Question

I’m trying to solve Kepler’s Equation as a step towards finding the true anomaly of an orbiting body given time. It turns out though, that Kepler’s equation is difficult to solve, and the wikipedia page describes the process using calculus. Well, I don’t know calculus, but I understand that solving the equation involves an infinite number of sets which produce closer and closer approximations to the correct answer.

I can’t see from looking at the math how to do this computationally, so I was hoping someone with a better maths background could help me out. How can I solve this beast computationally?

FWIW, I’m using F# — and I can calculate the other elements necessary for this equation, it’s just this part I’m having trouble with.

I’m also open to methods which approximate the true anomaly given time, periapsis distance, and eccentricity

Answer 1

You could check this out, implemented in C# by Carl Johansen

Represents a body in elliptical orbit about a massive central body

Here is a comment from the code

True Anomaly in this context is the
angle between the body and the sun.
For elliptical orbits, it’s a bit
tricky. The percentage of the period
completed is still a key input, but we
also need to apply Kepler’s
equation (based on the eccentricity)
to ensure that we sweep out equal
areas in equal times. This
equation is transcendental (ie can’t
be solved algebraically) so we
either have to use an approximating
equation or solve by a numeric method.
My implementation uses
Newton-Raphson iteration to get an
excellent approximate answer (usually
in 2 or 3 iterations).