Determination of the Earth’s figure
A historical review
Spherical era
The ancients

Credit for the idea that the Earth is spherical is usually given to Pythagoras (flourished 6th century BC) and his school, who reasoned that, because the Moon and the Sun are spherical, the Earth is too. Notable among other Greek philosophers, Hipparchus (2nd century BC) and Aristotle (4th century BC) came to the same conclusion. Aristotle devoted a part of his book De caelo (On the Heavens) to the defense of the doctrine. He also estimated that the circumference of the Earth is about 400,000 stadia (a Greek stadium varied in length locally from 154 to 215 metres). Since the length of his stadium is not known with certainty, the accuracy of his estimate cannot be established. This seems to be the first scientific attempt to estimate the size of the Earth. Eratosthenes (3rd century BC), however, is considered to be one of the founders of geodesy because he was the first to describe and apply a scientific measuring technique for determining the size of the Earth (see the figure). He used a simple principle of estimating the size of a great circle passing through the North and South poles. Knowing the length of an arc (l) and the size of the corresponding central angle (a) that it subtends, one can obtain the radius of the sphere from the simple proportion that length of arc to size of the great circle (or circumference, 2πR, in which R is the Earth’s radius) equals central angle to the angle subtended by the whole circumference (360°):


The introduction of triangulation



Ellipsoidal era

The period from Eratosthenes to Picard can be called the spherical era of geodesy. A new ellipsoidal era was begun by Newton and the Dutch mathematician and scientist Christiaan Huygens. In Ptolemaic astronomy it had seemed natural to assume that the Earth was an exact sphere with a centre that, in turn, all too easily became regarded as the centre of the entire universe. However, with growing conviction that the Copernican system is true—the Earth moves around the Sun and rotates about its own axis—and with the advance in mechanical knowledge due chiefly to Newton and Huygens, it seemed natural to conceive of the Earth as an oblate spheroid. In one of the many brilliant analyses in his Principia, published in 1687, Newton deduced the Earth’s shape theoretically and found that the equatorial semiaxis would be 1230 longer than the polar semiaxis (true value about 1300).

Experimental evidence supporting this idea emerged in 1672 as the result of a French expedition to Guiana. The members of the expedition found that a pendulum clock that kept accurate time in Paris lost 212 minutes a day at Cayenne near the Equator. At that time no one knew how to interpret the observation, but Newton’s theory that gravity must be stronger at the poles (because of closer proximity to the Earth’s centre) than at the Equator was a logical explanation.





The expression deduced by Clairaut is

where m = centrifugal acceleration at Equator / attraction at Equator.




The concept of the geoid



Determining the form of the geoid with Stokes’s formula



Combining both effects, therefore,


The contribution of orbiting satellites

The development of artificial satellites whose orbits could be observed from Earth totally revolutionized man’s ability to define the shape of the Earth and its gravity field. A value for the flattening of the ellipsoid that superseded all previous values was obtained within weeks after the launching of the Soviet Sputnik I in 1957. Since that time, scientists have repeatedly refined the geoid with observations from a succession of Earth-orbiting satellites.


It is usual to start with an expression for the potential U of the Earth’s gravitational field, in spherical coordinates (r,θ,λ), with origin at the mass centre of the Earth. The gravitational potential U satisfies Laplace’s equation, a widely used second-order partial differential equation named after the 18th-century French mathematician and astronomer Pierre-Simon Laplace. Accordingly, it can be expressed as a sum of spherical harmonics (a series of terms by which a variation of a quantity over the surface of a spherical or nearly spherical body such as the Earth can be expressed mathematically to any desired degree of accuracy):




Physically, the term represents the effect on the potential of the ellipsoidal shape of the Earth, and it is not surprising, therefore, that J2, o, known as the dynamical form factor, is closely related to the flattening f. In fact,

where m is the quantity introduced in equation (3).

As a satellite in an inclined orbit passes over the equatorial region of the Earth, it experiences a force toward the Equator as a result of the mass in the equatorial bulge. This force represents a torque about the origin, and as in the case of the spinning top or gyroscope, the application of the torque causes the rotation axis of the satellite (normal to the plane of the orbit) to precess about the Earth’s rotation axis. The plane of the orbit therefore precesses, resulting in changes in the satellite’s path that can be observed from Earth with a high degree of accuracy.





The tesseral harmonics in equation (7), those terms involving Jl, m, present an additional difficulty. Because the terms represent contributions to the potential having a longitude dependence, their effect in general is averaged out as the Earth rotates under a satellite. The only exception is if the orbit and period of the satellite are such that the satellite tracks over points on the Earth equally separated in longitude, repeating each track precisely after an integral number of cycles. Such a satellite is said to be resonant to a particular value of m. Values of m for which resonant satellites can be found are limited, lying between 9 and 15. For other tesseral harmonics, observations of surface gravity must again be used.





Radar altimetry of the ocean surface




Earth dimensions—radius, mass, and density

As previously noted, terrestrial arc measurements are capable of yielding a value of the equatorial radius of the Earth, but satellite measurements are greatly superior for determining the flattening. After 10 years of satellite observations, the International Union of Geodesy and Geophysics adopted the Geodetic Reference System 1967, defining aequatorial, MG, and J2, o. Minor revisions to the numerical values were made in 1983. The revised values are as follows:

The adopted value of J2, o corresponds to a flattening of 1298.257.

While satellite observations determine the value of the product MG to eight significant figures, they cannot determine M and G separately. Because satellites orbit (in general) above the atmosphere, the value of M includes the mass of the atmosphere, but, as shown above, the contribution of the latter to MG is extremely small. The gravitational constant G, measured in the laboratory, is known much less accurately; it is G = 6.67259 67428 × 10−11 m3s−2kg−1 (with uncertainty in the last place two places of decimals). The combination of the laboratory value of G and the adopted value of MG results in a value for the mass of the Earth (including the atmosphere) of M = 5.98 97 × 1024 kg. With the volume determined by aequatorial, the flattening, and the portions above sea level, this value of the mass gives the mean density ρ = 5,517 kg/m3.

There is some indication that J2, o, the dynamical form factor, varies slowly with time. There also have been suggestions that G has varied with time throughout the history of the universe and that it is scale-dependent. In the latter case, values determined in the laboratory would not be appropriate for terrestrial or astronomical problems. Evidence for either a time- or scale-dependence of G remains inconclusive.

For many years there has been speculation about the extent to which the actual flattening of the ellipsoid coincides with the theoretical form of a mass of fluid—of the same mass and rotation rate as the Earth—in hydrostatic equilibrium under its own attraction and rotational acceleration. In the presatellite era, neither the actual flattening nor the theoretical form was known with sufficient accuracy to permit a meaningful comparison. Recent estimates of the flattening, in the case of hydrostatic equilibrium, for a body free of lateral density variations are close to 1299.5; the actual flattening, 1298.257, is therefore slightly greater. Although some investigators have suggested that the discrepancy represents an inheritance from the time when the Earth was rotating more rapidly on its axis, the most probable explanation is that it is simply one effect of the recognized lateral heterogeneity in density of the real Earth.