= Projected velocity dispersion of a Keplerian cusp
{title2=$\sigma_{\rm los}^2=\frac{GM_\bullet}{(k+1-2\beta)R}\frac{B(k/2,1/2)}{B((k-1)/2,1/2)}\left(1-\frac{\beta k}{k+1}\right)$}
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 $\sigma_r^2=GM_\bullet/[(k+1-2\beta)r]$ 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 $\sigma_{\rm los}^2=2GM_\bullet/(3\pi R)$; 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.
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