Projected velocity dispersion of a Keplerian cusp (${\sigma }_{\mathrm{los}}^2=\frac{G{M}_{\bullet }}{(k+1-2\beta )R}\frac{B(k/2,1/2)}{B((k-1)/2,1/2)}\left(1-\frac{\beta k}{k+1}\right)$)

For a scale-free tracer $\nu \propto r^{-k}$ in a point-mass gravitational potential, with $k>1$ and constant velocity-anisotropy parameter, the Spherical Jeans equation gives ${\sigma }_r^2=G{M}_{\bullet }/[(k+1-2\beta )r]$ when the outer pressure term vanishes. Project the tracer density and its second stellar velocity moment along the line of sight. The substitution $r=R\sec \chi$ reduces both integrals to powers of $\cos \chi$ and hence the beta function, yielding the displayed expression. The ratio of the two numerator angular integrals is $k/(k+1)$. This makes the mass-anisotropy ambiguity explicit. At $k=2$ the velocity-anisotropy parameter dependence cancels and ${\sigma }_{\mathrm{los}}^2=2G{M}_{\bullet }/(3\pi R)$; that special slope does not remove the degeneracy in general. Physical use also requires a nonnegative tracer distribution and control of outer departures from the power law.