7 Feb 98 The following paper by J. W. Spencer was circulated to the Sundial List on 3 Feb 98 by John Pickard. Since some of the symbols used in the original caused ambiguity in the message as received after transmission via email attachment, I have edited out the offending symbols, replacing them with more normal characters. Otherwise, the paper is intact, at least as far as I can tell.
Mac Oglesby * * * * * * * * * * Fourier series representation of the position of the sun J. W. Spencer CSIRO Division of Building Research Melbourne, Victoria [Originally published in May 1971 in Search 2 (5), 172. See note at end of paper.] In the computer calculation of air-conditioning loads on buildings, and in the estimation of solar position and solar radiation intensity, it is desirable to have a method for calculating basic astronomical data to avoid storing the data in tabular form. The most convenient data are the solar declination (D) and the equation of time (E), both expressed in radians, and the reciprocal of the square of the radius vector of the Earth (1/r^2). To the accuracy required for building purposes, these quantities may be considered cyclic with a period of one year and may be usefully expressed in terms of Fourier series. Such a representation for sin D and 1/r^2 has been used by Berry (1964) in the computer evaluation of a natural evaporation formula. An angle T was defined by T = 2 * PI * D * d / 365, where the day number d is given in the Nautical Almanac and ranges from 0 on 1 January to 364 on 31 December. Values of D, E and r were taken from the nautical Almanac and Astronomical Ephemeris for the year 1950. It is important that these data be taken for a year in the middle of a 4-year leap cycle, since this cycle causes changes in the declination varying from the order of + or - 10' at the equinoxes, where the effect is greatest, to less than 1' at the solstices, where the effect is least. Other, longer-period variations in the declination are negligible compared to this. The maximum change in declination in 24 hours, which occurs at the equinoxes, is less than 1/2 degree, while the minimum change in 24 hours, which occurs at the solstices is less than 1'. If accuracy to the nearest degree in calculated solar altitude and azimuth is sufficient, a single value of D can thus be used for each day. Although the maximum error in doing so approaches 1/2 degree, for much of the year the error is considerably less, and it is worth including sufficient terms in the series to reduce errors at these times. In the three series given below the error as each additional pair of harmonic terms was added has been investigated, and in each case the series has been truncated at a point where the addition of further terms gives an uneconomically slow increase in precision. For the declination D in radians, Fourier analysis yielded the following series D = 0.006918 - 0.399912 cos T + 0.070257 sin T - 0.006758 cos 2*T + 0.000907 sin 2*T - 0.002697 cos 3*T + 0.001480 sin 3*T This series estimates D with a maximum error of 0.0006 radians (<3'), or if the final two terms are omitted, with a maximum error of 0.0035 radians (12'). For the equation of time E in radians the Fourier series obtained was E= 0.0000075 + 0.001868 cos T - 0.032077 sin T - 0.014615 cos 2*T - 0.040849 sin 2*T The maximum error with this series is 0.0025 radians, equivalent to about 35 seconds of time. The equation of time is here defined, as in the Nautical Almanac, in the sense of the apparent time minus mean time. For 1/r^2, the Fourier series obtained was 1/r^2 = 1.000110 + 0.034221 cos T + 0.001280 sin T + 0.000719 cos 2*T + 0.000077 sin 2*T for which the maximum error was 0.0001. The above series have been used for the computation of solar positions and radiation tables (Spencer 1968a, b) and for air-conditioning load calculations where accuracy to the nearest degree in solar altitude and azimuth was required. Received 11 January 1971 Resubmitted 19 April 1971 References BERRY, G. (1964) Aust. J. appl. Sci., 15 16. SPENCER, J.W. (1968a) Solar position and radiation tables for Sydney (Latitude 34 S). Rep. Div. Bldg Res. CSIRO Aust. (draft edition) SPENCER, J.W. (1968b) Solar position and radiation tables for Perth (Latitude 32 S). Rep. Div. Bldg Res. CSIRO Aust. (draft edition) Note added 3 February 1998 In the early 1970s, I contacted Dr Spencer about this method because I was trying to use a hand calculator for calculating solar positions, etc. He was extremely helpful and gave me a reprint of this paper. He also pointed out an error in the original: in the series for E, the constant was printed as 0.000075 rather than 0.0000075. I have corrected the error in this version. The CSIRO Division of Building Research no longer exists, a victim of the anti-intellectual and anti-science Thatcherite economic policies of successive Australian governments. Dr John Pickard Graduate School of the Environment Macquarie University NSW 2109 Australia Phone +61 2 9850 7981 Fax +61 2 9850 7972 email [EMAIL PROTECTED]