TIME AND FREQUENCY STANDARDS*

G. Clemence

Until recently there existed one fundamental standard of time and frequency, which was used on an equal footing for both cases, both for civil and for scientific purposes. This standard was the second, which was defined as \(1/86400\) of a mean solar day. At the present time a new official definition is being used, expressed no longer in terms of the day but in terms of the whole year, and the second is taken to be \(1/31\,556\,925\,975\) of the tropical year for 1900.0. However, it is now also possible to point to a frequency standard of an entirely different kind, which has appeared thanks to achievements in the measurement of the frequencies of natural oscillations occurring in atoms and molecules. For brevity we shall call these new standards atomic standards.

Thus, for the first time in the history of science, there exist simultaneously at least two fundamental standards of time and at least three fundamental standards of frequency. Such an abundance of standards is not often encountered, and in order to avoid confusion and misunderstandings it is desirable to examine these standards closely from the point of view of our basic ideas about measurement operations, and to determine, insofar as we can do so today, the advantages and shortcomings of each standard separately. The danger of confusion and misunderstanding is great also because the practical determination of time and frequency is in the hands of a small group of astronomers. Astronomers who are not engaged in time service, and scientists working in other branches of science, occupied with their own affairs, devote little attention to the question under consideration. Thus it is the direct duty of those few astronomers who know something about standards of time and frequency to tell a broad audience what is actually known to them. This is the task of the present article.

DEFINITIONS

We should begin with a survey of certain basic definitions. Several different kinds of definitions may be indicated, and those concerned with semantics devote much attention precisely to this side of the matter; but for our purposes it is sufficient to keep in mind two kinds of definitions—operational definitions, first, and all other definitions, second. Operational definitions directly indicate what must be done, whereas in other definitions there are no such indications. For example, the operational definition of the meter consists in saying that it is the distance between two marks on a platinum bar kept in a certain—

* Science 123, 567 (1956).

in a particular place in France. Another, earlier definition of the meter implied that the meter is the \(1/10\,000\,000\) part of the path of the meridian passing from the North Pole to the equator through Paris. In addition, the meter is equal to 39.37 inches. If we wish to conduct our reasoning in the strictest manner, then it is always preferable to use operational definitions. This, for the most part, is what I shall do. Therefore one should not be surprised that some of the definitions given below will differ from the definitions found in reference books.

Any recurring phenomenon whose recurrence can be determined is a measure of time. Examples are the passage of railway trains through a definite point, the ticking of clocks, the vibrations of a quartz crystal or vibrations in an atom, the passage of a star across a meridian, the revolution of the Moon about the Earth or of the Earth about the Sun.

The interval between two successive repetitions of a phenomenon is a unit of time.

A clock may be any mechanism that counts such repetitions. Clocks also often serve to count off intervals of time that are part of the basic unit. Thus, our ordinary clocks divide the day into hours and minutes, while special clock devices make it possible to count off \(1/1000\) and \(1/10\,000\) parts of a second.

Frequency is the ratio between two different units of time, usually expressed as a number showing how many times a unit of one kind occurs during a unit of time of another kind. As an example, let us consider alternating current. The unit of time is specified by one period of the current. If we choose the second as the other unit of time, then the frequency of the current may be expressed, for example, as 60 periods per second. Strictly speaking, any two units of time can be used as a basis for defining frequency; however, in practice only two kinds of frequencies occur: 1) Frequency may be used as the definition of one unit of time in terms of another. For example, when we define the (old) second as the \(1/86400\) part of a mean solar day, we may with equal justification say that the frequency of the second in a mean solar day is 86,400. Frequencies used as definitions are exactly determined and invariable. 2) Frequency may be used to establish the relation between a unit of time obtained from experiment and some basic unit of time. Frequencies of this kind are either nominal or actual. Thus, for example, the nominal frequency of an alternating current may be equal to 60 periods per second, whereas the actual frequency may turn out to be 59.9998 periods per second. Let us note that, since the actual frequency must be determined experimentally, it is certainly subject to the influence of measurement errors.

The rate accuracy of clocks is the difference between their normal frequency and their actual frequency, conventionally assumed to be equal to the difference between the nominal and the actual frequency. For example, a seconds pendulum has a nominal frequency of 86,400 periods per day. If its actual frequency is 86,401 periods per day, then the accuracy of its rate is equal to \(-1\) period per day. Usually one says that the clock gains 1 second per day, and this statement should be understood in the sense just explained, although such a mode of expression is less precise, since the word second is used here in two meanings: first, as a unit of time determined by the clock itself, and, second, as a unit obtained by dividing the mean solar day into 86,400 parts.

stability. The accuracy of a clock’s rate may also be expressed in words: “the clock gains 1 part in 86,400 parts,” or “the actual frequency of the clock is greater than the nominal by 1 in \(8.64\cdot 10^4\).” We see that frequency and rate accuracy are closely related to one another, but by no means coincide.

We note that, although the period of an alternating current and that of a seconds pendulum are both measures of time, nevertheless only the seconds pendulum can serve as a clock. The seconds pendulum is a clock because its period can be directly determined, which is not the case with alternating current. Of course, one can devise a mechanism for counting the period of an alternating current, but such a mechanism is a clock. This distinction is extremely essential, because clocks indicate time or, in technical language, establish an epoch—by an epoch is meant some definite moment of time—whereas alternating current by itself cannot do this. I shall note in passing that the word “epoch” is often used to denote a more or less vaguely defined interval of time (historical epoch, glacial epoch). We shall not use the word “epoch” here in this sense.

If the ratio of two different units of time, i.e. the frequency, changes from epoch to epoch, then one says that one of the units of time is accelerating relative to the other. Acceleration of this kind is the rule rather than the exception. Consequently, as a general rule, some clocks not only accelerate relative to others, but the acceleration itself also changes from epoch to epoch. Most work concerned with the practical determination of time and frequency has as its aim the determination of clock acceleration and frequency change.

The preceding definitions can be explained by means of an analogy with the usual terms used in measurements of length. This comparison is made in two parallel columns of Table 1.

Table 1

Comparison of the terms used in connection with measurements of time and length

If we imagine a time scale as an independent sequence of units of time put into one-to-one correspondence with the real numbers, then an epoch is a definite moment on the time scale, just as a definite point is on a scale of lengths.

If two different units of time are denoted by \(t_1\) and \(t_2\), then frequency is defined by the ratio \(t_1/t_2\), whereas velocity is defined, say, by the ratio \(L/t_2\).

Analogously, a change of frequency, or the acceleration of \(t_1\) relative to \(t_2\), can be expressed as \(t_1/t_2^2\), whereas ordinary acceleration is expressed in the form \(L/t_2^2\).

It is obvious that the reciprocal of a frequency is also a frequency, but this is not the case for velocity. Likewise the ratio of two frequencies is again a frequency, provided that among the frequencies entering into the ratio one of the time units is common. For example, the ratio of \(t_1/t_2\) to \(t_3/t_2\) is equal to

\(t_1/t_3\); whereas the ratio of two velocities is a dimensionless number only if the unit of time for these velocities has been chosen to be one and the same. Thus, the analogy between frequency and velocity is far from complete.

It is important to remember that a change of frequency is in itself an acceleration of one time scale relative to another and that it cannot, properly speaking, be called an acceleration of frequency. The acceleration of frequency, if such a term is to be introduced at all, is a change in the actual acceleration, i.e. it must have the form \(t_1/t_2^3\).

Everything that has been said so far about measures of time applies equally to any measures of time that have ever been used. It is practically obvious that some measures are preferable to others. We, for example, are not going to establish the fundamental units of time according to the intervals between the passage of railway trains; in practice we do the opposite: we dispatch trains more or less according to clocks, instead of setting clocks by the passage of trains. Let us ask ourselves what requirements should be imposed on measures of time so that they may serve as standards. Of course, we must require of them continuous operation. Their accessibility must also, naturally, be kept in mind; of two measures having essentially equal value, we shall always give preference to the more accessible one. There is, however, still another essential requirement, the formulation of which is in itself already difficult. We usually express it by saying that the standard measure of time must be invariable, but in fact we have no absolute criterion of invariability. Suppose that we compare two measures of time with one another and find that one of them is accelerating relative to the other; we are entitled to conclude that one of them, or both at once, are changing, but the question of which of them is changing remains open. If, on the other hand, in another case no acceleration is observed, we still cannot conclude that both measures are invariable (within the limits of errors of measurement); in reality both measures may be changing in some way relative to a third. Nevertheless, a quite definite meaning can be given to the term invariability of a measure of time. In order to understand this question fully, it will be very useful for us to dwell on a small chapter from the history of astronomy, which will explain to us how and why, quite recently, the second was redefined through the tropical year instead of the old definition through the mean solar day.

CELESTIAL MOTIONS

The equations of motion of any member of the solar system can be obtained from Newton’s law of gravitation. (I am not speaking of those refinements which are required by the general theory of relativity and which have no bearing on the questions under consideration.) For each body these equations constitute three differential equations determining the second derivatives of the three coordinates with respect to time as functions of the values of the masses and mutual distances of all the bodies present. Three such equations for every body can be solved by successive approximation if we possess the necessary information about the other bodies and if we know six constants of integration, which may be the three coordinates and the three components of velocity of a rectangular Cartesian coordinate system at some suitable epoch. The solution determines the three coordinates of the body as functions of the time counted from the adopted epoch, as

both forward and backward; such a mathematical expression of the solution is called the theory of the motion of a body. Since in the solar system there are nine planets possessing significant masses capable of influencing the motion of each of the other planets, it is easy to understand that the problem of constructing a theory for any one of these planets presents considerable difficulties. Nevertheless, this problem has been successfully solved, and with a very high degree of accuracy. The coordinates of Jupiter, for example, have been computed to the tenth significant digit for the interval of time from 1653 to 2060, and strict agreement with the initial conditions has been achieved[^1].

ASTRONOMICAL UNITS OF MASS, TIME, AND DISTANCE

Let us examine carefully the exact definitions of mass, the time scale, and distance adopted in astronomical theories. The unit of mass is the mass of the Sun, and the masses of all other celestial bodies are expressed in terms of the mass of the Sun. It is customary to say that the unit of time is the mean solar day, but, as we shall see below, this definition is not quite exact; the unit actually used is the mean value of the mean solar day, taken over the eighteenth and nineteenth centuries. The unit of distance is a purely astronomical unit, obtained from the adopted units of mass and time by means of Newton’s law of gravitation. For our purposes it will suffice to take as the unit of distance that distance from a unit mass at which a body of negligible mass, moving in a circular orbit, describes an angle of 0.01720209897 radians in a unit of time. The astronomical unit of length is close to, but by no means equal to, the mean distance of the Earth from the Sun.

To a physicist, the units of mass and length used by astronomers may seem strange. The reason for the choice of these units, however, lies in the extremely restricted nature of astronomical observations. Apart from measurements of velocity directed along the line of sight, everything astronomers know about the motion, distances, and masses of celestial bodies is derived from direct measurements of one single kind: measurements of the angle between two lines that are the directions along which the astronomer makes observations. The result of an observation is a value of an angle and an epoch. In some cases the angle is the angular distance between two celestial objects, and in other cases it is the angular distance between some celestial object and the direction to the zenith. From such measurements alone one can, with the aid of theory, obtain ratios of masses and ratios of distances, but in any case by astronomical methods it is impossible to express these masses and distances in grams and centimeters. It is true, of course, that we are able, to an accuracy of up to five or somewhat fewer significant digits, to express the masses of celestial bodies in grams and the distances between them in centimeters; but in order to obtain these numbers we must trust the physicists and geodesists who have determined the mass and dimensions of the Earth. Astronomers themselves do not need such information for any astronomical purposes at all; these data are used in those cases when it is necessary to answer questions posed by representatives of other specialties.

Even after mechanical clocks had been invented, observational astronomers continued to use mean solar days as the unit of time. There are several reasons for such a choice. First of all, an astronomer must have a natural unit

time, and by no means an arbitrary one. An arbitrary unit of time cannot be moved from place to place as easily as an arbitrary unit of length. Furthermore, all arbitrary units of time—so to speak, the handiwork of human hands, such as, for example, mechanical or electric clocks—are unstable and short-lived. It is impossible to construct a pair of mechanical or electric clocks that would always show the same time, or a clock that would run uniformly and without stopping all the time. Consequently, a natural unit of time is necessary in order that astronomical observations made in different places and in different epochs may be compared with one another. Of all the natural units of time available to astronomers, the period of the Earth’s rotation is the most accessible and can be measured with a very high degree of accuracy. It is only necessary to observe the passage across the meridian of some star on two successive nights, and thereby the period of the Earth’s rotation is obtained directly. Although the stars cannot be observed in inclement weather, clocks made by human hands are quite suitable for keeping count of the time from one clear night to another, and also for dividing the period of the Earth’s rotation into 86,400 parts. It is curious to note in passing that the manufacture and improvement of clocks reached its greatest development in England—a country distinguished by an abundance of cloudy days.

The mean solar day does not quite coincide with the period of the Earth’s rotation, and its observation is less accessible, since the Sun is more difficult to observe than the stars. Nevertheless, astronomers readily made this small sacrifice, which was necessary so that the unit of time would also be suitable for the organization of their life by day. To pass from the period of the Earth’s rotation to the mean solar day, it is merely necessary to introduce the factor 1.0027378118868. This number, which is an actual frequency, is one of the most precisely determined constants in physics and, perhaps, the very best constant in terms of accuracy of determination: only the thirteenth digit after the decimal point is doubtful. It should be emphasized that the mean solar day is defined in such a way that the factor just mentioned is an absolute constant; this means that the mean solar day is rigidly connected with the rotation of the Earth, and any cause that disturbs the mean solar day disturbs the speed of the Earth’s rotation, and conversely.

OBSERVATION AND THEORY

Astronomers have always sought to obtain theories of the motion of planets and satellites that would give complete agreement with the results of actual observations. With every significant step forward in our basic theoretical constructions, the hope of achieving complete success revived, but the continuous growth in the accuracy of observations doomed all hopes of this kind to failure, at least for our generation; it is difficult, of course, to predict what will be in the future. After the law of universal gravitation had been established, it seemed that all practical problems of celestial mechanics had been reduced to purely computational problems. However, the advance of Mercury’s perihelion was discovered, and it did not receive its explanation during the fifty years following its discovery. Consequently, it is one thing to reduce a certain problem to computations, and quite another thing to solve the posed problem completely. The computations of planetary theory or lunar theory constitute one of the most significant tasks among those known

in science, at least in those cases when it is necessary to obtain the values of coordinates with an accuracy of eight or more significant figures. The task consists not only in having to multiply millions of numbers—which, up to the very last years, in itself already constituted a problem—but also in having confirmation that the solution is correct; perhaps the most difficult thing is to organize the whole work in such a way that it is actually brought to completion. A decade of intensive work was required in order to obtain the best planetary theories, while the two best lunar theories each required about twenty-five years for their development. The motion of the Moon requires special attention; it is not impossible that no less effort has been devoted to the Moon than to all the other celestial objects taken together.

About eighty years ago it began to be suspected that the discrepancies in comparing theory and observation might be attributed, at least in part, rather to defects in the chosen measure of time than to inadequacy of the theory. To make it easier for us to imagine the consequences of defects in the chosen measure of time, let us assume certain things which in fact did not occur. Let us assume, first, that we had at our disposal theories of the motion of all celestial bodies which agreed completely with experiment; let us assume further that the speed of the Earth’s rotation is decreasing imperceptibly, in the sense that this decrease cannot be detected by clocks made by man; let us also imagine that observations of the Sun, the Moon, and the planets are postponed until such a moment when mean solar time lags by one hour. What will be found when observations are resumed? Obviously, the Sun, the Moon, and the planets will all be observed ahead of their precomputed positions by the angles through which they move in the course of one hour. For example, the Moon’s hourly motion during a month amounts to an angle of from 0.48 to 0.68 degrees, and its variations are known with great accuracy; thus it will at once become clear that the Moon is ahead of its theoretical motion, and we shall at once begin to suspect an error in the lunar theory, until we turn to observations of the Sun. The Sun will prove to be ahead of its precomputed position by an amount varying during the year from 136 to 166 seconds of arc; the magnitude of this discrepancy and its variations will easily be detected. Moreover, an eclipse of the Moon will be observed exactly 1 hour earlier than the predicted time. Mercury will sometimes be observed east, and sometimes west, of the position which it ought to occupy according to theory, depending on whether it is making direct or retrograde motion; the magnitude and variations of this discrepancy would indicate that at some epoch Mercury was in the position which, by calculation, corresponded to an epoch 1 hour later. Analogous observations and conclusions could also be made on the basis of observations of other planets. The inevitable conclusion at which the observers would have to arrive would be that either our clocks were 1 hour slow, or the Moon and the other planets had acquired accelerations in their orbits, so that they obtained an advance of one hour and then again acquired their former speed. In other words, either something was wrong with the measures of time, or something was wrong in the theories of the motion of the Moon and the planets, and these theories contained errors of a very strange character.

Something similar to the hypothetical example described in fact occurred, only this event had to do not with clocks and not with the rotation of the Earth losing exactly 1 hour of time. In order to reconcile the observations with the theory, it was necessary to suppose that clocks sometimes run faster and sometimes more slowly; it was necessary to suppose that

The Earth may rotate at a greater speed than the average speed of rotation for several years, and then rather sharply change its rate of rotation, so that it begins to rotate more slowly or even more quickly. Two thousand years ago, according to observations of solar eclipses made at that time, clocks corresponding to mean solar time must have been 2.6 hours slow; around 1750 clocks corresponded to true time; in 1850 they were 2 seconds slow; in 1900 they were 3.9 seconds fast; in 1940 they were 24.5 seconds slow.^2 The relatively large deviation at the beginning of our era cannot cause particular surprise if we recall that we adjusted our unit of mean solar time so that it would be close to the mean length of the solar day, averaged over the eighteenth and nineteenth centuries; in very remote epochs one may expect considerably larger deviations than in our own day.

Astronomers, without hesitation, attributed the observed discrepancy to errors in establishing the units of time, and not to errors in the theory of motion. It is quite appropriate to ask why they acted in this way. This question may be answered in different ways, proceeding from various points of view. For those who share Ockham’s convictions (economy of hypotheses), the answer is obvious. Either we must suppose that the speed of the Earth’s rotation changes in some manner that we cannot foresee, or we must suppose that the Moon and the planets undergo certain changes in the time of their orbital motion, likewise in an unpredictable manner, but all together, and that the astronomical unit of length also changes correspondingly; a change in the unit of length is accompanied by a change in the speed of light. For those who share the ideas of the general theory of relativity, the last of these consequences is sufficient, since the constancy of the speed of light is a fundamental basis of the general theory of relativity. For those who agree to accept hypotheses only if they have some suitable model to accompany them, only a few years ago nothing could have been said in justification of the assumption mentioned; but even at the present time the mechanism leading to a change in the speed of the Earth’s rotation has hardly been at all reliably disclosed. It is supposed, for example, that turbulence in the liquid core of the Earth, accompanied by electromagnetic interaction between the core and the Earth’s shell, is sufficient to cause the corresponding changes. Finally, for those who practically perform time service and who remember the days when there were no radio signals and when three chronometers were taken aboard a ship and the two that ran alike were considered correct, it is enough to point out that the rotation of the Earth and the revolution of the Moon and planets have in fact always served as clocks. Four clocks—the lunar revolution, the revolutions of Mercury, Venus, and the Earth itself—agree with one another, whereas the rotation of the Earth agrees with none of them.

The outer planets can serve as clocks to the same extent as the three inner ones, and in principle all celestial bodies are clocks, but most of them have such small angular motion because of their colossal distance from the Earth that their displacement over a second or two cannot be measured with the accuracy needed to verify the character of the Earth’s rotation.

Thus we find that there is yet another requirement which a practical unit of time must satisfy: in addition to its having to be constantly operating, readily accessible, and “invariable,” it is necessary that the recurring phenomenon which is to be observed should recur moderately often. The precise

meaning of the expression “moderately frequent” depends on the accuracy of astronomical observations. For modern astronomy the most natural unit of time having practical significance is the year. If the accuracy of astronomical observations increases tenfold, then the revolution of Jupiter, which takes place in approximately 12 years, can provide us with a measure of time whose accuracy will correspond to the accuracy of the measure of time determined at present by the revolution of the Earth.

EPHEMERIS TIME

We are now in a position to understand why the second was recently redefined as a definite part of the tropical year, and not as a definite part of the mean solar day.^3 At the same time, we can already answer the question of the sense in which one may speak of an invariable measure of time; an invariable measure of time is simply such a measure of time as permits the theories of the motion of celestial bodies to be brought into complete agreement with observations. To put it even more definitely, one may say that it is the independent variable in the accepted equations of motion. Generally speaking, there is no practical necessity at all to raise the question of an invariable measure of time, and we shall see later why it should even be recommended not to do so. The very word “invariable” will suggest to many people the idea of something absolute, and therefore it should be avoided if we wish to express ourselves with the utmost rigor. All that is in fact required for civil and scientific purposes consists in the choice of a suitable measure of time and in its precise definition. For convenience, a special name has been introduced for that measure of time which serves as the independent variable in the equations of motion; it is called ephemeris time, in contrast to mean solar time. Ephemerides are tables of the positions of celestial bodies at various epochs, computed according to accepted theories of motion; ephemeris time, therefore, is simply a measure of time determined by the ephemerides.

When the second was redefined, it was accepted that the tropical year for 1900.0 would be used; the zero in the first decimal place indicates the beginning of the year 1900. The reason for such a choice was that the tropical year (which corresponds to the seasons of the year) is decreasing at a rate of 0.530 ephemeris seconds per century, or by one unit in \(5.95 \cdot 10^9\) years. Knowing these variations, it is easy to find the relation between the year of interest to us and the tropical year for 1900.0 with an accuracy up to \(1 \cdot 10^{-13}\).

Ephemeris time is in practice determined by observations of the Moon; the Moon moves considerably faster than the planets and, consequently, with the aid of the Moon time can be determined with considerable accuracy. Of course, a single observation of the Moon does not permit ephemeris time to be determined with the required accuracy. Until very recently, in order to accumulate observational material of value in this respect, it was necessary to observe the Moon over an entire year. The processing of the observational data also required some time; hence these determinations of ephemeris time were made with a delay of the order of two years. The recent invention of a new photographic technique for observing the Moon^4 has led to a sharp increase in the accuracy of observations; it can now be thought that in the near future we shall already be able to determine ephemeris time from month to month as accurately as we now do it from year to year. But even in the best case ephemeris time cannot be determined

with the same accuracy as mean solar time. The rotation of the Earth takes place approximately 27 times faster than the revolution of the Moon, which is a substantial advantage. An error of 0.1 arc second in observing an equatorial star corresponds to an error of 0.007 second of mean solar time, whereas the very same error in observing the Moon corresponds to an error of 0.18 second of ephemeris time. Ephemeris time is less accessible than mean solar time. Thus, in redefining the second as a derivative of the year, instead of taking it as a derivative of the day, we replaced a less accessible invariable unit by a more accessible, variable one.

ACCURACY OF TIME DETERMINATION

The practical determination of mean solar time consists in using some mechanical or electrical clock to mark the instant at which a star crosses the local meridian. The true mean solar time of the star’s meridian transit is known in advance through a whole series of observations and computations, which we shall not discuss here. The discrepancy between the true time of the star’s meridian transit and the clock readings determines the error of the clock. The clock used in this case need not at all be set to the correct reading; instead, it is sufficient to keep a record of its errors; these errors change continuously. Such a record makes it possible to set all other clocks, after comparing them with the clock whose error is known, to the correct time, preparing them for checking the radio time signal. But radio time signals cannot be absolutely correct because of errors in astronomical observations, and also because of errors in extrapolating clock errors.

The most accurate instrument for determining mean solar time is the photographic zenith tube and the astrolabe[^6]. With the photographic zenith tube at the U.S. Naval Observatory, the practice has been established of observing about 15 stars on every clear night, and the probable error of the mean result of a night’s work is about 3 milliseconds. (By probable error is meant a quantity which, on the one hand, exceeds half of the actual errors but, on the other hand, is smaller than the other half of these errors.) The best quartz-crystal clocks run with greater accuracy than this, and they are used to smooth out the random errors in astronomical observations from night to night, so that within a single observatory possessing the best examples of instruments and clocks, mean solar time can be determined with a probable error, say, of two milliseconds. The International Bureau of Time in Paris, belonging to the International Astronomical Union, compares data obtained by various national time services and thereby has the possibility (with a delay of about a year) of making further improvements in our knowledge of mean solar time. At present the error in determining solar time is probably less than 1 millisecond. In the present article I shall assume a probable error equal to 2 milliseconds.

Everything that has been said so far concerns the establishment of an epoch; physicists and engineers are for the most part interested in determining frequencies or intervals of time. For these purposes quartz clocks are more accurate than astronomical observations, up to intervals of time of at least several weeks, whereas for longer ...

in intervals of time astronomical observations make it possible to attain greater accuracy than clocks. Since, insofar as only astronomical observations are concerned, any interval of mean solar time can be determined with an absolute error whose probable value is 3 milliseconds; this value is obtained by multiplying the probable error in determining an epoch by the square root of two. The relative error in determining intervals of time is governed by a different law. If the observations are confined to both ends of the interval, then the relative error varies inversely as the duration of the interval; however, if the interval of time is sufficiently large, so that many observations were made within this interval, then the accuracy increases and becomes approximately proportional to the duration of the interval taken to the power \(3/2\). Table II gives some approximate estimates of the magnitudes of the probable relative errors with which various intervals of mean solar time can be determined; they were obtained on the basis of the indications of the best quartz clocks and the best astronomical observations.

Table II

Approximate estimates of the probable relative error in the determination of various intervals of mean solar time

The high degree of accuracy indicated for a time interval of 356 days, of course, has no special significance, since the length of the day changes in a way unknown to us every several years by an amount of \(1 \times 10^8\) or even more. For example, if the length of the day in 1936 is taken as unity, then in 1923 the length of the day was greater by 1 in \(10^8\) and decreased at a rate of 2 in \(10^9\) per year, whereas in 1940 it was again greater by 1 in \(10^8\), but was already increasing at a rate of 2 in \(10^9\) per year. In practice the question is reduced to the accuracy with which, by means of astronomical observations, the redefined second—or, equivalently, the length of the year—can be determined. As has already been mentioned, the accuracy of determination in this case is lower than when determining the old second, chiefly because of the relatively slow motion of the Moon. In the past it was necessary to collect observations of the Moon over an entire year in order to determine the ephemeris time with a probable error of the order of 100 milliseconds; moreover, such accuracy has been achieved only in recent times. It corresponds to a probable error of 140 milliseconds in measuring the length of an individual year, or 1 in \(2 \times 10^8\).

The last two improvements of the U.S. Naval Observatory have sharply increased the accuracy of the determinations. One of these improvements is the most precise investigations in the marginal zone of the Moon, making it possible to determine satisfactorily the features of the lunar surface; it should be recalled that all observations of the Moon refer to the bright edge of the visible disk, and therefore any elevation at the point at which the measurements are made introduces a certain uncertainty into the determination of ephemeris time. The second improvement is a new observational technique, a report on which has already been published: the position of the Moon is referred to several stars among those in its nearest surroundings in order to eliminate observational errors associated with the motion of the Moon relative to the stars. At present, with the aid of one

telescope made it possible to determine the length of the year with a probable error of \(4\) in \(10^8\). Although at present only one telescope is operating, during the International Geophysical Year in 1957–1958 it is planned that twenty telescopes will be in operation, so that one may hope that the length of this particular year will be determined with a probable error of \(1 \div 2\) in \(10^9\). It should be assumed that at least four telescopes will continue their work in the future, independently of one another. On the basis of this assumption, one can estimate the probable relative error in determining intervals of ephemeris time. For certain intervals of time this error is given in Table III.

Table III

Probable relative error in determining intervals of ephemeris time

The probable error for \(1/12\) year or less has been calculated on the assumption that quartz clocks are used for the temporal subdivision of the year; for such intervals astronomical observations give less accuracy. It should be emphasized that determinations possessing the indicated degree of accuracy are possible only with some delay; this delay may amount to as much as a year. Although this delay does not reduce the value of the tabular value obtained, the absence of data for the recent past is often an unpleasant circumstance.

ATOMIC STANDARDS

In the last few years, progress in technology has made it possible to approach the processes of natural oscillations occurring in atoms and molecules. There is no reason to doubt that these oscillations have as high a degree of stability as the revolution of the planets around the Sun and of the Moon around the Earth. The practical difficulty lies in counting the number of these oscillations per second, which is of the order of magnitude \(10^{10}\). For this purpose special methods were developed, and the natural resonant frequency of cesium was measured at the National Physical Laboratory in England\(^7\) with a guaranteed accuracy of \(1\) in \(10^9\). The authors assert that the potential accuracy of the measurements is considerably higher, but in order for higher accuracy to be attained, further development of special electronic equipment is necessary.

Thus we have obtained a new standard of frequency, which can be used for calibrating other frequencies; moreover, within a few minutes the same degree of accuracy can be attained as is attained by annual astronomical observations. This substantial advantage suggests that for certain purposes atomic frequency standards will be used more readily than astronomical standards. Atomic standards in fact constitute a natural unit of time that does not depend on the second and differs from the second in its very nature, since atomic standards do not depend on the motion of celestial bodies, at least in the operational sense.

A very important question for the foundations of all scientific activity is whether atomic frequencies are constant or variable if they are expressed in astronomical (ephemeris) seconds. Physicists do not

can give a definite answer to the question posed, which must be resolved experimentally. Møller and his collaborators have constructed a very detailed physical theory, called by them kinematic relativity[^8], in which there are two natural time scales, one of them continually accelerating relative to the other, so that the ratio of the two units of time continuously increases; this change in the present epoch is, by assumption, less than 1 part in \(10^9\) per year. Dirac, Møller and Jordan, as well as some others, have expressed the view that one of these units may be identified with atomic clocks, and the other with astronomical clocks. If this is indeed the case, then over approximately five years it would be possible to measure this acceleration, and our former ideas about the invariability of units of time would have to be decisively revised. We have no grounds for giving preference to one unit over the other by calling either of them invariant; consequently, the very word “invariant” will pass out of our usage. It is also clear that, if such an acceleration is observed, this circumstance will have profound and far-reaching consequences for the fundamental constructions of physics.

ATOMIC UNIT OF TIME

Once atomic frequency standards can be used with greater success than astronomical ones, it is necessary to determine the frequency of the atomic standard itself by expressing it in terms of the second. As already mentioned, this operation has already been carried out for the old mean solar second; however, it is desirable to perform the same determination for the new ephemeris second, which, of course, will require some time. After this has once been done, in all probability the astronomical second will soon be forgotten by everyone who works daily with an atomic standard; at the same time there will arise the danger of a certain confusion, especially in the event that the ratio of atomic and astronomical units is constant, or if its change is so insignificant that a considerable interval of time will be needed to detect it. Such confusion already exists to a certain extent with units of length.

The meter is defined as the distance between two marks on a specified platinum bar. However, some physicists have come to the conclusion that it is more convenient, instead of actually using the meter as the unit of length, to take as the unit of length the wavelength of a specified spectral line of cadmium or mercury. In many cases wavelengths can be compared with one another with greater accuracy than a wavelength can be compared with the meter. The number of standard wavelengths contained in a meter can be measured with a certain degree of accuracy. Such a definition is entirely suitable for temporary use. However, the number of wavelengths in a meter is a quantity determined experimentally and subject to revision and refinement. But what is to be done each time after such a revision has been made? Should all measurements in which the standard wavelength has at some time entered be subjected to revision? It is quite obvious that such a procedure is practically unacceptable. In order to avoid repeated confusion, it is only necessary to adopt a new standard of length, say the ångström, which will now no longer be regarded as a quantity equal to \(10^{-10}\) meter, but will be defined by a definite and constant relation to the standard wavelength; the number of ångströms in a meter will then be determined experimentally and may be subject to revision. Thus, if measurements

if lengths are expressed in angstroms, this should be understood to mean that they are referred to the atomic wavelength, whereas if they are expressed in meters, they are referred to the standard meter.

It would also be highly desirable to proceed in the same way with units of time. We propose to retain the term second for the astronomical second, and at the same time to adopt a new unit, which I shall here call the “essen” (essen), and which will have a definite and constant relation to the frequency of the cesium atom, amounting to approximately a small fraction of a second; the exact number of essens in a second will be subject to experimental determination and revision. In this case, any frequency expressed in periods per essen is understood to be referred to the atomic standard of time, while a frequency expressed in periods per second is referred to the astronomical second. The distinction between the atomic unit of time and the astronomical one is considerably more substantial than the distinction in standards of length, since both units of time will certainly coexist for a long time.

CONSEQUENCES OF ADOPTING AN ATOMIC UNIT OF TIME

Two important consequences follow from the adoption of an atomic unit of time, i.e., from using it alongside the astronomical second. The first consequence follows from the circumstance that the atomic unit of time is not independent of the atomic unit of length in the same sense in which the meter is independent of the second. Wavelengths and frequencies are connected by a definite physical relation: their product is equal to the speed of light. Consequently, if the speed of light is known with a sufficient degree of accuracy, the wavelength can be calculated from the frequency, and vice versa. In fact, the speed of light (expressed with the aid of the astronomical second) has been determined with an error amounting to some part in \(10^4\) (the nominal values of recent determinations, whose probable error is indicated as 1 in \(3 \cdot 10^5\), should not yet be taken into account until the reasons for their discrepancy with earlier determinations have been clarified), and such an operation cannot be carried out. But it may happen that the frequency of the cesium atom (expressed in terms of the astronomical second) will be combined with a wavelength (expressed in meters) in order to obtain a refined value of the speed of light.

In discussing this situation, one must be extremely careful not to fall into a vicious circle of reasoning. Suppose that we have adopted an atomic unit of time alongside an atomic unit of length; let us ask ourselves in what sense (if indeed there is any sense at all) the product of the wavelength and the frequency, expressed in these units, can be regarded as the speed of light. The answer is that such a product of the indicated numbers does not give the value of the speed of light and in fact has no physical meaning whatever; it is simply a certain number that is already predetermined as soon as the atomic units of length and time have been adopted. Therefore, when we say that the speed of light is so many meters per second, we are expressing a definite experimental result; but to say that the speed of light is so many angstroms per essen (using these words in the sense I have indicated) is simply a tautology.

It is well known that the constancy of the speed of light is a fundamental postulate of the general theory of relativity. But what will become of the general theory of relativity if it is found that atomic time accelerates relative to astronomical time? Without doubt, we shall hear

would come to the opinion that thereby the general theory of relativity had been refuted; but such a conclusion is by no means necessary. There are three legitimate ways of expressing the speed of light: in meters per second, in meters per ephemeris second, and in angstroms per second. All three of these modes of expression reflect experimental facts. The questions that arise here are as follows: (a) in which of the three modes of expressing the speed of light is the speed of light variable, and (b) to which of the three modes does the fundamental postulate of the general theory of relativity refer?

The second consequence of adopting the atomic unit of time follows from the fact that the atom, being a natural standard of frequency, and in this respect considerably superior to arbitrary standards (such, for example, as quartz crystals, which must be continuously compared with natural standards), nevertheless cannot be brought under the control of natural clocks. In other words, atomic oscillations have no property that would allow them to be used for determining an epoch, as is done by the passage of a star across the Greenwich meridian. The assertion that atomic clocks can be constructed is perfectly correct, and it is correct that they can be used as secondary standards of time, but it does not seem possible to choose them as primary standards. They will not run indefinitely long, but will stop for accidental electrodynamic reasons. When they start again, then for establishing the time during which they were stopped there is no other method than comparison with other clocks that meanwhile have been running without stopping. The only way to check clocks separated from one another by some distance is by means of time radio signals; but the velocity of propagation of these signals is variable. Consequently, after the adoption of an atomic frequency standard, two units of time will be in circulation—one for frequencies and the other for measuring time itself.

QUALITY FACTOR \(Q\)

There exists a quantity \(Q\), which I mention here only because it is commonly used (and for the most part incorrectly) as a measure of the perfection of clocks. Among the most important correct applications of the quantity \(Q\) is its use as a measure of the quality factor of a resonant electrical circuit. In this application \(Q\) is a measure of the sharpness of tuning—the sharper the tuning, the higher \(Q\). In this case \(Q\) is equal to the ratio of the resonant frequency to the width of the resonance curve, taken at the point where the amplitude of the forced oscillations in the circuit differs from the amplitude of the oscillations at resonance by three decibels. In clocks with a quartz crystal, an electrical circuit tuned to resonance with the frequency of the crystal is used. Obviously, the larger the \(Q\) of such a clock, the more accurately the clock will count the oscillations of the crystal. In the technique used in work with atomic standards, the change in the amplitude of forced oscillations with frequency is very similar to the picture observed in a resonant circuit, and therefore it was considered convenient to define the ratio of the frequency at the maximum of the forced oscillations to the width of the resonance curve, taken 3 decibels lower, as an equivalent \(Q\). This quantity effectively determines one of the characteristics of an atomic standard.

In addition to the quantity \(Q\), there is another quantity, \(\delta\), called the logarithmic decrement of damping, which is very important when considering damped oscillations, such as, for example, a spark discharge. This quantity determines the rate of damping and is equal to the natural—

to the natural logarithm of the ratio of the amplitudes of two successive oscillations. It turns out that for the not-yet-established discharge through a resonant circuit

\[ Q=\pi/\delta \quad (\pi=3.14159). \]

This ratio makes it possible to measure damping through \(Q\) in the same degree as through \(\delta\); large \(Q\)’s correspond to small \(\delta\)’s.

In applying the quantity \(Q\) to damped electrical oscillations as a measure of damping, we have taken only the first step toward applying it, by analogy, as a measure of the damping of any damped oscillation, whatever its nature may be—for example, to the oscillations of an excited tuning fork or pendulum. It is asserted, in the form of proof of the superiority of atomic standards over pendulum and quartz clocks, that the latter have a value of \(Q\) only of the order of \(10^6\), whereas for the former the values of \(Q\) are much greater; various numbers between \(10^7\) and \(10^{18}\) are given. Of these numbers, \(10^{18}\) corresponds approximately to the value of \(Q\) in molecular transitions, while \(10^7\) is a value that can be counted on in practical applications. These entirely disorienting comparisons have introduced considerable confusion, since the quantity \(Q\) has approximately the same relation to the merit of clocks as the capacity of storage batteries has to the merit of an automobile; a certain capacity of storage batteries is, of course, necessary, but by no means sufficient for the high quality of an automobile.

The fact that the quantity \(Q\) cannot be used as a measure of the merit of clocks is, in the main, determined by two reasons. The first reason is that in clocks the natural damping of the pendulum or quartz crystal is opposed by an applied force, and in such a way as to maintain oscillations with an approximately constant amplitude. Consequently, \(Q\) has a reasonable meaning only in the interval between successive applications of the external force; and this external force may be applied as often as desired. It is precisely the manner in which this force is applied that is the most essential, and even the sole, factor determining the merit of pendulum clocks. The second reason is that the amplitude of the oscillations has nothing in common with the accuracy of clocks, provided that changes in the amplitude do not affect the frequency, or affect the frequency in a quite definite manner. Clocks based on the rotation of the Earth, for example, are presumably slowed by tidal friction; the corresponding value of \(Q\) is of the order of \(10^{13}\), while the change of frequency is \(1\) in \(5.3\cdot 10^9\) per year. Such a slowing, provided it is known and a correction has been made for it, does not in the slightest degree diminish the perfection of the clock. Far more unpleasant are unforeseen changes of frequency, which lead to a redefinition of the second through the value of the tropical year.

All that is appropriate to say about the quantity \(Q\) in connection with standards of time and frequency, and also in connection with the capacity of the storage battery in an automobile, is that they must not be too small, so as not to create substantial limitations in the qualities of these designs.

CARBON CLOCKS

For completeness of exposition one should mention an entirely special method of measuring time, unlike everything that has been described up to now: the measurement of time based on the radioactive decay of isotopes of various elements. In recent years the carbon isotope has been studied intensively for this purpose: the content of the isotope at a certain moment of time, referred to its content at the initial moment of time, is a measure of the time that has elapsed since the beginning of the decay. The principal use of this measure is the determination of the age of fossilized

ties and geological deposits. It is not very essential that the carbon isotope is not in fact a clock in the definition that we gave earlier. The recurring phenomenon that is counted in this case is the decay of atoms of the isotope. It is true that decaying atoms can be counted only statistically, and not individually, but in principle this is not a defect, although this circumstance sharply limits the accuracy of the determination. If the technique is developed to the point where individual decays can be counted at the moment when they occur, these clocks may become quite exact. The carbon isotope is also a natural clock. The epoch that it establishes is the epoch of the beginning of the decay (or of the deposition) of the carbon. However, carbon is not suitable as a primary standard, since it does not give uniqueness; there are as many epochs as there have been depositions of carbon.

REQUIREMENTS IMPOSED ON TIME STANDARDS

We can now formulate anew the requirements imposed on acceptable time standards. We have already convinced ourselves that there is no need to require invariability of standards, because, as was explained earlier, there are no ways of establishing whether the unit of time is changing or not. Instead of invariability we must require that, for two acceptable standards, the acceleration of one of them relative to the other be constant (in particular, equal to zero). Thus, an acceptable time standard must operate continuously, must be accessible, must have a constant acceleration with respect to another acceptable standard, must be based on a unit that is neither too large nor too small, and must establish the epoch unambiguously. Of two acceptable time standards that have mutual acceleration, the one adopted as the primary standard is the one that does not lead to a contradiction between observations and physical theories. If it were to turn out, for example, that the atomic measure of time agrees with quantum mechanics, whereas the astronomical measure agrees with the general theory of relativity, and that these measures accelerate with respect to one another, this would be very unfortunate, at least for one of these theories; contradictions in these theories would then have to be removed before it could be decided which of these two measures, if either of them at all, is the primary one.

The requirements imposed on an acceptable frequency standard and on a fundamental frequency standard are the same as for a time standard, with the sole exception that the frequency standard is not obliged to establish the epoch unambiguously. The requirement of continuous operation may also be somewhat relaxed; the frequency standard need operate continuously only during the interval of time that it is necessary to measure.

REFERENCES

1. W. J. Eckert, D. Brouwer, G. M. Clemens, Astron. Papers Amer. Ephemeris, v. 12 (Government Printing Office, Washington, D. C., 1951).
2. D. Brouwer, Astron. J. 57, 125 (1952).
3. Trans. Intern. Astron. Union (in press).
4. W. Markowitz, Astron. J. 59, 69 (1954).
5. G. M. Clemence, Am. Scientist 40, 260 (1952).
6. A. Danjon, Compt. Rend. Acad. Sci. Paris 227, 320 (1948).
7. L. Essen and J. V. L. Perry, Nature 176, 280 (1955).
8. E. A. Milne, Kinematic Relativity; a Sequel to Relativity, Gravitation and World Structure (Clarendon Press, Oxford, 1948).