Astron. Astrophys. 205, L8-L10 (1988)

Letter to the Editor

ELP 2000-85 and the Dynamical Time – Universal Time relation

K.M. Borkowski

Torun Radio Astronomy Observatory, Nicolaus Copernicus University,
ul. Chopina 12/18, PL-87-100 Torun, Poland

Received June 21, Accepted July 14, 1988


Summary. The new ephemeris of the Moon, ELP 2000-85, 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/cy2 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 (Chapront-Touzé and Chapront, 1983) recently emerged relatively simple, yet very accurate, semianalytical ephemeris of this body named ELP 2000-85 (Chapront-Touzé 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 2000-85 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 2000-85 on passing suggested (in Chapront-Touzé 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 2000-85) 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 "/cy2, while more commonly used value reads –26 "/cy2.

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 2000-85 (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   0.993-1.036
 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 Nan-ching(?)     3046  4678  4295    0     >1      1.018
  516 04 18 Nan-ching(?)     3146  4730  4343    0    >0.939   0.953
  120 01 18 Lo-yang          7604  8474  7967    0     >1      1.014
   65 12 16 Kuang-ling       8334  8838  8547    0     >1      1.010
 -135 04 15 Babylon         10336 11272 10878    0     >1      1.021
 -180 03 04 Ch'ang-an       10990 11932 11441    0     >1      1.020
 -197 08 07 Ch'ang-an        5650 11716 11651    0    >0.950   0.951
 -321 09 26 Babylon***      13200 13322 13285    0             0.087
 -548 06 19 Chu-fu          15751 19789 16560    0     >1      1.010
 -600 09 20 Ying(?)         17323 18105 17357    0     >1      1.001
 -708 07 17 Chu-fu          19091 20045 19082    9     >1      0.999 
-1374 05 03 Ugarit          30638 31760 31512    0     >1      1.004 
-2136 10 22 An-yi****       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)
where t is time measured in Julian centuries since 2000.0. Table 1 and Fig. 1 contain details of the analysed data and final residuals. Eq. (1) differs from that of Morrison and Stephenson (1982) by varying amount and up to about 25 minutes of time in the considered historical period (the maximum is reached near 700 BC). Though, due to limited observational material, I am not in position to claim this equation to be definite, I believe it is good to a few
 
ELP-F.gif  

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)

minutes over the entire period and thus may prove useful for researchers who need such relation immediately. The numerical coefficient of 35.0 s/cy2 in the formula for ΔT is interpreted as one half of the secular deceleration of the Earth rotation. It will be noted that our value of 70 s/cy2 or, in relative angular units, 22.10–9/cy does not differ much from other authors' determinations which range from about 60 to more than 90 s/cy2. In fact, it is enveloped by two recent results of 65 and 72.82 s/cy2 (Morrison and Stephenson, 1982, and Li Zhisen and Yang Xihong, 1985). It is difficult to assess accuracy to our result but the two following observations may be of some use. If the first eclipse listed in Table 1 is omitted in the analysis then the deceleration rises to 72.8 s/cy2. Judging by the formal standard deviations of the modified least squares fit alone one arrives at the uncertainty of about 1.4 s/cy2, which certainly is still too optimistic in view of the assumed modified definition of residua. In any case, the near future requires further work to be done encompassing also other historical eclipses (solar and lunar) and occultations of stars.

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 (Chu-fu) contributed to the solution described by Eq. (1), the remaining 12 eclipses happened not to be critical.

The ephemeris ELP 2000-85 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 Chapront-Touzé 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 "/cy2 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 'two-acceleration' model of Stephenson and Morrison (1984) performed much better in this respect. It only failed to predict the 'observed' (Table 1) phases at Kuang-ling, Ugarit and An-yi, which are considered to be less reliable.


Acknowledgements. I would like to thank the authors of the ELP 2000-85 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. Chapront-Touzé 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]

Chapront-Touzé, M., Chapront, J., 1983, Astron. Astroph. 124, 50

Chapront-Touzé, M., Chapront, J., 1987, Notes Scientifiques et Techniques du Bureau des Longitudes S021, Paris

Chapront-Touzé, M., Chapront, J., 1988, Astron. Astroph. 190, 342

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

Stephenson, F.R., Morrison, L.V., 1984, Phil. Trans. R. Soc. London A 313, 47

Stumpff, P., 1979, Astron. Astrophys. 78, 229

Stumpff, P., 1980, Astron. Astrophys. Suppl. Ser. 41, 1

Wang, P.K., Siscoe, G.L., 1980, Solar Phys. 66, 187


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