms of a generalized Schrödinger equation (Nottale 1993):

[EQUATION]

This equation becomes, for a Kepler potential and in the time-independent case:

[EQUATION]

Since the imaginary part of this equation is the equation of continuity, [FORMULA] can be interpreted as giving the probability density of the particle positions.

Even though it takes this Schrödinger-like form, this equation is still in essence an equation of gravitation, so that it must keep the fundamental properties it owns in Newton's and Einstein's theories. Namely, it must agree with the equivalence principle (Nottale 1996b; Greenberger 1983; Agnese & Festa 1997), i.e., it must be independant of the mass of the test-particle and GM must provide the natural length-unit of the system under consideration. As a consequence, the parameter [FORMULA] takes the form:

[EQUATION]

where w is a fundamental constant that has the dimension of a velocity.

The solutions of Eq. (3) are given by generalized Laguerre polynomials (see e.g. Nottale et al. 1997). We now assume that such a description can be applied to the distribution of planetesimals in the protoplanetary nebula. We expect them to fill these "orbitals", then to form a planet by accretion as in the standard models of planetary formation. But the new point here is that only some particular orbitals are allowed, so that the semi-major axes of the orbits of the resulting planets are quantized according to the law:

[EQUATION]

where n is an integer. In an equivalent way, using Kepler's third law that relates the semimajor axis a to the orbital period P, [FORMULA], the average velocity of the planet, [FORMULA], is expected to have a distribution peaked at [FORMULA], i.e. in Solar System units (AU, year, [FORMULA] and Earth velocity):

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: September 5, 2000
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