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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.

mass from motion. A resolved Keplerian orbit gives $M=4{\pi }^2{a}^3/(G{P}_{\mathrm{orb}}^2)$ when the orbiting object has negligible mass. At the Galactic centre, repeated infrared astrometry and spectroscopy measure sky motion, acceleration and line-of-sight velocity of stars around Sagittarius A*. Resolving a compact enclosed mass and fitting complete orbits avoids reliance on a single projected velocity. Source confusion, reference-frame stability, distance, inclination and possible extended mass must be included in the fit.^[ref]

For an unresolved stellar-mass binary in a circular orbit, the donor's velocity semiamplitude $K_2$ and orbital period give the binary mass functionIt provides a lower mass bound; donor mass and orbital inclination are needed for an actual mass estimate. For an eccentric orbit, multiply the observational combination $P_{\mathrm{orb}}{K}_2^3/(2\pi G)$ by $(1-e^2{)}^{3/2}$ when evaluating the binary mass function. Resolved gas or maser rotation and stellar-dynamical models also measure masses, provided pressure support, noncircular motion and line-of-sight projection are modeled. Resolving the black-hole sphere of influence improves separation of central mass from the host galaxy.

For active nuclei, reverberation mapping measures a lag $\tau$ between variable continuum and broad-line response. With a line velocity scale $\Delta v$,The factor $f_{\mathrm{geom}}$ accounts for geometry and kinematics and is not known simply from the lag. Cadence, line responsivity, inclination and winds require control. Rapid variability demonstrates compactness but is not by itself a unique mass measurement.

black-hole spin from strong-field emission. Define $a_{\ast }=cJ/(G{M}^2)$. Large-radius Keplerian motion is principally a mass probe and usually insensitive to black-hole spin. In the Kerr model, the radius and binding energy of the innermost stable circular orbit vary with $a_{\ast }$ and orbital orientation, making inner-disk radiation useful.

black-hole continuum fitting infers an inner emitting radius from the disk's thermal spectrum and flux normalization, then identifies it with the innermost stable circular orbit in a relativistic thin-disk model. Independent mass, distance and inclination, together with atmospheric spectral hardening, absorption and contamination, are needed. Spectra should be selected in a disk-dominated regime where truncation and strong advection are limited. A 2011 Cygnus X-1 analysis combined such spectra with independently constrained binary parameters and reported a near-extreme black-hole spin conditional on black-hole spin–orbit alignment and its disk model.^[ref]

black-hole reflection spectroscopy instead models a relativistically broadened iron-line and reflection continuum: orbital Doppler shifts and beaming broaden the profile, while gravitational redshift contributes to the extended red wing. Its innermost emitting radius can constrain black-hole spin if reflection extends to the stable inner edge. Ionization, abundance, absorption, coronal illumination, disk thickness and possible truncation are competing parameters; a broad line is not a direct black-hole spin reading independent of these assumptions. Timing or precession models can add information but introduce their own dynamical identification.

mass is principally inferred from gravitating dynamics; black-hole spin is inferred from strong-field effects with a specified spacetime and accretion model. Agreement between independent methods is especially valuable because disk inner radius, inclination and radiation-transfer assumptions can otherwise mimic a change of black-hole spin.