When the satellite is moving in circular orbit after launching the satellite, then centripetal force should be equal to the gravitational force,

$F_{\mathrm{c}}=F_{\mathrm{G}}$

$\frac{1}{2}\mathrm{m}{\left(v_0\right)}^2=G\frac{Mm}{{\left(r\right)}^2}$

where $G, M, m, h$ are universal gravitational constant, the mass of the planet, the mass of satellite and height where the satellite is projected, 

$\Rightarrow \frac{m{\left(v_0\right)}^2}{r}=G\frac{Mm}{{\left(R+h\right)}^2}$

$\Rightarrow {\left(v_0\right)}^2=G\frac{M}{r}.$

Now the time period of the satellite,

$T=\frac{2\mathrm{\pi }r}{v_0}=\frac{2\mathrm{\pi }r}{{\left(G\frac{M}{r}\right)}^{{\displaystyle \frac{1}{2}}}}$

$\Rightarrow T^2={\left(\frac{2\pi r}{{\left(G\frac{M}{r}\right)}^{{\displaystyle \frac{1}{2}}}}\right)}^2$

$\Rightarrow T\propto \sqrt{\frac{r^3}{GM}} .$