Gerber, as early as 1895, give the discrepancy in the advance of the perihelion of Mercury as the angle
6 pi GM/c^2 1/a(1-e^2), (1)
where a is the semi-major axis of the ellipse and e is the eccentricity of the orbit. Einstein in more than two decades later, came to the same expression. But, what does this mean in terms of the forces that are acting.
The equation of the orbit from which Einstein obtained the expression contains a perturbation term that implicates and inverse 4th radial force. The unperturbed force goes as the inverse square, and the inverse cube makes the elliptical orbit precess. So that is the lowest order perturbation that could give rise to an observable effect.
Schrodinger attempted to derive the angle from purely classical mechanics without the recourse to Einstein's theory in apparent dig. His expression is
(2 pi/T)^2 2 pi g a^2/(1-e^2), (2)
where T is the period of the orbit and g is a numerical factor that Schrodinger used to fit it to
equation (1). Based solely on dimensional arguments it must be an inverse square in some velocity.
Now using Kepler's area law
h= (2 pi/T) ab, (3)
where h is the angular momentum per unit mass and ab is the area of the ellipse, i.e. b=a(1-e^2)^1/2, is the semi-minor axis. In Einstein's theory h is NOT conserved, but here it is and rigorously. Introducing (3) into (2) gives
h^2 2pi g a^2/b^4= h^2 (4pi g/3) (3a/b^5) ab. (4)
Observing that the average of the inverse square
and that of the inverse fourth,
it is not difficult to see that (4) is the ratio
(4 pi/3)h^2 g <1/r^4>/<1/r^2> (5)
represents the advance of the perihelion of Mercury.
Unlike (1) it does not depend on the particular properties of Mercury or that of the central body around which the advance is occurring. It is simply the ratio of the average of the perturbed force to that of the unperturbed force. In quantum mechanics the numerator is the polarization energy in which the core atoms create an induced dipole moment.
The rotation of the orbit, discovered by Newton, is the average of the inverse cubic. In quantum mechanics this corresponds to the fine structure, or spin-orbit interaction. If the analogy with quantum mechanics bears out, there must be a gravitational analog of the <1/r^6> average force which is the polarizable energy due to core electrons that produce an inverse quadrupole moment. All these effects are directly calculable from only the semi-major and semi-minors axes of the ellipse.