Kepler.html

Expansion in eccentricity

Take the old-fashioned approach and expand the solution in a polynomial in eccentricity.

Stick this back into the Kepler equation, then expand on eccentricity to get

Solve successively for the coefficients. We'll write a quickie procedure to do this automatically.

Check:

Hence, the approximate solution is

Plot just the coefficients.

plot

Plot the difference between the approximate solution and the mean anomaly M (which is the zeroth-order approximation), as a function of eccentricity and M.

plot

Plot only the highest-order term of the approximate solution.

plot

Create a function to numerically solve for the eccentric anomaly, then plot the error of the approximate solution.

plot