Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 3

At fixed radius, use spherical coordinates in velocity space with polar angle $\alpha$ measured from the radial direction:The galactic distribution function and volume element give angular weightAll dependence on $v$, $r$, and $g(E)$ cancels from ratios of second moments. Symmetry in the tangential plane givesWriting the angular integrals as beta function integrals and using the Gamma function recurrence,Therefore, for every admissible energy factor $g$,Equivalently, the velocity-anisotropy parameter is the constant $\beta =1-p/2$.

Put $\Psi =-\Phi =GM(r^p+a^p{)}^{-1/p}$. The spherical Poisson equation gives the hypervirial model density

For the proposed hypervirial distribution function, write $\mathcal{E}=-E=\Psi -v^2/2$ and $n=(3p+1)/2$. Direct velocity integration givesMatching this expression to the density fixesThis coefficient is positive for $0<p\le 2$, so the distribution function self-consistently generates the stated density and potential.

The density has the formInsert the ansatz ${\sigma }_r^2=k\Psi$ into the Spherical Jeans equationThe radial-power derivative cancels the anisotropy term because $p-2+2\beta =0$, leaving $k(2p+2)=1$. ThusTheir sum is independent of $p$:Therefore the local kinetic-energy density and gravitational potential-energy density areand they satisfy the local virial relation of the hypervirial modelat every radius. This pointwise identity is not generic. The ordinary virial theorem constrains suitable global integrals, with boundary terms when the system is truncated, but does not normally impose a virial balance shell by shell.