# Solar declination ds--Algorithm 1.11

Solar declination is the angle between the sun's rays and the earth's equatorial plane, the latitude at which the sun is directly overhead at midday. Declination values are positive when the sun is north of the equator (March 21 to September 23) and negative when the sun is south of the equator. Maximum and minimum values of ds are +0.409 radians (+23.45 degrees) and -0.409 radians (-23.45 degrees).

The greatest error in one-year algorithms for solar declination tends to occur at the equinoxes in leap years. Declination varies by 0.3 degrees from year to year during the leap year cycle on a given day near the equinox, and by about half of this in midsummer and midwinter. There is also a limit to the accuracy in which the continuously-varying declination can be calculated from the day number. The maximum change in solar declination through twenty-four hours is about 0.4 degrees and occurs at the equinox, when the declination is close to zero. Very precise values of declination are rarely required in daylighting calculations but can be obtained, for a particular time of day and year, from almanacs such as the Nautical Almanac(1). Roy and others(2) compare the accuracy of several algorithms, and also list a four-year algorithm. The formula below is correct within 0.0007 radians (0.04 degrees) except near the equinoxes in leap years, when the error is 0.0014 radians (0.08 degrees)

Input
Day number, J
J=1 on 1 January, J=365 on 31 December. February is taken to have 28 days.
Equation

Source
Spencer(3)
References
1. H.M. Nautical Almanac Office The nautical almanac (London: HMSO) (published annually)
2. Roy G G, Rodrigo M and King W K A note on solar declination and the equation of time Architectural Science Review 32 (2) 43-51 (1989)
3. Spencer J W Fourier series representation of the position of the sun Search 2 (5) 172 (1971)

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