## ⓘ ΔT

In precise timekeeping, Δ T is a measure of the cumulative effect of the departure of the Earths rotation period from the fixed-length day of atomic time. Formally it is the time difference obtained by subtracting Universal Time from Terrestrial Time: Δ T = TT − UT. The value of ΔT for the start of 1902 is approximately zero; for 2002 it is about 64 seconds. So the Earths rotations over that century took about 64 seconds longer than would be required for days of atomic time.

## 1. Calculation

The Earths rotational speed is ν = 1 / 2π dθ / dt, and a day corresponds to one period P = 1 / ν. A rotational acceleration dν / dt gives a rate of change of the period of dP / dt = − 1 / ν 2 dν / dt, which is usually expressed as α = ν dP / dt = − 1 / dν / dt. This has units of 1/time, and is commonly quoted as milliseconds-per-day per century written as ms/day/cy, understood as ms/day/cy). Integrating α gives an expression for Δ T against time.

### * 1.1. Calculation * Universal time

Universal Time is a time scale based on the Earths 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 10 8. However, a larger, more consistent effect has been observed over many centuries: Earths rate of rotation is inexorably slowing down. This observed change in the rate of rotation is attributable to two primary forces, one decreasing and one increasing the Earths rate of rotation. Over the long term, the dominating force is tidal friction, which is slowing the rate of rotation, contributing about α = +2.3 ms/day/cy or dP / dt = +2.3 ms/cy, which is equal to the very small fractional change +7.3 × 10 −13 day/day. The most important force acting in the opposite direction, to speed up the rate, 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, an effect that is still occurring today and will continue until isostatic equilibrium is reached. This "post-glacial rebound" brings mass closer to the rotational axis of the Earth, which makes the Earth spin faster, according to the law of conservation of angular momentum, similar to an ice skater pulling their arms in to spin faster. Models estimate this effect to contribute about −0.6 ms/day/cy. Combining these two effects, 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 matches the average rate derived from astronomical records over the past 27 centuries.

### * 1.2. Calculation * 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 with the intent of forming a gravitationally uniform time scale as far was feasible at that time, and depending for its definition on Simon Newcombs Tables of the Sun 1895, interpreted in a new way to accommodate certain observed discrepancies. Newcombs 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, but later, in respect of the period 1960–1983, they were treated as expressed in terms of ET, in accordance with the adopted ET proposal of 1948–52. ET, in turn, can now be seen in light of modern results 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 Newcombs 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 10 14.

## 2. Earths rate of rotation

Earths rate of rotation must be integrated to obtain time, which is Earths angular position specifically, the orientation of the meridian of Greenwich relative to the fictitious mean sun. Integrating +1.7 ms/d/cy and centering the resulting parabola on the year 1820 yields to a first approximation 32 × year − 1820 / 100 2 - 20 seconds for Δ T. Smoothed historical measurements of Δ T 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 Δ T. Δ T 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 in January 2000 and +68.97 s in January 2018. 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. 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 3 ⁄ 4 h corresponds to 71.625°E. This means that in the year −500 501 BC, Earths faster rotation would cause a total solar eclipse to occur 71.625° to the east of the location calculated using the uniform TT.

## 3. Values prior to 1955

All values of Δ T before 1955 depend on observations of the Moon, either via eclipses or occultations. The angular momentum lost by the Earth due to friction induced by the Moons tidal effect is 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 Keplers laws of planetary motion causes the Moon to revolve around the Earth at a slower rate. The cited values of Δ T assume that the lunar acceleration actually a deceleration, that is a negative acceleration due to this effect is d n / dt = −26″/cy 2, where n is the mean sidereal angular motion of the Moon. This is close to the best estimate for d n / dt as of 2002 of −25.858 ± 0.003″/cy 2 so Δ T 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.

## 4. Geological evidence

Tidal deceleration rates have varied over the history of the Earth-Moon system. Analysis of layering in fossil mollusc shells from 70 million years ago, in the Late Cretaceous period, shows that there were 372 days a year, and thus that the day was about 23.5 hours long then. Based on geological studies of tidal rhythmites, the day was 21.9±0.4 hours long 620 million years ago and there were 13.1±0.1 synodic months/year and 400±7 solar days/year. The average recession rate of the Moon between then and now has been 2.17±0.31 cm/year, which is about half the present rate. The present high rate may be due to near resonance between natural ocean frequencies and tidal frequencies.

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