MadSci Network: Astronomy

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

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.


Jim Guinn

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