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The intracluster medium is diffuse hot plasma between the galaxies of a galaxy cluster. Its X-ray radiation includes thermal bremsstrahlung and atomic line emission. Measured temperature and density profiles supply a cluster hydrostatic mass estimator when approximately thermal hydrostatic equilibrium holds. The medium supplies much of a cluster's baryonic mass, but this need not equal its total gravitating mass.

The three probes constrain the same deep gravitational potential by different observations. A large member-galaxy velocity dispersion requires a large binding mass. Applying the virial theorem, with a gravitational scale radius $R$ and approximately isotropic orbits, gives a cluster velocity-dispersion mass estimate of order $M\sim R{\sigma }_v^2/G$, with a profile-dependent numerical coefficient. The inferred mass-to-light ratio is much larger than the stellar value. Membership, orbital anisotropy and equilibrium assumptions affect the precision.

The hot intracluster medium radiates through thermal bremsstrahlung and line emission. Its X-ray spectrum measures temperature, while its surface brightness constrains gas density. Confining this gas requires a potential with characteristic $GM/R\sim k_{B}T/(\mu m_p)$. Combining resolved profiles through the cluster hydrostatic mass estimator gives the total mass, not just the gas mass. The detected gas contributes substantial baryonic mass but does not normally account for the full gravitating mass. Nonthermal pressure or a merging cluster can invalidate a purely thermal hydrostatic equilibrium estimate.

Finally, gravitational lensing measures mass through light deflection. Strong gravitational lensing supplies multiple images and arcs; weak gravitational lensing measures coherent background-image distortion over larger radii. The lensing convergence is projected surface mass density divided by the critical surface density, irrespective of whether the material emits light. Cluster gravitational-lensing mass estimates agree broadly with the high masses implied by galaxy motions and hot gas, while having different equilibrium assumptions. Together these observations support a dominant nonluminous component in galaxy clusters, after accounting for their stars and observed gas.

Let ${\rho }_g(r)$ be the gas density and $M(<r)$ the total gravitating mass. For a spherically symmetric gas in hydrostatic equilibrium, radial force balance isUsing the ideal gas law $P={\rho }_g{k}_{B}T/(\mu m_p)$, with constant mean molecular weight $\mu$, differentiate the product rather than setting $T$ constant:Substitution and cancellation of ${\rho }_g$ give the cluster hydrostatic mass estimatorHere $k_B$ is the Boltzmann constant, denoted $K$ in the question. The prefactor has units of mass; a pressure decreasing outward makes the bracket negative and the inferred mass positive. For an isothermal gas with ${\rho }_g\propto r^{-\beta }$, this reduces to $M(<r)=\beta k_{B}Tr/(\mu m_{p}G)$. The density in the logarithmic derivative is the gas density, not the total density. A varying $\mu$ or nonthermal pressure would require the corresponding extra terms.

Three tracers give complementary estimates of a cluster's gravitating mass. First, galaxy redshifts measure a line-of-sight velocity dispersion ${\sigma }_{\mathrm{los}}$. For a roughly stationary bound cluster, the virial theorem gives a cluster velocity-dispersion mass estimatewith 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,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,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.

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.^[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.