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

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

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

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

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]

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