Solar Declination Angle (Cooper's Equation)

Declination is the angle between the sun's rays and the plane of the equator, and it is the reason seasons exist. The Earth's axis leans 23.45° from the perpendicular to its orbit, so through the year the sub-solar point migrates from 23.45° north to 23.45° south and back. Cooper published a one-line approximation in 1969 — $\delta = 23.45^\circ \sin[360^\circ(284+n)/365]$, where $n$ is the day of the year — and it has been the starting point of flat-plate collector calculations ever since, reproduced in Duffie and Beckman and in essentially every solar text since.

Four days anchor the curve. Declination is zero at the equinoxes, around 22 March (day 81) and 22 September (day 264); it peaks at $+23.45^\circ$ at the June solstice (day 172) and bottoms at $-23.45^\circ$ at the December solstice (day 355). Between them the sine is steep at the equinoxes and flat at the solstices, which is exactly why day length changes fastest in March and September and hardly at all in late June — a fact more people notice than can explain.

Cooper's equation is an approximation and an honest one about it. It assumes a circular orbit and a 365-day year, so it errs by up to about half a degree at some times of year. That is comfortably inside the accuracy any collector calculation needs and hopeless for astronomy, where the ephemeris and the equation of time are used instead. If the calculation must be right to a minute of arc, this is the wrong tool.

Where declination bites hardest is the sun's altitude at noon, which is $90^\circ - \phi + \delta$. At 45° north that swings from 68.45° in June down to 21.55° in December — a 47° range, the same 47° that separates the two solstice declinations. The low winter sun is doubly penalised: its energy is spread over more ground by the cosine of the angle, and it travels through a much longer path of atmosphere to arrive. That compound effect, not the modest change in Earth–sun distance, is what winter is.