Updated by Steve Carlip 1996.
Original by Michael Weiss.
If the Sun were not a perfect sphere, but had an equatorial bulge (i.e., was oblate), that would cause Mercury's orbit to precess. Well, surprise: the Sun isn't a perfect sphere; it is oblate! So how good is the agreement between Mercury's orbital precession and GR?
Clifford Will devotes a chapter to this in his 1986 book "Was Einstein Right? Putting General Relativity to the Test". Dicke and Goldenberg claimed to have detected a much larger bulge than solar models predicted, large enough to destroy the agreement between GR and Mercury's orbit, but not large enough to permit a Newtonian explanation. Specifically, the data looks like this:
Mercury's perihelion precession: 574 arcseconds/century
Newtonian perturbations from
other planets: 531 arcseconds/century
GR correction: 43 arcseconds/century
Newtonian correction from Dicke bulge: 3 arcseconds/century
So no hope for Newton, but a problem for GR if the Dicke-Goldenberg value for solar oblateness held up. The Brans-Dicke scalar-tensor theory of gravity could handle the discrepancy, via its adjustable parameter.
Subsequent observations by other groups detected much smaller solar bulges. However, the other measurements disagreed with each other. Here's the data, again from Will's book:
Amount predicted by conventional 0.2 km, 0.1 ppm solar models Dicke-Goldenberg (1966) 52 km, 40 ppm Hill (1973) 2 km, 1 ppm Hill (1982) 10 km, 7 ppm Dicke (1985) 24 km, 12 ppm
These aren't all direct measurements, and I haven't given any of the error bars, but here's Will's bottom line: "No one has been able to resolve or understand the discrepancies between all these values for solar oblateness, other than to say that the observations are difficult to make and subject to many errors." (In case you're wondering about the Brans-Dicke theory, that ran into a bunch of other problems during the 70's and 80's.)
However, At least two important new developments have occurred since Will's book was written.
1. A new and accurate direct measurement of Solar oblateness has been performed, using the balloon-borne Solar Disk Sextant. The result is a value that agrees with general relativity (a quadrupole moment on the order of 2 x 10-7). See Lydon and Sofia, Phys. Rev. Lett. 76, 177 (1996).
2. A new, slightly indirect but quite powerful, method has been developed to measure Solar quadrupole moment, by using helio-seismography to determine the rate of rotation of the Sun. The results again give a quadrupole moment on the order of 10-7, too small to affect the agreement between general relativity and the observed advance of Mercury's perihelion. The best reference I know is Brown et al., Astrophys. J. 343, 526 (1989), which is cited by Will in the 1993 revision of his more technical book, Theory and experiment in gravitational physics. There may be more recent results.
While it's too early to say the issue is completely settled, the most recent and most accurate results seem to be converging towards a value that makes the GR predictions agree well with observation.