Is there a closed form solution to Kepler's problem?

[Will Flannery]

TL;DR Summary

Kepler's equation is M = E - e*sinE
According to wiki "Kepler's equation is a transcendental equation because sine is a transcendental function, meaning it cannot be solved for E algebraically. Numerical analysis and series expansions are generally required to evaluate E."
However in the paper "A Solution of Kepler's Equation" by Tokis we read " Solution of the universal Kepler’s equation in closed form is obtained with the help of the two-dimensional Laplace technique, ... "

Kepler's equation is M = E - e*sinE

According to wiki - Kepler's equation - "Kepler's equation is a transcendental equation because sine is a transcendental function, meaning it cannot be solved for E algebraically. Numerical analysis and series expansions are generally required to evaluate E."

However in the paper "A Solution of Kepler's Equation" by Tokis we read " Solution of the universal Kepler’s equation in closed form is obtained with the help of the two-dimensional Laplace technique, ... "

I have an MS in math, but ... Tokis' paper is beyond me. Can anyone clarify the situation?

Tokis' paper is also available as a pdf at A Solution of Kepler's Equation

 

[Dale]

Have you tried plugging the proposed solution into the problem to see if it is a indeed a solution?

 

[Will Flannery]

Have you tried plugging the proposed solution into the problem to see if it is a indeed a solution?

I haven't. It would be a good thing to try, but even that would require a bit of effort, and I currently don't have MATLAB available since I switched computers. They've plotted graphs using the equation, so I assume it works to some degree of accuracy. However, the question remains, is the derivation valid, and that I can't begin to figure out. And, if it is, why hasn't wiki been updated, etc.?

Also, the paper includes the sentence "In virtually every decade from 1650 to the present, there have appeared papers devoted to the solution of thisKepler’s equation. Its exact analytical solution is unknown, ... ", so it's a little unclear to me what the paper is claiming.

 

[Will Flannery]

Well, I studied the paper a bit, and still cannot follow the details (and I'm unfamiliar with Kepler's equation)* The paper claims "we have a new exact solution of the present problem for the universal anomaly as a function of the time." This solution is eq (31). However the universal anomaly is a constructed variable, which isn't the usual eccentric anomaly. The paper continues and we have an expression for the eccentric anomaly as a function of the mean anomaly, eq (40), and I read this to be an exact solution. So, I retrieved my copy of MATLAB and set out to test the formula with the following code:
eps = 0.3; % eccentricity
E = 1; % eccentric anomaly
Me = E - eps*sin(E) % Kepler's eq., mean anomaly as a function of E (2a)
% now we use eq 41 and 40 to calculate E as a function of Me
phi = sqrt(eps^2-[1-(1-eps)*sqrt(1+eps*Me^2/(1-eps)^3)]^2); % (41)
Ec = Me + phi % eccentric anomaly as function of Me (40)
which prints out ...
Me = 0.7476
Ec = 1.0097
An exact solution would not be off by 1 part in 100, so, it doesn't seem to work, or ... I made a mistake :)

Note that there is a restriction (32) on Me in formula (31) and hence (40), that is Me <= 2*sqrt(1-eps^2) which is met in the example above. In a complete elliptical orbit Me goes from 0 to 2*pi, so this is a weird restriction.

*but since solving Kepler's equation arguably represents the beginning of modern physics and mathematics, I'm interested.

I'm still working on this ... may get Celestial Mechanics by Danby (pdf $10.00) - any other suggestions?