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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.

Motion can weigh a system whose mass is otherwise inaccessible. For a self-gravitating isolated system in virial equilibrium, let $K$ and $U$ denote equilibrium averages and use its gravitational radius $R_g=-G_N{M}^2/U$ and a mass-weighted mean squared speed relative to its centre of mass. Then $K=M⟨v^2⟩/2$ gives $M=R_g⟨v^2⟩/G_N$. If random velocities are isotropic and mean streaming has been removed, $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$, where ${\sigma }_{\mathrm{los}}$ is the one-dimensional line-of-sight velocity dispersion. Observations sample many objects at one epoch; replacing a time average by that sample requires a representative, approximately stationary population, beyond the trajectory identity itself.^[ref]

A practical measurement route uses an optical telescope to locate members and a spectrograph to measure Doppler shifts of identified spectral lines. Their shifts relative to the system's mean give radial velocities; angular separations plus a distance estimate constrain its size. Subtracting instrumental broadening and measurement variance, testing membership, and looking for coherent rotation or subgroups help control the inference. The projected luminous size need not equal $R_g$, and luminous tracers need not have the same distribution as all gravitating mass. Unknown velocity-anisotropy parameters, interlopers, mergers, external tidal forces and appreciable boundary stresses can bias a naive estimate. A large velocity dispersion alone does not establish virial equilibrium.