A spherically symmetric gravitational system of particles has a mass density
$\begin{array}{l}\rho ={\rho }_{0}forr\le R\\ =0forr>R\end{array}$ where ${\rho }_0$  is a constant.
A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed v as a function of distance $r(0<r<\infty )$  from the center of the system is represented by:

Gravitational field intensity for a given mass distribution as function of distance r can be written as:
$\begin{array}{l}\mathrm{E}=\frac{4{\mathrm{\rho }}_0\mathrm{\pi G}}{3}\mathrm{r}:(\mathrm{r}\le \mathrm{R})\\ =\frac{4{\mathrm{\rho }}_0{\mathrm{\pi GR}}^3}{3{\mathrm{r}}^2};(\mathrm{r}>\mathrm{R})\\ \mathrm{so},\mathrm{for}\mathrm{r}\le \mathrm{R},\mathrm{force}\mathrm{experienced}\mathrm{by}\mathrm{particle}\mathrm{is},\mathrm{F}=\frac{4{\mathrm{\rho }}_0\mathrm{\pi Gr}}{3}\mathrm{m}\\ \mathrm{which}\mathrm{is}\mathrm{providing}\mathrm{the}\mathrm{necessary}\mathrm{centripetal}\mathrm{force}.\\ \mathrm{So},\frac{{\mathrm{mv}}^2}{\mathrm{r}}=\frac{4{\mathrm{\rho }}_0\mathrm{\pi Gr}}{3}\mathrm{m}\Rightarrow \mathrm{v}\propto \mathrm{r}\\ \mathrm{similarly},\mathrm{for}\mathrm{r}>\mathrm{R}.\\ \mathrm{v}\propto \frac{1}{\sqrt{\mathrm{r}}}\end{array}$