Sunrise & Sunset Calculator

Formula

How this is calculated

You provide a calendar date, the latitude φ and longitude L of the location (north and east positive), and the time-zone offset from UTC. The date is converted to a day-of-year N, which sets the fractional year γ = 2π/365·(N−1). From γ the NOAA Fourier series give the equation of time (the offset between clock noon and true solar noon, in minutes) and the solar declination δ (the latitude where the sun is directly overhead).

The sunrise/sunset hour angle comes from ha = acos( cos(90.833°)/(cosφ·cosδ) − tanφ·tanδ ). The 90.833° zenith (slightly past 90°) bakes in atmospheric refraction and the sun’s apparent radius, so the result matches the moment the disc’s upper edge touches the horizon. Times are then sunrise = 720 − 4(L + ha) − eqTime + offset·60 and sunset = 720 − 4(L − ha) − eqTime + offset·60, expressed as minutes after local midnight; solar noon is 720 − 4·L − eqTime + offset·60, and day length equals 8·ha minutes.

Results are accurate to about a minute for typical latitudes. Near the poles the acos argument can leave [−1, 1]: above the Arctic/Antarctic circles in summer the sun never sets (polar day, length 24 h) and in winter it never rises (polar night, length 0 h), which the calculator flags instead of returning a time. The elevation curve plots the sun’s angle above the horizon across the 24-hour local day.