= Virial radius from turnaround energy
{title2=$R_{\rm vir}=R_{\rm ta}/2$}
For a homologous, isolated collapsing sphere, write its <Newtonian gravitational potential energy> as $U=-aGM^2/R$. At <spherical-collapse turnaround> the coherent <kinetic energy> vanishes. Conserving total energy and applying the <virial theorem> to the final equilibrium gives $U_{\rm ta}=U_{\rm vir}/2$ and hence $R_{\rm vir}=R_{\rm ta}/2$. If the gravitational structure coefficient changes, the ratio is $a_{\rm vir}/(2a_{\rm ta})$ instead. Mass loss, surface pressure, external tides and energy exchange also change this estimate.
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