The uncertainty in the value of ΔT is of particular interest in the calculation of eclipse paths in the distant past and future. Unfortunately, estimating the standard error in ΔT prior to 1600 CE is a difficult problem. It depends on a number of factors which include the accuracy of determining ΔT from historical eclipse records and modeling the physical processes producing changes in Earth's rotation. Morrison and Stephenson [2004] propose a simple parabolic relation to estimate the standard error (σ) which is valid over the period 1000 BCE to 1200 CE:
σ = 0.8 * t^2 seconds
where: t = (year-1820)/100
Table 1 gives the errors in ΔT along with the corresponding uncertainties in the longitude of an eclipse path.
| Table 1 - Uncertainty of ΔT (-500 to +1200) | ||
| Year | σ (seconds) |
Longitude |
| -1000 | 636 | 2.65° |
| -500 | 431 | 1.79° |
| 0 | 265 | 1.10° |
| +500 | 139 | 0.58° |
| +1000 | 54 | 0.22° |
| +1200 | 31 | 0.13° |
The decade fluctuations in ΔT result in an uncertainty of approximately 20 seconds (0.08°) for the period 1300 CE to 1600 CE.
During the telescopic era (1600 CE to present), records of astronomical observations pin down the decade fluctuations with increasing reliability. The uncertainties in ΔT are presented in Table 2 [Stephenson and Houlden, 1986].
| Table 2 - Uncertainty of ΔT (+1700 to +1900) | ||
| Year | σ (seconds) |
Longitude |
| +1700 | 5 | 0.021° |
| +1800 | 1 | 0.004° |
| +1900 | 0.1 | 0.0004° |
The estimation in the uncertainty of ΔT prior to 1000 BCE must rely on a certain amount of modeling and theoretical arguments since no measurements of ΔT are available for this period. Huber (2000) has proposed a Brownian motion model including drift to estimate the standard error in ΔT for periods outside the epoch of measured values. The intrinsic variability in the LOD (length of day) during the 2500 years of observations (500 BCE to 2000 CE) is 1.780 ms/cy with a standard error of 0.56 ms/cy. This rate is not due entirely to tidal friction but includes a drift in LOD from imperfectly understood effects such as changes in sea level due to variations in polar ice caps. Hopefully, the same mechanisms operating during the present era also operated prior to 1000 BCE as well as one millennium into the future.
Huber's derived estimate for the total standard error (fluctuations plus drift) in ΔT is as follows.
σ = 365.25 * N * SQRT [ ( N * Q / 3) * ( 1 + N / M ) ] / 1000
where:
N = Difference between target year and calibration year
M = 2500 years (-500 to +2000);
this covers period of observed ΔT measurements
Q = 0.058 ms^2/yr
The calibration year is taken as -500 for target years before 500 BCE, while the calibration year is 2005 CE for target years in the future. Evaluation of this expression at 500 year intervals is found in Table 3. It shows estimates in the standard error of of ΔT along with the equivalent shift in longitude.
The values of ΔT have been taken from Morrison and Stephenson (2004). 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.
| Table 3 - Uncertainty of ΔT (estimated) | ||
| Year | σ (seconds) |
Longitude |
| -4000 | 16291 | 67.9° |
| -3500 | 12378 | 51.6° |
| -3000 | 8978 | 37.4° |
| -2500 | 6094 | 25.4° |
| -2000 | 3732 | 15.6° |
| -1500 | 1900 | 7.9° |
| -1000 | 622 | 2.6° |
| - | - | - |
| +2500 | 612 | 2.6° |
| +3000 | 1885 | 7.9° |
| +3500 | 3711 | 15.6° |
| +4000 | 6068 | 25.3° |
| +4500 | 8946 | 37.3° |
| +5000 | 12341 | 51.4° |
References
Huber, P. J., "Modeling the Length of Day and Extrapolating the Rotation of the Earth", Astronomical Amusements, Edited by F. Bonoli, S. De Meis, < A. Panaino, Rome, (2000).
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.
Acknowledgment
This page is adapted from the the Five Millennium Canon of Solar Eclipses: -1999 to +3000 [Espenak and Meeus].