Jean-François COLONNA

jean-francois.colonna@polytechnique.edu

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France

france telecom, France Telecom R&D

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(CMAP28 WWW site: this page was created on 03/22/2003 and last updated on 01/28/2014 15:13:17 -CET-)

- Many centuries ago, the Earth was the center of everything.
But unfortunately, the perfect God's order was disturbed with
the retrogradations of Mars as can be seen on this
simulation of the Solar System ,
the Earth -the dark blue body- being as the origin of the coordinates.

Please, note that due to scale problems, it is impossible to visualize actual distances and sizes in the Solar System. Then, after computations using actual astronomical measures, it is the square root of the distances to the Sun that are displayed; moreover, the diameter of each celestial body is multiplied by a huge arbitrary factor.

As can be seen at the end of this simulation , each celestial body (with the exception of the Sun -the big yellow body-) seems to follow a complex trajectory that can be understood approximatively as the superposition of circular motions. - Since the work of Nicolas Copernic
(
*De revolutionibus orbium caelestium libri VI*, 1543) it is known that in fact the Sun is the center of the Solar System . - In the year 1609, Johannes Kepler (
*Astronomia Nova*) described the planet trajectories as ellipses whose one of the two focus is the Sun. - Finally, in the year 1687, Sir Isaac Newton
published his famous
*Philosophiae Naturalis Principia Mathematica*, where the laws of the classical mechanics can be found.

Then, each main body of the Solar System is subject to the following acceleration:2 ----> i=N d OC ----- M k \ i -----> --------- = / G ----------- C C 2 ----- 3 k i dt i=1 |----->| (i#k) | C C | k i

By means of a numerical integration method, the preceding system of differential equations is integrated. The trajectory of each body is then known as a set of points:-------> ----------> ----------> ----------> S = { OC (0) , OC (1.dt) , OC (2.dt) ,..., OC (n.dt) ,...} k k k k k

'dt' denoting the integration time step. This allows us to display the motion of the 9 planets using all the computed sets S(k) for the N=9+1 main bodies of the Solar System:/ -------> ----------> ----------> ----------> | S = { OC (0) , OC (1.dt) , OC (2.dt) ,..., OC (n.dt) ,...} | 1 1 1 1 1 | | -------> ----------> ----------> ----------> | S = { OC (0) , OC (1.dt) , OC (2.dt) ,..., OC (n.dt) ,...} < 2 2 2 2 2 | | (...) | | -------> ----------> ----------> ----------> | S = { OC (0) , OC (1.dt) , OC (2.dt) ,..., OC (n.dt) ,...} \ N N N N N

Instead of using a fixed point O (in the preceding lines, this point was in the close neighbourhood of the Sun) in order to visualize the trajectories, it is possible to use a moving one and in particular, the arbitrary body number R. For the body number k, the set of points to be visualized becomes:-------> -------> ----------> ----------> ----------> ----------> ----------> ----------> S' = {[ OC (0) - OC (0) ] , [ OC (1.dt) - OC (1.dt) ] , [ OC (2.dt) - OC (2.dt) ] ,..., [ OC (n.dt) - OC (n.dt) ] ,...} k k R k R k R k R

Using this simple idea will give us very new visualizations of the Solar System as exhibited in the following lines... - Now, let's try to visualize the epistemological revolution that led us from geocentrism
to heliocentrism. To do that, let's start with a 3-body system:

- the Sun (yellow),
- Mars (red),
- and a virtual planet dubbed "the Earth" (blue).

The Earth is moved from the Sun -top right- to Mars -bottom left-. Each of the displayed trajectories has a duration equals to one martian year and the Sun is the origin of the coordinates.

Now, let's display the same computation using the Earth as the origin of the coordinates .

Very interesting experiments can be conducted from there. For example, it is possible to study what happens if the plane of the Earth is chosen to be perpendicular to the one of Mars . - Now, we will use the following 11-body system:

- the Sun (yellow),
- the 9 planets,
- and a virtual planet dubbed "the Wanderer" (dark blue).

The Wanderer is moved from the Sun -top right- to Pluto -bottom left-. Each of the displayed trajectories has a duration equals to one plutonian year and belongs to the plane of Pluto and the Sun is the origin of the coordinates.

Now, let's display the same computation using the Wanderer as the origin of the coordinates .

All trajectories look non periodical and even chaotic (at the exception of the ones of Pluto and of the Sun), hence the... Do not forget that the trajectories of the Wanderer and of the Earth do not belong to the same plane (the plane of Pluto for the former and the ecliptic plane for the latter).*Virtual (or Subjective) Chaos*

Here again, very interesting experiments can be conducted from there. For example, it is possible to study what happens if the plane of the Wanderer is 2.pi rotated and to display an artistic view of this motion .

It is worth noting that the same phenomenon can be observed with with pure uniform circular motions . All that depends only on the notion of relative motion as it can be observed when the rain is falling...

**In such a non regular context, it is then highly probable that Science (and Astronomy in particular) would have followed a very different way than the one we know (and the same would apply certainly to the religions). But this chaos is only apparent; it suffices to change the point of view (using the Sun) to obtain the magnificent order we all know**.

**Thus, in certain cases, the notions of order and disorder can be relative**. - At last, let's display ten different points of view (from the Sun and the nine planets)
:

and two various point of view interpolations :

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