Question

Let the period of revolution of a planet at a distance R from a star be T. Prove that if it was at a distance of 2R from the star, its period of revolution will be $\sqrt{8}$ T.

Answer 1

From Kepler's third law of planetary motion, we have
$T^2\propto r^3$ .....(i)
Thus, when the period of revolution of planet at a distance R from a star is T, then from (i), we have
$T^2\propto R^3$ .....(ii)
Now, when the distance of the planet from the star is 2R, then its period of revolution becomes
$$\begin{gathered} T_1^2\propto (2R{)}^3 \\ \mathrm{or} \\ {T}_1^2\propto 8R^3.....(\mathrm{iii}) \end{gathered}$$
Dividing (iii) by (ii), we get
$$\begin{gathered} \frac{T_1^2}{T^2}=\frac{8R^3}{R^3} \\ \Rightarrow T_1=\sqrt{8}T \end{gathered}$$