= Virial mass estimate
{title2=$M\simeq3R_g\sigma_{\rm los}^2/G_{\!N}$}
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 $K=M\langle v^2\rangle/2$ gives $M=R_g\langle v^2\rangle/G_{\!N}$. If random <velocities> are <isotropic> and mean streaming has been removed, $\langle v^2\rangle=3\sigma_{\rm los}^2$, where $\sigma_{\rm los}$ 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.<Cosmology by David Tong/1 The Expanding Universe>[Cosmology by David Tong, §1.4.3, “The Virial Theorem and Galaxy Clusters”: self-averaging assumption and mass estimate (1.75)]{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>.
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