# Kepler's 3rd law/law of periods

Kepler's 3rd law of planetary motion is

The square of the time period of a planet orbiting a sun is proportional to the cube of the semi major axis of the elliptical orbit.

$$(T_1)^2 = (A_1)^3,$$

But in many other sites it (the square of the time period) is said to be proportional to the mean radius of the elliptical orbit.

$$(T_1)^2 = (R_1)^3,$$

which one is correct? Or is there a relation between the mean radius and the semi major axis which cancel out because of proportionality?

It is often said that the semi-major axis is the "average" distance between the primary focus of the ellipse and the orbiting body. This is not quite accurate, because it depends on what the average is taken over.

averaging the distance over the eccentric anomaly indeed results in the semi-major axis.

averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis ${\displaystyle b=a{\sqrt {1-e^{2}}}}$.

averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average ${\displaystyle a\left(1+{\frac{e^{2}}{2}}\right)}$.

The time-averaged value of the reciprocal of the radius, ${\displaystyle r^{-1}}$, is ${\displaystyle a^{-1}}$.