Horizontal Coordinates

It is convenient to specify the positions of celestial objects in the sky, when viewed from a particular observation site, $X$, on the earth's surface, using a pair of angular coordinates, $a$ and $A$, which are measured with respect to the local horizon. Let $R$ denote a celestial object, and $XR'$ the projection of the line $XR$ onto the horizontal plane, $NESW$--see Fig. 11. The coordinate $a$, which is known as altitude, is the angle subtended between $XR'$ and $XR$. Objects above the horizon have positive altitudes, whereas objects below the horizon have negative altitudes. The zenith has altitude $90^\circ$, and the horizon altitude $0^\circ$. The coordinate $A$, which is known as azimuth, is the angle subtended between $XN$ and $XR'$. Azimuth increases from the north towards the east. Thus, the north, east, south, and west compass points have azimuths of $0^\circ$, $90^\circ$, $180^\circ$, and $270^\circ$, respectively. Note that the basis vectors in the horizontal coordinate system are ${\bf e}$, ${\bf n}$, and ${\bf z}$, whereas the corresponding basis vectors in the celestial coordinate system are ${\bf v}$, ${\bf u}$, and ${\bf p}$--see Figs. 3 and 8. By analogy with Eqs. (1)--(3), we can write

$${\bf r} = \cos a\,\sin A\,{\bf e} + \cos a\,\cos A\,{\bf n} + \sin a\,{\bf z},$$ (27)

$$\sin a = {\bf r}\cdot {\bf z},$$ (28)

$$\tan A = \left(\frac{{\bf r}\cdot {\bf e}}{{\bf r}\cdot {\bf n}}\right),$$ (29)

where ${\bf r}$ is a unit vector directed from $X$ to $R$. Hence, it follows from Eqs. (1), and (22)-(23), that

$$\sin a = \sin L\,\sin \delta + \cos L\,\cos \delta\,\cos(\alpha-\alpha_0),$$ (30)

$$\tan A = \frac{\cos \delta\,\sin (\alpha-\alpha_0)}{\cos L\,\sin \delta - \sin L\,\cos\delta\,\cos(\alpha - \alpha_0)}.$$ (31)

These expressions allow us to calculate the altitude and azimuth of a celestial object of declination $\delta$ and right ascension $\alpha$ which is viewed from an observation site on the earth's surface of terrestrial latitude $L$ at an instant in time when celestial objects of right ascension $\alpha_0$ are culminating at the meridian. According to Eqs. (8), and (25)-(26), the altitude and azimuth of a similarly viewed point on the ecliptic (i.e., $\beta = 0$) of ecliptic longitude $\lambda$ are given by

$$\sin a = \cos L\,\cos\lambda\,\cos \alpha_0+ \sin L\,\sin\epsilon\,\sin \lambda + \cos L\,\cos\epsilon\,\sin\lambda\,\sin \alpha_0,$$ (32)

$$\tan A = \frac{\cos\epsilon\,\sin\lambda\,\cos \alpha_0-\cos\lambda\,\sin ... ...\,\cos\lambda\,\cos \alpha_0-\sin L\,\cos\epsilon\,\sin\lambda\,\sin \alpha_0}.$$ (33)

Figure 11: Horizontal coordinates. $R$ is a celestial object, and $R'$ its projection onto the horizontal plane, $NESW$.