The average radius of an ellipse I have recently been studying the 3rd Kepler´s law which uses the average radius of an orbit. I would love to know how to find the geometric average radius of an ellipse. Would you please show me or at least help me with the process of integration?
 A: Hints:
The polar equation of the ellipse with one focus at the origin and the other at the positive real axis ($e = $ eccentricity) is:
$$r = \frac{a(1 - e^2)}{1 - e\cos(\theta)}.$$
And the average radius can be calculated using the integral
$$\frac{1}{2\pi}\int_0^{2\pi}r(\theta)d\theta.$$
Can you continue?
Update: already asked at Astronomy SE. The important fact:

It's the semi-major axis that defines the period, not the average distance.

Bottom line: the "simple average" $\ne$ the integral average. Also interesting (and different): the time average.
Update 2: doing the cov $z = \tan(\theta/2)$,
$$\int\frac{1}{1 - e\cos(\theta)}d\theta = \int\frac{1}{1 - e\frac{1 - z^2}{1 + z^2}}\frac{2}{1 + z^2}dz = \int\frac{2}{(1 + e)z^2 + (1 - e)}dz =$$
$$ = \frac{2}{\sqrt{1 - e^2}}\arctan\left(\frac{\sqrt{1 + e}}{\sqrt{1 - e}}z\right) =
\frac{2}{\sqrt{1 - e^2}}\arctan\left(\frac{\sqrt{1 + e}}{\sqrt{1 - e}}\tan(\theta/2)\right).$$
A: @Martin-Blas has already suggested two possible notions of average


*

*Take an average with respect to $\theta$, the central angle

*Take an average with respect to time, i.e.,$$
\frac{1}{T}\int_0^T \| u(t) \| dt,
$$
where $T$ is the period, and $u(t)$ is the position of the planet at time $t$, and the sun is located at the origin of the coordinate system. 
There's a third notion, which is 


*An average with respect to arclength, i.e., $$
\int_0^L \| u(t) \| ds$$
where $L$ is the total arclength of the ellipse, and $ds = \|u'(t)\| dt$ is  arclength integrand. 


I'm pretty certain that one could even hoke up some others, but the key thing here is the one mentioned by others: this question doesn't have an answer until you know the measure with respect to which the "average" is being computed. 
A: It depends on your definition of that average radius means in this context. E.g. if we define average radius of an ellipse as the radius $r$ of a circle which has the same area an an ellipse whose length of the semi-major and semi-minor axis are $a$ and $b$ then
$$
Area = \pi r^2 = \pi ab
$$
which gives $r = \sqrt {ab}$.
Alternative definition is to literally take the average of $a$ and $b$ or $r = \frac{a+b}{2}$.
