\frac{365 \cdot cos^{-1}(\frac{\delta}{-23.45})}{360}
(365 * acos(declination/-23.45)) / 360
An approximation in days from the winter solstice in any year. Good enough for calculating on which days of the year, say, the sun sets at 8pm. Not good enough for eclipses.
NB that acos has two results, one positive and one negative, or to put it another way, the Sun has the same declination on two days each year (eg Mar 21 and Sep 21).
Calculators and programming languages will only return one result (the positive one, ie the day in the first half of the year) - to get the other one, substitute -1*acos(declination/-23.45) in the formula.
(People have asked me to say when the winter solstice is. I assume it's Dec 21, using any more accuracy together with this approximation formula seems pointless). (People have also been asking me to say what I mean by 'An approximation in days from the winter solstice in any year'. Well, if you put a declination (eg -23.45) into the formula above, and get the result 0, then the answer is '0 days from Dec 21' or 'Dec 21'. If the result was 1, then the answer is '1 days from Dec 21' or 'Dec 22', and so on and so forth.)
