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The equation which relates Terrestrial [Dynamical] Time (TT, or TDT) and International Atomic Time (TAI) is the following, according to the U.S. Naval Observatory:

$TDT = TAI + 32.184$

As I have understood, both TT and TAI are continuous time systems (differently from UTC, which has leap-second gaps) and described by the same unit (the SI second), but counted from a different epoch (the origin T0 of one is 32.184 seconds after the other).

It appears that both time systems are essentially the same. So what is the actual difference between them? If the difference between them is the nature of how they are computed (e.g., TT is a theoretical timescale; and TAI is a statistical timescale based on several clocks), why the 32.184 shift?

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It's historical. TT is a successor to The International Astronomy Union's ephemeris time, and the offset between TT and TAI is a result of getting TT to match ET when they switched.

Excerpts from https://en.wikipedia.org/wiki/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 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.

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.

(Bold emphasis mine).

For more on International Atomic time see https://en.wikipedia.org/wiki/International_Atomic_Time . Basically, it's a weighted average of multiple atomic clocks - and the altitude effect mentioned in the middle paragraph above is why they added the gravitational time dilation corrections. From the Wikipedia link above:

n the 1970s, it became clear that the clocks participating in TAI were ticking at different rates due to gravitational time dilation, and the combined TAI scale therefore corresponded to an average of the altitudes of the various clocks. Starting from Julian Date 2443144.5 (1 January 1977 00:00:00), corrections were applied to the output of all participating clocks, so that TAI would correspond to proper time at mean sea level (the geoid). Because the clocks had been on average well above sea level, this meant that TAI slowed down, by about one part in a trillion. The former uncorrected time scale continues to be published, under the name EAL (Echelle Atomique Libre, meaning Free Atomic Scale).[7]:215

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    $\begingroup$ No - the 32.184 seconds is just an offset so that the time number stayed the same when they switched from ET to TT. So at IAT 1977-01-01T00:00:00.000, which was ET 1977-01-01T00:00:32.184 they set TT to be the same value as ET so the ET/TT times were continuous with no jump - that's where the 32.184 second offset comes from. Adding the gravitational compensation to TAI was to get the atomic clocks to agree with other better, since the altitude effect was big enough to make noticeable.differences. $\endgroup$
    – JerryTheC
    Aug 27, 2017 at 0:08
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    $\begingroup$ And since TT is apparently currently measured using TAI (and presumably, therefore, the Atomic clock adjustments used are tweaking things to the geoid altitude value), the offset is fixed and won't change. $\endgroup$
    – JerryTheC
    Aug 27, 2017 at 0:13
  • $\begingroup$ Hmm. Are those compensations a kind of "weighted mean" (or other statistical device) which converts the "tick speed" of the several atomic clocks which computes TAI into a single rate? Or do they apply/multiply a factor of time dilation in each clock? $\endgroup$
    – Seninha
    Aug 27, 2017 at 1:45
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    $\begingroup$ They apply a correction based on altitude above mean sea level to each clock (So that should give a single rate) then do a weighted average to work out the official value, and publish a (retrospective) monthly list of deltas between the official value and the individual clocks. $\endgroup$
    – JerryTheC
    Aug 27, 2017 at 10:26
  • $\begingroup$ I'd like to quote this funny, but quite true, paragraph I read right now, from here: "All ephemeris time scales are defined to be 32.184 seconds ahead of atomic time on 1977-01-01 at 00:00:00 TAI. There is no real reason behind this number: it is merely due to a lack of sufficient communication between astronomers and physicists." $\endgroup$
    – Seninha
    Sep 6, 2017 at 4:25

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