The solar system structured by the golden ratio. Golden Law of Souriau Evocation of the work of Jean-Marie Souriau
on the dynamics of the solar system.
...This work was presented by the author at a conference held at the Geneva Observatory, in 1989, whose theme was:
"Resonances and non-resonances in the solar system"
...The starting point of Souriau is the analysis of the orbital periods of the different planets. He then takes that of the Earth: 365 days and that of Venus: 225 days and calculates, both downstream and upstream, the corresponding Fibonacci sequence (or of the Fibonacci type, where each term is the sum of the two preceding ones). It is known that under these conditions the ratio of two consecutive numbers in this sequence tends towards the golden ratio.
...Souriau then obtains this:
30 Sun (29 days)
55 Nothing
85 Mercury (88 days)
140 Nothing
225 Venus
365 Earth
590 (1 year and 7 months) Mars (1 year and 10 months)
955 Nothing
1545 (4 years and 3 months) Ceres-Pallas (asteroid belt)
2500 Nothing
4045 (11 years) Jupiter (11 years and 10 months)
6545 Nothing
10590 (29 years) Saturn (29 years and 5 months)
17135 Nothing
27725 (76 years) Uranus (84 years)
44860 Nothing
72585 (199 years) Neptune (165 years), Pluto (248 years)
...A rather surprising coincidence, let's agree. Souriau then studies the resonances between the planets. To do this, it is necessary to have a test that measures if the ratio x of two periods, between zero and 1, is "close" to an irreducible fraction:
...This test has long been developed by mathematicians (Liouville, Hurwitz, Borel, etc). It is the number:
q ( x , q) = (denominator)2 x I x - q I
...By denoting by q(x) its lower bound when q describes the set of rational numbers, q is zero if x is rational, small if x is close to a rational; it therefore measures the irrationality of x. The numbers "most irrational" are then the golden ratio:
and its square: w2 = 1 - w = 0.3820...
One can see this on the diagram giving the function q
**Fig.1: Diagram of the function **q with its two peaks, corresponding to the numbers the "least resonant": **the golden ratio and its square. **
...This function q (which has nothing to do with observational data) is a pure "mathematical object", a property arising from the sequence of real numbers. This continuous sequence then secretes this strange spectrum, filled with sort of gaps (where there are ratios of integers, rational numbers, where q = 0).
...Below are the orbital periods of the main planets in the solar system, in years:
Mercury: 0.2408425
Venus: 0.6151866
Earth: 1.0000000
Mars: 1.8808155
Ceres-Pallas: 4.604
Jupiter: 11.86178
Saturn: 29.45665
Uranus: 84.0189
Neptune: 164.765
Pluto: 247.68
Note that the ratio between the periods of Pluto and Neptune is:
...The ratio of one of these terms to the next remains between 1/3 and 2/3. Five of these nine ratios are between 0.35 and 0.40. Souriau then undertakes to study the ratios between the periods of different planets. Two planets in perfect resonance would lead to a ratio of their periods that would be a rational number, the quotient of two integers.
...Souriau decides to analyze the different resonances in the current solar system. To do this, he takes the ratios of the rotation periods of the main planets, two by two, and applies the test mentioned above.
...A simple calculation allows him to establish a list of resonances between large planets (Ceres and Pallas are the largest of the "small planets" and their periods differ by only 3 days and are located in the asteroid belt), whose q test is less than 0.1 (denominator ? 6). :
Neptune-Pluto: x = 2/3 x 0.9980 q = 0.01
Uranus-Neptune: x = 1/2 x 1.0199 q = 0.04
Uranus-Pluto: x = 1/3 x 1.0176 q = 0.05
Venus-Mars: x = 1/3 x 0.9812 q = 0.06
Jupiter-Saturn: x = 2/5 x 1.0067 q = 0.07
...This table shows that the two most distant planets, Neptune and Pluto, show particularly marked resonances. They therefore form a "special" couple compared to the others and Souriau decides to ignore them in the following analysis, performing a Fourier analysis of the periods:
...Pj being the periods of the planets, from Mercury to Uranus. The successive ratios of the periods are between 1/3 and 2/3. The following figure suggests the shape of the curve IF(a)I for a varying between 1/3 and 2/3. For better readability, Souriau has plotted IF(a)I4 on the graph.
Figure 2: Function F(a)
...Two significant peaks appear for the values 0.615 and 0.380, in precise coincidence with the peaks of Figure 1 (w = 0.618 and w2 = 0.380). Souriau then superposes this spectrum with the function q:
Figure 3.
and concludes to a global effect of non-resonance, except for the resonant couple Neptune-Pluto. The phase shift of F between the two peaks can be interpreted by the inverse Fourier transform: starting from a certain number of lines ak selected in the spectrum F, one constructs the function F:
...The values of Pj are then close to certain maxima of the real part of F. Souriau then limits this spectrum to the two lines a1 = w and a2 = w2 and obtains the curve of the following figure, where the real periods of the planets are also indicated.
Figure 4: Probable positions P of the planets based on a spectrum constructed from the two lines w and w2
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