As Earth rotates on its axis, tidal friction is imposed on it through the gravitational attraction with the Moon and, to a lesser extent, the Sun. This secular acceleration gradually transfers angular momentum from Earth to the Moon. As Earth loses energy and slows down, the Moon gains this energy and its orbital period and distance from Earth increase.

Stephenson and collaborators have produced a number of seminal works in the field of Earth's rotation covering the past several millennia. In particular, they have identified hundreds of eclipse and occultation observations in early European, Middle Eastern and Chinese annals, manuscripts, canons and records. In spite of their relatively low precision, these data represent our only record to the value of delta-T during the past several millennia.

In *Atlas of Historical Eclipse Maps East Asia 1500 BC - AD 1900*, Stephenson and Houlden (1986) present two empirically derived expressions to describe the behavior of delta-T prior to telescopic records (pre-1600):

(1) prior to 948 CE delta-T (seconds) = 1830 - 405*t + 46.5*t^2 (t = centuries since 948 CE) (2) 948 CE to 1600 CE delta-T (seconds) = 22.5*t^2 (t = centuries since 1850 CE)

These expressions are actually based on work of Stephenson and Morrison (1984) and were derived from an analysis and fit of hundreds of historical eclipse and occultation records from 700 BCE to 1600 CE.
More recently, Stephenson (1997) presents a new analysis of most if not all known historical records of solar and lunar eclipses that occurred during the same period (*Historical Eclipses and Earth's Rotation*, 1997).
The new analysis uses a spline to fit the observations.

The following table lists values of delta-T (seconds) derived from Stephenson and Houlden (1986), and Stephenson (1997) for comparison. The last column gives the difference (seconds) between the values derived from 1986 and 1997 analyses of delta-T.

Year delta-T delta-T difference (1986) (1997) -1500 39610 - (all values in seconds) -1400 36975 - -1300 34433 - -1200 31984 - -1100 29627 - -1000 27364 - -900 25194 - -800 23117 - -700 21133 - -600 19242 - -500 17444 16800 644 -400 15738 15300 436 -300 14126 14000 126 -200 12607 12800 -193 -100 11181 11600 -419 0 9848 10600 -752 100 8608 9600 -992 200 7461 8600 -1139 300 6406 7700 -1294 400 5445 6700 -1255 500 4577 5700 -1123 600 3802 4700 -898 700 3120 3800 -680 800 2531 3000 -469 900 2035 2200 -165 1000 1625 1600 25 1100 1265 1100 165 1200 950 750 200 1300 680 470 210 1400 455 300 155 1500 275 180 95 1600 140 110 30

Since historical records for deriving values of delta-T only cover the period from about 700 BCE to 1600 CE, any values of delta-T before this period are based on a smooth extrapolation from known values to a theoretical model based entirely on pure tidal braking of Earth's rotation. Stephenson and Morrison (1995) determined an expression due to lunar and solar tides with constant deceleration of Earth's rotation as follows:

delta-T (seconds) = -20 + 31*t^2 (t = centuries since 1820 CE)

In the eclipse predictions presented on this web site, the above expression has been used to evaluate delta-T before 700 BCE as well as for future values. In either case, the eclipse predictions use a weighted average between a delta-T from this expression and an extrapolated value from observations so as not to introduce any abrupt discontinuities. For the period 700 BCE to 1600 CE, delta-T from Stephenson (1997) is used, while direct observations of delta-T are implemented from 1600 CE to present.

Dickey, J.O., "Earth Rotation Variations from Hours to Centuries", in: I. Appenzeller (ed.), Highlights of Astronomy: Vol. 10 (Kluwer Academic Publishers, Dordrecht/Boston/London, 1995), pp. 17-44.

Meeus, J., "The Effect of Delta T on Astronomical Calculations", Journal of the British Astronomical Association, 108 (1998), 154-156.

Morrison, L.V. & Stephenson, F.R., "Historical Values of the EarthÕs Clock Error Delta T and the Calculation of Eclipses", Journal for the History of Astronomy, Vol. 35 Part 3, August 2004, No. 120 (2004), pages 327Š336.

Morrison, L.V. and Ward, C. G., "An analysis of the transits of Mercury: 1677-1973", Mon. Not. Roy. Astron. Soc., 173, 183-206, 1975.

Spencer Jones, H., "The Rotation of the Earth, and the Secular Accelerations of the Sun, Moon and Planets", Monthly Notices of the Royal Astronomical Society, 99 (1939), 541-558.

Stephenson, F.R. & Morrison, L.V., "Long-Term Changes in the Rotation of the Earth: 700 BC to AD 1980", Philosophical Transactions of the Royal Society of London, Ser. A, 313 (1984), 47-70.

Stephenson F.R and Houlden M.A., *Atlas of Historical Eclipse Maps: East Asia 1500 BD - AD 1900*, Cambridge Univ. Press., 1986.

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, Ser. A, 351 (1995), 165-202.

Stephenson F.R., *Historical Eclipses and Earth's Rotation*, Cambridge Univ. Press, 1997.