The Azimuth and Altitude Calculator converts equatorial coordinates (RA and Dec) to horizontal coordinates (azimuth and altitude) for any observer location and local sidereal time. The core transformation for telescope pointing, radio antenna alignment, and astronomical observation planning.
43.92
deg
168.037
deg
-0.575
h
-8.625
deg
43.92
deg
168.037
deg
-0.575
h
-8.625
deg
You want to point your telescope at the Andromeda Galaxy tonight at 10 PM from your backyard in Denver. The catalog gives RA = 0h 42m 44s, Dec = +41° 16'. Your telescope understands altitude and azimuth — how many degrees above the horizon, and in which compass direction. The calculator for azimuth and altitude performs the spherical trigonometry coordinate transformation that bridges the catalog system (equatorial) and the observer's local horizon system, accounting for geographic latitude and the rotation of the Earth through local sidereal time.
The transformation from equatorial (RA, Dec) to horizontal (Az, Alt) requires three inputs: the object's right ascension α and declination δ, the observer's geographic latitude φ, and the local mean sidereal time (LMST). The hour angle H = LMST − α is the key intermediate variable. The spherical trigonometry formulas:
sin(Alt) = sin(φ)sin(δ) + cos(φ)cos(δ)cos(H)
tan(Az) = sin(H) / [sin(φ)cos(H) − cos(φ)tan(δ)]
Altitude (Alt) ranges from −90° (below the horizon) to +90° (zenith); azimuth (Az) is measured 0°–360° clockwise from north. Objects with Alt < 0° are below the observer's horizon and unobservable. The sign convention for azimuth requires careful handling of the arctan quadrant: use atan2 to recover the correct quadrant. Use this online calculator for any RA/Dec/latitude/LMST combination. The hour angle converter computes the LMST-to-HA conversion needed as input.
Local sidereal time (LMST) is the right ascension currently on the meridian — the imaginary north-south line passing through the observer's zenith. LMST changes at a rate slightly faster than solar time (24 hours 3 minutes 56 seconds per sidereal day), advancing roughly 4 minutes per solar day. An object with RA equal to the current LMST is on the meridian — at its highest altitude for that night. LMST at any location = GMST + (longitude in hours, with east positive). GMST can be computed from the Julian Date; most planetarium software and smartphone apps display current LMST directly. For accurate calculations, use LMST to at least 0.1-minute precision; 1 minute of LMST error corresponds to approximately 0.25° of azimuth error for objects near the equator.
The azimuth/altitude calculation is the computational core of several practical applications:
The constellation visibility calculator and observational astronomy calculators provide complementary tools for sky observation planning.
The calculated geometric altitude differs from the apparent (observed) altitude due to atmospheric refraction — the bending of light rays as they pass through the atmosphere at a grazing angle. Near the horizon (Alt < 10°), refraction can elevate the apparent position by more than 0.5°; at the horizon itself, refraction lifts objects approximately 0.5° above their geometric position, which is why the Sun appears above the horizon for several minutes after it has geometrically set. The standard refraction correction is: R ≈ 1.02 / tan(Alt + 10.3/(Alt + 5.11)) arcminutes, where Alt is in degrees. For precision astrometry and professional telescope scheduling, atmospheric refraction must be applied to convert between calculated and observed positions.
1. Compute HA = LMST - RA (hours), convert to degrees, then radians. 2. Compute altitude: sin(alt) = sin(Dec)*sin(Lat) + cos(Dec)*cos(Lat)*cos(HA). 3. Compute azimuth: cos(Az) = (sin(Dec) - sin(alt)*sin(Lat)) / (cos(alt)*cos(Lat)). If sin(HA) is positive, Az = 360 - Az; otherwise Az is as computed. All angles in radians during computation, converted to degrees for output.
Altitude above 20 degrees is generally good for observing (less atmospheric distortion). Altitude 10-20 degrees is marginal. Below 10 degrees the atmosphere absorbs and distorts significantly. Azimuth tells you which direction to look: 0/360 = north, 90 = east, 180 = south, 270 = west. Combine altitude and azimuth to point your telescope or binoculars precisely.
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When LMST equals Betelgeuse's RA, the hour angle is 0 and the star transits the meridian, due south at 49.3 degrees altitude.
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Sirius is low in the west-southwest, about 2.75 hours past its meridian transit, near the horizon.
Altitude is the angular height of an object above the horizon, measured in degrees. 0 degrees = on the horizon, 90 degrees = zenith (straight up). Negative values mean the object is below the horizon and not visible from your location.
Azimuth is the compass bearing of an object, measured clockwise from north. North = 0 degrees, East = 90 degrees, South = 180 degrees, West = 270 degrees. It tells you which horizontal direction to face.
The geometry of the sky-dome depends on your latitude. At the equator, all objects rise nearly straight up from the horizon. At the poles, objects circle the horizon at constant altitude. Latitude determines how steeply objects move through the sky.
This calculator gives geometric altitude. Near the horizon, Earth's atmosphere bends light upward, making objects appear slightly higher than their true position. The correction is about 0.5 degrees at the horizon, decreasing to near zero above about 20 degrees altitude.
Look up the object's RA and Dec in any star atlas or planetarium app. Enter those values along with your latitude and current LMST (from the Sidereal Time Calculator). The result tells you exactly where to point your telescope.
The hour angle measures how far an object is from your meridian, in hours. HA = 0 at transit (highest point). Positive HA means the object is west of the meridian; negative HA means it is east. Objects with HA between -6h and +6h are generally above the horizon.
Yes, if you know the Sun's or Moon's current RA and Dec. These change rapidly (especially the Moon) so you would need to obtain current ephemeris values from a source like NASA Horizons or an astronomy app.
The zenith is the point directly overhead, at altitude 90 degrees. Any object at the zenith has an altitude of 90 degrees and its azimuth is undefined. Objects near the zenith cross the meridian at their highest altitude.
Very close to the celestial poles, small changes in RA and Dec produce large changes in azimuth. The formula remains mathematically valid but becomes numerically sensitive. For objects within a few degrees of the celestial pole, use a full spherical trigonometry implementation.
An object is circumpolar if its declination is greater than (90 - your latitude) degrees. For example, at latitude 51.5 degrees (London), objects with Dec > 38.5 degrees never set. These objects are always above the horizon, though they still change altitude and azimuth throughout the night.
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