Newton vs Einstein in the Precession of Mercury
The saying that "Let Newton be and all was light" is not far from its mark. Newton left little space for others to build in in such diverse fields as mechanics and optics. It left people with digging into minute differences between what Newton predicted and what was observed.
One such instance occurred in 1859 when Le Verrier noticed that Mercury's orbit did not close exactly after one orbital motion but seemed to precess causing an add to the 360 degrees in order for it to return to its initial position along the orbit. In Proposition 45 of Book I of Principia published in 1687, Newton used the lack of precession, or quiescence of the apsides in Newton's terminology, to establish his inverse square law. If the transit between the two apses of the orbit were not 180 degrees, then Newton gave the apsidal angle as 180/the square root of a certain number n which appear in the exponent of the potential as n-2. For n=2, the apsidal angle is 180 as it should be, but for n different than 2, it meant that his inverse square law was not being obeyed.
Parenthetically, there is another value of n for which the apsidal angle does not depend on initial conditions. That value is 4, and corresponds to a harmonic oscillator potential. The orbit, rather than being elliptical, is radial and in time 360 degrees returns to its initial position. In the one pound note on my home page, the artist put the sun at the center of the ellipse instead at one of its foci. The image should not have been Newton, but rather that of Hooke, his archrival enemy. Newton certainly wouldn't have appreciated that.
However, there is the possibility of transmuting a linear restoring force into an inverse square force simply by squaring the original origin-centered ellipse into another ellipse where one of its foci is moved to the origin. The act of squaring sends the left-hand focus to the origin. This possibility was derived by Bertrand in 1875 where the only two laws that do not involve precession are the inverse-square law and the linear law. This bonded Hooke to Newton for all time.
Now getting back to Le Verrier, if Mercury's orbit did not close, then Newton's square law needed to be modified. And the modification could be obtained by the apsidal angle. The gap between all known factors that could influence the orbit and its observed value was 38 arc seconds, which Newcomb later increased to 43 arc seconds. On this basis Hall in 1895 modified the 2 in the inverse square law to 2.00000016.
Newton believed his expression for the apsidal angle was good for small values of the eccentricity of the orbit. Mercury's eccentricity is 0.205 which indicates that the greatest distance from the sun is 1.5 times its smallest distance. And the apsidal angle should be a function slowly of this ratio.
The elliptical orbit has two unknown parameters, the total energy, which must be negative, and the angular momentum, both of which are conserved quantities. However, since both these parameters can be expressed in terms of the perihelion and aphelion (extreme distances), the apsidal angle should be a function of these distances, and more specifically their ratio. Now, the ratio of their difference and sum is the definition of the eccentricity, e, of the orbit. So the ratio of these distance is simply the ratio (1+e)/(1-e). And since the apsidal angle depends only on their ratio, it must be a sole function of e, the eccentricity.
Enter Einstein. Using the Schwarzschild solution to his field equations, Einstein obtained a small modification of the elliptic orbit. From this he was able to calculate the apsidal angle as the square of the ratio of the gravitational parameter, GM, to the angular momentum. For a constant mass of the sun, M, this mean that the apsidal angle should decrease with increasing angular momentum, which seem counter-intuitive. Using properties of the ellipse, this expression could be written as the negative ratio of the total energy to the product of the extreme distances, not as their ratio. As the extremes become further apart, the apsidal angle decreases, which is also counter-intuitive. And it seems very unlikely that it can be generalized to the apsidal angles of the other planets.
On the contrary, the relativists, like Kevin Brown, claim, when referring to Einstein's formula for the apsidal angle that "this sometimes misleads people into thinking that the lowest order approximation includes an effect due to the eccentricity, but in fact the eccentricity appears in this formula only to convert the semi-latus rectum into the semi-major axis."
This claim implies that the apsidal angle cannot be obtained in terms of the eccentricity alone, but needs the square of the speed of light to make the dimensions come out right. For the claim is that the apsidal angle is the square of the ratio of the gravitational parameter, GM, and the angular momentum. In order that the ratio be dimensionless, it must be divided by the square of velocity, and the only one around is that of light. This "parameter" is completely extraneous to the revolving orbits where the sole parameter is the eccentricity, or the ratio of the extreme radii.