No Simple Formula for Circumference of an Ellipse

Did you know there is no simple formula for the circumference of an ellipse?

Earlier I wanted to calculate (don't ask) the average speed of each planet in its orbit. So I wanted to calculate the circumference of its elliptical orbit around the Sun. For the Earth the eccentricity is small, 0.0167, so it's almost a circle. But what of the other planets?

This is probably common knowledge, but I don't think I've encountered it before. To my surprise, there is no simple formula for the distance p around the perimeter of an eclipse. The best you can do is an infinite series, which can be quickly truncated in the case of the Earth, but not necessarily for other planets.

 where

where a is the semi-major axis of the ellipse and b the semi-minor axis. (These aren't the orbit's perihelion and aphelion, but those can be easily calculated from a, b (=a*sqrt(1-e²)), and the eccentricity e: perihelion=a(1+e) and aphelion=a(1-e). The function in parentheses behind the sum for p is the combinatorical function, or binomial coefficient

(Here's how to take the factorial of a noninteger.) Anyway, for the Earth e << 1, so h << 1, so

which reduces to the circumference of a circle when e=0 (a=b), as it must.

Nothing so remarkable here -- I just never knew there was no simple formula for the circumference of an ellipse! Or if I did know, I'd forgotten.

PS: But there is a simple formula for the area enclosed by an ellipse: