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It is often said that he used Tycho's data.

But the apparent motion of a planet is chaotic, right? The reason is simply that we are living on and observing from the earth, but not the sun.

So, how did he manage to recover the elliptic closed regular orbits of the planets from the original data of Tycho? Any good reference?

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The movement of planets is not "chaotic". Otherwise prediction would be impossible. The planets move according to Kepler Laws, with very good accuracy.

Strictly speaking the description of the motion as regular or chaotic depends on the time scale. On the scale of few thousand years, it is very regular, obeying the Kepler laws. Deviations from this regular motion are small. At the time of Kepler they were not observed.

In 18-19 centuries these deviations were observed, explained on the basis of Newton's laws, and permitted the discovery of new planets.

But they may accumulate with time. There was a long discussion whether they actually accumulate or not (that is whether the mathematical model of the Solar system based on the Newton laws is stable or not. This discussion is not finished).

If we look at the scale of billions years instead, it may look much more complicated. But whether it is really chaotic or not (on the scale of billions years) we do not really know, and the question has little practical meaning, because our solar system lifespan is only few billion years.

The question how Kepler derived his laws based on Brahe's observations was discussed on this site:

What data did Kepler work out his laws from?

The best source for this is Kepler's Astronomia Nova itself. There is an English translation. In it Kepler described in the great detail, how after VERY many trials he figured the correct law. Unlike ost other authors, he describes the whole process, including all his mistakes, not only the final result.

EDIT. Explaining what and how Kepler did, even very roughly, needs some background in history of astronomy, and a bit more space than is allowed here. So I link a file with a non-technical explanation with essential background:

http://www.math.purdue.edu/~eremenko/dvi/kepler.pdf

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  • $\begingroup$ By chaotic, I actually mean irregular, or simply non-closed. For example, the apparent motion of Mars show prograde and retrograde. It should never close. I guess this is what Tycho's original data is about. How can you arrive at the Kelper's laws from this kind of data? $\endgroup$ – John Dec 16 '16 at 14:24
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    $\begingroup$ This complicated motion actually decomposes into simpler ones: a periodic, almost circular motion of Earth and similar motion of Mars around the Sun. This was already achieved by Ptolemy and Copernicus. The problem was that the paths are not exactly circles, and the motion is not exactly uniform. The available data were essentially many angles Sun-Earth-Marth measured at different times. It is possible to figure from these data the shape of the two trajectories (they turned out to be ellipses) and the law of motion (equal areas law). But this required insights and a lot of computation. $\endgroup$ – Alexandre Eremenko Dec 16 '16 at 14:45

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