Historical Values of Delta T

Fred Espenak

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.

References for Delta-T

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.

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2012 Aug 26