# Gravitational Field of the Earth

## Gravitational Field of the Earth

a gravitational field; a force field caused by the attraction (gravitation) of the earth and by the centrifugal force that results from its diurnal rotation. This field also depends slightly on the attraction of the moon, the sun, and other heavenly bodies and masses in the terrestrial atmosphere.

The gravitational field of the earth is characterized by the force of gravity and by the potential force of gravity and various derivatives of it. The potential has the dimension cm^{2} • sec^{−2}. The milligal (mgl), which is equal to 10^{−3} cm •sec^{−2}, is used in gravimetry as the unit of measurement of the first derivatives of potential, including the potential force of gravity, and the Eötvös unit (E), which is equal to 10^{−9}sec^{−2}, is used for second derivatives. The part of the potential force of gravity that is due solely to the attraction of the mass of the earth is called the gravitational potential of the earth, or the geopotential.

For practical problem-solving the gravitational potential of the earth is represented in the form of the series

where ρ is the geocentric distance, Φ and λ are the geographic latitude and longitude of the point at which the potential is being examined, P _{nm} are associated Legendre functions, *GE* is the product of the constant of gravitation and the mass of the earth, equal to 398.603 x 10^{9} m^{3} • sec^{−2}, *a* is the major semiaxis of the earth, and C _{nm} and S _{nm} are dimensionless coefficients that depend on the figure of the earth and the internal distribution of the mass within it. The main member of the series, *GE/p*, corresponds to the gravitational potential of a sphere having the mass of the earth. The second largest member (which contains C_{20}) takes into consideration the compression of the earth. The following members, whose coefficients are three or more orders less than C_{20}. reflect details of the figure and structure of the earth.

Because of the lack of precise data on the figure of the earth and on the true distribution of masses within it, the direct computation of the coefficients C_{nm} and S_{nm} is impossible. Therefore they are determined indirectly on the basis of aggregate measurements of the force of gravity at the surface of the earth and on the basis of observations of perturbations in the motion of nearby artificial earth satellites. The results of a determination of the resolution factors established on the basis of observations of the motions of artificial satellites are presented in Table 1. The gravitational field of the earth is described by similar series.

Table 1. Coefficients of resolution (multiplied by 10^{6}) of the gravitational potential of the earth according to spherical functions as determined from observations of the motion of artificial earth satellites (based on data of the Smithsonian Astro-physical Observatory in the USA, published in 1970) | ||||||
---|---|---|---|---|---|---|

m | 0 | 1 | 2 | 3 | 4 | 5 |

C_{2m} | –1,082.63 | — | 2.41 | — | — | — |

S_{2m} | — | — | –1.36 | — | — | — |

C_{3m} | 2.54 | 1 97 | 0.89 | 0.69 | — | — |

S_{3m} | — | 0.26 | –0.63 | 1.43 | — | — |

C_{4m} | 1.59 | –0.53 | 0.33 | 0.99 | –0.08 | — |

S_{4m} | — | –0.49 | 0.71 | –0.15 | 0.34 | — |

C_{5m} | 0.23 | –0.05 | 0.61 | –0.43 | –0.27 | 0.13 |

S_{5m} | — | –0.10 | –0.35 | –0.09 | 0.08 | –0.60 |

For the sake of convenience in solving various problems, the gravitational field of the earth is arbitrarily divided into normal and anomalous parts. The main part (the normal part), which is described by several first expansion terms, corresponds to an idealized (“normal”) earth, having a simple geometric form and simple density distribution within it. The anomalous section of the field is smaller in magnitude but has a complex structure. It reflects details of the configuration and distribution of the density of the real earth. The normal part of the gravitational field is computed on the basis of the formulas of the distribution of normal gravitational acceleration *y*. In the USSR and other socialist countries the Helmert formula (1901–09) is used most often:

*γ* = 978,030 (1 + 0.005302 sin^{2} Φ - 0.000007 sin^{2} 2 Φ) mgl

The Cassini formula (1930), called the international formula, has the form

γ = 978.049 (1 + 0.0052884 sin ^{2}Φ - 0.0000059 sin^{2} 2Φ mgl

There are other, less widely used formulas, which take into consideration the slight longitudinal change *γ*, as well as the asymmetry of the northern and southern hemispheres. Preparations are under way to change over to a new standard formula, which takes into account the refined absolute value of the force of gravity. By using formulas for the distribution of normal gravitational force and with knowledge of the altitudes of the observation posts and of the structure of the surrounding terrain and the density of the rocks in it, it is possible to compute gravitational anomalies that may be used to solve most problems of gravimetry.

The gravitational potential is used in studying the figure of the earth, which is close to the level surface of the gravitational field of the earth, and in astrodynamics in the study of the motion of artificial satellites in the earth’s gravitational field (a surface on all of whose points the potential has an identical value is said to be level; the force of gravity is directed against it at the normal). One of the level surfaces that coincides with the unperturbed mean surface of the oceans is called the geoid.

The perpendicular is established and the position of the astronomical zenith is determined on the basis of the direction of the force of gravity. Since deflections of the perpendicular are roughly equal to the ratio of the horizontal component of attraction to the force of gravity, knowledge of their magnitudes in one sense helps in assessing the gravitational fields of the earth as well.

The second derivatives of gravitational potential are used in solving prospecting and geodetic problems. The vertical gradient of the force of gravity, which corresponds to the normal portion of the earth’s gravitational field, changes from pole to equator by only 0.1 percent of its total magnitude, which is equal to an average of 3,086 Eôtvôs units for the entire earth. The normal horizontal gradients of the force of gravity and the second derivatives of gravitational potential, which characterize the curvature of the level surface of the earth, are much smaller in absolute magnitude. The anomalous portion of the second derivatives of potential makes it possible to assess nonuniformities of density in the upper parts of the earth’s crust. It reaches dozens of Eötvös units on level ground and hundreds of Eötvös units in mountainous areas. In addition to the second derivatives of gravitational potential, the third derivatives, which are obtained by recalculation on the basis of gravitational anomalies, are used in gravimetric prospecting. The force of gravity is measured with gravimeters and pendulum instruments, and the second derivatives of gravitational potential are measured with torsion balances.

### REFERENCES

Zhongolovich, I. “Vneshnee gravitatsionnoe pole Zemli i fundamental’nye postoiannye. sviazannye s nim.”*Tr. In-ta teoreticheskoi astronomii*, 1952, part 3.

Brovar. V. V., V. A. Magnitskii. and B. P. Shimbirev.

*Teoriia figury Zemli*. Moscow. 1961.

Grushinskii. N. P.

*Teoriia figury Zemli*. Moscow. 1963.

M. U. SAGITOV and V. A. KUZIVANOV