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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 Δ &tidle;ω, including both relativistic PPN contributions and the Newtonian contribution resulting from a possible solar quadrupole moment, is given by

( 2) Δ &tidle;ω = 6πm-- 1(2 + 2γ − β) + 1(2α − α + α + 2ζ ) μ-+ J2R-- , (51 ) p 3 6 1 2 3 2 m 2mp
where m ≡ m1 + m2 and μ ≡ m1m2 ∕m are the total mass and reduced mass of the two-body system respectively; 2 p ≡ a(1 − e ) 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 J2 is a dimensionless measure of its quadrupole moment, given by J2 = (CA) / 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 [281Jump To The Next Citation Point]). We have ignored preferred-frame and galaxy-induced contributions to Δω&tidle;; these are discussed in TEGP 8.3 [281Jump To The Next Citation Point].

The first term in Equation (51View Equation) is the classical relativistic perihelion shift, which depends upon the PPN parameters γ and β. The second term depends upon the ratio of the masses of the two bodies; it is zero in any fully conservative theory of gravity (α1 ≡ α2 ≡ α3 ≡ ζ2 ≡ 0); it is also negligible for Mercury, since μ ∕m ≈ mMerc∕M ⊙ ≈ 2 × 10−7. 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 [207211230184]. Substituting standard orbital elements and physical constants for Mercury and the Sun we obtain the rate of perihelion shift ω˙&tidle;, in seconds of arc per century,

( ) ˙ ′′ 1- −4-J2-- &tidle;ω = 42. 98 3 (2 + 2 γ − β) + 3 × 10 10− 7 . (52 )
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 −3 |2γ − β − 1| < 3 × 10.
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