The <hypervirial model> is generated self-consistently by a <constant-anisotropy distribution function> proportional to
$$
L^{p-2}\mathcal E^{(3p+1)/2}.
$$
Its <velocity-anisotropy parameter> is $\beta=1-p/2$.

Hypervirial distribution function

For the <hypervirial model>, $\langle v^2\rangle=\Psi/2=-\Phi/2$. The local kinetic- and potential-energy densities
$$
K=\frac12\rho\langle v^2\rangle,
\qquad
W=\frac12\rho\Phi
$$
therefore obey $2K+W=0$ at every radius, a stronger statement than the integrated <virial theorem>.

Local virial relation of the hypervirial model

Put $\Psi=-\Phi=GM(r^p+a^p)^{-1/p}$. The spherical <Poisson equation> gives the <hypervirial model> density
$$
\boxed{
\rho(r)=\frac{(p+1)Ma^p}{4\pi}
\frac{r^{p-2}}{(r^p+a^p)^{2+1/p}}}
=\frac{(p+1)Ma^p}{4\pi(GM)^{2p+1}}
r^{p-2}\Psi^{2p+1}.
$$

For the proposed <hypervirial distribution function>, write $\mathcal E=-E=\Psi-v^2/2$ and $n=(3p+1)/2$. Direct velocity integration gives
$$
\begin{aligned}
\rho
&=A r^{p-2}
\int_0^{\sqrt{2\Psi}}v^p
(\Psi-v^2/2)^n\,dv
\int_0^{2\pi}d\varphi
\int_0^\pi\sin^{p-1}\alpha\,d\alpha\\
&=A\,2^{(p+1)/2}\pi^{3/2}
\frac{\Gamma(p/2)\Gamma(3(p+1)/2)}
{\Gamma(2p+2)}
r^{p-2}\Psi^{2p+1}.
\end{aligned}
$$
Matching this expression to the density fixes
$$
\boxed{
A=\frac{(p+1)a^p\Gamma(2p+2)}
{2^{(p+5)/2}\pi^{5/2}G^{2p+1}M^{2p}
\Gamma(p/2)\Gamma(3(p+1)/2)}}.
$$
This coefficient is positive for $0<p\leq2$, so the distribution function self-consistently generates the stated density and potential.

Solution

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

The hypervirial family has relative potential
$$
\Psi(r)=\frac{GM}{(r^p+a^p)^{1/p}},
\qquad 0<p\leq2,
$$
and density
$$
\rho(r)=\frac{(p+1)Ma^p}{4\pi}
\frac{r^{p-2}}{(r^p+a^p)^{2+1/p}}.
$$
Its name comes from satisfying the scalar virial relation pointwise.

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