**Francis Berthomieu**

**"EAAE Summerschools" Working Group**

**CLEA, France**

# 1 - Before Kepler

**Geocentric models**

Obviously, the first models of the solar system were geocentric. As the perfection of the universe was seen in its spherical shape, Plato assumed that all the celestials bodies moved with uniform movement along circles. His pupil Aristotle constructed such geometric model of the solar system. Yet, as they developped better instruments and methods, the astronomers had to admit that this model did not describe the reality very well.

Hipparchus added the geometric devices of eccentrics, epicycles and deferents to explain some strange anomalies: let's see a simple example explaining the retrograde motion of Mars!

The planet is attached to a small circle (epicycle) whose center rides on a larger circle (deferent). The earth lies in the center of the deferent... The odd thing about it is that the period of the movement on the epicycle is the same as that of the Sun: one year exactly...

Ptolemy improved this model to explain the nonuniform movements of the celestial bodies, adding a lot of new devices. However his very complex system could predict planetary motion quite accurately.

**Heliocentric models**

Long before Copernicus, the greek astronomer Aristarchus proposed a heliocentric model of the cosmos. But this model contradicted the physical ideas of that time and was forgotten for centuries until Copernicus revive it.

What does the intellectual revolution initiated by Copernicus improve?

Copernicus stayed with the idea that the celestial motions must be uniform and circular. And after many years of hard work, the new model did not give better predictions than the Ptolemaic one ... and included the same complex devices than its rival.

Actually the Copernician model has better aesthetics than the geocentric one but both models are cinematically equivalent.

The "PTOLEMEE" application of the software shows convincingly this equivalence.

**Tycho Brahé's model**

As it is covenient in his epoch, Tycho Brahe is prudent and diplomatic.

He conceives a mixed system : the earth still lies in the center of the universe. The sun moves around it. The planets move around the sun. This elegant solution explains the strange equality of the sun and epicyclic periods but does not improve the predictions: it is a third form of the same cinematical model.

Definitively, none of these three models matches with the observed reality and each improvement of its accuracy comes with new complicated elements.

# 2 - Kepler and the orbit of Mars

**The method**

Kepler is one of the young Tycho Brahe's assitants and gets the challenge to solve the inextricable problem of Mars motion: How to know exactly its trajectory?

Kepler already admits the Copernican model and shapes it into a physical one, including an interaction between the sun and the planets. And he has the numberless Brahe's measurements at his command.

So, he knows the periodicity of the retrograde motion of Mars: There are 780 days between two "oppositions" of the red planet... A short calculation allows us to deduce its sideral period P_{0}.

You'll find that P_{0} = 687 days : This is the time planet Mars lasts to do its revolution around the sun and come back to the same point of its orbit.

And Kepler is a lucky man : in the crowd of the measurements he can find 5 couple of observations done with a 687 days interval!

You'll find here the dates of these observations, the geocentric ecliptic longitude of the sun LS (allowing us to calculate the heliocentric ecliptic longitude of the earth LT) , and the geocentric ecliptic longitude of Mars LM, as observed by Tycho Brahe.

Kepler draws a circular representation of the earth orbit (that is quite correct...), and locate on it the position of the earth at each date. Then he can draw in the plane of the sheet (that of the ecliptic), the line that aims at Mars.

Aiming at Mars on two days (with a 687 days interval), the position of the earth is not the same... but the two lines meets on one point of the orbit of Mars.

Another "easy to use" application of the software ("RETROGRADATION") will allow you to do these constructions in a few minutes.

Dates | LS | LT | LM |

17/02/1585 | 339°23' | 159°23' | 135°12' |

05/01/1587 | 295°21' | 115°21' | 182°08' |

10/03/1585 | 359°41' | 179°41' | 131°48' |

26/01/1587 | 316°06' | 136°06' | 184°42' |

28/03/1587 | 16°50' | 196°50' | 168°12' |

12/02/1589 | 333°42' | 153°42' | 218°48' |

19/09/1591 | 185°47' | 5°47' | 284°18' |

06/08/1593 | 143°26' | 323°26' | 346°56' |

07/12/1593 | 265°53' | 85°53' | 3°04' |

25/10/1595 | 221°42' | 41°42' | 49°42' |

**Some strange results**

As Kepler did, you will discover that these five points of the orbit belong to a circle... whose center is far from the sun! Such a mystic spirit as that of Kepler cannot conceive it: the sun MUST have a special position with a mathematical meaning.

Kepler searches... and find a figure that matches with his quest: ellipses have two foci! One of them can receive the sun...

Figure 5 shows the basic properties of ellipses: the line through the foci to both sides of the ellipse is called major axis. At its closest to the sun, the planet is at its perihelion point. When at its greatest distance the planet is at its aphelion point. The eccentricity of the ellipse indicates its "flatness".

On this copy you will find the aphelion and perihelion points of Mars and calculate the values of the parameters that give the complete definition of the ellipse : the perihel distance (*q*) and the semimajor axis (*a*). Then you will deduce easily the value of the eccentricity (*e*).

**Kepler's Laws**

1 - The orbit of each planet is an ellipse, with the sun at one focus.

2 - As a planet moves around the sun a line drawn from the sun to a planet sweeps out equal areas in equal time.

3 - Larger the orbit and slower the movement : The orbital period (*P*) and the semimajor axis (*a*) fulfill formula:

*P ^{2}/a^{3} = k*

where *k* is a constant that caracterize the solar system.

# 3 - From Kepler to... Hale Bopp

**Law of ellipse**

**How to construct an ellipse?**

You can use two methods: one of them is used by gardeners, the other one uses modern technology... and mathematics.

- The gardener's method

It is well known but rarely used by our pupils: two pins and a string are usefull! You can also see what happens with the software "methode du jardinier"...

- A computer allows some easy construction using the polar coordinates : even young pupils can use the "ellipse" application and experiment the geometrical implication of each parameter.

**The Hale Bopp orbit according to NASA**

Generaly, the orbital parameters of a comet get computed through the measurements of its positions in the sky. They can be downloaded from the NASA server on the INTERNET. For Hale Bopp you can find:

*q* = 0,9141189 AU

*e* = 0,9950842

These values are enough to draw the trajectory.

It is some more difficult to know at what time the comet passed on a special point of the orbit. The NASA ephemeris indicate for each day the distance from the sun to the comet. So we'll locate correctly the comet on its orbit for some dates choosen between 01/03/1997 and 30/06/1997.

**Law of Equal Area**

You can use the former construction to define and paint areas corresponding to equal times: for example 30 days. Then it will be possible to compare the areas by several means (cutting paper with scissors and comparing their shapes, measuring areas with millimetric grid, or... weighing).

**Harmonic Law**

As you know the parameters of the comet orbit, you can calculate its semimajor axis ac . We know the datas for the Earth:

*a* = 1 AU and *P* = 1 year

So *a ^{3}/P^{2}* = 1 AU

^{3}year

^{-2}

Using the same units, please, deduce the period *P _{c}* of the comet...

On what year will Hale Bopp return?

dates (à 0h T.U.) | r (U.A.) |

01/03/1997 | 1,067 |

11/03/1997 | 0,989 |

21/03/1997 | 0,936 |

31/03/1997 | 0,914 |

10/04/1997 | 0,925 |

20/04/1997 | 0,974 |

30/04/1997 | 1,047 |

10/05/1997 | 1,141 |

20/05/1997 | 1,248 |

30/05/1997 | 1,364 |

09/06/1997 | 1,486 |

19/06/1997 | 1,610 |

**Cut and paste game...**

By chance, the orbit plan of comet Hale Bopp is quite perpendicular with that of the earth orbit. This fact permits us to make a very simple paper device that shows the respectve trajectories of the comet and the earth, from March to June 1997.

This ready-to-use "cut and paste game" shows plainly most of the astronomics concepts used in this workshop.