Question

A planet of mass m moves around the Sun of mass M in an elliptical orbit. The maximum and minimum distance of the planet from the Sun are $r_1$ and $r_2$, respectively. Find the relation between the time period of the planet in terms of $r_1$ and $r_2$.

Answer 1

Given the periheleon$=r_2$

distance of planet from sun$=r_1$

Planet's mass$=m$

mass of sun$=M$

Length of major axes of no elliptical orbit$=r_1+r_2$

Semi-major axis$=R=\left(\frac{r_1+r_2}{2}\right)$

by Kepler's law

$T^2\propto R^{3}T\propto R^{\frac{3}{2}}T\propto {(r_1+r_2)}^{\frac{3}{2}}T=2\pi \sqrt{\frac{{(r_1+r_2)}^2}{8GM}}$