How sunrise and sunset times are calculated — latitude, the seasons, and the equation of time

Sunrise tomorrow can be computed for any point on Earth to within a minute with nothing but the date, latitude, longitude and a page of trigonometry — no data feed, which is why the sunrise and sunset tool runs entirely on your device. This guide explains what goes into that page and the things about daylight that surprise people every year.

What 'sunrise' means

Sunrise is the moment the upper edge of the Sun appears on a flat horizon — not the centre, and not the geometric position. Two corrections apply: the Sun's disc is about half a degree across, so its top edge shows a quarter-degree before its centre; and atmospheric refraction bends light over the horizon by about 0.57°, so we see the Sun before it has geometrically risen. Together, published times use the Sun's centre at −0.833° below the horizon. Hills, buildings and altitude change what you actually see by minutes; the published figure is the standard flat-horizon one.

Latitude and the seasons

Earth's axis is tilted 23.4° to its orbit, so the Sun's position against the sky moves between 23.4° north (June solstice) and 23.4° south (December solstice) of the celestial equator — its declination. Day length follows from your latitude and that declination: on the equator every day is close to 12 hours; at 50° north, from about 8 hours in December to 16 in June; above the Arctic Circle (66.6°) the Sun doesn't rise at midwinter or set at midsummer. Equinoxes give near-12-hour days everywhere — slightly more, because of the refraction and disc-size corrections. Leap years, explained properly covers why the calendar has to keep pace with the tilt's cycle.

Why the earliest sunset isn't the shortest day

In the northern mid-latitudes the earliest sunset comes around 10–13 December and the latest sunrise in early January, straddling the 21 December solstice. The cause is the equation of time: because Earth's orbit is elliptical and its axis tilted, solar noon drifts up to 16 minutes earlier or later than clock noon over the year. In December it is drifting later by about half a minute a day, which pushes both sunrise and sunset later — enough to move the sunset minimum before the solstice and the sunrise maximum after it. The same effect makes the analemma, the figure-of-eight the Sun traces if photographed at the same clock time all year.

Twilight: civil, nautical, astronomical

After sunset the sky stays light while the Sun is just below the horizon. Civil twilight lasts until the Sun is 6° below — enough light for outdoor activity; it's what "lighting-up time" and many legal definitions use. Nautical twilight ends at 12°, when the horizon is no longer visible for sextant sights. Astronomical twilight ends at 18°, when the sky is fully dark. Near the equator these pass in under an hour; at high latitudes in summer, astronomical twilight never ends and the sky never goes truly dark. The tool lists all three.

How the calculation works

The standard algorithm: compute the day of the year; from it the Sun's declination and the equation of time (from series approximations accurate to a minute or so); then solve the sunrise equation, cos(ω) = (sin(−0.833°) − sin(φ) sin(δ)) ÷ (cos(φ) cos(δ)), where φ is latitude and δ declination, for the hour angle ω; sunrise and sunset are solar noon ∓ ω, and solar noon is 12:00 corrected by the longitude's offset from the time zone's meridian and the equation of time. Convert to local time with the time zone and daylight saving (the time zone converter). The whole thing is a few lines of code; the hard parts are the time zone and what counts as the horizon. The world clock shows where the Sun is up right now; How moon phases work — the geometry behind the 29.5-day cycle the night's other timetable.