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 is a dimensionless measure of its quadrupole moment, given by , 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 Eq. (35) 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 . 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 to be . 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 , 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 . 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 .
|The Confrontation between General Relativity and
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
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