Hypervirial model

The hypervirial family has relative potentialand densityIts name comes from satisfying the scalar virial relation pointwise.

In unit scales $I=\int \rho \psi d^{3}x=(p+1)\Gamma (1+1/p{)}^2/[p\Gamma (2+2/p)]$. The local virial relation of the hypervirial model gives global energies $T=I/4$ and $W=-I/2$. Restoring physical scales multiplies each integral by $G{M}^2/a$.

For $p>0$, the relative potential $GM/(a^p+r^p{)}^{1/p}$ generates density $(p+1)M{a}^p{r}^{p-2}/[4\pi (a^p+r^p{)}^{2+1/p}]$. In unit scales the separable augmented density is $(p+1)r^{p-2}{\psi }^{2p+1}/(4\pi )$. Its Jeans moments from an augmented density give constant anisotropy $\beta =1-p/2$.

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

In units $G=M=a=1$, velocity integration of $F=C{L}^{p-2}{E}^{(3p+1)/2}$ gives $C=(p+1)\Gamma (2p+2)/[2^{(p+5)/2}{\pi }^{5/2}\Gamma (p/2)\Gamma ((3p+3)/2)]$. The beta function separates the radial-speed and angular integrals. This constant is positive for every $p>0$.

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.