The predicted advance per orbit , including both relativistic PPN contributions and the Newtonian contribution resulting from a possible solar quadrupole moment, is given bya and the eccentricity e; R is the mean radius of the oblate body; and J2 is a dimensionless measure of its quadrupole moment, given by J2 = (C–A) / m1R2, 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 ). We have ignored preferred-frame and galaxy-induced contributions to ; these are discussed in TEGP 8.3 .
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 J2. 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 J2 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 J2 = (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,. 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 .
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