= Solution
Three tracers give complementary estimates of a cluster's gravitating <mass>. First, <galaxy> <redshifts> measure a line-of-sight <velocity dispersion> $\sigma_{\rm los}$. For a roughly stationary bound cluster, the <virial theorem> gives a <cluster velocity-dispersion mass estimate>
$$
M(<r)\sim\frac{3\sigma_{\rm los}^2r}{G},
$$
with a coefficient depending on the <density> profile and orbital anisotropy. The <galaxy> <luminosity> and plausible stellar <mass-to-light ratio> supply too little <mass> to bind typical rich clusters. Foreground interlopers, substructure, velocity anisotropy and a recent merger can bias the dynamical estimate; a large dispersion is not by itself proof of equilibrium.
Second, diffuse X-ray emission measures a hot intracluster <plasma>, not merely unresolved starlight. Spectral continuum and lines constrain its <temperature>, while surface-brightness profiles constrain its <density> after <line-of-sight projection> and emissivity modeling. If the gas has constant composition and is approximately spherical and in <hydrostatic equilibrium>,
$$
\frac{dP_g}{dr}=-\rho_g\frac{GM(<r)}{r^2}
\quad\Longrightarrow\quad
\boxed{M(<r)=-\frac{k_BT(r)r}{G\mu m_p}
\left(\frac{d\log\rho_g}{d\log r}+\frac{d\log T}{d\log r}\right)}.
$$
This <cluster hydrostatic mass estimator> uses $\mu m_p$ as the mean <mass> per pressure-supporting particle. The inferred potential requires substantially more <mass> than the <stars> plus measured gas, although the gas contains much of the visible baryonic <mass>. Turbulence, bulk motion, <temperature> structure and departures from <hydrostatic equilibrium> are important <systematic errors>.
Third, strong-lensing arcs and multiple images, together with coherent weak-lensing shear of background <galaxies>, measure projected gravitational deflection. In <general relativity>,
$$
\kappa=\frac{\Sigma}{\Sigma_{\rm crit}},\qquad
\Sigma_{\rm crit}=\frac{c^2}{4\pi G}\frac{D_s}{D_lD_{ls}}.
$$
This <cluster gravitational-lensing mass estimate> does not require the cluster <galaxies> or gas to be in dynamical equilibrium. It does require source-distance information, a lens model and control of point-spread-function-induced shapes, <line-of-sight projection>, unrelated line-of-sight structure and the <mass-sheet degeneracy>.
\b[The convergence of dynamical, hydrostatic and lensing <masses>, exceeding the baryonic inventory, is the principal evidence.] The Bullet Cluster adds a spatial test: collisional X-ray gas lags behind the <galaxies> during the merger, while the inferred lensing concentrations lie nearer the collisionless <galaxy> component. An additional weakly collisional gravitating <mass> accounts naturally for that separation.<A direct empirical proof of the existence of dark matter by Douglas Clowe and others>[L109–L113, optical/X-ray/lensing comparison]{ref} The conclusion assumes a gravitational framework and does not identify the microscopic dark-matter particle; the different tracer systematics should be evaluated rather than treating three <mass> estimates as completely assumption free.
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