Basics of Positional Astronomy

by
Juegen Giesen

The observer is located at the center of his "celestial sphere" with zenith Z above his head and the horizon N-E-S-W. The Sun, Moon or any other celestial body can be identified by the two coordinates altitude h and azimuth alpha (horizontal coordinates). Altitude is the angular distance above the horizon (0 < h < 90°), and azimuth the angular distance, measured along the horizon, westwards from the south point S (in astronomy) or eastwards from the north point N in nautics (0 < alpha <360°).

The daily movement of an object - resulting from the rotation of the Earth on its axis - starts when it rises at (1). At (2) it passes across the observer's meridian NZS, reaching its maximum altitude above horizon (transit, culmination), and it sets at (3).

Please watch the Java applet "Apparent Movement of a Star" (with kind permission of Walter Fendt).

The horizontal coordinates of an object depend on the location of the observer on the Earth (and on time). In astronomy equatorial coordinates are commonly used when giving the position of an object on the celestial sphere. The equatorial system is based on the celestial equator, which is the great circle obtained by projecting the Earth's equator on to the celestial sphere, the equatorial plane being perpendicular to the Earth's axis of rotation.

The first equatorial coordinate is declination delta, measured in degrees north and south of the celestial equator (N: 0° < delta < 90°, S: 0° > delta > - 90°. The second coordinate, may be the hour angle tau, measured along the equator from the meridian S-NP-N ob the observer to the hour circle SP-St-NP of the star St. The hour angle corresponds to the length of sidereal time elapsed since the body St last made a transit of the meridian.
A screen shot of Walter Fendts applet shows the relationship of the two systems:

To convert equatorial coordinates hour angle and declination (tau, delta) to horizontal coordinates azimith and altitude (az, h), the "nautical triangle" NP-Ze-St is used (NP-Ze = 90° - beta, NP-St = 90° - delta, Ze-St = 90° - h). From spherical trigonometry we get:

Example:
Observer O at geogr. lat. beta=50° N and long 10° E, on 1991/05/19 at 13:00 UT
will see a star of right ascension RA=55.8° and declination delta=19.7°
at azimuth az=43.6° and altitude h=53.4°
(Sidereal time is 81.7°, hour angle is 25.9°)

The second equatorial coordinate may also be right ascension RA, measured in hours, minutes and seconds of time, taking into account the rotation of the celestial sphere once in 24 hours of sidereal time. The zero point for right ascension is taken as the northern vernal equinox.
This is one of the two points at which the celestial equator intersects the ecliptic (the plane of the Earth's orbit around the Sun).

Right ascension RA, hour angle tau and sidereal time theta are related by: