The celestial equator, , is the intersection of the earth's equatorial plane with the celestial sphere, and is therefore perpendicular to the celestial axis. The so-called vernal equinox, , is a particular point on the celestial equator that is used as the origin of celestial longitude. Furthermore, the autumnal equinox, , is a point which lies directly opposite the vernal equinox on the celestial equator. Let the line lie in the plane of the celestial equator such that it is perpendicular to , as shown in the figure.
It is helpful to define three, right-handed, mutually perpendicular, unit vectors: , , and . Here, is directed from the earth to the vernal equinox, from the earth to point , and from the earth to the north celestial pole--see Fig. 3.
Consider a general celestial object, --see Fig. 4. The location of on the celestial sphere
is conveniently specified by two angular coordinates, and . Let be the
projection of onto the equatorial plane. The coordinate , which is known as declination, is the angle subtended between and . Objects north of the celestial equator have positive
declinations, and vice versa. It follows that objects on the celestial equator have declinations of ,
whereas the north and south celestial poles have declinations of and , respectively.
The coordinate , which is known as right ascension, is the angle subtended between
and . Right ascension increases from west to east (i.e., in the opposite direction to the
celestial sphere's diurnal rotation). Thus, the vernal and autumnal equinoxes have right ascensions of and , respectively. Note that lies in the range to
. Right ascension is sometimes measured in hours, instead of degrees, with one hour corresponding to
(since it takes 24 hours for the celestial sphere to complete
one diurnal rotation). In this scheme, the vernal and autumnal equinoxes
have right ascensions of hrs. and hrs., respectively. Moreover, lies in the
range to 24 hrs. (Incidentally, in this treatise, is measured
relative to the mean equinox at date, unless otherwise specified.)
Finally, let be a unit vector which is directed from the earth to --see Fig. 4. It is easily