Jean Meeus's "Astronomical Algorithms" gives five abbreviated coefficients
for terms up to sin 5M. He references
"Annales de l'Observatoire de Paris," Vol. I, pages 202-204, and only says
that it is derived from a series expansion.
The abbreviated coefficients are
2e - (e^3}/4 + (5/96)e^5
(5/4)e^2 - (11/24)e^4
(13/12)e^3 - (43/64)e^5
(103/96)e^4
(1097/960)e^5
I hope that the more-complete coefficients will give you additional
insight. Good luck with your derivation.
Gordon
At 05:25 PM 4/6/02 +0200, Anselmo Pérez Serrada wrote:
Hi dialists,
Maybe this is an off-topic, but I found it in some gnomonics books and I'd
like to know more about it:
It is well known that due to Kepler's Second Law the Earth (and any
satellite) does not follow a circular
uniform movement but an elliptical non-uniform one. So the longitude of
the Earth across the ecliptic is
not exactly proportional to time: the difference (True Longitude minus
Mean Longitude) is called equation
of center 'c' and can be derived either solving Kepler's equation ( M = E
- e*sin(E) ) or using an approximate
formula, something like
c = (2e- 1/4*e^3)*sin(M) + 5/4*sin(2*M) + 13/12*e^3*sin(3*M)+ ...
where M, the mean anomaly, is the fraction of area swept by the Earth and
e is the eccentricity of the orbit
(Yes, it's pre-copernican, but it works!).
OK, but which is the general expression of this formula (for the n-th
term) and where does it come from? Is
it a Fourier series or a Taylor series? I've tried both and wasn't able
to reach to any result like this.
Could anyone point me at some website where this series is derived?
Anselmo Perez Serrada
Gordon Uber [EMAIL PROTECTED] San Diego, California USA
Webmaster: Clocks and Time: http://www.ubr.com/clocks
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