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Basics of Positional Astronomy and Ephemerides
The observer is located at the centre 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 is the angular distance, measured along the horizon, eastward from the North point N (as 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 (on this server, 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 of 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.
 To convert equatorial coordinates hour angle and declination (tau, delta) to horizontal coordinates azimuth and altitude (az, h), the "nautical triangle" NP-Ze-St is used: NP-St = 90° - delta, Ze-St = 90° - h. From spherical trigonometry we get: tan az = (- sin tau) / (cos beta tan delta - sin beta cos tau) Example: 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. Right ascension RA, hour angle tau and sidereal time theta are related by: tau = theta - RA Animation: |
1.
Conversion of date and time: local
time to universal time UT 15
h CEST = 13 h UT convert
time: Julian Day of 1991/ 5/19
at 13 UT Julian
day of 2000/01/01
at 12 UT number
of Julian days since 2000/01/01 at 12 UT number
of Julian centuries since 2000/01/01 at 12 UT
JD
= 2448396.04167 JD = 2451545.0 -3148.95833 2. Astronomical
algorithms: compute the
ecliptic latitude B B
= -1.87° convert ecliptic
latitude and longitude to right ascension RA and declination delta
RA
= 133.44° compute sidereal time (degrees) at
Greenwich local
sidereal time at longitude 10° E local
hour angle theta0
= 71.698° theta
= theta0 + 10° = 81.698° tau
= theta - RA = 81.698° - 133.46° 3. Final results
convert (tau, delta)
to horizon coordinates (h, az) of the observer (50° N, 10° E)
The
function atan2(numerator,denominator) should be used to avoid ambiguity.
true
(geocentric) altitude: h = 36.1° azimuth
angle: az = 110.6° from N p
= 0.80°
The
refraction R is calculated by Saemundsson's formula (Meeus,
Astronomical Algorithms): h
is the true (airless) altitude in degrees, R is in minutes of arc. The
apparent (measured) altitude is h+R. R
= 0.0004° apparent
altitude h h
= 36.1° - 0.80° = 35.3°
compute the position
of the Moon on 1991/05/19 at 15:00 CEST
Berechnung der Mondposition
used by the algorithm for B and L
T
= - 3148.95833/36525
= - 0.0862137805
and the longitude L
of the Moon
L = 131.52°
delta = 15.53°
= -51.74° = 308.25°
(corrected for parallax and refraction)
