Johannes Kepler: the laws of planetary motion

The planets' orbits around the Sun are described by three laws first formulated by Johannes Kepler (1571-1630), a great astronomer of the sixteenth century. These empirical laws have been initially formulated on the basis of astronomical observations of Mars' orbit, made by Tycho Brahe (1546-1601).
Later, Isaac Newton (1642-1727) demonstrated that these three laws can be obtained as an application of the law of universal gravitation in an approximated case (a 2-body problem where the bodies are point-like, the Sun is still and the planets don't interact ). To formulate this demonstration, Newton invented the infinitesimal calculus.
Here are Kepler's three laws and their main consequences:

Kepler's first law :

The orbits of the planets are ellipses with the Sun at one focus.

The picture is just an illustration, but in reality, the planets' orbits are ellipses much less eccentric (much more similar to a circle). Some important consequences of this law need to be enumerated:
  • the orbit of a planet is planar, this means that it all lies on a single plane;
  • the orbit is also closed and periodic;
  • the distance between Sun and planet is not constant during the orbit.  


Kepler's second law:

The line joining the planet to the Sun sweeps equal areas in equal times 

A very important consequence of this second law is that the planet's speed will not be constant during the orbit. In fact, due to the elliptical shape of the orbit (first law) the line joining the Sun and the planet will not be a constant distance. Since, as the second law says, the area swept by this line must be constant, this means that the speed of the motion will vary during the orbit. So, as a consequence of the first and second Kepler's laws, the planet will go faster near the perihelion and slow down when it comes near the aphelion.

Kepler's third law:

The ratio of the squares of the revolutionary
period and the cube of the semimajor axis is constant for all planets: 

This third law implies that the period of a planet increases rapidly with the radius of its orbit: the more the planet is distant from the sun, the bigger is its major semiaxis, and the longer is its period P. This means that the farther planets of the solar system are much slower than the inner planets, and therefore have much longer years (the time it takes them to make a complete orbit around the Sun).