f703 J-M Souriau: On the dynamics of the solar system (p2).
...The planets are quite accurately positioned at the maxima of this curve, except for Neptune and Pluto. The Earth is also located near a maximum, but on an intermediate arc. Mercury, Venus, Jupiter, Saturn, Uranus and the Ceres-Pallas pair (asteroid belt) are "quite well positioned". Mars and Earth are "less well". Neptune and Pluto are... shifted.
What about Bode's law?
...The figure above immediately provides a new law, proposed by Souriau, which he calls the "Golden Law". The orbital radii then fit into a geometric progression with a ratio of:
which corresponds to the exponential (Golden Law): 1.9n
Below are the two curves: Bode's law and the Golden Law. Bode's law is:
2.4 (0.4 + 0.3 2n)
Fig. 5: Comparison of the two laws giving the orbital radii (in logarithmic coordinates)
...The Sun also follows this Golden Law (regarding its period of revolution). Indeed, its average rotational movement has adapted, just like the other movements, due to dissipative processes. Thus, one would find a justification for the weakness of the solar angular momentum compared to that of the planets, the effect being the consequence, as always, of dissipative processes, through tidal effects.
Souriau then reuses his method, applying it to the satellites of Saturn.
Fig. 6: Fourier transform result - Periods of Saturn's satellites.
...The inverse Fourier transform, filtering with these two lines, gives a sequence of probable periods for the satellites. Some are "well positioned", others "less well", we find a phenomenon similar to that affecting the orbits of the resonant pair Neptune-Pluto, which are "being explained", at the edge of the solar system.
Fig.7: Probable positioning of the periods P of Saturn's satellites based on a spectrum constructed from the two lines w and w2
...In this diagram, the Sun is also "positioned as a satellite of Saturn". The same will be true for the diagram concerning Jupiter's satellites.
...By plotting this function in regions closer to the planet, we find the rings, which fit remarkably with this other "Golden Law".
Fig.8: Positioning of the periods P of Saturn's rings based on a spectrum constructed from the two lines w and w2 For Jupiter, the situation is similar, with a more detailed spectrum.
Fig.9: Fourier transform result - Periods of Jupiter's satellites.
Some satellites then follow the new Golden Law, others do not.
Fig.10: Probable positioning of the periods P of Jupiter's satellites based on a spectrum constructed from the two lines w and w2
Note again the presence of the Sun, "as a satellite of Jupiter".
...In a later work, to be published, in a book entitled "Grammar of Nature", Souriau has combined non-resonant and resonant situations, applied to the trajectories of the planets. By reusing the spectrum from the analyses of resonances and non-resonances, he creates this time the sequence of probable positions of the planets by selecting non-resonant and resonant lines. He then manages to construct a curve where all the planets are located at the maxima (just like the satellites of Saturn and Jupiter), and concludes that the solar system, as it is, is the combination of non-resonances and resonances, like a musical note which is the combination of consonances and dissonances.
Pythagoras is not dead.
...According to Souriau, both the resonant and non-resonant subsystems are dissipative. They have their own stability and it takes energy to maintain them in this state.
...If a planet is located, with respect to the Sun, on a non-resonant orbit (Golden Law), it will continue to exchange energy with it, simply during its annual passage. A planet like Earth raises the surface of the Sun by one centimeter. One might tend to think immediately that the large planets should produce more significant tidal effects. However, these effects are in 1/r3. Therefore, the tiny Mercury has the same effect on the Sun as the Earth, Jupiter or Saturn.