Earth’s figure and gravitation - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 325

erential equation, governing the variation
of the ellipticity of the level surfaces in the Earth, remained unused, owing to the
apparent need to first specify the density variation with depth. In 1885, Radau
found a transformation of the second-order Clairaut equation that reduced it to first
order. G. H. Darwin (1899), in the course of extending the theory of the Earth's
figure to second-degree terms in small quantities, used the Radau transformation to
show that, to good approximation, the internal density distribution enters the theory
only through the ratio of the radius of gyration to the equatorial radius. Therefore,
little about the density distribution was to be learned from geodetic measurements,
but the approximation allowed the ellipticity of figure to be determined from the
precessional constant.
The advent of the artificial Earth satellite has put the study of the figure and
gravitational field on a new level of sophistication. Since the orbit of a satellite
is determined by the gravitational field of the Earth, apart from perturbing e

For more than a century Clairaut's di ff

ects
such as atmospheric drag and radiation pressure, inversion of tracking data can be
performed to produce a very detailed description of the gravitational field. Soon
after the first satellites were launched, a much more accurate direct measure of
the dynamical ellipticity became available (O'Keefe et al. , 1958). This led to a
value for the ellipticity of figure, independent of the hydrostatic assumption, with
far greater accuracy than had been available from conventional geodesy. It also
altered the moment of inertia constraint on Earth models computed from seismo-
logical information, requiring a consequent revision of the density profile (Bul-
len, 1975). Further, the ellipticity of figure, determined through solution of the
Clairaut equation on the hydrostatic assumption, was shown to be significantly
smaller than that based on the directly observed dynamical ellipticity. Satellites,
in addition, have permitted the techniques of geometrical geodesy to be used over
great distances, tying together reference systems between continents; and, since
their orbits are nearly about the geocentre, they give direct access to true geocentric
co-ordinates.

ff

5.2 External gravity and figure
Description of the Earth's figure in mathematical terms is usually accomplished, as
Gauss suggested, by specifying the shape of a level reference surface. It is possible
in principle to measure topography directly, and it may even become practical in
the future to describe topography in detail in geocentric co-ordinates, through the
application of evolving space measurement techniques; however, the level refer-
ence surface is not only more convenient in theoretical development but also arises
naturally in surveying, gravimetry and astronomic observations.