An earlier put up mentioned the way to resolve Kepler’s equation
M = E − e sin(E)
utilizing a sine collection. You would additionally resolve Kepler’s equation utilizing an influence collection, which Lagrange did in 1771. Each approaches categorical E as a perform of e and M, however from totally different views. Bessel considered his answer as a sum of sines in M, with coefficients that rely upon e. Lagrange considered his answer as an influence collection in e whose coefficients contain sines in M. You possibly can rearrange the phrases of both answer into the opposite.
Probably the most attention-grabbing factor concerning the energy collection answer, in my view, is that it solely converges for e lower than roughly 2/3 whereas the sine collection answer is legitimate for all e < 1. In astronomical phrases, this implies the facility collection answer works for the orbit of some planets however not others!
In our photo voltaic system, the planets all have eccentricity nicely beneath 2/3, however not all minor planets do. For instance, the orbit of Eris has eccentricity 0.4407 however the orbit of Sedna has eccentricity 0.8549. And in different photo voltaic methods there are planets with eccentricity a lot larger than 2/3.
The Laplace restrict
The radius of convergence for Lagrange’s energy collection answer is known as the Laplace restrict. Its worth is eL = 0.6627…. There’s no apparent purpose why there’s something particular about this worth. There’s no astronomical purpose for this worth. It’s an artifact of the facility collection type of the answer.
If the collection works for e = 0.66, you’d moderately suppose it really works for e = 0.67, however that’s not the case. And should you’re observant, you may discover that though the collection works for e = 0.66, it takes longer to converge than for smaller values of e; the speed of convergence is slowing down, warning you of hazard forward.
The precise worth of eL is the distinctive actual answer to the equation
There’s no apparent purpose for this both. It has to do with discovering the biggest circle that may slot in a lens-shaped area of convergence. Extra on that right here.
We are able to calculate eL with the next Python code.
from math import exp
from scipy.optimize import root_scalar
def f(x):
t = (1 + x*x)**0.5
return x*math.exp(t) - 1 - t
sol = root_scalar(f, bracket=[0, 1], methodology='brentq')
print(sol.root)
This prints 0.6627434193491817.
Collection particulars
We are able to use the Lagrange inversion method to seek out the collection, simply as Lagrange did two and a half centuries in the past.
The powers of sine could be expanded into the sum of sines of assorted frequencies and differentiated, resulting in the equation
