RE: About the equation of center

[Roger Bailey]

Hello Anselmo,   The derivation of the equation of the center shown below is given in W. M. Smart's classic "Textbook on Spherical Astronomy". This topic is covered on pages 116 to 120 of the Sixth Edition (1977) that I picked up in a used book store years ago.   The proof is too long to include in the margin of this e-mail but it is based on the following logic.   1. Express Kepler's equation E = M + eSin E as a series of successive approximations. This is a power series for e.   2. Express the true anomaly  v as a series in terms of e and the eccentric anomaly E. This starts with the usual _expression_  Tan v/2 = [(1+e)/(1-e)]^.5  Tan E/2.  The approximate solution involves complex number representations of trig functions and a logarithmic series expansion.   3. Bring the two series equations together and eliminate the higher order terms to obtain the equation of the center that you have given below.   Please, no questions. I am simply providing the reference. Although I use the equations, I don't really understand the math. .   I can scan the relevant pages and provide TIF files as e-mail attachments.   Roger Bailey Walking Shadow Designs N 51  W 115     Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]On Behalf Of Anselmo Pérez Serrada Sent: April 6, 2002 8:26 AM To: Sundial, Mailinglist Subject: About the equation of center 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