The observer's own frame
Horizontal coordinates describe the sky as you see it. The reference plane is your local horizon. Altitude (h) is the height above the horizon, from 0° on the horizon to +90° at the zenith, and it is negative for objects below the horizon. Azimuth (A) is the direction around the horizon. The zenith distance z = 90° − h is sometimes used instead of altitude.
These are the coordinates of an altazimuth telescope mount and of the sky disk on AstroAlgos, where the centre is the zenith and the outer circle is the horizon. Unlike right ascension and declination, they change all the time as the Earth turns, and they are different for every observer.
Two ways to count azimuth
Navigators, surveyors and most planetarium software measure azimuth from north, turning east: north is 0°, east 90°, south 180° and west 270°. Meeus, following an older astronomical tradition, measures it from south, turning west: south is 0°, west 90°, north 180° and east 270°.
The two conventions differ by exactly 180°. Always check which one a source uses before comparing numbers. Compass directions such as SW or ESE, used in the sky disk panel, avoid the ambiguity.
From equatorial to horizontal coordinates
Given the declination δ of an object, its hour angle H = LST − α and the observer's latitude φ, the horizontal coordinates follow from two formulas (azimuth measured from south, as in Meeus):
tan A = sin H / (cos H sin φ − tan δ cos φ)sin h = sin φ sin δ + cos φ cos δ cos HMeeus' example 13.b computes Venus as seen from the US Naval Observatory in Washington (φ = +38° 55′ 17″) on 10 April 1987 at 19h 21m UT. With δ = −6° 43′ 11.61″ and H = 64.352133°, the result is A = 68.0337° from south and h = +15.1249°. In the north-based convention the azimuth is 248.03°: Venus is low in the west-south-west.
Use the two-argument arctangent (atan2) with sin H as the first argument to get the azimuth in the right quadrant.
Useful relations
- The altitude of the celestial pole equals your latitude. In the northern hemisphere, Polaris stands about as high as your latitude, within a degree.
- An object transits your meridian, due south or due north, when its hour angle is zero. That is also when it is highest in the sky.
- Near the horizon, atmospheric refraction lifts objects by about 34′. Calculated altitudes are true altitudes, and refraction must be added to get what you see.
- The Moon is so close that its horizontal coordinates depend on where you stand on the Earth. Its parallax can shift it by up to 1° compared with a calculation made for the centre of the Earth.
Measuring angles with your hand
| At arm's length | Approximate angle |
|---|---|
| Little finger | 1° |
| Three middle fingers | 5° |
| Closed fist | 10° |
| Spread hand, thumb to little finger | 20° to 25° |
| From horizon to zenith | 90°, or nine fists |
With these, an altitude from the sky disk panel, such as "Jupiter SW · 45°", becomes easy to find: face south-west and count four and a half fists up from the horizon.
Explore it on AstroAlgos
Frequently asked questions
What is the difference between altitude and azimuth?
Altitude is how high an object is above the horizon, from 0° to 90°. Azimuth is its compass direction along the horizon, usually measured from north through east, from 0° to 360°.
Why do horizontal coordinates keep changing?
Because they are tied to your horizon, which turns with the Earth. A star's altitude and azimuth change continuously during the night, while its right ascension and declination stay fixed.
How high is Polaris in my sky?
About as high as your latitude. From Paris, at 48.9°N, Polaris stands about 49° above the northern horizon. From the equator it sits on the horizon.
