Equatorial coordinates
By extending the lines of latitude and longitude outward from the
Earth and onto the inside of the celestial sphere we get the equatorial
coordinate system. The coordinates of stars, planets, and other
celestial objects corresponding to latitude and longitude are declination
(DEC) and right ascension (RA).
The declination of an object is its angle in degrees, minutes,
and seconds of arc above or below the celestial equator. The right
ascension is the angle between an object and the location of the
vernal equinox (First Point in Aries) measured eastward along the
celestial equator in hours, minutes, and seconds of sidereal time.
Since the location of the vernal equinox changes due to the precession
of the Earth's axis of rotation, coordinates must be given with
reference to a date or epoch.
Right ascension is given in time units. One hour corresponds to
1/24 of a circle, or 15° of arc. As the Earth rotates, the sky moves
to the West by about 1 hour of right ascension during each hour
of clock time or exactly one hour of sidereal time. The Earth makes
one full revolution in about 23 hours and 56 minutes of clock time
or 24 hours of sidereal time. Sidereal time corresponds to the right
ascension of the zenith, the point in the sky directly overhead.
For example, the coordinates of the star Regulus (Leo a) for epoch
J2000 are:
RA: 10h 08m 22.3s
DEC: +11° 58' 02"
When the local sidereal time is 10h 08m 22.3s, it would be on the
local meridian.
Horizon coordinates: azimuth and altitude
This is a local coordinate system to use for locating objects in
the night sky as seen from a point on the Earth's surface. Azimuth
is the angle of a celestial object around the sky from north. It
is measure along the horizon in from North 0° through East 90°,
South 180°, West 270° and back to North. Altitude is the complement
of the zenith angle, which is the angle from the local meridian
to the hour circle of object being observed. An object directly
overhead would have an altitude of 90°. An object with a calculated
altitude of 0° may not appear exactly on the horizon due to the
refraction of light through the atmosphere. Generally, refraction
makes objects near the horizon appear higher than their computed
altitude.
Coordinate transformation
The azimuth (AZ) and altitude (ALT) of an object
in the sky can be calculated easily using the date, universal time
(UT), and the latitude (LAT) and longitude (LON)
of the observing site and the right ascension (RA) and
declination (DEC) of the object. All coordinates are expressed
in degrees in the range 0° to 360°, so that trigonometric functions
can be used for coordinate conversion.
Local Mean Sidereal Time
The mean sidereal time (MST) is calculated from a polynomial
function of UT since epoch J2000. This formula gives MST,
the sidereal time at the Greenwich meridian (at longitude 0°) in
degrees. To get local mean sidereal time (LMST), add longitude
if East or subtract longitude if West.
MST = f(UT)
LMST = MST + LON
Hour Angle
The hour angle (HA) is the angle between an observer's
meridian projected onto the celestial sphere and the right ascension
of a celestial body. It is used in coordinate conversion.
HA = LMST  RA
Conversion of HA and DEC into ALT and
AZ
Using the RA, DEC and HA for the object,
and the latitude (LAT) of the observing site, the following
formulas give the ALT and AZ of the object at
the time and longitude that was used to calculate HA.
sin(ALT) = sin(DEC)·sin(LAT) + cos(DEC)·cos(LAT)·cos(HA)
sin(DEC)  sin(ALT)·sin(LAT)
cos(A) = 
cos(ALT)·cos(LAT)
If sin(HA) is negative, then AZ = A, otherwise AZ = 360  A
This gives the computed horizon coordinates without correction
for atmospheric refraction.
Copyright © 2004, Stephen R. Schmitt
