Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 13C (Cosmology)

With constant masses, differentiate the moment of inertia twice:For each pair, $F_{ij}=-G{m}_i{m}_j(r_i-r_j)/|r_i-r_j{|}^3$ and $F_{ji}=-F_{ij}$. Hence its contribution is $(r_i-r_j)\cdot F_{ij}=-G{m}_i{m}_j/|r_i-r_j|=V_{ij}$. Summing over unordered pairs gives $\sum r_i\cdot F_i=V$. Time-averaging over $[0,T]$ yieldsFor a bound system with $I^{\prime }(T)=o(T)$, or a stationary equilibrium with zero average left-hand derivative, the limit is the virial theoremBounded spatial extent together with bounded velocities suffices for the derivative condition; possible gravitational collisions must be excluded or otherwise regularized.

For the initially homogeneous sphere, $V_i=-3G{M}^2/(5R_i)$ 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, $V\propto -1/R$, soThese are the conventional collapse estimates. A final virial equilibrium is not, by itself, guaranteed to remain homogeneous: if $V_f=-{\alpha }_{f}G{M}^2/R_f$, 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. Baryonic cooling can destabilize the collisionless virial state; baryons are not inevitably unstable if cooling is slow or heating and pressure support maintain equilibrium.