What are the Best Approximate Algorithms for Solving Kepler's Equation?

[cptolemy]

Good evening,

I do realize that is relatively easy to solve Kepler's equation, M = E - e*sin(E) by iteration in a PC, for instant.
But I'm more interested in knowing better the available aproximative algorithms for its computation.
I've studied so far the Meeus proposal for the true anomaly of a planetary body computed from the mean anomaly, and Laplace / Fourier series on the equation. My tests show, unfortunatly, that they are not indeed as good as they seem... :(
Does anybody knows of another improved formulas for this calculation? Or even old ones. E series, or M sine series?

Kind regards,

CPtolemy

 

[AndyW]

Dear CPtolemy,

Thank you for your interest in the computation of Kepler's equation. As a scientist familiar with this topic, I can provide some insight into the available approximate algorithms for its computation.

In addition to the Meeus proposal and the Laplace/Fourier series, there are a few other methods that have been proposed for solving Kepler's equation. One approach is the Newton-Raphson method, which involves iteratively solving for the root of a function using its derivative. This method can be quite accurate and efficient, but may require a significant number of iterations in some cases.

Another method is the Householder's method, which involves using a sequence of linear approximations to solve for the root of a function. This method can also be accurate and efficient, but may require more iterations than the Newton-Raphson method.

There are also some older methods that have been used in the past, such as the Laguerre's method and the Halley's method. These methods are based on similar principles as the Newton-Raphson and Householder's methods, but may have different convergence properties.

In terms of series expansions, there are indeed some E series and M sine series that have been proposed for solving Kepler's equation. However, their accuracy may vary depending on the specific problem being solved.

I would recommend exploring these different methods and seeing which one works best for your specific problem. It may also be helpful to consult with other scientists or researchers in the field for their insights and experiences with these algorithms.

I hope this information helps in your pursuit to better understand and compute Kepler's equation.