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Andrea M. Ghez measures stellar orbits around the supermassive black hole at the center of the Milky Way. Her team's Keck observations combined astrometry with radial velocities for the short-period star S0-2.

An angular orbit alone leaves a mass-distance degeneracy of an angular stellar orbit. Adding Doppler spectroscopy supplies a physical velocity scale, allowing central mass and Euclidean distance to be fitted together. Their 2008 analysis inferred roughly four million solar masses at a Euclidean distance of eight kiloparsecs, and examined how source confusion and assumptions about central motion affected the result.^[ref]

An astrophysical black hole is an astronomical object interpreted as a black hole through evidence about its mass, compactness and interaction with surrounding matter or gravitational waves. A black hole of stellar mass can form after stellar core collapse; a supermassive black hole can grow through black-hole accretion and mergers. The event horizon prevents light from escaping from within, but matter outside can radiate and reveal the central gravity. For Sagittarius A*, astrometry and radial velocities of nearby stars constrain a very large mass inside their small orbits. Such dynamical evidence complements observations of accretion and of binary black holes, which probe different aspects of the black-hole interpretation.^[ref]

Rainer Schödel, Reinhard Genzel and collaborators combined NTT imaging with VLT adaptive-optics infrared observations. Silicon-monoxide masers registered stellar images to the radio position of Sagittarius A*. Their 2002 observations included S2's closest approach; a fitted Kepler orbit constrained a concentrated central mass. Rejecting a particular extended cluster also required its density and survival assumptions.^[ref]

Andrea Ghez and collaborators combined Keck speckle imaging and laser-guide-star adaptive optics with infrared spectra. Image fitting supplied angular positions; spectral lines supplied velocities. The 2008 analysis treated optical distortion, coordinate alignment and blended sources: a shifting photocentre can imitate motion. Joint angular and spectral fitting constrained both distance and mass, with assumptions about central radial motion affecting the result. Apparatus calibration and confusion tests are therefore part of the dynamical inference.^[ref]

Orbital parallax infers a system's Euclidean distance by comparing the angular size of an orbit with its physical size. If an orbital fit gives a deprojected angular semimajor axis $\alpha$ in radians and spectroscopy gives the corresponding physical semimajor axis $a$, the small-angle approximation gives $d=a/\alpha$.

The scales must describe the same orbit. In a binary star, radial-velocity amplitudes, orbital period and eccentricity give physical projected orbital sizes; the astrometric geometry can supply the orbital inclination needed to remove that projection. The mass-distance degeneracy of an angular stellar orbit gives a worked example of why adding a physical velocity measurement changes what can be inferred.^[ref]

A quasar is a luminous active galactic nucleus powered by gas accretion onto a supermassive black hole. The hot disc produces much of the optical/ultraviolet continuum; surrounding gas gives broad and narrow emission lines, while some systems have relativistic jets. A small emitting region is inferred from rapid variability, $R\lesssim c\Delta t_{\mathrm{obs}}/(1+z)$ when beaming is absent. Sustaining a high luminosity from such a compact region favours accretion over ordinary stellar power. For approximately spherical, ionized hydrogen with electron-scattering opacity,Comparing luminosity with this Eddington scale constrains mass only after assumptions about anisotropy, opacity and accretion state. The emitted power obeys $L=\eta \dot{M}{c}^2$; high binding-energy efficiency makes growth of a massive compact accretor energetically plausible.

Broad-line reverberation mapping estimates a gas radius from a continuum-to-line time lag, $R\sim c\tau$, and combines it with line velocity width to infer $M\sim f_{\mathrm{vir}}R{v}^2/G$. Geometry, inclination and nongravitational gas motions enter the factor $f_{\mathrm{vir}}$. Independent evidence for compact massive nuclei comes from resolved stellar or gas orbits in nearby galaxies. Andrea M. Ghez and collaborators' Galactic-centre stellar-orbit study fitted a common central potential and distance, demonstrating a compact dark mass of order four million solar masses; the Galactic centre is a low-luminosity example rather than a quasar.^[ref]

quasars are an accretion phase of massive nuclear black holes; luminosity, variability and dynamical evidence constrain different aspects of that interpretation. A finite-resolution dark-mass measurement by itself is not a direct image of an event horizon.

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.

The strongest pre-2014 evidence combines large gravitating mass with extreme compactness. In an X-ray binary, companion-star radial velocities determine a binary mass function; orbital inclination, donor properties and distance complete the mass inference. An unseen component above conventional neutron-star masses is a black-hole candidate. For example, a 2011 Cygnus X-1 analysis combined optical photometry, spectroscopy and distance information to infer a compact-object mass about $14.8M_{\odot }$, with model and observational uncertainties.^[ref]

Near Sagittarius A*, infrared stellar astrometry and radial velocities trace nearly Keplerian orbits around a dark compact mass. The concentration of several million solar masses in a small volume strongly challenges an extended cluster of ordinary remnants; source confusion, reference-frame errors, distance and possible extended mass must be fitted.^[ref] Gas and maser dynamics provide related compact-mass tests in other nuclei.

accretion explains luminous X-rays, rapid variability and broad relativistic disk features. These constrain size and the strong-field environment, but emission alone is not unique to an event horizon. Lack of surface-like signatures is supporting, model-dependent evidence. Mass, compactness and consistent accretion phenomenology together support the black-hole interpretation; large-radius mass measurements alone do not directly demonstrate a horizon.

Stellar dynamics treats stars as gravitating bodies whose collective motions reveal and reshape their environment. In a collisionless stellar system, stars mainly follow the smooth collective gravitational potential; in a sufficiently dense star cluster, accumulated gravitational encounters produce two-body relaxation. These encounters need not involve physical contact between stars. Observers combine astrometry with radial velocities to constrain orbits and infer unseen mass. The stellar orbits at the Galactic centre provide a particularly direct example: following a short-period star around Sagittarius A* measures the compact central mass, rather than estimating it from the light emitted by its surroundings.^[ref]