Virial radius from turnaround energy ($R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$)

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