Question

If a satellite is revolving close to a planet of density $p$ with period $T$, show that the quantity $p{T}^2$ is a universal constant.

Answer 1

Let the angular velocity of planet be $\omega$. Hence,

$m{\omega }^{2}R=\frac{GMm}{R^2}$

$\omega =\sqrt{\frac{GM}{R^3}}$

Thus, time period is

$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{R^3}{GM}}$

Also,

$\rho =\frac{M}{V}=\frac{M}{\frac{4}{3}\pi R^3}$

Thus, $\rho T^2=\frac{M}{\frac{4}{3}\pi R^3}4{\pi }^2\frac{R^3}{GM}=\frac{3\pi }{G}=constant$

Hence Proved.