Ecliptic Coordinates

It is convenient to specify the positions of the sun, moon, and planets in the sky using a pair of angular coordinates, $\beta$ and $\lambda$, which are measured with respect to the ecliptic, rather than the celestial equator. Let $R$ denote a celestial object, and $GR'$ the projection of the line $GR$ onto the plane of the ecliptic, $VR'V'$--see Fig. 6. The coordinate $\beta$, which is known as ecliptic latitude, is the angle subtended between $GR'$ and $GR$. Objects north of the ecliptic plane have positive ecliptic latitudes, and vice versa. The coordinate $\lambda$, which is known as ecliptic longitude, is the angle subtended between $GV$ and $GR'$. Ecliptic longitude increases from west to east (i.e., in the same direction that the sun travels around the ecliptic). (Again, in this treatise, $\lambda$ is measured relative to the mean equinox at date, unless specified otherwise.) Note that the basis vectors in the ecliptic coordinate system are ${\bf v}$, ${\bf s}$, and ${\bf q}$, whereas the corresponding basis vectors in the celestial coordinate system are ${\bf v}$, ${\bf u}$, and ${\bf p}$--see Figs. 3 and 5. By analogy with Eqs. (1)-(3), we can write

$${\bf r} = \cos\beta\,\cos\lambda\,{\bf v} + \cos\beta\,\sin\lambda\,{\bf s} +\sin\beta\,{\bf q},$$ (8)

$$\sin\beta = {\bf r}\cdot {\bf q},$$ (9)

$$\tan\lambda = \left(\frac{{\bf r}\cdot{\bf s}}{{\bf r}\cdot {\bf v}}\right),$$ (10)

where ${\bf r}$ is a unit vector which is directed from $G$ to $R$. Hence, it follows from Eqs. (1), (4), and (5) that

$$\sin\beta = \cos\epsilon\,\sin\delta - \sin\epsilon\,\cos\delta\,\sin\alpha,$$ (11)

$$\tan\lambda = \frac{\cos\epsilon\,\cos\delta\,\sin\alpha+\sin\epsilon\,\sin\delta}{\cos\delta\,\cos\alpha}.$$ (12)

These expressions specify the transformation from celestial to ecliptic coordinates. The inverse transformation follows from Eqs. (2), (3), and (6)-(8):

$$\sin\delta = \cos\epsilon\,\sin\beta +\sin\epsilon\,\cos\beta\,\sin\lambda,$$ (13)

$$\tan\alpha = \frac{\cos\epsilon\,\cos\beta\,\sin\lambda-\sin\epsilon\,\sin\beta }{\cos\beta\,\cos\lambda}.$$ (14)

Figures 15 and 16 show all stars of visible magnitude less than $+6$ lying within $15^\circ$ of the ecliptic. Table 1 gives the ecliptic longitudes, ecliptic latitudes, and visible magnitudes of a selection of these stars which lie within $10^\circ$ of the ecliptic. The figures and table can be used to convert ecliptic longitude and latitude into approximate position in the sky against the backdrop of the fixed stars.