J.M. Souriau's work on the solar system.
...This work was presented by J.M. Souriau in 1989 at a scientific conference on gravitation held in Geneva, Switzerland. The title of the paper was:
Resonant and non-resonant phenomena in the solar system
...Souriau begins with an analysis of the orbital periods of the different planets. The Earth orbits the Sun in 365 days. The duration of the Venusian year is 225 days. From these two numbers, Souriau constructs a Fibonacci sequence (in which each term is the sum of the two preceding terms). We know that the ratio of successive terms tends toward the golden ratio. He compares these values to the orbital periods.
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)
...He then studies resonances between pairs of planets. Mathematicians (Liouville, Hurwitz, Borel) have developed a mathematical test, a "measure of the degree of irrationality of a given number," indicating how "far" it deviates from a rational fraction, i.e., the ratio of two integers.
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Borel introduces the number:
q ( x , q) = (denominator)² * | x - q |
q(x)
if the lower limit, when q takes rational values.
q tends to be zero if x is close to a rational number. We obtain a curve showing the measure of irrationality q(x) of a given number x. Among all possible values, two numbers are the most irrational:
-
The golden ratio:
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-
and its square: w² = 1 - w = 0.3820...
as can be seen in the following diagram.
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Fig. 1: Diagram q(x) showing its two characteristic peaks corresponding to the golden ratio and its square.
This function q(x), which has nothing to do with any observational data, is a purely mathematical object. The visible gaps correspond to rational fractions (q = 0).
Next: the orbital periods, where the unit is the Earth year.
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
Calculate the ratio of the orbital periods of Neptune and Pluto.
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...If one calculates the ratio of two successive periods, one observes that these ratios lie between 1/3 and 2/3. Five ratios lie between 0.35 and 0.40. The Neptune-Pluto pair is a resonant one.
Souriau applies the aforementioned test to pairs of planets.
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
...We see that two distant planets, Neptune and Pluto, exhibit an exceptionally strong resonance. Souriau decides to neglect this peculiar pair of planets in the following analysis, based on Fourier analysis of the periods Pj:
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In the next figure, IF(a)I⁴ is plotted.
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Figure 2: Function F(a)
...Souriau finds two significant peaks for values 0.615 and 0.380, fitting very well with the q(x) curve of Figure 1. See Figure 3:
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Figure 3.
...He concludes that, as a whole, the solar system is a non-resonant, or weakly resonant, system. He performs an inverse Fourier transform (reciprocal) in order to construct the probable values of the orbital periods. The reciprocal Fourier transform
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can be constructed from selected lines ak. He selects the two peculiar lines:
a1 = w
a2 = w²
He then obtains the following results. The true values of the orbital periods are indicated.
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Figure 4: Probable period P for planets, based on a spectrum limited to the two peculiar lines w and w²
