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Measurement of gravitational force.



a branch of science that deals with the measurement of the quantities that characterize the earth’s gravitational field and with their use in determining the figure of the earth, for the study of its general internal structure and the structure of its upper parts, and for the solution of certain navigation problems. A future task of gravimetry is the study of the moon and the planets according to their gravitational fields.

In gravimetry, the earth’s gravitational field is usually defined as a field of gravitational force (or an acceleration due to gravity that is numerically equal to it) that is the resultant of two main forces, the earth’s attractive force and a centrifugal force caused by the diurnal rotation of the globe. The centrifugal force, directed away from the axis of rotation, reduces the force of gravity (to the greatest extent at the equator). The decrease in gravitational force from the poles to the equator is also influenced by the compression of the earth. As a result of these two factors, the force of gravity at the equator is about 0.5 percent less than at the poles. The change in gravitational force caused by the attractions of the sun and moon does not exceed a few ten-millionths of the total, and the variations caused by shifts of masses in the bowels of the earth and of masses of air are still smaller.

Gravity values on the earth’s surface depend on the figure of the earth and its internal density distribution. Therefore, the study of the earth’s gravitational field provides valuable material for estimates of its shape and internal structure, particularly in prospecting for minerals.

Determination of the force of gravity The force of gravity is determined by the relative method and by measurements, made with gravimeters and pendulum instruments, of the difference in the force of gravity at the points being studied and at reference points. The network of gravimetric reference points over the entire earth is connected, in the final analysis, with a point at Potsdam (German Democratic Republic). where, at the beginning of the 20th century, the absolute value of the acceleration due to gravity (981.274 milligals) was determined by means of reversible pendulums.

Absolute determinations of the force of gravity involve considerable difficulties and are less accurate than relative measurements. New absolute determinations, carried out at more than ten points on the earth’s surface, show that the above value of the acceleration of gravity at Potsdam is apparently 13–14 milligals too high. When this work was completed, a new gravimetric system was set up. However, in many problems of gravimetry, this error is of no real significance. since their solution depends not on absolute values but on differences. The most accurate determination of the force of gravity depends on the free fall of bodies in a vacuum chamber. Progress in the technique of time and distance measurement facilitates the success of the experiments.

Relative determinations of the force of gravity are made with pendulum instruments that are accurate to a few hundredths of a milligal. Gravimeters provide somewhat greater measurement accuracy than pendulum instruments and are portable and simple to use. Special gravimetric apparatus exists for measuring gravity from moving objects (submarine and surface craft, airplanes). The instruments continuously record the change in the acceleration due to gravity over the path taken by the ship or airplane. In connection with such measurements it is difficult to protect the instrument readings from the disturbing influence of accelerations and inclinations of the instrument’s base caused by rolling. There are special gravimeters for making measurements at the bottom of shallow bodies of water and in boreholes. Torsion balances measure the second derivatives of the potential of the force of gravity.

The study of a steady spatial gravitational field is used in solving the main range of gravimetry problems. Gravity variations with time are continuously recorded to study the earth’s elastic properties. Since the earth is of nonuniform density and irregular shape, its external gravitational field is characterized by a complex structure.

For the solution of various problems it is convenient to regard the earth’s gravitational field as consisting of two parts: the main part, called the normal, which varies with the latitude of a location according to a simple law, and the anomalous part, which is of small size but complex distribution and is caused by inhomogeneities in the density of rock in the upper layers of the earth. The normal gravitational field corresponds to an idealized model of the earth that is simple in form and internal structure (an ellipsoid or a spheroid close to an ellipsoid). The difference between the observed force of gravity and the normal distribution of the normal force of gravity as found from one or another formula and reduced by the appropriate corrections to the assumed altitude level is called the gravitational anomaly. If only the normal vertical gravity gradient, which is 3,086 Eötvös units, is considered when making the reduction (that is, assuming that there are no masses between the point of observation and the reduction level), the resultant anomaly is called the free-air effect. The anomalies thus calculated are most often used in studying the shape of the earth. If allowance is also made for the attraction of a layer of masses between the observation and reduction levels that is considered to be uniform, the anomalies produced are called the Bouguer effect. They reflect the nonuniformity in density of the upper parts of the earth and are used in solving problems in geological prospecting.

Isostatic anomalies, which take into account in a special way the effect of masses between the earth’s surface and the level of the surface at a depth at which the overlying masses exert an identical pressure, are also considered in gravime-try. In addition to these anomalies, gravimetry deals with a number of others (Prey anomalies, the modified Bouguer effect, and so on). Gravimetric maps with isolines of gravity anomalies have been compiled from gravimetric measurements. Anomalies of the second derivatives of the gravitational potential are found analogously as the difference between the observed value (corrected for the terrain of the locality) and the normal value. Such anomalies are mainly used in prospecting for minerals.

Study of the figure of the earth In problems connected with using gravimetric data to study the figure of the earth, the ellipsoid that best represents the geometrical shape and external gravitational field of the earth is usually sought. In the mid-18th century the French scientist A. Clairaut found the general law of variation of force of gravity y with geographic latitude Φ. assuming the mass inside the earth to be in hydrostatic equilibrium:

where γe is the force of gravity at the equator, q = ω2α/γe, is the ratio of centrifugal force to the gravitational force at the equator, y is the compression of the earth’s ellipsoid, ω is the angular velocity of the earth’s diurnal rotation, and a is the major semiaxis of the earth. Having determined ω and a from astronomical and geodetic observations and having measured the force of gravity at various latitudes, the above formulas are used to obtain the earth’s compression α. In the mid-19th century the British scientist G. Stokes generalized Clairaut’s deduction, having shown that given the shape of the level surface, the direction of the axis, and the velocity of diurnal rotation, as well as the total mass of the earth contained inside the level surface, with any density distribution, the gravitational potential and its derivatives are uniquely determined in all external space. To solve the inverse problem— determining the level surface, a particular case of which is the geoid, from a given gravitational field—Stokes derived a formula that made possible the calculation of the height of the geoid relative to the ellipsoid, if the distribution of gravity over the entire earth is known. Theory and experiment show that the geoid closely approximates an ellipsoid, with deviations not exceeding dozens of meters. The Dutch scientist F. Vening Meinesz derived a formula for determining the Eötvös deviations from the gravitational anomalies. The theories of Stokes and Clairaut were replaced in the mid-1940’s by a theory of the physical surface of the earth, the idea of which was first formulated by the Soviet scientist M. S. Molodenskii. His theory is free from hypotheses regarding mass distribution below the surface of observation. It makes possible the calculation, with any accuracy desired, of the desired elements of the earth’s gravitational field; the accuracy is determined only by the accuracy of the measurements made at the earth’s surface. Instead of the geoid, an auxiliary surface called a quasi-geoid is used.

Inhomogeneities of density Gravimetric measurements are used to study inhomogeneities of density in the upper parts of the earth for geological prospecting purposes. The analysis of gravity anomalies is the basis for qualitative conclusions regarding the positions of the masses causing the anomalies; under favorable conditions, quantitative calculations are made. The gravimetric method makes possible the more rational arrangement of drilling and geological-prospecting work. It aids in the study of strata of the earth’s crust and upper mantle that are inaccessible to drilling and ordinary geological observations. The study of the earth’s gravitational field forms a basis for investigating the questions of whether the earth is in a condition of hydrostatic equilibrium and what kind of stresses exist in the body of the earth. By comparing observed changes in the force of gravity under the attraction of the moon and sun with the theoretical values calculated for a completely solid earth, conclusions are drawn about the earth’s internal structure and elastic forces. Knowledge of the detailed composition of the earth’s gravitational field is also necessary for calculating the orbits of artificial earth satellites. There the main effect is exerted by the inhomogeneity of the gravitational field caused by the earth’s compression. The inverse problem—that is. calculating the components of the gravitational field from observations of perturbations in the motion of artificial satellites—is also being dealt with. Theory and experiment show that this method gives with a very high degree of reliability the gravitational fields derived with the least accuracy by gravimetric measurements. Therefore, satellite and gravimetric observations, as well as geodetic measurements on the earth, are combined in studying the figure of the earth.


Shokin. P. F. Gravimetriia. Moscow, 1960.
Brovar, V. V.. V. A. Magnitskii, and B. P. Shimbirev. Teoriia figury Zemli. Moscow, 1961.
Grushinskii, N. P. Teoriia figury Zemli. Moscow, 1963.
Kaula, W. M. Kosmicheskaia geodeziia. Moscow. 1966. (Translated from English.)
Veselov, K. E.. and M. U. Sagitov. Gravimetricheskaia razvedka. Moscow. 1968.


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