3.6 Tests of the strong 3 Tests of Post-Newtonian Gravity3.4 Tests of the parameter

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 tex2html_wrap_inline4711, including both relativistic PPN contributions and the Newtonian contribution resulting from a possible solar quadrupole moment, is given by


where tex2html_wrap_inline4713 and tex2html_wrap_inline4715 are the total mass and reduced mass of the two-body system respectively; tex2html_wrap_inline4717 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 tex2html_wrap_inline4725 is a dimensionless measure of its quadrupole moment, given by tex2html_wrap_inline4727, 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 [147Jump To The Next Citation Point In The Article]). We have ignored preferred-frame and galaxy-induced contributions to tex2html_wrap_inline4711 ; these are discussed in TEGP 8.3 [147Jump To The Next Citation Point In The Article].

The first term in Eq. (35Popup Equation) is the classical relativistic perihelion shift, which depends upon the PPN parameters tex2html_wrap_inline4351 and tex2html_wrap_inline4353 . The second term depends upon the ratio of the masses of the two bodies; it is zero in any fully conservative theory of gravity (tex2html_wrap_inline4739); it is also negligible for Mercury, since tex2html_wrap_inline4741 . We shall drop this term henceforth. The third term depends upon the solar quadrupole moment tex2html_wrap_inline4725 . 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 tex2html_wrap_inline4725 to be tex2html_wrap_inline4747 . 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). Substituting standard orbital elements and physical constants for Mercury and the Sun we obtain the rate of perihelion shift tex2html_wrap_inline4749, 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 [116]. 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 tex2html_wrap_inline4751 .

3.6 Tests of the strong 3 Tests of Post-Newtonian Gravity3.4 Tests of the parameter

image The Confrontation between General Relativity and Experiment
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de