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
upstream and downstream, 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 successive 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 seven
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)
Quite an astonishing coincidence, let's agree. Souriau then studies the resonances between the planets. For this, it is necessary to have a test measuring if the ratio x of two periods, between zero and one, 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
IBy denoting
q
(x) its lower bound
when
q
ranges over the set of rationals,
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:
w 2
= 1 -
w
= 0,3820...
This can be seen on the diagram showing 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 the
observation 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 of 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 begins
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
solar system, as it is now. 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 test
q
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 farthest 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 analysis
that follows, by 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
on the graph IF(a)I
4
.
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
w 2
= 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 pair
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 =
w 2
and obtains the curve of the following figure, where are also
marked the real periods of the planets.
Figure 4: Probable positions P
of the planets
based on a spectrum constructed
from the two lines
w
and
w 2