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Astrometry measures the positions and motions of celestial objects. Repeated position measurements can distinguish stellar parallax, caused by the observer's motion, from proper motion of the source across the sky.

An astrometry measurement needs a date and a reference frame, as well as an angle. In an equatorial coordinate system, right ascension and declination label two viewing directions. A catalogue's reference date states when its quoted position applies; comparing entries at different dates without accounting for proper motion can make the same star appear misplaced. The observational problem is to infer a consistent history from successive incoming light directions, while accounting for the observer's changing position and measurement error.

To see the separation clearly, consider just one angular coordinate $\theta$, with $t$ the numerical elapsed time in years. Assume a simplified circular orbit of the Earth, a fixed source Euclidean distance, constant proper motion in this coordinate and no other accelerations. Choose the direction and the starting date so that the annual stellar parallax projection is the cosine termHere ${\theta }_0$ is the coordinate without the annual stellar parallax offset at the reference date; the observed $\theta (0)$ is ${\theta }_0+p$. The proper motion coefficient $\mu$ is quoted in arcseconds per year, and $p$ is the annual stellar parallax in arcseconds. This is an illustrative choice of a full-amplitude projection. General sky directions have different projected amplitudes and phases in their two coordinates; their apparent paths need not be circles. Real astrometry uses the observer's calculated position rather than imposing this single cosine. ^[ref].

Set ${\theta }_0=0$, $\mu =100\mathrm{mas}\mathrm{y}{\mathrm{r}}^{-1}$ and $p=50\mathrm{mas}$, where $1\mathrm{mas}$ is one thousandth of an arcsecond. The model gives $\theta (0)=50$, $\theta (0.5)=0$ and $\theta (1)=150\mathrm{mas}$. Same-season observations one year apart differ by $100\mathrm{mas}$ because the annual stellar parallax contribution repeats. Subtracting that fitted straight trend leaves an annual signal between $-50$ and $+50\mathrm{mas}$. The existing stellar parallax distance relation then gives $20$ parsecs.

Observing only at the same annual phase would recover the proper motion trend but leave $p$ indistinguishable from a constant offset. Sampling several seasons as well as several years separates the repeating signal from the accumulating displacement. The two signatures answer different questions using the same measurements.

Astronomy studies celestial objects and phenomena, including stars, planets and galaxies. Astrometry measures their positions and motion; astrophysics uses physics to explain their properties and behaviour.

Much of astronomy begins with incoming light. An image tells us which direction the light arrived from, rather than directly providing the source's three-dimensional position. Two stars close together on the sky may be very far apart in space. The celestial sphere organizes these viewing directions; locating an object on it is a different task from measuring how far away that object is. Repeated observations turn this limitation into an opportunity: the changing direction of light contains evidence about both Euclidean distance and motion.

The particularly satisfying combination is stellar parallax with proper motion. The first supplies a geometric Euclidean distance, while the second measures how rapidly the viewing direction changes after accounting for the observer's annual motion. Together they give a transverse speed, connecting tiny angles on an image to kilometres travelled each second. This bridge from a flat sky map to moving stars is one reason astrometry matters to the study of the Milky Way.

Angular astrometry determines the combination displayed by substituting $a=D\alpha$ into Kepler's third law. Keeping the fitted shape, orientation and orbital period fixed, the rescalingleaves the entire angular time series unchanged. A longer physical orbit around a larger mass can trace precisely the same angular path. More angular samples improve the fitted combination; by themselves they cannot break this scaling symmetry of the model.

The radial-velocity semi-amplitude isFor $\sin i\ne 0$, combining it with the angular fit gives the orbital parallax distanceThe spectral velocity scale supplies the missing physical ruler. For a face-on orbit, $\sin i=0$, this radial signal vanishes and that method cannot supply a distance. In a partial or noisy data set, fitted orientation and systemic motion also correlate with the inferred scale.

Methods of detecting exoplanets infer or resolve an exoplanet through its effects on the host star's light or through its own light. The observable determines what can be learned:

An exoplanet secondary eclipse measures the loss of the exoplanet's light when it passes behind its star, complementing exoplanet transit photometry. A detection method also selects which systems are easiest to find: deeper exoplanet transits favour larger radius ratios, and larger stellar radial-velocity semi-amplitudes favour more massive companions, shorter orbital periods, and larger $\sin i$. These preferences must be accounted for when inferring the underlying exoplanet population.

Atmospheric refractive-index fluctuations impose a changing phase screen on the pupil. For a telescope much larger than the Fried parameter $r_0$, long-exposure resolution is approximately $0.98\lambda /r_0$, rather than the diffraction scale $\lambda /D$. The scaling $r_0\propto {\lambda }^{6/5}$ makes the seeing width proportional to ${\lambda }^{-1/5}$ for fixed turbulence. An atmospheric coherence time sets the exposure or correction cadence needed to freeze the phase, while the isoplanatic angle sets the useful angular separation from a reference source. These are atmospheric quantities, not improvements automatically supplied by a larger mirror.

First reduce avoidable turbulence: choose a good high site, control mirror and enclosure temperatures, ventilate the dome, and observe when the line of sight crosses less turbulent air. Guiding removes telescope drift, and a fast tip-tilt correction removes image motion. Slow active optics corrects mirror figure and alignment; it cannot by itself follow all atmospheric phase fluctuations. These measures improve the input wavefront but do not guarantee diffraction-limited performance.

In adaptive optics, a wavefront sensor measures the reference-star wavefront, a reconstructor estimates its aberration, and a fast control loop commands a deformable mirror and tip-tilt element. A Shack-Hartmann sensor obtains local slopes from the displacement of focused lenslet spots. The science beam receives the corresponding optical-path correction. Residual phase variance limits the Strehl ratio approximately as $S\simeq e^{-{\sigma }_{\varphi }^2}$ when the residual is small. Enough corrected spatial modes, adequate reference photons and a response faster than atmospheric changes are all needed. infrared correction is easier because a given path error has smaller phase in radians and the atmospheric coherence length is larger. A faint or distant guide star leaves larger errors and a core-plus-halo point spread function. The NACO manual documents these reference-star and anisoplanatic limitations.^[ref]

A laser guide star supplies a wavefront reference in fields lacking a bright suitable star, but a finite-altitude beacon samples a cone rather than the full stellar cylinder. It also does not by itself supply the absolute celestial tip-tilt reference; a natural source is still required for that measurement. Multiple guide stars and conjugate correction planes can improve wider-field compensation, with greater hardware and reconstruction demands. The point spread function can vary across the field and with time, so reference stars and telemetry are needed to calibrate astrometry and photometry; a sharper image alone is not a guarantee of accurate science.^[ref]

Short-exposure methods provide another route. Exposures shorter than the atmospheric coherence time freeze speckle patterns. speckle interferometry uses calibrated Fourier statistics of many frames to recover diffraction-scale information; phase information requires methods such as bispectrum reconstruction, not just averaging the blurred frames. lucky imaging selects the best short frames, registers them and combines them. It sacrifices photons through selection and can be read-noise limited for faint targets, so it is most effective with enough reference photons. Ordinary shift-and-add mainly corrects motion; it does not remove every high-order phase error. Deconvolution using a measured point spread function can recover supported structure but amplifies noise and cannot reliably invent spatial frequencies absent from the observations. Good-site engineering, fast wavefront correction and short-exposure reconstruction address different limitations and can be combined.

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.

In general relativity, light follows null geodesics. A weak gravitational field outside an isolated spherical mass gives the Schwarzschild light deflectionfor impact parameter $b\gg GM/c^2$. At the solar limb this is approximately $1.75$ arcseconds. A Newtonian corpuscular calculation gives half this value; the full result depends on spatial curvature as well as the gravitational time dilation. The parametrized post-Newtonian formalism predicts $\alpha =2(1+{\gamma }_{\mathrm{PPN}})GM/(b{c}^2)$, with ${\gamma }_{\mathrm{PPN}}=1$ in general relativity.

The Eddington experiment compared photographic star positions near the eclipsed Sun with reference exposures taken without the Sun in that field. Rays bend toward the Sun, but their backward extrapolations place the apparent stellar images farther from it. Plate scale, focus changes, atmospheric refraction and comparison-field alignment must be separated from a radial displacement proportional to $1/b$. The original report discusses the instruments, plate reductions and competing predicted deflections; the test is a quantitative comparison with uncertainty, not a unique proof of a gravitational theory.^[ref]

radio interferometry permits an independent test without an eclipse. A 2005 Very Long Baseline Array experiment observed four compact sources at 43, 23 and 15 GHz. Differential astrometry suppressed common antenna errors, while a fit $\mathbf{p}(\nu )={\mathbf{p}}_0+\mathbf{I}{\nu }^{-2}$ separated refraction in the solar corona from achromatic gravitational deflection. Tropospheric errors and radio-source structure also required calibration. The reported ${\gamma }_{\mathrm{PPN}}=0.9998\pm 0.0003$ was consistent with general relativity.^[ref]

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.

Proper motion describes the change in a celestial object's direction on the sky due to its motion relative to the observer. Its angular velocity differs from the annual stellar parallax signal caused by the observer's orbit.

Thus proper motion is an angular velocity, rather than a speed in kilometres per second. It describes the relative transverse motion of source and observer in a specified reference frame. A star moving towards us can have little proper motion, while a nearer star can show larger proper motion than a faster but more distant one. In the example above, the observed coordinate actually decreases over the first half-year, although $\mu$ is positive: the annual stellar parallax excursion temporarily outweighs the steady drift. A short observed displacement alone therefore need not estimate proper motion well.

Real catalogues give two proper motion components. For right ascension $\alpha$ and declination $\delta$, the angular velocities in perpendicular directions on the local sky areThe cosine factor accounts for the shorter east-west arc associated with a given right ascension increment away from the celestial equator. ESA's Gaia DR3 fields pmra and pmdec already contain these two components; pm is their total proper motion. Do not multiply pmra by another cosine. Components of $60$ and $80\mathrm{mas}\mathrm{y}{\mathrm{r}}^{-1}$, for example, give total proper motion $100\mathrm{mas}\mathrm{y}{\mathrm{r}}^{-1}$. ^[ref].

For nearby stars in the ordinary nonrelativistic approximation, Euclidean distance $d$ converts total proper motion into transverse speed $v_{\perp }$:The small-angle approximation gives transverse speed as Euclidean distance times angular velocity in radians per unit time. With arcseconds and parsecs, the parsec convention turns their product into astronomical units per year. One astronomical unit per Julian year of $365.25$ days is approximately $4.74047\mathrm{km}{\mathrm{s}}^{-1}$. The coefficient is a unit conversion, rather than a property of a particular star.

Taking the model's $100\mathrm{mas}\mathrm{y}{\mathrm{r}}^{-1}=0.100\mathrm{arcsec}\mathrm{y}{\mathrm{r}}^{-1}$ as the total proper motion, with the other component zero, gives $v_{\perp }=4.74047\times 0.100\times 20\simeq 9.48\mathrm{km}{\mathrm{s}}^{-1}$. At the same total proper motion, Euclidean distances of $10$, $20$ and $100$ parsecs instead give transverse speeds of $4.74$, $9.48$ and $47.40\mathrm{km}{\mathrm{s}}^{-1}$, respectively. Their stellar parallax values are $100$, $50$ and $10\mathrm{mas}$. The same annual angular displacement can therefore describe quite different journeys through space.

The transverse speed still omits the line-of-sight velocity, which can be measured through the Doppler effect in spectral lines. Combining these measurements permits a three-dimensional velocity estimate in a consistent reference frame. ^[ref].

Historically, Edmond Halley's 1718 paper compared contemporary and ancient stellar directions and argued that some supposedly fixed stars had moved. Its title, "Considerations on the change of the latitudes of some of the principal fixt stars", captures the conceptual surprise: long-separated measurements could reveal motion invisible in an ordinary night's watching. Modern astrometry retains that insight, using repeated precise measurements to turn the apparently static sky into a record of changing positions. ^[ref]. ^[ref].

The useful idea is to weigh an unseen object through its effect on a visible tracer. A curved stellar orbit carries more information than a single fast-moving star: its changing position and velocity constrain a common centre, a length scale and a clock. The Galactic-centre stellar-orbit experiments combine astrometry with Doppler spectroscopy; the Galactic-centre point-mass orbit model below shows how the two measurements complement one another.

Stellar parallax is the apparent change in a star's viewing direction caused by the observer's changing position. Nearby stars shift more against a distant reference background. In an ideal circular Earth orbit, the conventional annual stellar parallax $p$ uses a baseline of one astronomical unit, approximately the Earth-Sun distance. Two opposite observing positions, about six months apart, are separated by approximately two astronomical units. Their maximum angular separation is $2p$, so $p$ is half that excursion, not the full January-to-July shift. The observed excursion depends on the projected baseline and observing dates. ^[ref].

For the perpendicular baseline in the diagram, a right triangle giveswhere $p$ in the final expression is in radians. Combining the small-angle approximation with the parsec convention givesThus $p=0.05$ arcseconds implies $d\simeq 20$ parsecs, while the ideal six-month excursion is $0.10$ arcseconds. Geometry supplies this distance without assuming a luminosity or a particular stellar atmosphere.

In practical astrometry, many observations separate the annual stellar parallax signal from proper motion, the star's own changing sky position. The Gaia DR3 catalogue reports stellar parallax and its uncertainty in milliarcseconds: $50\mathrm{mas}=0.05$ arcseconds. A catalogue angle is a fitted measurement with measurement error. ^[ref].

For a positive stellar parallax with small standard deviation ${\sigma }_p$, first-order propagation gives ${\sigma }_d/d\simeq {\sigma }_p/p$. Thus $p=50\mathrm{mas}$ with ${\sigma }_p=1\mathrm{mas}$ gives $d\simeq 20$ parsecs with ${\sigma }_d\simeq 0.4$ parsecs, assuming systematic offsets are negligible. This linear estimate requires ${\sigma }_p\ll p$; a reciprocal is nonlinear and does not generally preserve symmetric uncertainty intervals.

The motion of a star around the astronomical barycenter caused by an orbiting companion. In an isolated two-body problem with relative semi-major axis $a$, the stellar semi-major axis is $a_{\star }=M_{p}a/(M_{\star }+M_p)$. Doppler spectroscopy measures its line-of-sight velocity and astrometry measures its projected position on the sky.

The Very Long Baseline Array is a radio-telescope array used for very-long-baseline interferometry. In a solar-deflection experiment, differential astrometry of nearby compact sources compares their apparent directions as their angular separation from the Sun changes. Measurements at several frequencies can distinguish dispersive refraction in the solar corona from achromatic gravitational deflection.^[ref]