Kepler's Laws of Planetary Motion - Computing Position As A Function of Time

Computing Position As A Function of Time

Kepler used his two first laws for computing the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.

The procedure for calculating the heliocentric polar coordinates (r,θ) to a planetary position as a function of the time t since perihelion, and the mean motion n = 2π/P, is the following four steps.

1. Compute the mean anomaly

2. Compute the eccentric anomaly E by solving Kepler's equation:

3. Compute the true anomaly θ by the equation:

4. Compute the heliocentric distance r from the first law:

The important special case of circular orbit, ε = 0, gives simply θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.

The proof of this procedure is shown below.