🌅 Sunrise Calculators

How Sunrise and Sunset Times Are Calculated

Sunrise and sunset times depend on three inputs: latitude, longitude, and date. The calculation begins with solar declination — the angle between the sun and the celestial equator, which varies from +23.45° at the summer solstice to −23.45° at the winter solstice. Next, the hour angle at sunrise is found using the formula: cos(H) = −tan(latitude) × tan(declination). Converting H to time gives the duration from solar noon to sunrise or sunset. Solar noon — when the sun crosses the observer's meridian — is then corrected for the observer's longitude within their time zone and the equation of time (a ±16 minute correction for Earth's elliptical orbit and axial tilt), yielding local clock times for sunrise and sunset.

Twilight: Civil, Nautical, and Astronomical

Twilight is the period before sunrise and after sunset when the sky is partially lit by indirect sunlight. Three definitions correspond to different sun angles below the horizon. Civil twilight (sun 0°–6° below horizon) is bright enough for most outdoor activities without artificial light — this is when the sun is below the horizon but the sky is still clearly illuminated. Nautical twilight (6°–12° below) is dim enough to see the brightest stars but the horizon is still visible — historically used for celestial navigation at sea. Astronomical twilight (12°–18° below) is when the sky becomes dark enough for serious astronomical observation; at 18° below the horizon, the sun contributes no detectable light to the night sky.

Solar Noon and the Equation of Time

Solar noon is the moment when the sun reaches its highest point in the sky — due south in the Northern Hemisphere, due north in the Southern Hemisphere. It does not correspond to clock noon except at specific longitudes and on specific dates. The equation of time accounts for two effects: Earth's elliptical orbit (Earth moves faster in January when closest to the sun, slower in July) and the obliquity of the ecliptic (Earth's axial tilt causes the sun to appear to speed up and slow down through the year). The combined effect means solar noon can be up to 16 minutes ahead of or behind clock noon (corrected for longitude). The analemma — the figure-8 path traced by the sun's position at the same clock time throughout the year — is a visualisation of the equation of time combined with the sun's changing declination.

Solar Elevation Angle

The solar elevation angle is the angle between the sun's rays and the horizontal plane — 0° at sunrise/sunset, 90° directly overhead (occurs only between the tropics). It determines the intensity of solar radiation reaching the surface: at low elevation angles the sunlight passes through more atmosphere, scattering blue light and producing the warm orange tones of golden hour. The formula is: sin(elevation) = sin(latitude) × sin(declination) + cos(latitude) × cos(declination) × cos(hour_angle). Maximum elevation at solar noon equals 90° − |latitude − declination|. At the Arctic Circle on the winter solstice, solar elevation at noon reaches only 0° — the sun grazes the horizon and technically never rises above it at the solstice.