Question

A planet of mass $M$ moves around the Sun along an ellipse so that its minimum distance from the Sun is equal to $r$ and the maximum distance to $R$. Making use of Kepler's laws, find its period of revolution around the Sun.

Answer 1

Semi-major axis $=\frac{(r+R)}{2}$
It is sufficient to consider the motion be along a circle of semi-major axis $\frac{r+R}{2}$ for $T$ does not depend on eccentricity.
Hence $T=\frac{2\pi {\left(\frac{r+R}{2}\right)}^{\frac{3}{2}}}{\sqrt{\gamma m_s}}=\pi \sqrt{\frac{{(r+R)}^3}{2\gamma m_s}}$
(again $m_s$ is the mass of the Sun)