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Kepler

Johaness Kepler readily accepted the Copernican model, but his first attempts to understand the motion of the planets were still tied to the Aristotelian idea that planets ``must'' move on spheres. Thus his first model of the solar system was based on the following reasoning: there are, he argued, six planets (Uranus, Neptune and Pluto would not be discovered for almost 300 years) which move on the surfaces of spheres. There are also five perfect geometric figures, the Platonic solids: cube, tetrahedron, octahedron, icosahedron and dodecahedron. Then, he argued that the relative sizes of the spheres on which planets move can be obtained as follows (see Fig. 3.6)


 

Figure 3.6: Illustration of Kepler's geometrical model of the solar system  
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Therefore the ordering is octahedron, icosahedron, dodecahedron, tetrahedron, cube (8-faces, 20-faces, 12-faces, 4-faces, 6-faces). He spent 20 years trying to make this model work...and failed: the data would just not agree with the model. Hard as this was, he dropped this line of investigation. This work, however, was of some use: he was recognized as ``someone'' and, in 1600, was hired by Tycho Brahe (then in Prague) as an assistant (at miserly wages). Tycho was very reluctant to share his data with Kepler (who was also made fun for being provincial); Tycho died in 1601 and the king appointed Kepler as successor (at a much smaller salary which was irregularly paid).

For many years thereafter Kepler studied Tycho's data using the heliocentric model as a hypothesis. In 1609 he determined that Mars does not move in a circle but in an ellipse with the sun in one of the foci and that in so moving it sweeps equal areas in equal times. This later blossomed into his first and second laws of planetary motion. Ten years after he discovered his third law: the cube of the average distance of a planet to the sun is proportional to the square of its period. All this was very important: Tycho's data, thanks to Kepler's persistence and genius, finally disproved the epicyclic theory and, on top of this, the idea that planets must move in circles.

This is a good example of the evolution of a scientific theory (see Sect. 1.2.1). The data required Kepler to modify the original hypothesis (planets move in circles with the sun at the center) to a new hypothesis (planets move in ellipses with a sun at one focus). He showed that this was the case for Mars, and then checked whether it was also true for the other planets (it was).




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The three laws obtained by Kepler are

1. Planets move in ellipses with the sun at one focus; see Fig. 3.7.
  
Figure 3.7: How to draw and ellipse (left) and the elliptical orbit of planets (right)
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2. Planets sweep equal areas in equal times in their motion around the sun; see Fig. 3.8.


  

Figure 3.8: The planet in a given time moves from a; to b sometime later it reaches c and it takes the same time to go from c to d. Kepler's second law states that the shaded areas are equal
\begin{figure} \centerline{\vbox to 1.7 truein{\epsfysize=5.17 truein\epsfbox[0 -100 612 692]{3.renais/keplerlaw2c.ps}}}\end{figure}


3. The average distance to the sun cubed is proportional to the period squared; see Table 3.1 for the data which led Kepler to this conclusion.

The first two laws describe the motion of single planets, the third law relates the properties of the orbits of different planets.

Kepler did not know why planets behaved in this way. It was only about 50 years later that Newton explained these laws in terms of his universal law of gravitation. In modern language these results imply the following (discovered by Newton): the planets move the way they do because they experience a force from the sun, this force is directed along the line from the planet to the sun, it is attractive and decreases as the square of the distance.


 

Table 3.1: Period and average distancs for the innermost five planets, a plot of the last two columns gives a straight line as claimed by Kepler's third law. 
Planet Period (years) Avg. dist. (AU) Period 2 Dist3
Mercury 0.24 0.39 0.06 0.06
Venus 0.62 0.72 0.39 0.37
Earth 1.00 1.00 1.00 1.00
Mars 1.88 1.52 3.53 3.51
Jupiter 11.9 5.20 142 141
Saturn 29.5 9.54 870 868



next up previous contents
Next: Galileo and Newton Up: From the Middle Ages Previous: Aristotle in the 16th

Jose Wudka
9/24/1998