### 3.5 The perihelion shift of Mercury

The explanation of the anomalous perihelion shift of Mercury’s orbit was another of the triumphs of GR.
This had been an unsolved problem in celestial mechanics for over half a century, since the announcement
by Le Verrier in 1859 that, after the perturbing effects of the planets on Mercury’s orbit had been accounted
for, and after the effect of the precession of the equinoxes on the astronomical coordinate system had been
subtracted, there remained in the data an unexplained advance in the perihelion of Mercury. The modern
value for this discrepancy is 43 arcseconds per century. A number of ad hoc proposals were made in
an attempt to account for this excess, including, among others, the existence of a new planet
Vulcan near the Sun, a ring of planetoids, a solar quadrupole moment and a deviation from the
inverse-square law of gravitation, but none was successful. General relativity accounted for
the anomalous shift in a natural way without disturbing the agreement with other planetary
observations.
The predicted advance per orbit , including both relativistic PPN contributions and
the Newtonian contribution resulting from a possible solar quadrupole moment, is given by

where and are the total mass and reduced mass of the two-body system
respectively; is the semi-latus rectum of the orbit, with the semi-major axis a and the
eccentricity e; R is the mean radius of the oblate body; and J_{2} is a dimensionless measure of its quadrupole
moment, given by J_{2} = (C–A) / m_{1}R^{2}, where C and A are the moments of inertia about the body’s
rotation and equatorial axes, respectively (for details of the derivation see TEGP 7.3 [281]). We
have ignored preferred-frame and galaxy-induced contributions to ; these are discussed in
TEGP 8.3 [281].
The first term in Equation (51) is the classical relativistic perihelion shift, which depends upon the PPN
parameters and . The second term depends upon the ratio of the masses of the two
bodies; it is zero in any fully conservative theory of gravity (); it is
also negligible for Mercury, since . We shall drop this term
henceforth.

The third term depends upon the solar quadrupole moment J_{2}. For a Sun that rotates uniformly with its
observed surface angular velocity, so that the quadrupole moment is produced by centrifugal
flattening, one may estimate J_{2} to be 1 10^{–7}. This actually agrees reasonably well with
values inferred from rotating solar models that are in accord with observations of the normal
modes of solar oscillations (helioseismology); the latest inversions of helioseismology data give
J_{2} = (2.2 0.1) 10^{–7} [207, 211, 230, 184]. Substituting standard orbital elements and physical
constants for Mercury and the Sun we obtain the rate of perihelion shift , in seconds of arc per century,

Now, the measured perihelion shift of Mercury is known accurately: After the perturbing effects of the other
planets have been accounted for, the excess shift is known to about 0.1 percent from radar observations of
Mercury between 1966 and 1990 [238]. Analysis of data taken since 1990 could improve the accuracy. The
solar oblateness effect is smaller than the observational error, so we obtain the PPN bound
.