MadSci Network: Astronomy Query:

### Re: how do physicists calculate the advance of perihelion of Mercury?

Date: Tue Aug 5 07:47:21 2008
Posted By: Jim Guinn, Staff, Science, Georgia Perimeter College
Area of science: Astronomy
ID: 1215420869.As
Message:
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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

Sincerely,

Jim Guinn

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