Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 45 / 3 / c

For a homogeneous potential contribution of degree $n$, the scalar virial theorem follows from the moment of inertia identity $\frac{1}{2}\ddot{I}=2K-{\sum }_i{\mathbf{r}}_i\cdot {\nabla }_{i}U$. Time averaging in equilibrium gives $2K=nU$. Gravitational self-energy has degree $-1$, whereas $U_{\Lambda }$ has degree $2$. Apply the theorem to the sum of the two contributions:With the uniform-sphere expressions this isThe cosmological-constant force is repulsive, so its virial contribution reduces the kinetic support required at a fixed radius. One must not apply the inverse-radius gravitational relation to the whole potential: the two terms have different homogeneity degrees.