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Astronomy/Fourier expansion of (w-M)

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Expert: - 2/8/2010


Question
I am interested in satellites with highly elliptical orbits like the Molniya. In that case I would need several more terms in the Fourier expansion of w-M. Can you provide a way to get additional terms in the Fourier expansion?

Answer
Hello,

Actually it is the Lagrange expansion theorem which is used to generate terms for (w - M) which is known as "the equation of the center".  But how many terms do you really need- and CAN your computing facilities support the accuracy associated with terms beyond say order four for e? I doubt it.

Anyway, the sequence of terms usually given up to order e^4 (in the Lagrange terms) is:

w = M + (2e - e^3/4)sin M  + 5e^2/4 {sin 2M}  + 13 e^3/12 {sin 3M}

Which is certainly adequate for nearly all computations unless you actually want to travel to a planet,....or in this case....a satellite.

If this isn't sufficient, you can find the detailed mathematical development for generating many more terms in Philip M. Fitzpatrick's 'Principles of Celestial Mechanics', pp. 93-95.

It's much too involved to give here and this interface isn't suitable for the notation.  

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Philip Stahl

Expertise

I have forty years of experience in Astronomy, specifically solar and space physics. My specialties include the physics of solar flares, sunspots, including their effects on Earth and statistics as applied to astronomical investigations.

Experience

Astronomy: more than forty years experience starting with construction of my own simple telescopes. Worked at university observatory in college, doing astrographic measurements. M.Phil. degree in Physics/Solar Physics and more than ten years as researcher.

Organizations
American Astronomical Society (Solar Physics and Dynamical Astronomy divisions), American Mathematical Society, American Geophysical Union

Publications
Solar Physics (journal), The Journal of the Royal Astronomical Society of Canada, The Proceedings of the Meudon Solar Flare Workshop (1986), The Proceedings of the Caribbean Physics Conference (1985). Books: 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level'.

Education/Credentials
B.A. Astronomy, M. Phil. Physics

Awards and Honors
American Astronomical Society Studentship Award (1984), Barbados Government Award for Solar Research

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