Day Length from Latitude and Solar Declination (Sunrise Equation)

Day length is the number of hours the sun is above the horizon, found from just two things: the latitude $\varphi$ of the place (north positive, south negative) and the declination $\delta$ of the sun that day, the latitude of the point on Earth directly beneath it. Over a year the declination swings between -23.44 and 23.44 degrees, and the declination for a date comes from the day of the year.

The formula is the sunrise equation. At sunrise and sunset the sun is on the horizon, and its hour angle $H_0$ there satisfies $\cos H_0=-\tan\varphi\tan\delta$. The Earth turns 15 degrees an hour, so the day is $2H_0/15$ hours long, which is $\frac{24}{\pi}\arccos(-\tan\varphi\tan\delta)$ with the angle in radians. Read it through the product $\tan\varphi\tan\delta$. It is zero at the equator, which has a 12 hour day on every date, and zero at an equinox, when the declination is zero and the day is 12 hours everywhere. At opposite latitudes the lengths of the day add up to 24 hours: with declination 10 degrees, 45 north has 13.3541 h and 45 south has 10.6459 h.

The worked example is 49.3 degrees north at the June solstice, with declination 23.44 degrees: the equation gives 16.0359 hours, about 16 hours 2 minutes, which is the day at North Vancouver. The equation treats the sun as a point and ignores the refraction of the air and the size of the disc. Those add about 10 minutes at mid-latitudes, and 13 minutes at 49.3 degrees north at the June solstice with a corrected altitude of -0.833 degrees.

North of the Arctic Circle in summer, where $\tan\varphi\tan\delta$ exceeds 1, the sun does not set (24 hours) and the equation has no answer; in winter at the same latitudes the sun does not rise. The checks here are worked by hand from the equation; Forsythe and co-authors (1995) compare models of day length as a function of latitude and day of year.