= Equatorial coordinate system
{wiki}
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. https://www.astro.uvic.ca/~tatum/celmechs/celm6.pdf[Jeremy Tatum: Celestial mechanics, chapter 6, the celestial sphere]{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> is
$$
\mathbf n=(\cos\delta\cos\alpha,\ \cos\delta\sin\alpha,\ \sin\delta).
$$
This 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 $\alpha=4^{\rm h}=60^\circ$, $\delta=30^\circ$ gives $\mathbf n=(\sqrt3/4,3/4,1/2)$; 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 $\operatorname{atan2}(n_y,n_x)$ 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.
\Image[/equatorial-coordinates-reference-geometry.png]
{title=Analytical geometry of right ascension, declination and the shrinking east-west arc toward a celestial pole}
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. https://aa.usno.navy.mil/faq/ICRS_doc[US Naval Observatory: Reference systems, catalogue epochs and fixed celestial axes]{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> $H$ obeys
$$
H=\Theta-\alpha\pmod{2\pi}.
$$
At 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. https://aa.usno.navy.mil/faq/GAST[US Naval Observatory: Mean and apparent sidereal time and the local longitude correction]{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. https://aa.usno.navy.mil/faq/ICRS_doc[US Naval Observatory: Transforming catalogue directions into observable directions]{ref}.
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