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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.

An astronomical epoch is a reference instant for astronomical data. A catalogue's position astronomical epoch says when the listed source direction applies; proper motion advances that direction to another instant. The reference date for an equatorial coordinate system's axes need not equal the position astronomical epoch. ESA's Gaia documentation explicitly associates both right ascension and declination with its reference-epoch field. ^[ref].

An equatorial coordinate system labels a viewing direction on the celestial sphere by right ascension $\alpha$, increasing eastward around the reference celestial equator, and declination $\delta$, increasing northward from that celestial equator. These two angles locate a direction, without specifying the object's Euclidean distance. In the traditional convention, zero right ascension points toward the reference equinox, the northward crossing of the Sun's annual path through the celestial equator. An hour circle through the direction supplies the north-south arc on which declination is measured. ^[ref].

For a concrete construction, choose Cartesian coordinates with $x$ pointing toward the reference equinox, $z$ toward the north reference celestial pole, and positive $y$ given by the cross product of the positive $z$ and $x$ directions. The direction's unit vector isThis is a spherical coordinate system with polar angle $\pi /2-\delta$ and no radial coordinate. The trigonometric functions require consistent angular units: convert right ascension in hours to radians before evaluating them. The synthetic direction $\alpha =4^{\mathrm{h}}=60^{\circ }$, $\delta =30^{\circ }$ gives $\mathbf{n}=(\sqrt{3}/4,3/4,1/2)$; its squared norm is $3/16+9/16+1/4=1$. A change of reference axes applies a rotation to this unit vector before recovering the two angles. Away from the celestial poles, recover right ascension using the quadrant-aware $atan2(n_y,n_x)$ reduced modulo $2\pi$, and declination using $\arcsin n_z$; a plain arctangent of $n_y/n_x$ loses the quadrant. Adding a common offset to every right ascension describes only the special case of a rotation about the same polar axis.

The reference specification matters. For an equatorial coordinate system tied to the Earth, precession changes the reference celestial equator and reference equinox, and nutation distinguishes their mean and true versions. The date specifying those axes is distinct from the astronomical epoch at which a catalogue's source directions apply: advancing a star's proper motion and changing reference axes are separate operations. A modern catalogue may instead use fixed reference frame axes, aligned approximately with the traditional J2000.0 axes, without a moving reference equinox. Thus a pair of numbers labelled merely "J2000" does not settle every convention. State the reference frame, the astronomical epoch, and whether a direction is catalogued or apparent. ^[ref].

During one night's elementary fixed-direction approximation, a star's right ascension and declination remain constant while its altitude, azimuth and hour angle change because the Earth turns. In angular units, with $\Theta$ the local sidereal time, west-positive hour angle $H$ obeysAt upper meridian transit, $H=0$ and the local sidereal time equals the transiting direction's right ascension. Pair apparent sidereal time with apparent right ascension referred to the true celestial equator and reference equinox of date; mean sidereal time refers to the mean reference equinox. Raw fixed-frame catalogue coordinates require conversion before precision pointing. ^[ref].

This separation makes the equatorial coordinate system useful: a catalogue can identify a target independently of the observer's current astronomical horizon, while local sidereal time tells the observer which hour circle is on the celestial meridian. Precision observations also account for proper motion, stellar parallax, aberration of light and atmospheric refraction; the fixed-direction construction above omits those effects. A reference-axis rotation preserves angles between directions, whereas changing the observer or correcting atmospheric refraction is a different operation. ^[ref].

A horizontal coordinate system describes a viewing direction using altitude $\gamma$ above the observer's astronomical horizon and azimuth $\eta$ around it. Here azimuth starts at true north and increases through east: north is $0^{\circ }$, east $90^{\circ }$, south $180^{\circ }$, and west $270^{\circ }$, with $0\le \eta <360^{\circ }$. True north is the northward intersection of the celestial meridian with the astronomical horizon, rather than an uncorrected magnetic-compass reading. Other origins and directions are legitimate, but their formulae must be converted consistently. ^[ref].

Construct a local orthonormal basis of east ${\mathbf{e}}_E$, north ${\mathbf{e}}_N$, and upward ${\mathbf{e}}_U$, with ${\mathbf{e}}_E\times {\mathbf{e}}_N={\mathbf{e}}_U$. The upward unit vector points to the zenith; the horizontal plane is perpendicular to it. For a ground instrument, a freely hanging weight establishes the local vertical through gravity. In an elementary spherical-Earth model, this vertical is radial; precision work distinguishes it from the normal to a reference surface. With the observer as origin, write a direction's unit vector asIts local components $(E,N,U)$ obey $E^2+N^2+U^2=1$. Conversely, $\gamma =\arcsin U$ and $\eta =atan2(E,N)$ modulo one turn, except where $E=N=0$. The quadrant-aware arctangent takes east as its first argument and north as its second. A synthetic direction with $\eta =60^{\circ }$ and $\gamma =30^{\circ }$ has $(E,N,U)=(3/4,\sqrt{3}/4,1/2)$. Thus its horizontal component has norm $\sqrt{3}/2$; its altitude is not the angle of the projected arrow in the horizontal plane.

A horizontal coordinate system depends on the observer's site and the observation time. The Earth's rotation changes the local axes relative to a fixed distant direction, while changing the site changes the zenith and the astronomical horizon. This makes the horizontal coordinate system directly useful for pointing an optical telescope and checking whether terrain blocks a target. A pair of angles without a site, time, azimuth convention, and geometric or refracted interpretation is an incomplete pointing instruction.

For a check against an equatorial coordinate system, let $\theta$ be the observer's geographical latitude, $\delta$ the direction's declination, and $h$ its west-positive hour angle, obtained from local sidereal time minus right ascension. In the spherical-Earth, unrefracted fixed-direction model, projection onto the local orthonormal basis givesTaking the dot product with the upward unit vector gives the last formula; the other two follow by the same projection. Recover altitude and azimuth from $(E,N,U)$ as above, using radians inside software trigonometric functions. The minus sign in $E$ is essential: a negative hour angle puts this synthetic direction east of the celestial meridian. ^[ref].

For illustrative geographical latitude $\theta =45^{\circ }$ and invented declination $\delta =15^{\circ }$, hour angle $h=-60^{\circ }$ gives $\gamma \simeq 31.636^{\circ }$ and $\eta \simeq 100.729^{\circ }$. At $h=+60^{\circ }$, the same altitude accompanies $\eta \simeq 259.271^{\circ }$: the direction is now westward. At $h=0$, it is due south with altitude $60^{\circ }$, agreeing with $90^{\circ }-|\theta -\delta |$ at upper meridian transit. These are analytical checks, not present positions of any star. For nearby objects, the direction must also be referred to the observer; simply rotating a geocentric direction omits the change of viewing position. Catalogue-to-observation refinements are described under equatorial coordinate system.

Let $\mathbf{n},\mathbf{w},\mathbf{u}$ be orthonormal unit vectors north, west and upward. A direction of altitude $\gamma$ and westward azimuth $\eta$ isThe north celestial pole and the upper equatorial meridian direction areIn the equatorial coordinate system, the same unit vector is $\mathbf{x}=\sin \delta \mathbf{p}+\cos \delta (\cos h\mathbf{e}+\sin h\mathbf{w})$. Taking dot products with $\mathbf{p},\mathbf{w},\mathbf{e}$ respectively givesThis is an orthogonal transformation between two bases, so it preserves the length of the direction vector. Recover $h$ from both its sine and cosine to keep the correct quadrant. At the zenith, azimuth itself is undefined; the vector equations remain meaningful by continuity. Using the more usual eastward azimuth changes the sign of the second equation.

Local sidereal time is the right ascension on the upper celestial meridian. A sidereal day is approximately $1436.07$ minutes; its angular rate is $2\pi /T_{\mathrm{sid}}$. Mean and apparent sidereal time must be paired with corresponding equatorial coordinate systems.