The Ecliptic Circle

During the course of a year, the sun's intrinsic motion causes it to trace out a fixed circle bisecting the celestial sphere. This circle is known as the ecliptic. The sun travels around the ecliptic from west to east. Moreover, the ecliptic circle is inclined at a fixed angle of $\epsilon = 23^\circ 26'$ to the celestial equator. This angle actually represents the fixed inclination of the earth's axis of rotation to the normal to its orbital plane.

Figure 5: The ecliptic circle. $P$, $P'$, $Q$, $Q'$, $V$, $V'$, $S$, and $S'$ denote the north celestial pole, south celestial pole, north ecliptic pole, south ecliptic pole, vernal equinox, autumnal equinox, summer solstice, and winter solstice, respectively. $VUV'U'$ is the celestial equator, $VSV'S'$ the ecliptic, and $PP'$ the celestial axis.

The vernal equinox, $V$, is defined as the point at which the ecliptic crosses the celestial equator from south to north (in the direction of the sun's ecliptic motion)--see Fig. 5. Likewise, the autumnal equinox, $V'$, is the point at which the ecliptic crosses the celestial equator from north to south. In addition, the summer solstice, $S$, is the point on the ecliptic which is furthest north of the celestial equator, whereas the winter solstice, $S'$, is the point which is furthest south. It follows that the lines $VV'$ and $SS'$ are perpendicular. Let $QQ'$ be the normal to the plane of the ecliptic which passes through the earth, as shown in Fig. 5. Here, $Q$ is termed the northern ecliptic pole, and $Q'$ the southern ecliptic pole. It is easily demonstrated that

$${\bf s} = \cos\epsilon\,{\bf u}+ \sin\epsilon\,{\bf p},$$ (29)

$${\bf q} = -\sin \epsilon\,{\bf u} + \cos\epsilon\,{\bf p},$$ (30)

where ${\bf s}$ is a unit vector which is directed from the earth to the summer solstice, and ${\bf q}$ a unit vector which is directed from the earth to the north ecliptic pole--see Fig. 5. We can also write

$${\bf u} = \cos\epsilon\,{\bf s} - \sin\epsilon\,{\bf q},$$ (31)

$${\bf p} = \sin\epsilon\,{\bf s} + \cos\epsilon\,{\bf q}.$$ (32)

Figure 6: Ecliptic coordinates. $G$ is the earth, $R$ a celestial object, and $R'$ its projection onto the ecliptic plane, $VR'V'$.