Kepler's Laws of Planetary Motion

Johannes Kepler, the great mathematician, who proved laws of planetary motion, had no fame till last years of 16th century, when he wrote a book which introduced him to other scientists. In his book he showed how we can measure the distance between the planets. He believed that distance of two planet depends on the geometrical shape that could be placed between them. Of course he thought wrong, but his book took attention of an astronomer, Tycho Brahe, who demanded a mathematician. Kepler began to assist Tycho and found that Tycho's beliefs does not agree with his observations. Also he derived that Copernic's theory about circular orbit of planets was wrong.

Three Laws
According to physic's laws about effect of gravitation force on circular motion, when two bodies, one with less mass and the other with much more, gravitate each other, the body with less mass orbits the other one. Kepler derived laws that describe this motion:
1. The orbit of every planet is an ellipse with the Sun at a focus.
2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3.The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.





Generality
These laws describe the motion of any two bodies in orbit around each other. The masses of the two bodies can be nearly equal, e.g. Charon—Pluto (~1:10), in a small proportion, e.g. Moon—Earth (~1:100), or in a great proportion, e.g. Mercury—Sun (~1:10,000,000).
Since Kepler stated these laws as they apply to the Sun and the planets, and did not know of their generality, this article discusses these laws as they apply to the Sun and its planets.




First Law
"The orbit of every planet is an ellipse with the Sun at a focus."
Symbolically:



where (r, θ) are heliocentric polar coordinates for the planet, p is the semi-latus rectum, and ε is the eccentricity.
The prevailing belief was that orbits should be based on perfect circles. This observation was very significant at the time as it supported the Copernican view of the Universe. This does not mean it loses relevance in a more modern context. A circle is just one form of an ellipse, but most of the planets follow an orbit of low eccentricity, meaning that they can be crudely approximated as circles. So it is not evident from the orbit of the planets that the orbits are indeed elliptic. However, Kepler's calculations proved they were.

Second Law
"The line joining a planet and the Sun sweeps out equal areas during equal intervals of time"
Symbolically:





where is the "areal velocity".
This is also known as the law of equal areas. To understand this let us suppose a planet takes one day to travel from
point A to point B. The lines from the Sun to points A and B, together with the planet orbit, will define an (roughly triangular) area. This same area will be covered every day regardless of where in its orbit the planet is. Now as the first law states that the planet follows an ellipse, the planet is at different distances from the Sun at different parts in its orbit. This leads to the conclusion that the planet has to move faster when it is closer to the Sun so that it sweeps an equal area.

Third Law
"The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."
Symbolically:




where P is the orbital period of planet and a is the semimajor axis of the orbit.
Third Law published by Kepler in 1619, This captures the relationship between the distance of planets from the Sun, and their orbital periods.