If G is universal gravitational constant and $\rho$ is the uniform density of a spherical planet,  the shortest possible period of rotation of the satellite can be $\sqrt{\frac{\beta \pi }{G\rho }}$ find the value of  $\beta$

The fastest possible rate of rotation of a planet is that for which the gravitational force on  a body at the equator just barely provides the centripetal force needed for the rotation.  Let M be the mass and R be the radius of the planet and m is the mass of a body on its surface.
Then, 
$\begin{array}{l}mR{W^2}_{\max }=\frac{GMm}{R^2}\\ W_{\max }=\sqrt{\frac{GM}{R^3}}\\ \left(OR\right)\\ W_{\max }=\sqrt{\frac{G}{R^3}*\frac{4}{3}\pi R^{3}P}=\sqrt{\frac{4GP\pi }{3}}=2\sqrt{\frac{\pi GP}{3}}\\ T_{\min }=\frac{2\pi }{W_{\max }}=\sqrt{\frac{3\pi }{GP}}\\ \beta =3\end{array}$