The orbital positions of the Sun and Moon required by eclipse predictions, are calculated using Terrestrial Dynamical Time (TD) because it is a uniform time scale. However, world time zones and daily life are based on Universal Time[1] (UT). In order to convert eclipse predictions from TD to UT, the difference between these two time scales must be known. The parameter delta-T (ΔT) is the arithmetic difference, in seconds, between the two as:

ΔT = TD - UT

Past values of ΔT can be deduced from the historical records. In particular, hundreds of eclipse observations (both solar and lunar) were recorded in early European, Middle Eastern and Chinese annals, manuscripts, and canons. In spite of their relatively low precision, these data represent the only evidence for the value of ΔT prior to 1600 CE. In the centuries following the introduction of the telescope (circa 1609 CE), thousands of high quality observations have been made of lunar occultations of stars. The number and accuracy of these timings increase from the seventeenth through the twentieth century, affording valuable data in the determination of ΔT. A detailed analysis of these measurements fitted with cubic splines for ΔT from -500 to +1950 is presented in Table 1 and includes the standard error for each value (Morrison and Stephenson, 2004).

Table 1 - Values of ΔT Derived from Historical Records | ||

Year | ΔT (seconds) |
Standard Error (seconds) |

-500 | 17190 | 430 |

-400 | 15530 | 390 |

-300 | 14080 | 360 |

-200 | 12790 | 330 |

-100 | 11640 | 290 |

0 | 10580 | 260 |

100 | 9600 | 240 |

200 | 8640 | 210 |

300 | 7680 | 180 |

400 | 6700 | 160 |

500 | 5710 | 140 |

600 | 4740 | 120 |

700 | 3810 | 100 |

800 | 2960 | 80 |

900 | 2200 | 70 |

1000 | 1570 | 55 |

1100 | 1090 | 40 |

1200 | 740 | 30 |

1300 | 490 | 20 |

1400 | 320 | 20 |

1500 | 200 | 20 |

1600 | 120 | 20 |

1700 | 9 | 5 |

1750 | 13 | 2 |

1800 | 14 | 1 |

1850 | 7 | <1 |

1900 | -3 | <1 |

1950 | 29 | <0.1 |

In modern times, the determination of ΔT is made using atomic clocks and radio observations of quasars, so it is completely independent of the lunar ephemeris. Table 2 gives the value of ΔT every five years from 1955 to 2005 (Astronomical Almanac for 2006, page K9).

Table 2 - Recent Values of ΔT from Direct Observations | |||

Year | ΔT (seconds) |
5-Year Change (seconds) |
Average 1-Year Change (seconds) |

1955.0 | +31.1 | - | - |

1960.0 | +33.2 | 2.1 | 0.42 |

1965.0 | +35.7 | 2.5 | 0.50 |

1970.0 | +40.2 | 4.5 | 0.90 |

1975.0 | +45.5 | 5.3 | 1.06 |

1980.0 | +50.5 | 5.0 | 1.00 |

1985.0 | +54.3 | 3.8 | 0.76 |

1990.0 | +56.9 | 2.6 | 0.52 |

1995.0 | +60.8 | 3.9 | 0.78 |

2000.0 | +63.8 | 3.0 | 0.60 |

2005.0 | +64.7 | 0.9 | 0.18 |

The average annual change of ΔT was 0.99 second from 1965 to 1980. However, the average annual increase was just 0.63 second from 1985 to 2000, and only 0.18 second from 2000 to 2005. Future changes and trends in ΔT can not be predicted with certainty since theoretical models of the physical causes are not of high enough precision. Extrapolations from the table weighted by the long period trend from tidal braking of the Moon offer reasonable estimates of +67 seconds in 2010, +93 seconds in 2050, +203 seconds in 2100, and +442 seconds in the year 2200.

Outside the period of observations (500 BCE to 2005 CE), the value of ΔT can be extrapolated from measured values using the long-term mean parabolic trend:

ΔT = -20 + 32 * t^2 seconds where: t = (year-1820)/100

A series of polynomial expressions have been derived from these data to simplify the evaluation of ΔT for any time during the interval -1999 to +3000. The uncertainty in ΔT over this period can be estimated from scatter in the measurements.

The information above was previously published in:

Five Millennium Catalog of Lunar Eclipses: -1999 to +3000 (NASA TP-2009-214173)

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[1] World time zones are actually based on Coordinated Universal Time (UTC). It is an atomic time synchronized and adjusted to stay within 0.9 seconds of astronomically determined Universal Time (UT). Occasionally, a "leap second" is added to UTC to keep it in sync with UT (which changes due to variations in Earth's rotation rate).

Morrison, L. and Stephenson, F. R., "Historical Values of the Earth's Clock Error ΔT and the Calculation of Eclipses", J. Hist. Astron., Vol. 35 Part 3, August 2004, No. 120, pp 327-336 (2004).

Stephenson F.R and Houlden M.A., *Atlas of Historical Eclipse Maps*, Cambridge Univ.Press., Cambridge, 1986.