# Terrestrial Time

Terrestrial Time (TT) is a modern astronomical time standard defined by the International Astronomical Union, primarily for time-measurements of astronomical observations made from the surface of Earth.[1] For example, the Astronomical Almanac uses TT for its tables of positions (ephemerides) of the Sun, Moon and planets as seen from Earth. In this role, TT continues Terrestrial Dynamical Time (TDT or TD),[2] which in turn succeeded ephemeris time (ET). TT shares the original purpose for which ET was designed, to be free of the irregularities in the rotation of Earth.

The unit of TT is the SI second, the definition of which is currently based on the caesium atomic clock,[3] but TT is not itself defined by atomic clocks. It is a theoretical ideal, and real clocks can only approximate it.

TT is distinct from the time scale often used as a basis for civil purposes, Coordinated Universal Time (UTC). TT indirectly underlies UTC, via International Atomic Time (TAI). Because of the historical difference between TAI and ET when TT was introduced, TT is approximately 32.184 s ahead of TAI.

## History

A definition of a terrestrial time standard was adopted by the International Astronomical Union (IAU) in 1976 at its XVI General Assembly, and later named Terrestrial Dynamical Time (TDT). It was the counterpart to Barycentric Dynamical Time (TDB), which was a time standard for Solar system ephemerides, to be based on a dynamical time scale. Both of these time standards turned out to be imperfectly defined. Doubts were also expressed about the meaning of 'dynamical' in the name TDT.

In 1991, in Recommendation IV of the XXI General Assembly, the IAU redefined TDT, also renaming it "Terrestrial Time". TT was formally defined in terms of Geocentric Coordinate Time (TCG), defined by the IAU on the same occasion. TT was defined to be a linear scaling of TCG, such that the unit of TT is the SI second on the geoid (Earth surface at mean sea level). This left the exact ratio between TT time and TCG time as something to be determined by experiment. Experimental determination of the gravitational potential at the geoid surface is a task in physical geodesy.

In 2000, the IAU very slightly altered the definition of TT by adopting an exact value for the ratio between TT and TCG time, as 1 − 6.969290134×10−10[4] (As measured on the geoid surface, the rate of TCG is very slightly faster than that of TT, see below, Relativistic relationships of TT).

## Current definition

TT differs from Geocentric Coordinate Time (TCG) by a constant rate. Formally it is defined by the equation

${\displaystyle TT={\bigl (}1-L_{g}{\bigr )}\times TCG\ +\ E}$

where TT and TCG are linear counts of SI seconds in Terrestrial Time and Geocentric Coordinate Time respectively, ${\displaystyle L_{g}}$ is the constant difference in the rates of the two time scales, and ${\displaystyle E}$ is a constant to resolve the epochs (see below). ${\displaystyle L_{g}}$ is defined as exactly 6.969290134×10−10 (In 1991 when TT was first defined, ${\displaystyle L_{g}}$ was to be determined by experiment, and the best available estimate was 6.969291×10−10).

The equation linking TT and TCG is more commonly seen in the form

${\displaystyle TT=TCG-L_{g}\times {\bigl (}JD_{TCG}-2443144.5003725{\bigr )}\times 86400}$

where ${\displaystyle JD_{TCG}}$ is the TCG time expressed as a Julian date (JD). This is just a transformation of the raw count of seconds represented by the variable TCG, so this form of the equation is needlessly complex. The use of a Julian Date specifies the epoch fully. The above equation is often given with the Julian Date 2443144.5 for the epoch, but that is inexact (though inappreciably so, because of the small size of the multiplier ${\displaystyle L_{g}}$). The value 2443144.5003725 is exactly in accord with the definition.

Time coordinates on the TT and TCG scales are conventionally specified using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with their predecessor Ephemeris Time (ET), TT and TCG were set to match ET at around Julian Date 2443144.5 (1977-01-01T00Z). More precisely, it was defined that TT instant 1977-01-01T00:00:32.184 exactly and TCG instant 1977-01-01T00:00:32.184 exactly correspond to the International Atomic Time (TAI) instant 1977-01-01T00:00:00.000 exactly. This is also the instant at which TAI introduced corrections for gravitational time dilation.

TT and TCG expressed as Julian Dates can be related precisely and most simply by the equation

${\displaystyle JD_{TT}=E_{JD}+{\bigl (}JD_{TCG}-E_{JD}{\bigr )}\times {\bigl (}1-L_{g}{\bigr )}}$

where ${\displaystyle E_{JD}}$ is 2443144.5003725 exactly.

## Realization

TT is a theoretical ideal, not dependent on a particular realization. For practical purposes, TT must be realized by actual clocks in the Earth system.

The main realization of TT is supplied by TAI. The TAI service, running since 1958, attempts to match the rate of proper time on the geoid, using an ensemble of atomic clocks spread over the surface and low orbital space of Earth. TAI is canonically defined retrospectively, in monthly bulletins, in relation to the readings shown by that particular group of atomic clocks at the time. Estimates of TAI are also provided in real time by the institutions that operate the participating clocks. Because of the historical difference between TAI and ET when TT was introduced, the TAI realization of TT is defined thus:

${\displaystyle TT(TAI)=TAI+32.184s}$

Because TAI is never revised once published, it is possible for errors in it to become known and remain uncorrected. It is thus possible to produce a better realization of TT based on reanalysis of historical TAI data. The BIPM has done this approximately annually since 1992. These realizations of TT are named in the form "TT(BIPM08)", with the digits indicating the year of publication. They are published in the form of table of differences from TT(TAI). The latest as of March 2020 is TT(BIPM19).[5]

The international communities of precision timekeeping, astronomy, and radio broadcasts have considered creating a new precision time scale based on observations of an ensemble of pulsars. This new pulsar time scale will serve as an independent means of computing TT, and it may eventually be useful to identify defects in TAI.

## Approximation

Sometimes times described in TT must be handled in situations where TT's detailed theoretical properties are not significant. Where millisecond accuracy is enough (or more than enough), TT can be summarized in the following ways:

• To millisecond accuracy, TT runs parallel to the atomic timescale (International Atomic Time, TAI) maintained by the BIPM. TT is ahead of TAI, and can be approximated as TT ≅ TAI + 32.184 seconds.[6] (The offset 32.184 s arises from the history[7]).
• TT also runs in parallel with the GPS time scale, which has a constant difference from atomic time (TAI − GPS time = +19 seconds),[8] so that TT ≅ GPS time + 51.184 seconds.
• TT is in effect a continuation of (but is more precisely uniform than) the former Ephemeris Time (ET). It was designed for continuity with ET,[9] and it runs at the rate of the SI second, which was itself derived from a calibration using the second of ET (see, under Ephemeris time, Redefinition of the second and Implementations).
• TT runs a little ahead of UT1 (a refined measure of mean solar time at Greenwich) by an amount known as ΔT = TT − UT1. ΔT was measured at +67.6439 seconds (TT ahead of UT1) at 0h UTC on 1 January 2015;[10] and by retrospective calculation, ΔT was close to zero around the year 1900. The difference ΔT, though somewhat unpredictable in fine detail, is expected to continue to increase, with UT1 becoming steadily (but irregularly) further behind TT in the future.

## Relativistic relationships

Observers in different locations, that are in relative motion or at different altitudes, can disagree about the rates of each other's clocks, owing to effects described by the theory of relativity. As a result, TT (even as a theoretical ideal) does not match the proper time of all observers.

In relativistic terms, TT is described as the proper time of a clock located on the geoid (essentially mean sea level).[11] However,[12] TT is now actually defined as a coordinate time scale.[13] The redefinition did not quantitatively change TT, but rather made the existing definition more precise. In effect it defined the geoid (mean sea level) in terms of a particular level of gravitational time dilation relative to a notional observer located at infinitely high altitude.

The present definition of TT is a linear scaling of Geocentric Coordinate Time (TCG), which is the proper time of a notional observer who is infinitely far away (so not affected by gravitational time dilation) and at rest relative to Earth. TCG is used so far mainly for theoretical purposes in astronomy. From the point of view of an observer on Earth's surface the second of TCG passes in slightly less than the observer's SI second. The comparison of the observer's clock against TT depends on the observer's altitude: they will match on the geoid, and clocks at higher altitude tick slightly faster.

## References

1. ^ The 1991 definition refers to the scale agreeing with the SI second "on the geoid", i.e. close to mean sea level on Earth's surface, see IAU 1991 XXIst General Assembly (Buenos Aires) Resolutions, Resolution A.4 (Recommendation IV). A redefinition by resolution of the IAU 2000 24th General Assembly (Manchester), at Resolution B1.9, is in different terms intended for continuity and to come very close to the same standard.
2. ^ TT is equivalent to TDT, see IAU conference 1991, Resolution A4, recommendation IV, note 4.
3. ^ IAU conference 1991, Resolution A4, recommendation IV, part 2 states that the unit for TT is to agree with the SI second 'on the geoid'.
4. ^ Resolution B1.9 of the IAU XXIV General Assembly, 2000
5. ^ "BIPM - Time Department FTP server". www.bipm.org.
6. ^ IAU conference 1991, Resolution A4, recommendation IV, note 9.
7. ^ The atomic time scale A1 (a predecessor of TAI) was set equal to UT2 at its conventional starting date of 1 January 1958 (see L Essen, "Time Scales", Metrologia, vol.4 (1968), 161-165, at 163), when ΔT (ET-UT) was about 32 seconds. The offset 32.184 seconds was the 1976 estimate of the difference between Ephemeris Time (ET) and TAI, "to provide continuity with the current values and practice in the use of Ephemeris Time" (see IAU Commission 4 (Ephemerides), Recommendations to IAU General Assembly 1976, Notes on Recommendation 5, note 2)
8. ^ Steve Allen. "Time Scales". Lick Observatory. Retrieved 13 August 2017.
9. ^ P K Seidelmann (ed.) (1992), 'Explanatory Supplement to the Astronomical Almanac', at p.42; also IAU Commission 4 (Ephemerides), Recommendations to IAU General Assembly 1976, Notes on Recommendation 5, note 2.
10. ^ US Naval Observatory (USNO) data file online at ftp://maia.usno.navy.mil/ser7/deltat.data (accessed 27 October 2015).
11. ^ For example, IAU Commission 4 (Ephemerides), Recommendations to IAU General Assembly 1976, Notes on Recommendation 5, note 1, as well as other sources, indicate the time scale for apparent geocentric ephemerides as a proper time.
12. ^ B Guinot (1986), "Is the International Atomic Time a Coordinate Time or a Proper Time?", Celestial Mechanics, 38 (1986), pp.155-161.
13. ^ IAU General Assembly 1991, Resolution A4, Recommendations III and IV, define TCB and TCG as coordinate time scales, and TT as a linear scaling of TCG, hence also a coordinate time.