The mass density of a spherical galaxy varies as $\frac{\mathit{K}}{\mathit{r}}$ over a large distance $r$ from its center. In that region, a small star is in a circular orbit of radius $R$. Then the period of revolution,$T$ depends on $R$ as:

A planet orbits in an elliptical path of eccentricity $e$ around a massive star considered fixed at one of the foci. The point in space where it is closest to the star is denoted by $P$ and the point where it is farthest is denoted by $A$. Let $v_{\mathrm{P}}$ and $v_{\mathrm{A}}$ be the respective speeds at $P$ and $A$. Then,