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 [147
]). We have ignored preferred-frame and galaxy-induced
contributions to
; these are discussed in TEGP 8.3 [147
].
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 [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
.
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The Confrontation between General Relativity and
Experiment
Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |