Past exam of the astrophysics course of the University of Oxford / 2022 / 5 / d / Solution

Use fixed masses in an isolated Newtonian $N$-body system, with origin at its centre of mass. Define the $xx$ moment $I_{xx}={\sum }_i{m}_i{x}_i^2$ and $K_{xx}=\frac{1}{2}{\sum }_i{m}_i{v}_{x,i}^2$. Twice differentiating givesInsert the gravitational equation of motion and pair each ordered term $(i,k)$ with $(k,i)$:Thus one component of the tensor virial theorem isA stationary or time-averaged bounded system has $⟨{\ddot{I}}_{xx}⟩=0$, yielding $2⟨K_{xx}⟩+⟨W_{xx}⟩=0$. If ordered and random motions are separated, $2K_{xx}=2T_{xx}+{\Pi }_{xx}$, where $T_{xx}=\frac{1}{2}\int \rho u_x^{2}dV$ and ${\Pi }_{xx}=\int \rho {\sigma }_{xx}^{2}dV$. Summing the three diagonal equations recovers the scalar virial theorem, with the pair gravitational potential energy counted once.