Dear John > Is there an approximate formula for the declination of the > sun vs day number?
This is a tantalising story which doesn't really have a happy ending! Only gluttons for punishment should read any further... Your solution is a good starting point: > I just tried the obvious > > 23.44*SIN[(day number)*360degrees/365.2422] You have taken the obliquity of the ecliptic as 23.44 degrees which is close enough. You implicitly start at the Vernal Equinox (day number = 0 gives declination = 0) and you have taken the length of the year as 365.2422 days. You can improve on this by looking at: http://www.sundialsoc.org.uk/glossary/frameset.htm This is the truly wonderful Glossary of the British Sundial Society (it is edited by John Davies) and you will find under Equations (look for number 9) the following Fourier transform: D = 0.006918 - 0.399912 cos w + 0.070257 sin w - 0.006758 cos 2w + 0.000907 sin 2w - 0.002697 cos 3w + 0.001480 sin 3w where D is the declination in radians. The parameter w is also in radians and represents a proportion of the year scaled to the range 0 to 2pi. Using your scaling, you could take w as: w = (day number)*2pi/365.2422] Here, though, day number = 0 corresponds to somewhere around 1 January. The maximum error is said to be 0.0006 radians (less than 3 arcminutes). If you want to do better than that, you can implement the appropriate algorithms described by Meeus and you will find yourself keying in over 500 constants. It is very rewarding to get these right but it takes quite a while! The real difficulty is what you mean by `day number'. If you are just interested in the fraction of the year from the Vernal Equinox then you need take in no more. If you want to relate `day number' to a date then you will be defeated by the Gregorian Calendar. You can see the problem by asking the reverse question, `What is the day number corresponding to a given declination?' Even if you take a nice easy declination, like 0 degrees, you find the date varies by over two days over the 400-year Gregorian cycle. On the Greenwich Meridian the instant of the Vernal Equinox varies from late afternoon on 21 March (e.g. 1903) to early afternoon on 19 March (e.g. 2096). If you are in a different time zone you may well be the other side of midnight so the date changes again. Worse still, counting days from 1 January involves having to include 29 February one year in four which throws out the count by one day for the rest of the year. I said there wasn't a happy ending but if you want some light relief you can read a nice article that alludes to this kind of thing in the latest, March 2004, Issue of the British Sundial Society Bulletin. I wrote it myself and it's about a sundial I did for the Queen a couple of years ago! Frank H. King Cambridge University England -
