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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.

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.

Let $\Psi (r)>0$ be the relative potential of a spherical stellar system, $\mathcal{E}=\Psi -v^2/2$ the specific binding energy, and $L=rv\sin \vartheta$ the specific angular momentum. Consider the galactic distribution functionwith $f=0$ for $\mathcal{E}\le 0$ and $f_0>0$. Velocity integration gives the mass densityFor this ordinary locally integrable distribution, the conditions are $\beta <1$ and $n>-1$. They ensure convergence at purely radial directions and at the escape-energy boundary, respectively.^[ref]

To derive the coefficient, write the velocity-volume element as $2\pi v^2\sin \vartheta dvd\vartheta$. The angular integral is $\sqrt{\pi }\Gamma (1-\beta )/\Gamma (3/2-\beta )$. In the radial integral set $x=v^2/(2\Psi )$, givingMultiplication proves the result using the gamma function normalization.

The angular moments give velocity-anisotropy parameter $\beta$, explaining the name constant-anisotropy distribution function. As a check, $\beta =0$, $n=7/2$ gives an isotropic distribution with $\rho \propto {\Psi }^5$, as in the Plummer model. The formula converts a phase-space ansatz into a density-potential relation; a self-consistent stellar model must additionally solve the Poisson equation for Newtonian gravity.