= Solution
For the initially homogeneous <sphere>, $V_i=-3GM^2/(5R_i)$ and $K_i=0$. <Conservation of energy> and the <virial theorem> imply $E_f=V_f/2=V_i$, hence $V_f=2V_i$. With the usual homologous-<sphere> approximation, $V\propto-1/R$, so
$$
\boxed{R_f=R_i/2,\qquad \rho_f=8\rho_i.}
$$
These are the conventional collapse estimates. A final virial equilibrium is not, by itself, guaranteed to remain homogeneous: if $V_f=-\alpha_f GM^2/R_f$, the actual relation is $R_f/R_i=\alpha_f/(2\alpha_i)$ with $\alpha_i=3/5$. Thus the factor of two applies exactly to the gravitational radius, and to the outer radius only under the stated structural approximation.
Here “non-interacting” means collisionless apart from gravity. Baryonic gas can collide, convert ordered motion to thermal <energy>, and radiate it. If <radiative cooling> is effective, the fixed-<energy> virial state loses <pressure> support and contracts further. \b[Baryonic cooling can destabilize the collisionless virial state]; baryons are not inevitably unstable if cooling is slow or heating and <pressure> support maintain equilibrium.
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