Dear friends,
I could read only few days ago the many messages that have been
sent to
the List regarding the development of the Equation of Time in Fourier series.
I thank in particular John Pickard to have taken up again the matter and
Fer de Vries and Luke Coletti to have recalled and explained my note of 1996.
>From the reading of the letters I have been pushed to return to my old
programs, to repeat all the calculations and to seek the value of the
errors that we find when we use the developments.
I try here to organize my results and apologize immediately for the length
of this message and for the repetition of some parts in comparison to that
I have written in 1996
THE REASON TO USE THE FOURIER SERIES
When, in 1984, I begun to write my first programs for the calculation of
sundials I used a ZX SPECTRUM and the speed of calculation was very low
(more than 2000 times inferior to my actual PC). Therefore the precise
calculation of the Right Ascension (RA) ,of the Declination (Dec.) of the
Sun and of the Equation of Time (TEq) took too much time.
To reduce this time of calculation I have need of more fast procedures than
those standards and so I have thought to use the Fourier series of RA, Dec
and TEq.
For the same reason I have used, in the series, the mode Modulus-Phase
instead of that Sine-Cosine reducing in this way to the half the number of
the trigonometric functions to calculate.
Since then I use in my programs this method even if, with the speed of
calculation of today, the reason for which I begun to use them, has fallen.
THE FOURIER SERIES
A periodic function (that is a function that repeat itself equal after a
certain period T), given analytically can be developed as an infinite sum
of sine and cosine terms.
If we limit the number of the terms of the sum (harmonics) we have some
errors (differences between the exact values and the sums) : greater it is
the number of the considered terms smaller is the error.
If the function is not known but of it we know only a certain number R of
values (that it assumes in different instants) we can, with opportune
methods, to get a Fourier series truncated to the term R/2 (that is a sum
of Fourier of R/2 terms)
The Fourier s. may be written in two forms.
It forms sine-cosine
y = V0 +A1*Cos(wt) +B1*Sen(wt)+A2*Cos(2wt)+B2*Sen(2wt) + A3*Cos(3wt) +.....
and the form Modulus-Phase
y = V0 +M1*Cos(wt+F1) + M2*Cos(2wt+F2) + M3*Cos(3wt+F3) +...
where
w = 2*pigrec/T rad = 360/T degrees
V0 = mean value of the function in the period
M are the amplitude of the harmonics and F the phases (in rad or degrees)
The passage from one form to the other can be done applying the formula
Cos(a+b)=Cos(a)*Cos(b)-Sen(a)*Sen(b)
THE CALCULATIONS
To get the Fourier series of the Equation of Time I have done in this way.
I have calculated, with great precision, for every day of the year (at
12h), the values of the TEq
I have repeated the calculation for 32 years (from 1990 to 2021) (I could
use any number of years but I have restricted the period to my life : I am
60 now).
I have calculated, for every day, the mean value finding in this way 365
values of the TEq .
With these values I have finally calculated the first 20 terms (20
harmonics) of the Fourier Sum that approach the mean Teq.
I have repeated the procedure for the RA and for the Dec.
To make the calculations I have written a program in which I use, in the
first part, the algorithms of J.Meeus.
For the RA it is necessary to make the development only of the difference
between the RA and the mean anomaly of the Sun. For this to the results of
the development it is necessary to add the value:
Mean Anomaly = w*(t-Tequinox) where Tequinox is the instant of the Spring
Equinox
THE RESULTS
I have used:
w = 360/Tropical year in days = 360/365.2421897=0.98564736 degrees / day
time = t = time from the beginning of the year to the midday
If N is the progressive number of the day (1 for 1 January, 32 for 1
February etc.) we have:
t = N - 0.5
For the calculation of the RA I have used as instant of the Spring Equinox
the value 78.82215 (gotten by the mean course)
The values of the coefficients (last calculation February 1998), rounded
off, are the followings:
Equation of Time TEq
V0 = 0
M1 = 7.3670 minutes F1 = 86.33 degrees
M2 = 9.9182 minutes F2 = 110.97 degrees
M3 = 0.3060 minutes F3 = 105.12 degrees
M4 = 0.2027 minutes F4 = 130.65 degrees
M5 = 0.0008 minutes F5 = -73.40 degrees
M6 = 0.0069 minutes F6 = -98.81 degrees
Declination Dec
V0 = 0.3838 degrees
M1 = 23.2623 degrees F1 = -169.390 degrees
M2 = 0.3552 degrees F2 = -174.537 degrees
M3 = 0.1342 degrees F3 = -146.899 degrees
M4 = 0.0326 degrees F4 = 4.904 degrees
M5 = 0.0358 degrees F5 = -5.565 degrees
M6 = 0.0324 degrees F6 = -6.757 degrees
Right Ascension RA
V0 = -1.9015 degrees
M1 = 1.8101 degrees F1 = -95.769 degrees
M2 = 2.4153 degrees F2 = -69.272 degrees
M3 = 0.0294 degrees F3 = -47.384 degrees
M4 = 0.0515 degrees F4 = -7.690 degrees
M5 = 0.0200 degrees F5 = 31.193 degrees
M6 = 0.0056 degrees F6 = 20.58 degrees
We can see that only the first 4 harmonicas are meaningful
ERRORS
It is not immediate what error is necessary to consider.
I have calculated, for every day, the greatest difference between the value
of the TEq gotten by the Fourier s. in that day and the 32 values of the
TEq calculated correctly.
Of these differences I have found the greater positive and negative values.
In this way I know what is the maximum error that I can have using the
Fourier s.
The values that I have found for the errors are the followings (I have used
the sign > for " from ...to "):
TEq with 4 harmonicas -0.232> +0.304 minutes Standard Deviation STD =0.17
mins
Dec with 4 harmonicas -0.266> +0.216 degrees STD = 0.14 degrees
RA with 4 harmonicas -0.664> +0.653 degrees STD = 0.36 degrees
I point out that in a given day of the year the values calculated at the
12h in the 32 years have differences from the middle value that are
inferior to:
Teq 0.15 minutes
Dec 0.2 degrees
RA 0.3 degrees
SPENCER's FORMULAS
I have read with a lot of pleasure the article of the 1971 of J.W.Spencer
since I have found in it a confirmation of the correctness of my calculations.
I have transformed the coefficients of Spencer in the form Modulus-Phase
considering the fact that for Spencer the time is t = N-1 days (for me it
is = N-0.5 days) and I have found the following values:
Equation of the time
V0 = 0.017189 minutes
M1 = 7.36396 minutes F1 = 87.16 degrees
M2 = 9.94306 minutes F2 = 110.67 degrees
Declination
V0 = 0.39637 degrees
M1 = 23.26418 degrees F1 = -169.542 degrees
M2 = 0.39068 degrees F2 = -171.370 degrees
M3 = 0.17624 degrees F3 = -149.764 degrees
as you can see they are little different from mine.
It doesn't seem to me that the maximum error for the Declination is correct
(three ')
Using Spencer's coefficients I have found the maximum error of
-0.27> +0.32 degrees
For ex. in the day with N=240 (I think 24 August) the difference between
exact value and value from Spencer is: in 1998 equal to 4.4 ' and in 2000
to 15 ' = 0.5 degrees
CONCLUSION
If have to calculate the TEq, the RA and the Dec of the Sun with precision
not too high (as it is in the calculation of sundials) I think that the use
of the Fourier series (limited to 4 harmonics) is practical and fast.
With my regards
Ing. Gianni Ferrari
Via Valdrighi, 135
41100 - MODENA ( ITALY )
EMail : [EMAIL PROTECTED]
