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. 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), which 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 based currently on the caesium atomic clock, 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 is indirectly the basis of 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", i.e. the rate approximately matched the rate of proper time on the Earth's surface at mean sea level. Thus the exact ratio between TT time and TCG time was formula_1, where formula_2 was a constant and formula_3 was the gravitational potential at the geoid surface, a value measured by physical geodesy. In 1991 the best available estimate of formula_4 was . In 2000, the IAU very slightly altered the definition of TT by adopting an exact  value, . Current definition. TT differs from Geocentric Coordinate Time (TCG) by a constant rate. Formally it is defined by the equation formula_5 where TT and TCG are linear counts of SI seconds in Terrestrial Time and Geocentric Coordinate Time respectively, formula_4 is the constant difference in the rates of the two time scales, and formula_7 is a constant to resolve the epochs (see below). formula_4 is defined as exactly . Due to the term formula_1 the rate of TT is very slightly slower than that of TCG. The equation linking TT and TCG more commonly has the form given by the  IAU, formula_10 where formula_11 is the TCG time expressed as a Julian date (JD). The Julian Date is a linear transformation of the raw count of seconds represented by the variable TCG, so this form of the equation is not simplified. The use of a Julian Date specifies the epoch fully. The above equation is often given with the Julian Date for the epoch, but that is inexact (though inappreciably so, because of the small size of the multiplier formula_4). The value is exactly in accord with the definition. Time coordinates on the TT and TCG scales are specified conventionally  using traditional means of specifying days, inherited 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 (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 formula_13 where formula_14 is exactly. Realizations. 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. TAI. The main realization of TT is supplied by TAI. The TAI service, performed since 1958, estimates TT using measurements from 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: formula_15 TT(BIPM). Because TAI is never revised once published, it is possible for errors in it to become known and remain uncorrected. Approximately annually since 1992, the International Bureau of Weights and Measures (BIPM) has produced better realizations of TT based on reanalysis of historical TAI data. BIPM's 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 a table of differences from TT(TAI), along with an extrapolation equation that may be used for dates later than the table. The latest is TT(BIPM21). Others. Researchers from the International Pulsar Timing Array collaboration have created a realization of TT based on observations of an ensemble of pulsars. This new pulsar time scale is an independent means of computing TT, and it may eventually be useful to identify defects in TAI. Approximation. Sometimes times described in TT are used 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 manners: 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). However, TT is now actually  defined as a coordinate time scale. 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 to date 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.