Definition
The elevation angle α is measured from the horizon up to the centre of the sun: 0° at sunrise and sunset, 90° when the sun is directly overhead. Its complement, 90° − α, is the zenith angle.
Formula
sin α = sin φ · sin δ + cos φ · cos δ · cos H
φ is latitude, δ the declination and H the hour angle. At solar noon H = 0 and the formula collapses to a subtraction:
αnoon = 90° − |φ − δ|
At 50° N on the June solstice (δ = +23.44°) the noon sun stands at 63.4°; at the equinoxes at 40°; on the December solstice at 16.6°.
Noon elevation through the year
| Date | Declination | Equation of time | Noon elevation at 40° N | at 50° N | at 35° S |
|---|---|---|---|---|---|
| 1 Jan | -22.98° | −3.6 min | 27.0° | 17.0° | 78.0° |
| 15 Jan | -21.07° | −9.4 min | 28.9° | 18.9° | 76.1° |
| 1 Feb | -17.02° | −13.5 min | 33.0° | 23.0° | 72.0° |
| 15 Feb | -12.58° | −14.1 min | 37.4° | 27.4° | 67.6° |
| 1 Mar | -7.48° | −12.3 min | 42.5° | 32.5° | 62.5° |
| 15 Mar | -2.02° | −8.9 min | 48.0° | 38.0° | 57.0° |
| 1 Apr | 4.65° | −3.8 min | 54.7° | 44.7° | 50.3° |
| 15 Apr | 9.87° | −0.0 min | 59.9° | 49.9° | 45.1° |
| 1 May | 15.17° | +2.9 min | 65.2° | 55.2° | 39.8° |
| 15 May | 18.94° | +3.7 min | 68.9° | 58.9° | 36.1° |
| 1 Jun | 22.09° | +2.2 min | 72.1° | 62.1° | 32.9° |
| 15 Jun | 23.32° | −0.5 min | 73.3° | 63.3° | 31.7° |
| 1 Jul | 23.08° | −3.9 min | 73.1° | 63.1° | 31.9° |
| 15 Jul | 21.47° | −6.0 min | 71.5° | 61.5° | 33.5° |
| 1 Aug | 17.94° | −6.4 min | 67.9° | 57.9° | 37.1° |
| 15 Aug | 13.95° | −4.5 min | 64.0° | 54.0° | 41.0° |
| 1 Sep | 8.17° | −0.0 min | 58.2° | 48.2° | 46.8° |
| 15 Sep | 2.91° | +4.8 min | 52.9° | 42.9° | 52.1° |
| 1 Oct | -3.30° | +10.3 min | 46.7° | 36.7° | 58.3° |
| 15 Oct | -8.63° | +14.2 min | 41.4° | 31.4° | 63.6° |
| 1 Nov | -14.52° | +16.4 min | 35.5° | 25.5° | 69.5° |
| 15 Nov | -18.56° | +15.4 min | 31.4° | 21.4° | 73.6° |
| 1 Dec | -21.84° | +11.0 min | 28.2° | 18.2° | 76.8° |
| 15 Dec | -23.28° | +4.9 min | 26.7° | 16.7° | 78.3° |
Refraction
The atmosphere bends sunlight downwards, so the sun appears higher than it geometrically is — by about 0.57° at the horizon and a few hundredths of a degree at 30°. That is why the sun is still visible for a couple of minutes after it has geometrically set, and why sunrise and sunset are defined at a geometric elevation of −0.833° (refraction plus the sun's radius). The elevations on this site include refraction.
Elevation and shadows
shadow length = height ÷ tan α
| Sun elevation | Shadow of a 1 m pole |
|---|---|
| 60° | 0.58 m |
| 45° | 1.00 m |
| 30° | 1.73 m |
| 20° | 2.75 m |
| 15° | 3.73 m |
| 10° | 5.67 m |
| 5° | 11.4 m |
The sun map draws the shadow of an object of any height to scale on the map, for any minute of the year.
Elevation and solar panels
A panel faces the sun head-on when its tilt equals the zenith angle, 90° − α. That is the reasoning behind steep off-grid tilts: a December noon sun at 16.6° wants a panel at 73.4°.