## Universal and atomic times

Universal time (UT) and earth spin

The modern equivalent of GMT is universal time. There are two principal versions of universal time, UT (= UT1), and UTC. Here we discuss the former; the latter will come soon. The UT time is obtained from the overhead passages of stars rather than the passage of the sun, but it is continually adjusted (in principle) to closely approximate mean solar time at Greenwich such that the mean sun is on the meridian at noon and one day equals precisely 86 400 s. This time standard thus uses the earth spin as the clock.

Formally, UT is defined in terms of the sidereal time at Greenwich. Specifically, 0 h UT (midnight in Greenwich) on Jan. 1 is defined to occur at Greenwich mean sidereal time (GMST) ~ 6.7 h. Examine Fig. 3.1 and note that on Jan. 1, which is 285 d after Mar. 21, the sun is about 18.7 h east of the equinox [(285 d/365 d) x 24 h ~ 18.7 h], that is, a0 ~ 18.7 h. At midnight in Greenwich on this date, the zenith will be on the side of the celestial sphere exactly opposite (in RA) to the sun, i.e., at 18.7 - 12 = 6.7 h. The sidereal time at Greenwich will thus be about 6.7 h at 0 h UT on Jan. 1, in accord with the definition.

Variations in the earth spin rate result in variations in the rate of time based on star transits, relative to an ideal Newtonian model. These variations can arise from tidal friction which slows the earth spin, varying angular momentum carried by oceans and the atmosphere which cause seasonal variations in the rotation rate and in the motion of the earth's spin axis relative to the earth's crust (polar motion). In large part these changes are unpredictable. UT has been corrected for polar motion (UT0 is the uncorrected version), but UT still slows and speeds up with the earth spin. The duration of a second or a minute actually grows and shrinks so that 86 400 s carries one from one mean solar transit to the next. For precise measurements, a variable second is not very useful. We use atomic time for that; see below. The precise relation between UT and GMST is presented in the next section.

Greenwich mean sidereal time (GMST) at 0 h UT

As stated above, one can set a UT clock by observing the stars passing over Greenwich at midnight on Jan. 1. When the appropriate meridian of the celestial sphere (sidereal time ~6 h 42 m = 6.7 h on Jan. 1) transits the zenith, it is exactly 0 h UT. The current relation used to set UT in terms of the star transits at Greenwich gives Greenwich mean sidereal time at 0 h UT in terms of the day of the year,

+ (8 640184.812 866 T (Definition of 0 h UT) (4.14)

+ 0.093104 T 2 - 6.2 x 10-6 T3)s where the parameter T specifies the day for which one desires to find the GMST at 0 h UT. Recall that the part of the sky that is overhead at midnight in Greenwich changes with the seasons. Here T is measured in Julian centuries of 36 525 d from epoch J2000.0 (Jan. 1,12 h UT, 2000). Specifically, T = d/(36 525) where d is the number of days from epoch J2000.0. The day count d is constrained to the values of ±0.5, ±1.5, ±2.5 ... because it is applied only at 0 h UT. (We discuss epochs below.)

The first term on the right of (14) specifies the ~6.7 h sidereal time discussed above. The second term is linear with time; in one (Julian) day, it advances the GMST at 0 h UT by 236.55 sidereal seconds to take into account the difference between the sidereal and solar days. (The difference between (12) and (13) is somewhat less, 235.91 s, because it is in solar seconds. The two kinds of seconds differ by one part in 365.) The numerical coefficient of this (second) term is the number of seconds of sidereal time that must be added for this correction in a Julian century. As an example, at 2000 Jan. 1, 0 h UT, the date is d = -0.5 d and this term subtracts 118.28 s (ST) (=236.55/2) from the first term as expected for the solar-day correction in 12 h of elapsed time. Twenty-four hours later, the correction is +118.28 s, the next day +384.83 s, and so on until in 1 yr the correction would correspond to 24 h bringing the GMST at 0 h UT back to ~6.7 h ST. Finally, the two non-linear terms correct for polar motion; in one century (T = 1) the correction is ~0.1 s.

### Ephemeris second

Beginning in the late 1920s, it was realized that there were better clocks than the variable earth spin period. Accordingly, the highly stable orbital motion of the earth about the sun was adopted as a reference. Thus in 1958, the ephemeris second was defined as a fixed fraction of the "tropical year 1900", namely 1/(31 556 925.9747). Recall that the tropical year is the time required for the sun to travel from vernal equinox to vernal equinox on the celestial sphere. The ephemeris second was free of the vagaries of earth spin.

Atomic time (TAI)

Atomic clocks introduced a new level of accuracy to time keeping in the late 1940s. Comparisons of different atomic clocks indicated that these uncertainties were about 1 second in 3000 yr, or 0.3 ms in 1 year, or better. This means that the time of a clock tick after one year would have this uncertainty. This is 1 part in 1011 since there are ~1 x 1011 seconds in 3000 yr. There are several types of atomic clocks such as cesium beams, hydrogen masers, rubidium vapor cells, and mercury ion frequency standards. They have differing accuracies ranging up to 1 part in 1014 in a day. The fractional accuracies can differ for different time intervals.

The accuracy of atomic clocks permitted the demonstration by 1972 that the earth's spin period (the traditional clock) is lengthening at a variable rate averaging —1.7 ms per century, most of which is attributable to tidal action. This is not as small as it sounds because each day the small changes accumulate. After only one year at this slowdown rate, the earth clock would be —3.1 ms retarded. (After six months, the day would be 1.7ms/200 longer than at the beginning. Take this as the average excess period during the year and multiply by 365.25 d in a year to obtain 3.1 ms.) Thus the earth clock is good to 1 part in (1 yr/3.1 ms) = 1.0 x 1010, or about 10 times worse than our rudimentary atomic clock with 1 part in 1011 accuracy. Thus atomic time became a better standard than the earth's spin.

In 1967, the atomic second (now called the SI second) was adopted as a fundamental unit of measurement,

1.0 atomic second = 9 192 631770 cycles of Cs133 (SI second) (4.15)

where the cycle count refers to the radiation from a ground-state transition between two hyperfine levels of Cs133. The number of cycles was chosen to agree with the ephemeris second. Atomic time was defined to agree with UT at 1958.0. Currently, the atomic time standard is TAI (Temps Atomique International). It is based on the average of —150 atomic clocks in 30 countries. Note that UT and TAI run independently of one another; the one tied to the transits of the stars (i.e., earth spin) and the other to an atomic standard. TAI is stable to about 30 |s in a century, or 1 part in 1014. Expected advances, e.g., with cold cesium in space, could yield stabilities 100 times better.

Universal coordinated time (UTC) and leap seconds In 1972, Universal Time Coordinated (UTC) was adopted to bring together the UT and TAI systems. It is based on the atomic second but it is occasionally adjusted by the addition of an extra second, a leap second, to maintain it within 0.9 s of UT. In principle, the leap second could be subtracted, if necessary to maintain the 0.9 s difference.

Leap seconds are inserted into UTC when needed at midnight on June 30 or Dec. 31. Sometimes a Dec. 31 will last 86401 s! Compare to the insertion of 3 months by Caesar in 46 BCE. This method allows UTC to reflect the changing spin period of the earth while maintaining the unit of 1 s as a very stable atomic unit of time.

One might ask why one adds an entire leap second every year or two when we argued above that the earth clock is retarded by only —3.1 ms in a year. Recall that this was the retardation relative to the tick rate at the beginning of the year. The leap-second corrections refer, though, to the date when the day was exactly 86 400 s long, which occurred in —1820, or —180 yr ago. Since that time, the day has lengthened by —1.7 ms/century x 1.8 centuries ~ 3.1 ms. (Do not confuse this "length of day"

with the 1-yr "clock retardation" value of 3.1 ms.) This would suggest that the day is now about 86 400.003 s in duration. In one year, after 365 rotations at this period, the discrepancy relative to the 86 400-s day would build up to 3.1 ms x 365.25 ~ 1.1 s. Hence, even if the earth spin were to remain constant at a period of 86 400.003 s, the daily accumulation of 3-ms intervals would require a leap second every year or so to keep our watches in step with the sun's overhead transits.

The irregular variations of the earth clock are huge; annual fluctuations alone change the earth period by ~1 ms or more. You can find a plot of the length of day excess over 86 400 SI seconds (LOD) at the US Naval Observatory "Earth Orientation" web site, http://maia.usno.navy.mil; click on "What is Earth Orientation?" The excess was 2 to 2.5 ms in ~1993, but now as I write this in January 2003, it is fluctuating around 0.6 ms. The earth spin has been speeding up (!) over the past decade, but now is relatively steady. At the current 86 400.0006 s period, the earth clock would lose only 0.6 ms x 365 d = 0.22 s in a year compared to UTC. No wonder there have been no leap second insertions since 1999 Jan. 1 (through 2002).

Terrestrial time (TT)

Astronomers and others have a need for a continuously running time standard that is never adjusted for irregularities of the earth's rotation. A natural choice for this would be atomic time TAI. In fact, since 1972 this has been the standard for such purposes, except for a constant offset of 32.184 s required to match it to the previous (discarded) free-running standard, ephemeris time (ET) described above. This corrected time is known as terrestrial time (TT),

For most purposes, the two time standards, ET and TT can be considered a single seamless time standard: ET before 1958 and TT from 1958 onward. It runs at the same rate as TAI. The TT day contains exactly 86 400 SI seconds.

An earlier name for TT was Terrestrial Dynamic Time (TDT). A dynamic time is one that is used as the independent variable in physical equations such as Newton's laws. Thus, in principle, an ideal TT could be obtained only from extended observations of the planetary motions in the solar system, and this time could in principle diverge from atomic time. Divergence could also arise from small inaccuracies in the rates of atomic clocks. At present, nevertheless, TT is defined to have the same rate as TAI, but with the offset given in (16). The nominal atomic clock is taken to be on the surface of the earth because experiments show that clock rates depend on their location in the earth's gravitational potential in accord with the GR model of time.

In practice, TT is a method of keeping track of the leap seconds inserted into UTC. This is important if one wants to calculate time intervals over several years as in the timing of pulsar pulses. The Astronomical Almanac published each year gives

Table 4.1. TT and TAI offsets relative to UTCa

Start (UTC)

1902 Jan. 1958 Jan. 1977 Jan. 1988 Jan.

1990 Jan.

1991 Jan.

1992 Jul.

1993 Jul.

1994 Jul.

1996 Jan.

1997 Jul. 1999 Jan.

1977 Dec.

1989 Dec.

1990 Dec.

1992 Jun.

1993 Jun.

1994 Jun.

1995 Dec.

1997 Jun.

1998 Dec. >2002

30 24 h

31 24 h

30 24 h

31 24 h