Mathematica in Celestial Mechanics: A Sampling
Let and be bipolar coordinates and the smaller mass.
Let R[t] be the radius vector, r[t] the length. Then we can write . f[t] and g[t] can be calculated iteratively by series expansion using the equation of motion.
Here is the actual computation.
The motion of the so-called center in Kepler motion cannot be expressed in closed form, but a series approximation can be calculated in the following way.
Here is the 2,2 approximation.
Here is the more accurate 6,6 approximation.
One of the most important equations in celestial mechanics is the Kepler equation:
Here is a picture of a part the Riemann surface of the function over the complex m-plane. The branch points at are clearly visible .