Kepler's Laws

One of the 17^th century’s greatest Astronomers, Johannes Kepler published his three laws in 1609 and 1618. Based on very precise observations of Mars’ orbit made by Tycho Brahe, and developed in the Copernican system, they describe the shape of the orbit of the planets around the Sun  as an ellipse, explain how the planets move around the Sun and give a relationship between the extent of the orbit and its period.

Kepler’s first law

• The path of each planet around the Sun is an ellipse, with the Sun at one of the foci.

Kepler’s first law, purely geometric, describes the shape of the orbit as a very simple figure: the ellipse. An ellipse being a plane figure, this already constitutes a remarkable result. From Tycho Brahe’s observations, Kepler proved that planets moving in three-dimensional space follow a path included in a fixed plane. Astronomers before him only supposed this fact. Furthermore, Kepler was able to exhibit the exact shape of the trajectory as well as the role of the Sun. The Sun occupies one of the foci of the ellipse, the other focus and the center of the ellipse playing absolutely no part. According to the predominating position of the Sun, this first law confirmed the relevance of the Copernican system. Kepler’s first law, when defining the trajectory of a planet, remains, however, incomplete. The trajectory gives all the possible positions, but no time indications. The time dependence, crucial for the construction of astronomical ephemerides predicting the planetary positions, is given by the second law.

Kepler’s second law

• Each planet moves in such a way along its orbit that a line drawn from the Sun to the planet sweeps an equal area in an equal amount of time.

The second law does not introduce the time dependence in a simple way, since, except in the particular case of a circular path, the position along the orbit cannot be expressed explicitly. It was therefore necessary for Kepler to consider not the position of the planet, but the area swept by the position vector around the Sun, in order to show that this area increases linearly with time. The first step was to identify that a planet moves faster when closer to the Sun. The precise expression of this second law was carefully derived from the analysis of a very large amount of data concerning the position of Mars.

In fact, this second law establishes the possibility of astronomical prediction: it makes it possible to follow step by step the progression of a planet along its orbit. According to his first and second laws, Kepler constructed timetables for the planets, which were used for more than one century. They were much more precise than the astronomical tables based on the Ptolemaic system. For the first time, predictions concerning the declinations of the planets were made possible (actually, the Ptolemaic system was unable to predict the declinations of the planets). More than the novelty of the approach, the accuracy of the predictions ensured the success of Kepler’s laws.

Kepler’s third law

• The ratio of the cube of the semi-major axis of orbit to the square of the orbital period is the same for all planets.

This third law introduces another concept compared with the two previous ones, which are restricted to one single object around the Sun. It gives a relation valid for all the objects orbiting the Sun. Kepler discovered the relation between the semi-major axis of the orbit and the sidereal period T. The ratio T² /a³ is constant for all objects in the solar system orbiting the Sun. This relation allows us to determine the semi-major axis of the orbit, knowing its period, or vice versa.

Questions to Ponder

• Can Kepler’s laws be applied to stellar motion around galactic centers?
• Why do planets have elliptical orbits?
• How did Tycho Brahe contribute to Kepler's laws? ‎