1. Horizon Coordinates
A: Azimuth. The angle measured beginning at cardinal pt. north eastward along the horizon to the base of the vertical circle of the object. Azimuth is measured in units of degrees.
h = 90o
h: Altitude. The angle measured upward from the horizon along the vertical circle of an object to the object itself. The angle is measured in degrees. Z
Vertical Circle
h = 30o
A = 270o W
A = 310o
45o = h
A = 180o N N
A = 0o S A
= 135o
h = 0o E
A = 90o

2. The Celestial Sphere Z
NCP
Meridian altitude of the C.E. = observer’s co-latitude
Altitude of the NCP = observer’s latitude W
(DVC)
52
38 S N N
Definition: Co-latitude = 90o - latitude
HORIZON
DIURNAL
CELESTIAL E
CIRCLE
EQUATOR
SCP

3. Celestial CoordinatesZ
R.A.: Right Ascension. The angle measured beginning at the vernal equinox eastward (counter-clockwise) along the celestial equator to the base of the hour circle of the object. R.A. is measured in units of hours, minutes and seconds.
V.E.
R.A. = 0h
HOUR CIRCLE
NCP
 = 90o (at NCP)
R.A.
= 5h W 
= 40o S N N
V.E.: Vernal Equinox. The point on the celestial equator from which Right Ascension is measured. The vernal equinox moves at the same rate as the stars (361o west every day with respect to the horizon).
 = 0o (on C.E.)
HORIZON
δ: Declination. The angle measured upward (if positive) or downward (if negative) from the celestial equator along the hour circle of an object to the object itself. The angle is measured in degrees.
CELESTIAL E
EQUATOR
SCP
 = -90o (at SCP)

4. Hour Angle
The angle in units of hours, minutes and seconds between the celestial meridian and the hour circle of an object. The angle is positive if measured westward, negative if measured eastward.
HOUR ANGLE (positive)
= 4h Z
HOUR ANGLE (negative)
= - 3h15m
NCP
HOUR CIRCLE W S N N
A positive hour angle states the amount of time since an object went through upper transit.
HORIZON
CELESTIAL
A negative hour angle states the amount of time until an object goes through upper transit. E
EQUATOR
SCP

5. Sidereal Time H.A. = S.T. – R.A.
The angle in units of hours, minutes and seconds between the celestial meridian measured westward to the hour circle of the vernal equinox. Z
R.A.
= 5h15m
HOUR ANGLE (negative)
V.E.
= - 3h15m
NCP
HOUR CIRCLE W S N N
Sidereal time states the amount of time since the vernal equinox went through upper transit. It equals the hour angle of the vernal equinox.
HORIZON
CELESTIAL E
EQUATOR
SCP
H.A. = S.T. – R.A.