Escape and circular speeds inside an isolated uniform sphere ($v_e^2=V_c^2(3r_0^2/r^2-1)$)

For a sphere of radius $r_0$ and uniform mass density ${\rho }_0$, with no surrounding mass and gravitational potential zero at infinity, the spherical shell theorem gives $V_c^2=4\pi G{\rho }_0{r}^2/3$ inside. Matching to the exterior point-mass potential gives $\Phi (r)=-(2\pi G{\rho }_0/3)(3r_0^2-r^2)$. Since $v_e^2=-2\Phi$, the displayed relation follows for $0<r\le r_0$. Exterior shells leave the interior circular speed unchanged but deepen the potential and change the escape velocity; a uniform core alone is insufficient to determine escape.