# Kepler's Laws

Three laws of planetary motion, published by Johannes Kepler using accurate observations by Tycho Brahe and shown by Isaac Newton to be a direct result of his theory of gravitation and his laws of motion:

1. The orbit of each planet is an ellipse which has the sun at one of its foci.
This corrected the simpler model of Copernicus, which assumed circles. More accurately, the focus is at the center of gravity of the Sun and orbiting body (discounting other planets) and non-periodic motions along parabolas or hyperbolas are also possible.

2. Each planet moves in such a way that the imaginary line joining it to the sun sweeps out equal areas in equal times.
The second law expresses the way a planet speeds up when approaching the Sun and the way it slows down when drawing away.

3. The squares of the periods of revolution of the planets about the sun are proportional to the cubes of their mean distances from it.
The third law gives the exact relation by which planets move faster on orbits which are closer to the Sun, e.g. Venus moves faster than Earth. For a more precise formulation, "mean distance" should be replaced by semimajor axis.

where
a = semi-major axis [m]
α = positive constant
A = area swept out [m2]
b = semi-minor axis [m]
e = eccentricity
E = total energy (constant) [J]
J = total angular momentum [kgm2rads-1]
m = mass of orbiting body [kg]
M = mass of central body [kg]