Definition
The altitude h of a celestial object is its angular distance above the horizon, measured along the vertical circle through the object. It is 0° on the horizon, 90° at the zenith, and negative for objects below the horizon. Its complement, 90° − h, is the zenith distance. With the azimuth, it forms the horizontal coordinate system.
In astronomy, altitude is always an angle. Elevation is a common synonym in radio astronomy and satellite tracking, while the height of the observer above sea level is called elevation or height, to avoid confusion.
Computing altitude
From the observer's latitude φ, the object's declination δ and its local hour angle H:
sin h = sin φ · sin δ + cos φ · cos δ · cos HThe hour angle grows by 15° per hour of sidereal time, so h changes all night. The highest point, the upper culmination, occurs when the object crosses the meridian (H = 0):
h(max) = 90° − |φ − δ|
Example: Vega (δ = +38.8°) culminates at 90° − |48.9° − 38.8°| = 79.9° over Paris, nearly overhead. From Sydney (φ = −33.9°), it culminates only 17.3° above the northern horizon.
The Sun at noon
Applying the meridian formula to the Sun, whose declination goes from −23.44° to +23.44° over the year, gives its noon altitude:
| Place | Latitude | June solstice | Equinoxes | December solstice |
|---|---|---|---|---|
| Singapore | 1.3° N | 67.9° (north) | 88.7° | 65.3° |
| Cairo | 30.0° N | 83.4° | 60.0° | 36.6° |
| Paris | 48.9° N | 64.6° | 41.1° | 17.7° |
| Oslo | 59.9° N | 53.5° | 30.1° | 6.7° |
The noon altitude explains the seasons better than the distance to the Sun: the higher the Sun, the more energy each square metre of ground receives, roughly in proportion to sin h. At Paris, the June Sun delivers almost three times more energy per square metre at noon than the December Sun.
Important negative altitudes
| Altitude of the Sun's centre | Event |
|---|---|
| −0.83° | Sunrise and sunset (upper edge on the horizon, with refraction) |
| −6° | End of civil twilight: brightest stars and planets visible |
| −12° | End of nautical twilight: the sea horizon disappears |
| −18° | End of astronomical twilight: the sky is fully dark |
The −0.83° value combines 34′ of refraction and the Sun's 16′ semidiameter. For the Moon, Meeus uses a value that depends on its parallax, about +0.125°, and for stars and planets −0.57°, refraction alone. At latitudes above 48.6° the Sun never goes below −18° around the June solstice: Paris has no full astronomical night for a few weeks in June.
Near the horizon: refraction and airmass
Low altitudes are where observations suffer most. Atmospheric refraction lifts objects by about 34′ on the horizon but only 1′ at 45°, so the apparent altitude must be corrected before comparing with a computed one. The light also crosses more air: the airmass, the thickness of atmosphere crossed relative to the zenith, is about 1 / sin h above 30° and grows quickly below.
| Altitude | 90° | 60° | 30° | 20° | 10° | 5° |
|---|---|---|---|---|---|---|
| Airmass | 1.0 | 1.15 | 2.0 | 2.9 | 5.6 | 10 |
Each airmass dims a star by roughly 0.2 to 0.3 magnitude in visible light under a clear sky, and the blurring by turbulence grows too. This is why observers schedule targets when they are high, ideally near culmination.
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Frequently asked questions
How high is the Sun at noon?
At local apparent noon its altitude is 90° minus your latitude plus its declination: 90° − φ + δ in the northern hemisphere. At Paris that gives 64.6° at the June solstice and 17.7° at the December solstice.
What is the difference between altitude and elevation?
As angles above the horizon they mean the same thing; altitude is the usual astronomical term, elevation is common in radio and satellite tracking. Elevation also means the observer's height above sea level, which is why astronomers prefer altitude for the angle.
At what altitude does the sky become dark?
When the Sun is 18° below the horizon, at the end of astronomical twilight. Most stars are visible once the Sun is 12° below, and the brightest ones from about 6° below.
