The Maths Behind Sun Azimuth and Altitude, in Plain English

If you read any sun-tracking documentation you will hit the words "azimuth" and "altitude" within thirty seconds. Most explanations are written by astronomers, for astronomers. Here is the same thing written for people who just want to know where the sun is.

Imagine standing on a flat plain

You are standing in the middle of nowhere. The ground is flat in all directions, and the sky is a hemisphere over your head. There is no obvious way to describe where the sun is, other than pointing. Azimuth and altitude are a system for describing that direction with two numbers.

• Azimuth is the compass direction along the ground that you would face to look toward the sun. Measured in degrees, with north as 0°, east as 90°, south as 180°, west as 270°. So an azimuth of 135° means the sun is in the south-east.
• Altitude (sometimes called "elevation") is the angle up from the horizon to the sun. Zero degrees means it is on the horizon (rising or setting). Ninety degrees means it is straight up. A typical mid-morning sun in summer might be at 50° altitude.

Combine the two — "azimuth 135°, altitude 30°" — and you have completely fixed where the sun is in the sky from your spot.

What changes them?

Both numbers change continuously throughout the day and across the year. The drivers are three: your latitude, the date (specifically the sun's declination, which is the season-dependent tilt), and the time of day.

Latitude effect

The further north you are, the lower the maximum altitude the sun ever reaches. In Berlin (52° latitude), the highest the sun can ever be is about 61° (at noon on the summer solstice). In Singapore (1° latitude), the sun can be 88° up, essentially overhead. In Reykjavik (64° latitude), the highest possible noon sun is about 49°.

Date effect

On the equinoxes (around 21 March and 22 September), the sun rises due east (azimuth 90°), sets due west (270°), and reaches a noon altitude of 90° minus your latitude. In northern summer, sunrise shifts to the north-east (azimuth less than 90°) and sunset to the north-west. In northern winter, both shift south.

Time-of-day effect

The sun traces an arc across the sky. Azimuth sweeps from sunrise (east-ish) through south (in the northern hemisphere) at noon, to sunset (west-ish). Altitude rises from zero at sunrise, peaks at solar noon, falls back to zero at sunset.

The formulas, in case you want them

For a given date, latitude, longitude, and time, the calculations are well-defined. The classic reference is Jean Meeus's Astronomical Algorithms, but for everyday use the simplified version is:

1. Calculate the sun's declination from the date (varies between -23.5° and +23.5° across the year).
2. Calculate the hour angle from the local solar time (15° per hour from noon).
3. Apply two trigonometric formulas: one for altitude, one for azimuth, using your latitude.

NOAA publishes this calculation in spreadsheet form on their solar calculator page if you want to reproduce it. Wikipedia has the azimuth and altitude formulas spelled out.

Why photographers should care

Two practical uses.

• Knowing where the sun will rise/set on a specific date. If you want to photograph the sun rising directly behind a landmark, you need the azimuth on that date to match the bearing from your camera to the landmark. Sunhour shows this as a compass-bearing readout.
• Knowing the sun's altitude when you arrive. If the altitude will be less than 6° when you arrive, you are in golden hour. If it is over 30°, you are in mid-day light. Knowing this without checking on site changes how you pack and pace.

The shorthand most pros use

You do not have to do the maths in your head. The shorthand is:

• Golden hour ≈ altitude between -4° and +6°.
• Blue hour ≈ altitude between -4° and -8°.
• Civil twilight ≈ altitude 0° to -6°.
• Nautical twilight ≈ altitude -6° to -12°.
• Astronomical twilight ≈ altitude -12° to -18°.

Once you read sun position in altitude rather than clock time, the whole planning game gets simpler.