 Solution

For a family of spherical profiles with a fixed dimensionless shape, write the gravitational potential energy as $\Omega =-\alpha G{M}^2/R$, where $\alpha >0$ depends on that shape. The virial theorem in b and the sphere volume $V=4\pi R^3/3$ giveFor a uniform density, $\alpha =3/5$; formally integrating $-{\int }_0^{R}GM(r)dM(r)/r$ for $M(r)=Mr/R$ gives $\alpha =1$. Neither the virial theorem nor the initial profile guarantees a specific final shape, so the proportionality requires the stated homologous-profile assumption. Surface pressure must be handled consistently with b before using either coefficient as an equilibrium model.