 Solution

The stated relation follows for an isolated uniform sphere ending at $r_0$, with the gravitational potential zero at infinity. Uniform density in a core alone does not fix the escape velocity: exterior shells add a constant interior gravitational potential while leaving the interior force unchanged.

Under the isolated-sphere assumption, the enclosed mass and circular speed areThe boundary value is $\Phi (r_0)=-Gm(r_0)/r_0$. Integrating ${\Phi }^{\prime }=Gm(r)/r^2$ inwards givesThus the escape and circular speeds inside an isolated uniform sphere satisfyThe finite central limit is $v_e^2(0)=4\pi G{\rho }_0{r}_0^2$, although $V_c(0)=0$. Adding an exterior spherical shell of mass $m_s$ at radius $s>r_0$ raises $v_e^2$ by $2G{m}_s/s$ and is a direct counterexample to deducing the boxed relation from a uniform core alone.