Frank, Checking your reply to John Bercovitz, I see that you used the Fourier series from the BSS Glossary. In my quick reading, I missed this (don't ask how!). Hopefully by now, you will have received a copy of the scanned and OCRed paper which was the first publication of this equation.
I recall that a few years ago, I posted a copy of Spencer's original paper on the list, and it looks like it found its way to the BSS Glossary. I wonder what Spencer would think if he knew that his research on air-conditioning of buildings would be helping diallists 30 years later? I am sure that he would be pretty pleased. BTW: As in so many countries, the Australian government has progressively gutted science and research organisations. The CSIRO Division of Building Research was disbanded many years ago. Pity. Cheers, John [EMAIL PROTECTED] ----- Original Message ----- From: "Frank King" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Cc: <[email protected]>; <[EMAIL PROTECTED]> Sent: Friday, March 19, 2004 8:46 PM Subject: Re: Declination approximation? > 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 > > - -
