How do you calculate the anomalous precession of Mercury? One of the three classic tests of general relativity is the calculation of the precession of the perihelion of Mercury's orbit.
This precession rate had been precisely measured using data collected since the 1600's, and it was later found that Newton's theory of gravity predicts a value that differs from the observed value. That difference, which I am calling the anomalous precession, was estimated to be about 43 arcseconds per century in Einstein's time.
I have heard that general relativity predicts an additional correction that is almost exactly sufficient to account for that 43"/century difference, but I've never seen that calculation done, at least not correctly. Can anyone supply the details?
 A: Well, it goes like this: consider Schwarzschild metric and testing particle (this is Mercury) with energy $E$ and momentum $L$. Because you have enough integrals of motion, the equations basically solve themselves and you obtain an effective potential that contains basic Newton potential, the centrifugal potential and a correction term. Then you apply Binet equation and you are left with some differential equation that is not easy to solve, but essentially is an equation for conic section (as in the classical case) plus some correction terms. So you make some approximation (based on the parameters of the problem) and are left with "conic section" that precesses a little.
Now, I wonder: how much more precise do you want the above argument to be made? Do you want a complete derivation, or just to clarify some confusing point? Also, how much are you familiar with GTR? I am asking just so that I know at what level should I be explaining.
Also, see the wikipedia pages, it looks like they have some derivation there (although I did not check it and you can't always trust wikipedia).
A: Try http://www.mathpages.com/rr/s6-02/6-02.htm .  Caveat, I haven't looked at it carefully yet.   
There is a detailed discussion at http://wapedia.mobi/en/Two-body_problem_in_general_relativity?t=3.
A: Sorry this is one year late - but there is a rather detailed calculation of the precession of Mercury's orbit using General Relativity in Cornelius Lanczos book The Variational Principles of Mechanics, Dover Publications. The first edition appeared in 1949, the Dover edition in 1986. 
A: A very detailed computation with a comparison between the classical and the relativistic solution: The Precession of Mercury’s Perihelion.
A: Einstein's general relativity states that, in a Schwarzschild spacetime (which approximates our solar system), gravity is slightly different from the inverse square law by an extra fourth power term. It is this extra small term that causes the perihelion motion of Mercury's orbit. If we take this extra term into account and follow the same logic of classical considerations, we find that planets' orbits aren't closed, but are precessing consistently. Mercury's orbit has an apprant precession about 43 arcs per century at its perihelion position which is exactly the amount of precession observed.
