Letter to the Editor
Torun Radio Astronomy Observatory,
Nicolaus Copernicus University,
ul. Chopina 12/18, PL87100 Torun, Poland
Received June 21, Accepted July 14, 1988
Summary. The new ephemeris of the Moon,
ELP 200085, has been used to analyse some of
more reliably documented historical eclipses
of the Sun. The conclusion is that the
presently available expressions of the
Dynamical Time – Universal Time (DT – UT)
difference are clearly inadequate for the use
with this ephemeris for historical studies.
The basic reason lies in the difference
between mean lunar longitude in the new and
older theories of the motion of the Moon. A
provisional relation between the two times
applicable presumably before about AD 1700
(back to about 2000 BC) reads in seconds of
time
DT – UT = 35.0 (t + 3.75)^{2} + 40,
where t is time in Julian centuries elapsed from the epoch 2000.0. This equation includes a magnitude of the secular deceleration of the Earth rotation of 70 s/cy^{2} or, in other units, 22 10^{–9} /cy.
Key words: Ephemerides – eclipses – Earth:rotation – time
With the appearance of the new theory of the motion of the Moon worked out in the Bureau des Longitudes (ChaprontTouzé and Chapront, 1983) recently emerged relatively simple, yet very accurate, semianalytical ephemeris of this body named ELP 200085 (ChaprontTouzé and Chapront, 1987, 1988), which seems very suitable for studies of historical observations of occultations and solar and lunar eclipses. Over a few thousands of years (since –1500) its internal accuracy is 10" or better, as compared to the JPL numerical integration LE51. Since the authors offer a ready FORTRAN coding of the ELP 200085 it may be expected that in the near future this ephemeris will become a sort of standard tool in analyses of ancient astronomical observations. Some of such analyses require the knowledge of the difference between the Dynamical Time (DT), which is the argument of the discussed ephemeris and is a direct extension of the Ephemeris Time (these two times can be considered equivalent for our purposes), and the Universal Time (UT).
The authors of ELP 200085 on passing suggested (in ChaprontTouzé and Chapront, 1988) to use the relation DT – UT derived by Morrison and Stephenson (1982), which is based on Babylonian observations of the eclipses of the Moon. I have used this relation in another study (Borkowski, 1988) and I found it to be indeed very acceptable, at least when the solar eclipses are concerned and older theories of the motion of the Moon used. However, in conjunction with the French ephemeris (ELP 200085) the suggested relation fails to give satisfactory predictions of ancient solar eclipses (and presumably also other astonomical phenomena, which I did not analyse). In fact, it should not be surprising because this ephemeris uses considerably different expression for the mean lunar longitude than does e.g. the Improved Lunar Ephemeris. In particular, it incorporates the secular tidal acceleration of the Moon of –23.895 "/cy^{2}, while more commonly used value reads –26 "/cy^{2}.
To estimate the quantity ΔT = DT – UT I have used the solar eclipses recorded in historical times between –2136 and 1715 of the common era, most of which were presented by Stephenson and Clark (1978). Firstly, I analysed in detail each eclipse with the aim to estimate a range of ΔT values that satisfy the stated condition of centrality. In two cases, concerning partial eclipses of 928 at Baghdad and –321 at Babylon, angular (altitude of the Sun) or timing information (duration of eclipse that occurred at sunset) from ancient records were used as landmarks for ΔT estimates. For this purpose I made use of the computer program described in Borkowski (1988) with original less accurate ephemerides replaced by ELP 200085 (a FORTRAN programmed version supplied by its authors which I slightly modified to fit my philosophy and to include the nutation theory) and the Stumpff's algorithm for the motion of the Earth (Stumpff, 1979, 1980; also this algorithm I extended by adding the nutation to get the apparent coordinates of the Sun). Then a least squares fit of the estimated ΔT values to the quadratic polynomial in time was performed. The usual method of fit I modified so as to carry minimization of the summed squares of deviations above or below the mentioned range of acceptable ΔT values, rather than deviations from certain point inside the range. This procedure effectively eliminates observations that are not critical for the determination of ΔT curve.
Table 1. Eclipses of the Sun used for derivation of DT – UT difference. The 'fit' denotes values obtained from Eq. (1) and 'res.' stands for the deviation outside the 'min.' (usually the second contact) to 'max.' (usually the third contact) range. The 'computed' data correspond to the 'fit' values of DT – UT. The 'observed' phases of 1 or >1 refer to a totality, while these of the form >0.9... to an annular eclipse 
_______________________________________________________________________ Date Place DT  UT [s] Eclipse phase Year M D min. max. fit res. observed computed _______________________________________________________________________ 1715 05 03 England* 23 11 69 80 1 0.997/1.003 1567 04 09 Roma 86 158 52 34 1 0.998 1560 08 21 Coimbra 523 179 55 0 >1 1.008 1485 03 16 Melk 4540 2040 108 0 >1 1.020 1415 06 07 Prague 1500 628 194 0 >1 1.016 1406 06 16 Braunschweig 198 1400 207 0 >1 1.014 1267 05 25 Constantinopole 810 736 488 0 >1 1.003 1241 10 06 Stade/Cairo(?) 521 1007 554 0 >1 1.000/1.017 1239 06 03 Southern Europe 718 1200 560 158 >1 0.9931.036 1221 05 23 Kerulen River <5400 1043 610 0 >1 1.011 1176 04 11 Antioch 348 1482 745 0 >1 1.022 1124 08 11 Novgorod 837 2571 916 0 >1 1.002 1133 08 02 Salzburg 146 1334 885 0 >1 1.025 975 08 10 Kyoto 954 4295 1516 0 >1 1.007 968 12 22 Constantinopole 1411 4680 1546 0 >1 1.001 928 08 18 Baghdad** 1357 1615 1737 122 0.218 912 06 17 Cordoba(?) 1007 2453 1817 0 >1 1.022 840 05 05 Bergamo <3780 6238 2195 0 >1 1.012 522 06 10 Nanching(?) 3046 4678 4295 0 >1 1.018 516 04 18 Nanching(?) 3146 4730 4343 0 >0.939 0.953 120 01 18 Loyang 7604 8474 7967 0 >1 1.014 65 12 16 Kuangling 8334 8838 8547 0 >1 1.010 135 04 15 Babylon 10336 11272 10878 0 >1 1.021 180 03 04 Ch'angan 10990 11932 11441 0 >1 1.020 197 08 07 Ch'angan 5650 11716 11651 0 >0.950 0.951 321 09 26 Babylon*** 13200 13322 13285 0 0.087 548 06 19 Chufu 15751 19789 16560 0 >1 1.010 600 09 20 Ying(?) 17323 18105 17357 0 >1 1.001 708 07 17 Chufu 19091 20045 19082 9 >1 0.999 1374 05 03 Ugarit 30638 31760 31512 0 >1 1.004 2136 10 22 Anyi**** 49261 50216 49528 0 >0.967 0.976 _______________________________________________________________________ 
Notes to Table 1: * — at Darrington and Lewes (the path of totality limits); ref.: Morrison et al. (1988) ** — DT – UT estimated from observed altitude of the Sun of about 12 (+/–0.5) deg; computed value is 11.0 deg. *** — DT – UT estimated from observed duration of this eclipse of about 12 (+/–1) min. (till sunset); computed value is 12.5 min. **** — Ref.: Wang and Siscoe (1980) and P.K. Wang (1988, private communication); assumed geographical coordinates were 35.1 (latitude) and 111.2 deg (longitude) ? — uncertain location 
As a result of the above described fit the following expression for DT (in seconds) has been established:
DT – UT = 35.0 (t + 3.75)^{2} + 40,  (1) 
Fig.1. Plot of allowed ranges of the DT – UT differences for eclipses of Table 1 relative to the mean ΔT curve [Eq. (1)] represented here by the abscissa axis. The abscissa axis is scaled every 100 years from –2200 to 1800, and the ordinate axis is marked every 500 s between the range of +/–2000 s. Each vertical bar is halved by a short horizontal bar to mark the centre of the allowed range. The broken curve shows the ΔT of Morrison and Stephenson (1982)

L.V. Morrison (1988, private communication) criticized my use of rather unreliable ancient (6th century and earlier) records of total solar eclipses. The reader will observe, however, that of them only the eclipse of –708 (Chufu) contributed to the solution described by Eq. (1), the remaining 12 eclipses happened not to be critical.
The ephemeris ELP 200085 can be modified to account for other adopted values of the lunar tidal acceleration. For comparison purposes I have used the lunar arguments listed in Table 8 (plus adequately changed expression for their L) of ChaprontTouzé and Chapront (1988) as the replacement of those originally included in the FORTRAN code of the ephemeris. The modification incorporates the tidal acceleration of –26.305 "/cy^{2} and makes the ephemeris to closely agree with the LE51. Then, I have computed the circumstances of all the eclipses listed in Table 1 using different expressions for the ΔT. The expression of Morrison and Stephenson (1982) gave satisfactory predictions only back to 912 (excluding the Baghdad eclipse). Similarly, unsuitable with the modified lunar ephemeris appears Eq. (1) above. In contrast, a 'twoacceleration' model of Stephenson and Morrison (1984) performed much better in this respect. It only failed to predict the 'observed' (Table 1) phases at Kuangling, Ugarit and Anyi, which are considered to be less reliable.
Acknowledgements. I would like to thank the
authors of the ELP 200085 for a copy of their
program and associated printed materials,
which made this work possible. I am also
grateful to Dr Dr L.V. Morrison and
M. ChaprontTouzé for helpful criticisms of, and
comments on an earlier version of this letter.
Prof. S. Gorgolewski kindly corrected my
English.
REFERENCES:
Borkowski K.M., 1988, in preparation
[Post. Astronaut., 22 (1989), 99 – 130]
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ChaprontTouzé, M., Chapront, J., 1987, Notes Scientifiques et Techniques du Bureau des Longitudes S021, Paris
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Li Zhisen, Yang Xihong, 1985, Scientia Sinica, Ser. A 28, 1299
Morrison, L.V., Stephenson, F.R., 1982, in Sun and Planetary System (Astrophys. Space Sci. Lib. 96), eds. W. Fricke, G. Teleki, Reidel, Dordrecht, p. 173
Morrison, L.V., Stephenson, F.R., Parkinson, J., 1988, Nature 331, 421
Stephenson, F.R., Clark, D.H., 1978, Applications of Early Astronomical Records, Adam Hilger, Bristol
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Wang, P.K., Siscoe, G.L., 1980, Solar Phys. 66, 187