= Pressureless homologous spherical collapse
{title2=$R^2\ddot R=-GM$}
A uniform spherical cloud with negligible <pressure> and <velocity> $\mathbf u=(\dot R/R)\mathbf r$ remains uniform: the <continuity equation> gives $\rho R^3=\rho_{00}R_0^3$. The <shell theorem> gives interior <gravitational acceleration> $-(4\pi G\rho/3)\mathbf r$, so its outer radius satisfies the displayed equation, with fixed mass $M=4\pi\rho_{00}R_0^3/3$. Collapse from rest has $\dot R^2=2GM(1/R-1/R_0)$. Writing $C^2=2GM/R_0$, its <cycloid> parametrization is $R=R_0(1+\cos\phi)/2$ and $Ct=R_0(\phi+\sin\phi)/2$, for $0\le\phi\le\pi$. Complete collapse occurs after $\sqrt{3\pi/(32G\rho_{00})}$. This Newtonian dust solution ceases to be a regular fluid configuration at zero radius.
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