# Rotation of the Earth

## Rotation of the Earth

one of the earth’s motions. The earth’s rotation is the basis for the explanation of day and night, of the apparent daily motion of celestial bodies, and of certain phenomena occurring on the earth’s surface, such as the rotation of the oscillation plane of a weight sus-pended on a string (Foucault pendulum) and the deflection of falling objects toward the east. Because of the earth’s rotation, a Coriolis force acts on objects moving on the sur-face; this force’s effect appears in the washing away of the right banks of rivers in the northern hemisphere and of the left banks in the southern hemisphere (Baer’s law) as well as in certain features of atmospheric circulation. The centrifugal force caused by the earth’s rotation partially explains the differences in the acceleration of gravity at the earth’s equator and poles.

To study the laws of the earth’s rotation, two systems of coordinate axes are used with a common origin *O* at the earth’s center of mass: (see Figure 1): one system moves with the earth (*X _{1}Y_{1}Z_{1}*) and the other is stationary

*(XYZ)*. The plane

*XOY*of the stationary system is made to coincide with the plane of the ecliptic at the initial epoch (a moment of time is taken as the initial moment); the

*OX*axis is directed toward the vernal equinox of this epoch. It is convenient to use the earth’s principal axis of inertia as the

*X*axes of the moving system, although, depending on the problem being studied, other axes can also be used. The position of the

_{1}Y_{1}Z_{1}*X*system relative to the

_{1}Y_{1}Z_{1}*XYZ*system is customarily denoted by the three Euler angles: ψ,and Φ

Most information about the earth’s rotation is provided by observations of the daily motions of celestial bodies. Observation has established that with respect to the vernal equinoxial point, the earth completes one rotation per sidereal day (approximately 23 hr 56 min 4 sec of mean solar time). The earth’s rotation proceeds from west to east, that is, counterclockwise as viewed from the north pole. The earth’s axis of rotation does not maintain a fixed direction in space. It moves so that the average inclination (9) of the equator to the ecliptic of the initial epoch is almost constant (in 1900, it was 23°27’8.26“, and during the 20th century it will increase by less than 0.1”). The line of intersection of the equator and the ecliptic of the initial epoch (nodal line) slowly moves along the ecliptic from east to west, shifting by 1°13’57.08” in one century, as a result of which angle ψ changes by 360° in 25,700 years. Thus, the *OP* axis describes a conic surface around the perpendicular to the plane of the ecliptic (precession). In addition, the *OP* axis performs a number of oscillations in space with periods ranging from several days to 18.6 years (nutation). Relative to the earth’s axis of rotation, the earth itself performs small oscillations. The instantaneous axis of rotation *OP* almost always coincides with the smallest axis of the earth’s ellipsoid of inertia 0Zi;-the angle between these axes has not exceeded 0.4” from observations made since the end of the 19th century.

Up to the beginning of the 20th century, it was assumed that the earth rotated uniformly, and its period of rotation was used as a natural time unit. The time interval between two successive coincidences of the *OX _{1}* axis with the nodal line

*Oy*, during which the angle Φ increases by 360°, was called the sidereal day. Owing to the rotation of the line

*Oy*itself, the sidereal day is 0.0084 sec shorter than the period of the earth’s rotation. However, precise analysis of position observations of the sun, moon, and planets has shown that the earth’s rotation proceeds nonuniformly and that the length of the sidereal day varies. Tidal friction slows the earth’s rotation, as a result of which the day’s length increases gradually; in the last 2,500 years, it has increased an average of 0.0024 sec per century. Periodic variations in the earth’s velocity of rotation also occur: annual and semiannual variations connected with seasonal meteorological phenomena and monthly and semimonthly variations caused by tidal deformations owing to the gravitational action of the moon. Because of the annual variations of the earth’s velocity of rotation, a day in January is approximately 0.001 sec longer than a day in July. “Abrupt” variations in the earth’s velocity of rotation have also been detected whereby the day’s length decreases or increases by several thousandths of a second over 1-3 years. The most significant of them occurred in 1864, 1876, 1898, and 1920. Their cause has not been definitively established.

The sun’s and moon’s attraction for the equatorial excess of mass (the result of the earth’s flattening) creates a moment of external force that influences the earth’s rotation. This influence was first used by I. Newton to explain precession, and J. L. D’Alembert gave a rigorous theory of it. L. Euler showed that the earth’s axis of rotation must also, in general, shift relative to the earth itself with a period of 305 days. The theory of the earth’s rotation developed by the foregoing scientists was based on the assumption that the earth is an absolutely solid object; however, this assumption was rejected at the end of the 19th century, when certain discrepancies between theoretical conclusions and observations were found. Later, in the theory of the earth’s rotation, other models of the earth were proposed: an ideally elastic spheroid and a spheroidal shell with a liquid core under various assumptions of the depth dependence of density and elastic properties. A theory of the earth’s rotation in which modern data on the earth’s internal structure are most completely used was developed by the Soviet geophysicist M. S. Molodenskii.

### REFERENCES

Molodenskii, M. S., and M. V. Kramer.*Zemnye prilivy i nutatsiia Zemli*. Moscow,

*1961*. (Collection of articles.)

Woolard, E.

*Teoriia vrashcheniia Zemli vokrug tsentra mass*. Moscow, 1963. (Translated from English.)

Munk, W., and G. MacDonald.

*Vmshchenie Zemli*. Moscow, 1964. (Translated from English.)

Zagrebin, D. V.

*Vvedenie v astrometriiu*. Moscow-Leningrad, 1966.

E. P. FEDOROV