Hyperbolic Envelopes and Parabolic Drapes: Solving Kepler's Equation for Fun and Poverty!
Part 0a: Dear JPL
If Kepler’s Equation
M = E - e*sin(E)
"Kepler's equation" - Wikipedia
is restated as
M(x) = M/x = 1 - e*sin(x)/x = 1 - e*sinc(x) = E_kO(x)
where x is the eccentric anomaly
solutions for the eccentric anomaly become easier as the eccentricity decreases.
This solution succeeds another iterated solution that I developed about 10 years ago exploring Kepler's Equation for a solar energy spreadsheet that used constraints on the geometry of the ellipse to find high-accuracy estimates of the eccentric anomaly, where the sinc function
"The Function sin(x)/x" (1991 George Pólya Writing Award) - Mathematical Assoc. of America
appeared on its own as the primary function of the solution steps and was unusual in that it always converged to 16 digits accuracy in *exactly* eight steps, while a Newton's Method solution that I used for comparison
"Newton's method"- Wikipedia
took a few more steps that varied a bit with eccentricity.
I was never completely happy with my first solar position project and spent some time earlier this year to find a simpler system to calculate daily sun positions but my original notes and spreadsheet weren't available.
With only the memory that the first solution used the sinc function I first noticed how the shape of M/x followed the the curve of E_kO
which is understandable since sin(x)/x is just the sine function being "scaled" by the reciprocal 1/x, and since 1/x is a simple hyperbolic function, "framing" E_kO with hyperbolas seemed like good constraints.