by Vladimir ZHAROV, Dr. Sc. (Phys. & Math.), Gravimetry Laboratory Head, P. Shternberg Astronomical Institute, Lomonosov Moscow State University

For ages man has been using rotation of the earth in time measurements. Found to be irregular since the atomic clock entered the stage in the mid-20th century, it depends on many processes unfolding within our planet, in its oceans and atmosphere. Relevant information goes far beyond pure science alone-it matters a good deal for hands-on problem solving, too.

Now why is our planet spinning irregularly and why is humankind spending so much for research into the phenomenon? Before answering these and other questions, let us first turn to the history of astronomy, a science as old as the human world. From the very outset it was concerned with the following objectives: fixing an observer's position; establishing the location of a place on earth; taking celestial fixes to obtain a position (and determine the time of religious and historic events). Economic considerations, too, made it necessary to have an exact calendar based on astronomical observations of the sun, moon and stars. Today these problems are being tackled by astrometry, one of the disciplines within astronomy.

From a large number of cuneiform-script clay tablets found in Mesopotamia (a flat country between the Tigris and the Euphrates) we know for sure: even at that time (4 - 3 thousand years B. C.) high priests indulged in star-gazing. They determined the periods of solar and lunar eclipses and learned how to predict these natural phenomena. Babylonians (Babylonia was an ancient empire in southern Mesopotamia that flourished early in the second millennium B. C. in what is now Iraq) developed a sexagesimal (base - 60) system of calculus and a lunar-solar calendar.

The nascent science of astronomy owed much to high priests of ancient Egypt, too. The welfare of that country depended on the Nile and its regular overflows that brought fertile silt onto cultivated fields. When such floods came late, the country faced harvest failures and famine. Small wonder that the priests were so much eager to spot Sirius, the brightest star otherwise known as the Dog Star, in the heavens-its appearance before sunrise presaged a major flooding of the Nile. In fact, astronomy emerged as a practical science capable of predicting Nile transgressions and regressions, ebbs and flows. Astronomical observations made it possible to obtain an exact solar calendar and calculate the year as equal to 365.25 days. Sundials and water hourglasses came to be employed as timepieces. That's why, owing to these achievements, we know many historical dates in the life of ancient Egypt.

Hellenes, well conversant with mathematics, have contributed a great deal for the further advancement of astronomy. Thus, Heraclitus of Ephesus ("the Weeping Philosopher" who lived ca. 540 - 480 B. C.) suggested that the visible movement of the celestial sphere was due to the rotation of the earth. Furthermore, Eratosthenes of Cyrene (ca. 276 - 194 B. C.) proved our planet to be spherical; and Hipparchus (ca. 180/190 - 127 B. C.) discovered the phenomenon of precession.*

So, from the very start astronomy was geared to both practical and theoretical problem solving. It studied configurations of stars and planets, the position of the sun in the sky and sought to tie in these phenomena with particular events on earth. Besides, it was involved with such things as orientation and plotting a coordinate system in the celestial sphere, both essential for studies into the structure of the earth and consequently, for navigation.

To attack these and other problems, we must begin with precise reference points, adequate observation methods and clocks. A proper model of the earth is also a must. In fact, our planet is also a clock, and a good one for that matter. The high-precision quartz-crystal and the atomic clock** came only in the 12th century; these timepieces proved to be better than the spinning earth, and enabled astronomers to detect the planet's

* Precession-the movement of autumnal and vernal equinox points from east to west due to the slow revolution of the earth axis whereby the sun returns to its initial point before completing its circuit relative to the stars. - Ed.

** Atomic (quantum) clock is a timepiece where atoms are assigned the role of a "pendulum". The radiation frequency obtained with the transition of the atoms from one level to another is used for frequency adjustment of a crystal oscillator which is an essential part of the clock. The relative error for the best atomic clocks is equal to 10^-14 - 10^-15 . - Ed.

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Our planet's orientation in space is determined by parameters of earth orientation, such as universal time UT1, polar coordinates, and precession/nutation angles.

Discrepancy of time scales.

rotational irregularities and construct the now adopted atomic time scale.

The rotation of our planet is determined by the condition of plutonic matter and the transfer of matter in the earth core, mantle and oceans. Other processes are likewise implicated, though at this stage we cannot describe them precisely. In the absence of an exact theory of rotation, astronomers have to carry out regular observations to compare universal time (UT1*, preassigned by earth rotation) with atomic time (TAI).

Thus, on one hand, rotation of the earth is used for the construction of the universal time scale which is compared with that of atomic time; and, on the other, for converting the coordinates of an object from the terrestrial system of calculus (orientation) to the celestial one (and vice versa). To do that we should know rotational parameters, or the parameters of earth orientation, i.e. the position of the poles of the earth, angular velocity, nutation (oscillatory motions of the axis) and precession.

Before measuring the coordinates of a particular object, we should be in the clear about our system of calculus (orientations), both terrestrial and celestial. The terrestrial system is assigned by the rectangular coordinates of about 300 stations, and these are radio telescopes and antennae picking up signals from the orbital navigation systems GPS and GLONASS**; rigidly connected with the earth's crust, these ground stations move together with it.

* Universal time UT1 is equal to the integral of the angular velocity of rotation by atomic time, i.e. to the angle of the earth's spin. Quite often we use the day length instead of universal time as a rotation parameter; the day length is equal to a derivative with respect to UT1 and is inversely proportional to rotational angular velocity. - Auth.

** See: A. Finkelstein, "Radiointerferometric Network QUASAR", Science in Russia, No. 5, 2001. - Ed.

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High-frequency variations of UT1; MJD-modified Julian date.

Nutation of the earth's rotation axis from 1982 to 2000 (without allowing for precession movement). Nutation movement components: Δψ - in longitude, and Δε - in obliquity.

The celestial system of orientation is immobile in space and is prescribed by the coordinates of about 600 extra-galactical radiation sources. So in order to pinpoint his position, a terrestrial observer should measure the coordinates of stars, satellites and other bodies in the celestial sphere. And should he know the moment of observation and the earth rotation parameters, our observer can fix his position by the coordinates of these bodies.

Now let us look into this matter. While choosing our reference systems for coordinates, we should simultaneously adopt a similar system for calculating observation moments and time intervals in between, i.e. the time scale.

Up until 1960 we used the astronomical day (25 hours), which is one of the extrasystemic units of time measurement, for determining the second, a basic unit equal to 1/86,400 of the mean solar day. Which means that the second is a function of the earth rotation rate. But now an atomic second, equal to 9,192,631,770 periods of radiation, has been adopted as a basic unit corresponding to the transition between two states of the superfine structure of a caesium atom (Cs¹³³ ) at sea level. Thus atomic time measured by the number of seconds that have passed from some arbitrary moment up to now does not depend on the earth rotation velocity; this is a very convenient arrangement since we obtain a high-precision time scale that could be quantified readily and relayed to consumers by radio and television signals or other communication media. For navigation, however, we must use universal time UT1, that is we should know the position angle of our planet relative to the celestial system. Regular observations are carried out for the purpose. Parameters of earth orientation (PEO) are calculated by competent international research services.

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Length-of-day measurements (ΔLOD): a - data supplied by international research bodies; b - variations due to zonal tides; c - decade, and d - second harmonics.

To tell the UT1 time, one should take the atomic clock and register the moments when the sun or stars travel through one and the same meridian. This done, we make a correction for the polar motion. What with the earth moving irregularly, such moments change from day to day. Collating the UT1 and TAI scales, we shall see that they are always at variance, that is the earth rotation rate slows down. This is caused by tidal friction in the earth-moon system; as a result, our natural satellite moves farther away from us by 1 - 2 cm a year.

The discrepancy between UT1 and TAI scales is estimated at 32s. Such data are not convenient for use, though. That is why the scale of universal coordinated time (UTC) has been in use since 1964. UTC takes clock readings, it functions within the TAI system and also allows for changes in the earth rotation velocity (the difference between UT1 and UTC should not be above 0.9 s). Therefore another second is added to atomic clock readings-this is done either on December 31 and/or on June 30. The latest adjustment like that was made on December 31, 1999, and so the TAI-UTC difference makes up exactly 32 seconds. UTC is the time shown by our clocks and watches.

Computing the UT1 spectrum or day length, we come up against harmonics* with periods ranging from several hours to fifteen hundred years with amplitudes from 2 - 3 (is to 2 ms. Since the length of the mean day is equal to 86,400 atomic seconds, the maximal change in the earth rotation rate should not exceed 2 ? 10^-8 . Due to deceleration of this rate, the terrestrial day gained about 1.5 ms in a hundred years between 1900 and 2000.

Next, we come to deal with changes in the earth rotation velocity that occur in decades, i.e. ten-year periods. Diurnal variations every ten to twenty years are attributed to motions in the liquid core of the earth. The energy of such motions migrates from the core to the mantle on account of factional forces and/or electromagnetic coupling. The relevant theory is still in its swaddling clothes because we have but a poor knowledge of specific plutonic parameters of our planet's structure. Incidentally, in these ten years the earth has increased its rotation rate, but we cannot tell why as yet. This rotational gain, though, does not compel us to correct the UTC scale.

Superposed on these decade variations are many harmonics with periods in a range from a few days to years and years. Some are caused by zonal tides deforming the globe at the poles. Seasonal variations in the rotation velocity are known best, in particular, a harmonic with an annual period. And thus we find that in winter the day (on the 24-hour scale) is by 86,400 seconds longer than in summer. The chief cause of seasonal irregularity in the earth rotation velocity resides in the atmospheric circulation actuated by

* Harmonics - oscillations of a composite signal with frequencies multiple of the basic (primary) frequency - Ed.

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the uneven heating of the air due to the absorption of solar radiation. Hence steady winds blowing from east to west in low latitudes, and from west to east in the middle and high ones. The forces thus generated travel to the earth core owing to the friction of the atmosphere against the ground surface and also because of air pressure differentials on the opposite slopes of mountain chains. A similar mechanism is at work in oceans, too: the energy of water currents is imparted to the earth crust; however, the amplitude of variations is far smaller here. This effect was discovered about 10 or 15 years ago. The basic harmonics have periods around 1 and 0.5 days; these are high-frequency harmonics.

But not only the angular velocity of our planet is change prone. The earth also rocks with respect to its axis of rotation. This phenomenon, known as the polar motion (polar wandering) was discovered late in the 19th century. It causes a shift in the coordinate grid whereby points on the terrestrial surface change their position.

The highest amplitude (-0.2", where 1" - 1 s of the arc) is observed in the Chandler oscillation (named after Subrahmanyan Chandler of the United States who discovered it in 1891); its period is about 1.2 years and its annual amplitude - 0.1". The cumulative effect of these two phenomena results in fluctuations with a period of six years; the polar motion is like that of a spiral, now winding, now unwinding. Its radius on the global surface is not above 15 meters.

In theory the Chandler oscillation is a free one, that is it occurs in a rotating body even in the absence of external forces (in contrast to the annual harmonics), the appearance of which is conditioned by seasonal oscillations of air pressure on the Earth mantle. And yet there ought to be some source of excitation after all-for otherwise this kind of oscillation would have come to a halt in 15 to 20 years, for our planet is a viscous body. But this does not happen. The nature of such excitation is still obscure. It is thought to be caused by changes of circulation in the atmosphere and oceans that convey their energy to the earth crust on the Chandler oscillation frequency.

Besides, the pole also travels towards longitude of 75 degrees south at a mean annual rate of 10 cm. Such perpetual polar motions must have been triggered by the melting of glaciers about 10,000 to 15,000 years ago when they covered a huge territory-most of Europe and North America. Free of an immense load like that, the earth crust started rising slowly. This process is still on, and it causes change in the moments of earth inertia and consequently, induces polar motion.

Now over the changes in the orientation of the earth's axis of rotation in space. The position of the vector of the planet's instantaneous angular velocity depends on the tidal action of the moon and the sun, a phenomenon known as lunar-solar precession. It occurs because of the attraction of the surplus mass of the earth at the equator by these nocturnal and diurnal luminaries. The gravitational pull forces are out to align the planes of the earth's equator and orbit (ecliptic), but this does not occur because of rotation. As a result the orientation of the earth's rotation axis is changed: it strikes a cone at a mean velocity of 50.3", with the angle between the axis of rotation and that of the cone equal to -23.5°. The period of movement approaches 26,000 years. Precession changes the aspect of the starry sky. Today the globe's north pole is near Polaris (North Star). Yet in about 8,000 years it will be replaced by alpha (a) Cephei star, and in 13.5 thousand years, by Vega (α Lyra).

Besides the slow precession movement, the axis of the earth's rotation experiences periodic oscillations in the form of nutation. The periods of basic nutational harmonics are equal to 13.7 and 27.6 days, 6 months, and 1 and 18.6 years. The latter harmonic has the highest amplitude (- 9"). The amplitudes of the other harmonics are much smaller. Thus the earth's rotational axis describes sophisticated loops in space. The angle between the equator and the ecliptic is changed thereby, and the line of equator/ecliptic intersection likewise keeps moving. This is why nutation is divided into two components Δε - in obliquity, and Δψ - in longitude.

Precession and nutation depend on the compression and inner structure of the earth, on the tilt of the axis of its rotation to the plane of its orbit and on many other factors like the position of the moon, sun and planets. So in order to make a theoretical construction of this phenomenon we should know the exact coordinates of the sun, moon and other planets of the solar system; and we should know the internal structure of the earth as well. The present precession/nutation theory is quite good: we can predict the position of the axis of the earth's rotation within an error margin of 0.0002" (this angle corresponds to 2 cm on ground surface). Therefore the value of some of the parameters relevant to the internal structure of the planet is selected in such a way so as to correlate best the nutational angles Δε and Δψ with the results of observation. Using this theory we can learn more about the viscosity of the earth's liquid core, the rotation rate of its solid core, and about electromagnetic coupling forces, and thus add to our knowledge of our planet's structure.

And so the precession/nutation theory enables us to compute the orientation of the earth's rotation axis in space and pinpoint the polar coordinates and, consequently, the position of the axis within the planet; the UT1 time helps us to determine the turning angle of the initial meridian with respect to an arbitrary point in the sky. Since the moment of observation is registered by the UTC atomic scale, we should make a correction for UT1-UTC. These parameters are a necessary condition for calculating coordinates on the surface of the earth by the coordinates of celestial bodies, and the other way round.

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