# ΔT

ΔT vs. time from 1657 to 2015.[1][2]

In precise timekeeping, ΔT (Delta T, delta-T, deltaT, or DT) is the time difference obtained by subtracting Universal Time (UT) from Terrestrial Time (TT): ΔT = TT − UT.

## Calculating ΔT

The Earth's rotational speed is ν = 1//dt, and a day corresponds to one period T = 1/ν. A rotational acceleration /dt gives a rate of change of the period of dT/dt = −1/ν2/dt, which is usually expressed as α = νdT/dt = −1/ν/dt. This has units of 1/time, and is commonly quoted as milliseconds per day per century (ms/day/cy).

### Universal time

Universal Time is a time scale based on the Earth's rotation, which is somewhat irregular over short periods (days up to a century), thus any time based on it cannot have an accuracy better than 1 in 108. But the principal effect is over the long term: over many centuries tidal friction inexorably slows Earth's rate of rotation by about dT/dt = +2.3 ms/cy, or α = +2.3 ms/day/cy. During one day, this results in a very small fractional change of ΔT/T = 7.3×10−13. However, there are other forces changing the rotation rate of the Earth. The most important one is believed to be a result of the melting of continental ice sheets at the end of the last glacial period. This removed their tremendous weight, allowing the land under them to begin to rebound upward in the polar regions, which has been continuing and will continue until isostatic equilibrium is reached. This "post-glacial rebound" brings mass closer to the rotation axis of the Earth, which makes the Earth spin faster, according to the law of conservation of angular momentum,: the rate derived from models is about −0.6 ms/day/cy. So the net acceleration (actually a deceleration) of the rotation of the Earth, or the change in the length of the mean solar day (LOD), is +1.7 ms/day/cy. This is indeed the average rate as observed over the past 27 centuries.[3]

### Terrestrial time

Terrestrial Time is a theoretical uniform time scale, defined to provide continuity with the former Ephemeris Time (ET). ET was an independent time-variable, proposed (and its adoption agreed) in the period 1948–52[4] with the intent of forming a gravitationally uniform time scale as far as was feasible at that time, and depending for its definition on Simon Newcomb's Tables of the Sun (1895), interpreted in a new way to accommodate certain observed discrepancies.[5] Newcomb's tables formed the basis of all astronomical ephemerides of the Sun from 1900 through 1983: they were originally expressed (and published) in terms of Greenwich Mean Time and the mean solar day,[6] but later, in respect of the period 1960–1983, they were treated as expressed in terms of ET,[7] in accordance with the adopted ET proposal of 1948–52. ET, in turn, can now be seen (in light of modern results)[8] as close to the average mean solar time between 1750 and 1890 (centered on 1820), because that was the period during which the observations on which Newcomb's tables were based were performed. While TT is strictly uniform (being based on the SI second, every second is the same as every other second), it is in practice realised by International Atomic Time (TAI) with an accuracy of about 1 part in 1014.

## Earth's rate of rotation

Earth's rate of rotation must be integrated to obtain time, which is Earth's angular position (specifically, the orientation of the meridian of Greenwich relative to the fictitious mean sun). Integrating +1.7 ms/d/cy and centring the resulting parabola on the year 1820 yields (to a first approximation) 31 × (year − 1820/100)2
seconds for ΔT.[9] Smoothed historical measurements of ΔT using total solar eclipses are about +17190 s in the year −500 (501 BC), +10580 s in 0 (1 BC), +5710 s in 500, +1570 s in 1000, and +200  s in 1500. After the invention of the telescope, measurements were made by observing occultations of stars by the Moon, which allowed the derivation of more closely spaced and more accurate values for ΔT. ΔT continued to decrease until it reached a plateau of +11 ± 6 s between 1680 and 1866. For about three decades immediately before 1902 it was negative, reaching −6.64 s. Then it increased to +63.83 s at 2000. It will continue to increase at an ever faster (quadratic) rate in the future. This will require the addition of an ever greater number of leap seconds to UTC as long as UTC tracks UT1 with one - second adjustments. (The SI second as now used for UTC, when adopted, was already a little shorter than the current value of the second of mean solar time.[10]) Physically, the meridian of Greenwich in Universal Time is almost always to the east of the meridian in Terrestrial Time, both in the past and in the future. +17190 s or about 4 34 h corresponds to 71.625°E. This means that in the year −500 (501 BC), Earth's faster rotation would cause a total solar eclipse to occur 71.625° to the east of the location calculated using the uniform TT.

## Values prior to 1955

All values of ΔT before 1955 depend on observations of the Moon, either via eclipses or occultations. Conservation of angular momentum in the Earth-Moon system requires that the angular momentum lost by the Earth due to tidal friction be transferred to the Moon, increasing its angular momentum, which means that its moment arm (its distance from the Earth) is increased (for the time being about +3.8 cm/year), which via Kepler's laws of planetary motion causes the Moon to revolve around the Earth at a slower rate. The cited values of ΔT assume that the lunar acceleration (actually a deceleration, that is a negative acceleration) due to this effect is dn/dt = −26″/cy2, where n is the mean sidereal angular motion of the Moon. This is close to the best estimate for dn/dt as of 2002 of −25.858 ± 0.003″/cy2[11] so ΔT need not be recalculated given the uncertainties and smoothing applied to its current values. Nowadays, UT is the observed orientation of the Earth relative to an inertial reference frame formed by extra-galactic radio sources, modified by an adopted ratio between sidereal time and solar time. Its measurement by several observatories is coordinated by the International Earth Rotation and Reference Systems Service (IERS).

## Notes

1. IERS Rapid Service/Prediction Center (c. 1986). Historic Delta T and LOD. Source attributed data to McCarthy and Babcock (1986). Retrieved December 2009.
2. IERS Rapid Service/Prediction Center. Monthly determinations of Delta T. Retrieved October 2015.
3. McCarthy & Seidelmann 2009, 88–89.
4. Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac, Nautical Almanac Offices of UK and US (1961), at pp. 9 and 71.
5. See G M Clemence's proposal of 1948, contained in his paper: "On the System of Astronomical Constants", Astronomical Journal (1948) vol.53 (6), issue #1170, pp 169–179; also G M Clemence (1971), "The Concept of Ephemeris Time", in Journal for the History of Astronomy v2 (1971), pp. 73–79 (giving details of the genesis and adoption of the ephemeris time proposal); also article Ephemeris time and references therein.
6. See Newcomb's Tables of the Sun (Washington, 1895), Introduction, I. Basis of the Tables, pp. 9 and 20, citing time units of Greenwich Mean Noon, Greenwich Mean Time, and mean solar day: and W de Sitter, on p. 38 of Bulletin of the Astronomical Institutes of the Netherlands, v4 (1927), pp.21–38, "On the secular accelerations and the fluctuations of the moon, the sun, Mercury and Venus", which refers to "the 'astronomical time', given by the earth's rotation, and used in all practical astronomical computations", and states that it "differs from the 'uniform' or 'Newtonian' time".
7. See p.612 in Explanatory Supplement to the Astronomical Almanac, ed. P K Seidelmann, 1992, confirming introduction of ET in the 1960 edition of the ephemerides.
8. See especially F R Stephenson (1997), and Stephenson & Morrison (1995), book and papers cited below.
9. A similar parabola is plotted on p. 54 of McCarthy & Seidelmann (2009).
10. (1) In "The Physical Basis of the Leap Second", by D D McCarthy, C Hackman and R A Nelson, in Astronomical Journal, vol.136 (2008), pages 1906–1908, it is stated (page 1908), that "the SI second is equivalent to an older measure of the second of UT1, which was too small to start with and further, as the duration of the UT1 second increases, the discrepancy widens." :(2) In the late 1950s, the caesium standard was used to measure both the current mean length of the second of mean solar time (UT2) (result: 9192631830 cycles) and also the second of ephemeris time (ET) (result: 9192631770 ± 20 cycles), see "Time Scales", by L. Essen, in Metrologia, vol.4 (1968), pp.161–165, on p.162. As is well known, the 9192631770 figure was chosen for the SI second. L Essen in the same 1968 article (p.162) stated that this "seemed reasonable in view of the variations in UT2".
11. J.Chapront, M.Chapront-Touzé, G.Francou (2002): "A new determination of lunar orbital parameters, precession constant, and tidal acceleration from LLR measurements" (also in PDF). Astronomy & Astrophysics 387, 700–709.

## References

• McCarthy, D.D. & Seidelmann, P.K. TIME: From Earth Rotation to Atomic Physics. Weinheim: Wiley-VCH. (2009). ISBN 978-3-527-40780-4
• Morrison, L.V. & Stephenson, F. R. "Historical values of the Earth's clock error ΔT and the calculation of eclipses" (pdf, 862 KB), Journal for the History of Astronomy 35 (2004) 327–336.
• Stephenson, F.R. Historical Eclipses and Earth's Rotation. Cambridge University Press, 1997. ISBN 0-521-46194-4
• Stephenson, F. R. & Morrison, L.V. "Long-term fluctuations in the Earth's rotation: 700 BC to AD 1990". Philosophical Transactions of the Royal Society of London, Series A 351 (1995) 165-202. JSTOR link. Includes evidence that the 'growth' in Delta-T is being modified by an oscillation with a wavelength around 1500 years; if that is true, then during the next few centuries Delta-T values will increase more slowly than is envisaged.