|MadSci Network: Astronomy|
Dear Thierry, This is a very interesting question and it involves some rather advanced classical mechanics. I looked through Newton’s “Principia” and Richard Westfall’s biography on Newton “Never at Rest”, but I couldn’t find a single reference to the determination of perihelion (periapsis, more generally) advance due to either the Sun or the presence of other planets. This is not to say Newton didn’t do these types of calculations, I just couldn’t find them if he did. He did do a number of other calculations dealing with the motion of the periapsis (see for example Section IX, Proposition XLIII, Problem XXX in the Principia) which you might want to look at. This problem may be solved by assuming that the “non-sphericalness” of the Sun, or the presence of another gravitating body like the Earth, causes a small perturbation in the 1/r^2 force that is acting on Mercury. (This approach is described in Herbert Goldstein’s book “Classical Mechanics”, second edition, on page 123 problem 14.) We can write this in terms of Mercury’s gravitational potential in the form V(r) = -k/r + h/r^2 , where we will assume that h is a small quantity. The resulting perihelion shift is given as d omega / dt = 2 pi m h / l^2 tau , where “omega” is the perihelion angle, “m” is Mercury’s mass, “h” is the small perturbation parameter in the potential, “l” is Mercury's orbital angular momentum, and “tau” is the period of the orbit. Well, I hope this is the equation you are looking for, Thierry. If you would like some more information, please let us know. Sincerely, Jim Guinn
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