Sunrise & Sunset Calculator

Sunrise and sunset are both, fundamentally, the moment the Sun's centre crosses the horizon — plus a small standard correction (about 0.833°) for atmospheric refraction bending light over the horizon, and the Sun's own apparent size, both of which make it visible slightly before true geometric sunrise and after true sunset. This calculates that crossing from the Sun's declination (how far north or south of the equator it appears, which changes through the year) and your latitude.

Longitude and the equation of time both shift the result away from a clean 12:00 solar noon. Longitude shifts it by simple time-zone-like arithmetic — four minutes per degree, since the Earth rotates 360° in 24 hours. The equation of time is smaller and stranger: even at a fixed longitude, true solar noon drifts up to about 16 minutes either side of clock noon through the year, because Earth's elliptical orbit and axial tilt both make its orbital speed and apparent solar motion uneven.

Near the poles, the whole framework breaks down for part of the year — at high enough latitude, the Sun's daily path barely dips toward the horizon during the local summer (it never sets: polar day) or barely rises above it in winter (it never rises: polar night). Both are detected and reported explicitly rather than returned as a nonsensical time.