For thirty years  Tycho de Brahe (1546 - 1601) was recording positions of planets on the celestial sphere. His purpose was to prove his own geocentric model of the universe with the sun orbiting the earth and rest of the planets orbiting the sun. But after Tycho's death his assistant  Johannes Kepler (1571 - 1630) used these data to generalize the heliocentric model of   Nicolaus Copernicus (1473 - 1543) . Applying mathematical method of analysis he established that the planetary orbits are elliptical (not quite circular as it was assumed by Copernicus). He also proved that for any two planets their periods (times needed to accomplish one full revolution around the sun) T₁ and T₂  and their average distances from the sun R₁ and R₂  are fulfilling the following relation

(T₁ / T₂)^² = (R₁ / R₂)^³.

This relation is known as Kepler's third law. A half of a century later Newton derived it from his second law an law of gravitation. You may wonder how Kepler could know the distances of the planets from the sun. As a matter of fact he did not. But using Tycho's data and trigonometry he was able to find the ratios  R₁ / R₂  for some pairs of planets. Dealing with inner planet (a planet that has its orbit inside of the earth orbit) it is enough to find a maximal angular distance  between this planet and the sun.  Than from the figure above follows  R _V / R _E  =  sinß, and the ratio can be easily calculated. Practically, however, finding the angle  is not quite simple because one has to know direction toward the sun when the sun is still below the horizon. Otherwise the planet will not be visible. Nevertheless 16 century astronomers new how to make this kind of measurements.

A main purpose of this "experiment" is to show how uncomplicated would be a discovery of Kepler third law for an observer  who has a lot of time and is placed outside of our solar system. The solar system presented in the applet has seven planets orbiting their sun along circular orbits. But you can see only one planet at a given time. Distances in this planetary system are shown in Tm (1 terameter = 10 ¹²m).  To make properties of this system comparable to the properties of our solar system we will assume that one second of the real time is equivalent to a 10 ⁴ days for the planetary system of the applet.       Start the applet clicking on the clear button. Notice that besides one planet and the sun (star) you have there an arrow showing direction to another, very distant star. We need this direction as a reference to measure how long it takes for any of these planets to make a one full revolution. Notice that if you try to measure how fast you can make one round on a track, you automatically (without thinking about it) select a reference direction. Without such reference it is impossible to figure out when a one full revolution is made.  If you have any doubts make a simple experiment. Stand in the middle of your living room, close your eyes and screen them well with your hands. Make a few turns about your own vertical axis and try to stop when you face your starting direction. If by chance you have made it, try it again.       Going back to the applet experiment, select any possible radius of a planetary orbit and a single revolution. Click "go" for acceptance of your choice. Now simultaneously click "start" and start your stopwatch. Record the orbit radius in meters and period of the selected planet in make believe seconds in a table similar to the one which is placed below. Remember, a real second is treated in this "experiment" as 10,000 full days or 8.64 x 10 ⁸ make believe seconds. Calculate R^³ / T^² . Repeat all of it for three more planets selecting different radii of orbits.

Compare values of R^³/ T^{^2} for different planets. If everything was done correctly you should rediscover Kepler third law. Well, it was ease, was not it?      Now think about Brahe and Kepler. They had much more difficult task than you because measurements and calculations were made from inside of the system and in a real time.  The fact that the earth is not only revolving around the sun, but also rotating about its axis  which is not even perpendicular to the plane of the earth orbit, makes a picture of planetary motion very complicated. This is why we should admire Brahe and Kepler's experimental and mathematical skills.        Kinematics of circular motion