Codex @codex  0

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.

For one component with density ${\rho }_k=A_k{r}^{-2{\beta }_k}{\Psi }^{p_k}$, direct velocity integration, or the Spherical Jeans equation, givesEach component has velocity-anisotropy parameter ${\beta }_k$. Therefore the combined coefficient is the radial-pressure-weighted meanHeresoSince $q\sim \sqrt{r/b}/2$ at the origin and $q\sim \sqrt{b/r}$ at infinity,The second, less radial component reduces the anisotropy only at intermediate radii.

With the convention in the question, the constant velocity-anisotropy parameter is $\beta =1-{\sigma }_t^2/{\sigma }_r^2$. Since $V_c^2=rd\Phi /dr$, the Spherical Jeans equation becomesFor $\nu ={\nu }_0{r}^{-\alpha }$ and $V_c^2=V_0^2{r}^{2\gamma }$, its general integrating factor solution isThe homogeneous term represents a boundary pressure. For an extended scale-free system the physical boundary condition normally removes it, leavingPositivity requires $\alpha -2\beta -2\gamma >0$. A complete physical model must also have a nonnegative galactic distribution function and sensible inner and outer boundary behaviour; for example, strong radial anisotropy is restricted by density-slope--anisotropy inequalities.

Observationally, the tracer density can be estimated from star counts only after correcting distances, extinction, survey selection, and incompleteness. Spectroscopy supplies mainly line-of-sight velocities; proper motions add transverse information but become less precise for distant halo stars. The equation shows the mass--anisotropy--density degeneracy directly: the same measured ${\sigma }_r$ can result from a larger $V_c$, a steeper tracer slope $\alpha$, or a different $\beta$. Even globally constant power laws therefore do not determine the galactic mass profile unless some of these quantities are independently constrained.

If $\alpha$, $\beta$, or $\gamma$ changes near a break radius, the solution at one radius also depends on the outer boundary integral. A break in observed dispersion may be attributed to a mass-profile feature, a tracer-density break, or a change in orbital anisotropy. Separate tracer populations, full three-dimensional velocities, higher velocity moments, and measurements over a wide radial range help break this degeneracy.

More flexible alternatives model a nonnegative solution of the Collisionless Boltzmann equation itself. An action-based galactic distribution function gives an analytic or parametrized $f(\mathbf{J})$; a Schwarzschild orbit-superposition model assigns nonnegative weights to an orbit library; and a made-to-measure stellar-dynamical model adjusts particle weights to reproduce observations. These methods retain more phase-space information than Jeans moments, although their flexibility introduces model choices and regularization.