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An observer’s celestial meridian is the great circle containing the zenith and the celestial poles. Its upper half passes through the local north and south directions.

The hour angle of a direction is the westward angle from the upper celestial meridian to its hour circle. In a consistent equinox convention, $h=\mathrm{LST}-\alpha$ modulo one turn, where $\alpha$ is right ascension and LST is local sidereal time.

Local sidereal time is the right ascension on the upper celestial meridian. Since right ascension increases eastward while hour angle increases westward, their difference isOne hour of either coordinate is $15^{\circ }$. For a fixed stellar direction, $h$ increases at the sidereal rotation rate. Mean sidereal time should be used with mean right ascension, or apparent sidereal time with apparent right ascension, for the same reference equinox.

Use azimuth measured from north toward west, and positive hour angle westward. These conventions are forced by the signs in the coordinate formulae. In the diagram the observer is at $O$, the local vertical meets the celestial sphere at the zenith $Z$, and the north celestial pole is $P$. The astronomical horizon is perpendicular to $OZ$; the celestial equator is perpendicular to $OP$. Their intersections with the celestial meridian give the north horizon point and the upper equatorial meridian point.

The north celestial pole has altitude $\theta$; thus the angle between $OP$ and $OZ$ is $90^{\circ }-\theta$. Project $X$ along its vertical great circle onto the astronomical horizon at $H$: the arc $HX$ is the altitude $\gamma$, and the westward horizon arc from north to $H$ is the azimuth $\eta$. Project $X$ along its hour circle onto the celestial equator at $Q$: the arc $QX$ is the declination $\delta$, and the westward equatorial arc from the upper celestial meridian to $Q$ is the hour angle $h$. Equivalently the triangle on the celestial sphere $PZX$ has sides $PZ=90^{\circ }-\theta$, $PX=90^{\circ }-\delta$ and $ZX=90^{\circ }-\gamma$. All five angles refer to arcs on their specified reference circles, not arbitrary angles in the projected drawing.

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.