The orbital period of a satellite in a circular orbit of radius r about a spherical planet of mass M and mean density $\mathrm{\rho }$ for a low altitude orbit $\left(\mathrm{r}\simeq {\mathrm{r}}_{\mathrm{p}}\right)$ will be:

Applying newton's second law to the circular orbit
We have,

$$\begin{gathered} \frac{{\mathrm{mv}}^2}{\mathrm{r}}=\frac{\mathrm{GMm}}{{\mathrm{r}}^2} \\ \mathrm{But} \mathrm{v}=\frac{2\mathrm{\pi r}}{\mathrm{T}} \\ \mathrm{m}\frac{4{\mathrm{\pi }}^2{\mathrm{r}}^2}{{\mathrm{rT}}^2}=\frac{\mathrm{GMm}}{{\mathrm{r}}^2};{\mathrm{T}}^2=\frac{4{\mathrm{\pi }}^2{\mathrm{r}}^3}{\mathrm{GM}};\mathrm{T}=\frac{2{\mathrm{\pi r}}^{\frac{3}{2}}}{\sqrt{\mathrm{GM}}} \\ \text{ but }\mathrm{M}=\frac{4}{3}{\mathrm{\pi r}}^3\mathrm{\rho }\text{ and }\mathrm{r}={\mathrm{r}}_{\mathrm{p}} \\ \mathrm{therefore}, \\ \mathrm{T}=\frac{2\mathrm{\pi }{\left({\mathrm{r}}_{\mathrm{p}}\right)}^{\frac{3}{2}}}{\sqrt{\mathrm{G}\frac{4}{3}\mathrm{\pi }{\left({\mathrm{r}}_{\mathrm{p}}\right)}^3\mathrm{\rho }}}=2\mathrm{\pi }\sqrt{\frac{3}{4\mathrm{\pi G\rho }}} =\sqrt{\frac{3\mathrm{\pi }}{\mathrm{G\rho }}} \end{gathered}$$