a702 J.M. Souriau's work on the solar system. ** **
...This work was presented by J.M. Souriau in 1989, during a scientific meeting devoted to gravitation, held in Genoa, Switzerland. The title of the paper was: Resonant and non-resonant phenomena in the solar system
...Souriau starts from the 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 series (where each term is the sum of the two previous terms). We know that the ratio of successive terms tends toward the golden number. 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)
...Then he studies resonances in pairs of planets. Mathematicians (Liouville, Hurwitz, Borel) have established a mathematical test, a "measure of the level of irrationality of a given number", indicating "how far" it is from a rational fraction, from the ratio of two integers. (a701)
Borel introduces the number: q (x, q) = (denominator)² * |x - q|
q(x) is the lower limit, when q takes rational values.
q tends to 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 number: (a702)
- and its square: w² = 1 - w = 0.3820...
as can be seen on the following diagram. (a703)
Fig.1: Diagram q(x) showing its two characteristic peaks corresponding to the golden number and to its square.
This function q(x), which has nothing to do with any observed data, is a purely mathematical object. The visible gaps correspond to rational fractions (q = 0).
Next: the orbital periods, the unit being 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. (a704)
...If one calculates the ratio of two successive periods, one sees that these ratios lie between 1/3 and 2/3. Five ratios lie between 0.35 and 0.40. The Neptune-Pluto pair is therefore resonant.
Souriau applies the test mentioned above to pairs of planets.
Neptune-Pluto: x = 2/3 × 0.9980 q = 0.01
Uranus-Neptune: x = 1/2 × 1.0199 q = 0.04
Uranus-Pluto: x = 1/3 × 1.0176 q = 0.05
Venus-Mars: x = 1/3 × 0.9812 q = 0.06
Jupiter-Saturn: x = 2/5 × 1.0067 q = 0.07
...We see that two distant planets, Neptune and Pluto, have an exceptionally strong resonance. Souriau decides to neglect this particular pair in the following analysis, based on a Fourier analysis of the periods Pj: (a705)
On the following figure, |F(a)|⁴ is plotted. (a706)
Figure 2: Function F(a)
...Souriau finds two significant peaks for the values 0.615 and 0.380, which fit very well with the q(x) curve of Figure 1. See Figure 3. : (a707)
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 (a708)
can be constructed from selected lines ak. He selects the two particular lines: a₁ = w a₂ = w²
He then obtains the following results. The actual values of the orbital periods are indicated. (a709)
Figure 4: Probable period P for the planets, based on a spectrum limited to the two particular lines w and w²