= Escape and circular speeds inside an isolated uniform sphere
{title2=$v_e^2=V_c^2(3r_0^2/r^2-1)$}
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 $V_c^2=4\pi G\rho_0r^2/3$ 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 $0<r\leq r_0$. 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.
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