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The hypervirial model is generated self-consistently by a constant-anisotropy distribution function proportional toIts velocity-anisotropy parameter is $\beta =1-p/2$.

J. An constructed analytic hypervirial models of stellar systems with N. W. Evans. Their 2005 family combines explicit gravitational potentials, cusped density profiles and simple phase-space distribution functions, with the special property that a local virial relation holds at every radius.^[ref]

Such models provide controlled examples for relating density cusps and velocity anisotropy. The phase-space distribution function must remain integrable and nonnegative; an attractive density formula alone does not guarantee a physical stellar system.

For the hypervirial model, $⟨v^2⟩=\Psi /2=-\Phi /2$. The local kinetic- and potential-energy densitiestherefore obey $2K+W=0$ at every radius, a stronger statement than the integrated virial theorem.

N. W. Evans developed analytic models of galaxies and dark haloes with J. An. Their hypervirial distribution function has the form $f\propto L^{p-2}{\mathcal{E}}^{(3p+1)/2}$, where $L$ is specific angular momentum and $\mathcal{E}$ is relative binding energy.^[ref]

The resulting hypervirial models connect an adjustable central density cusp to an explicit dynamical population, extending useful features of the Plummer model beyond its finite-density core.

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 spherical Plummer model has $\psi =GM/\sqrt{a^2+r^2}$ and $\rho =3M{a}^2/[4\pi (a^2+r^2{)}^{5/2}]$. It is the $p=2$ hypervirial model. In unit scales its isotropic distribution is $24\sqrt{2}{E}^{7/2}/(7{\pi }^3)$ for positive relative energy.