Sunrise Sunset Calculator

Sunrise and sunset are not geometric events but observational conventions. The Sun is defined as risen when the upper limb of its disc touches the ideal horizon, which happens when the geometric centre sits 0.833 degrees below it. That figure is the sum of two corrections: about 16 arcminutes for the solar semidiameter and about 34 arcminutes for mean atmospheric refraction, which bends light over the horizon and lets you see the Sun before it has physically arrived. The hour angle H solves cos H = (sin h - sin(lat) sin(dec)) / (cos(lat) cos(dec)), where h is the target altitude, and the same equation with h set to -6, -12 and -18 degrees produces the civil, nautical and astronomical twilight boundaries defined by national almanac offices. Solar declination, which drives the seasonal swing, ranges between plus and minus 23.44 degrees, the current obliquity of the ecliptic. It is derived from the ecliptic longitude, which in turn needs the mean anomaly plus an equation-of-centre term of up to about 1.9 degrees arising from Earth orbital eccentricity of 0.0167. The same eccentricity and the axis tilt together produce the equation of time, which shifts true solar noon by up to roughly 16 minutes from clock noon across the year. Two limits matter in practice. Above about 66.5 degrees latitude the cosine argument exceeds 1 near the solstices and no rise or set exists, giving polar day or polar night. And the 0.833 degree refraction allowance assumes standard conditions; temperature inversions over cold ground routinely shift observed times by several minutes, so results are typically good to 1 or 2 minutes at mid-latitudes and noticeably worse near the poles.

Computing the Sun rising point is one of the oldest problems in mathematical astronomy, driven first by calendars and ritual. Babylonian and later Greek astronomers tabulated daylight lengths for fixed latitudes, and Ptolemy set out rising-time tables in the Almagest in the second century. The problem became far sharper in the medieval Islamic world, where the daily prayer times are defined by solar altitude and by twilight, so astronomers including al-Biruni and Ibn Yunus developed and tabulated the spherical trigonometry needed to convert declination and latitude into hour angles for arbitrary places. The modern apparatus arrived with practical navigation. Nevil Maskelyne began publishing the British Nautical Almanac in 1767, standardising precomputed solar positions, and Simon Newcomb tables of the Sun, issued in the 1890s, supplied the ephemeris that underpinned almanac work for most of the twentieth century. Twilight itself was codified into the familiar civil, nautical and astronomical bands of 6, 12 and 18 degrees by almanac offices such as the US Naval Observatory. The compact series approximations used by nearly every modern sunrise routine, including this one, descend from Jean Meeus and his 1991 book Astronomical Algorithms, which reduced the full ephemeris to a handful of polynomial and sine terms accurate to well under a minute. The NOAA Solar Calculator popularised that same approach for everyday use.