Kepler's Laws

The first true laws of planetary motion were proposed by Johannes Kepler between 1609 and 1619 as a result of his work on the twenty years of planetary observations made by Tycho Brahe, the astronomer royal to the King of Denmark. Tycho Brahe worked on the island of Uraniborg and his work is all the more impressive when you remember that the telescope was not invented and certainly not used for astronomy until after Brahe's death.

Kepler was Brahe's pupil and worked on his data to propose three laws of planetary motion

1. The planets move in ellipses with the Sun at one focus
2. A line drawn from the planet to the Sun sweeps out equal areas in equal times
3. The ratio of the square of the period (T) of the planet about the Sun to the cube of the mean orbit radius (r) is a constant.

1. The planets move in ellipses with the Sun at one focus
You can prove this by measuring the diameter of a projected image of the Sun at different times of the year.
(NB although the Earth does move in an elliptical orbit this orbit is every nearly circular. The only planet with a markedly elliptical orbit is Pluto.)
Distance from the Earth to the Sun at its closest (perihelion) = 1.471x10¹¹ m
Distance from the Earth to the Sun when at its furthest (aphelion) = 1.521x10¹¹m

2. A line drawn from the planet to the Sun sweeps out equal areas in equal times (see Figure 1)
This means that bodies move faster when closer to the Sun – the best example of this being the long period comets. They spend only a few months close to the Sun before moving off into deep space – not returning for maybe many hundreds of years.

3. The ratio of the square of the period (T) of the planet about the Sun to the cube of the mean orbit radius (r) is a constant. This means that:     T²/r³ = constant. (Notice that r³/T² is also constant)

The Earth is 150x10⁹ m from the Sun and it takes one year (3.16x10⁷ s) to orbit the Sun.
Therefore Kepler's constant ( r³/T²) for the Solar System is (150x10⁹)³/(3.16x10⁷)² = 3.37x10¹⁸m³s^-2

schoolphysics: Kepler's laws animation

To see an animation of Kepler's second law please click on the animation link.