Uncertainty in Delta T (ΔでるたT)

Five Millennium Canon of Lunar Eclipses [Espenak and Meeus]

The uncertainty in the value of ΔでるたT is of particular interest in the calculation of geographic regions of eclipse visibility in the distant past and future. Unfortunately, estimating the standard error in ΔでるたT prior to 1600 CE is a difficult problem. It depends on a number of factors which include the accuracy of determining ΔでるたT 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しぐま) which is valid over the period 1000 BCE to 1200 CE:

                              σしぐま = 0.8 * t^2  seconds
                                  where:  t = (year-1820)/100

Table 1 gives the errors in ΔでるたT along with the corresponding uncertainties in the longitude of the zones of eclipse visibility.

Table 1 - Uncertainty of ΔでるたT (-500 to +1200)
Year	σしぐま (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 ΔでるたT 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 ΔでるたT are presented in Table 2 [Stephenson and Houlden, 1986].

Table 2 - Uncertainty of ΔでるたT (+1700 to +1900)
Year	σしぐま (seconds)	Longitude
+1700	5	0.021°
+1800	1	0.004°
+1900	0.1	0.0004°

The estimation in the uncertainty of ΔでるたT prior to 1000 BCE must rely on a certain amount of modeling and theoretical arguments since no measurements of ΔでるたT are available for this period. Huber (2000) has proposed a Brownian motion model including drift to estimate the standard error in ΔでるたT 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 ΔでるたT is as follows.

     σしぐま = 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 ΔでるたT 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 ΔでるたT along with the equivalent shift in longitude.

The values of ΔでるたT have been taken from Morrison and Stephenson (2004). A series of polynomial expressions have been derived from these data to simplify the evaluation of ΔでるたT for any time during the interval -1999 to +3000.

Table 3 - Uncertainty of ΔでるたT (estimated)
Year	σしぐま (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°

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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 ΔでるたT 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.

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