Delta T
The greatest uncertainty in any ephemeris concerns delta T = ET - UT, which affects primarily the moon, but also, at earlier epochs, the faster planets. Some of the formulas given for delta T are based upon uncertain assumptions, such as the reliability and even the reality of early eclipse reports, and must be used with caution. Delta T must be used for ephemeris calculation, but should always be regarded with caution, for there is no reason to place great confidence in any formula. At earlier epochs, the uncertainty of delta T increases rapidly and the differences in the formulas can amount to hours; there are differences of nearly an hour in AD 0 and differences of a few minutes in 1500 and even 1600. These differences are far greater than the uncertainties in lunar theory apart from delta T. The following table compares delta T for 1 Jan 0h ET at various epochs from -2999 to 1620, the last from tabular interpolation with only the lunar acceleration coefficient from the formula, in hours, minutes, and seconds (hh:mm:ss), although in truth before 1500 or 1600 the seconds are without significance. The sources of the formulas are:
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Tuckerman & Goldstine (1962, 1964, 1973)
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Morrison & Stephenson (1982)
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Stephenson & Morrison (1984)
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Stephenson & Houlden (1986)
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Borkowski (1988)
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Chapront-Touzé & Chapront (1991)
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Chapront, Chapront-Touzé & Francou (1997)
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JPL Horizons
Source -2999 -2000 -1000 -500 0 500 1000 1500 1600 1620 1 24:03:04 15:09:55 8:18:49 5:39:14 3:30:20 1:52:05 0:44:29 0:07:33 0:03:51 0:02:04 2 20:52:26 13:06:02 7:07:27 4:48:47 2:57:12 1:32:42 0:35:17 0:04:57 0:02:08 0:02:04 3 24:27:08 14:46:09 7:32:11 4:50:35 2:45:53 1:18:07 0:27:12 0:03:49 0:01:42 0:02:04 4 25:04:17 15:03:01 7:36:05 4:50:44 2:44:08 1:16:18 0:27:06 0:04:36 0:02:21 0:02:04 5 20:47:55 12:47:12 6:42:37 4:24:05 2:34:42 1:14:30 0:23:27 0:01:35 0:00:42 0:01:42 6 23:09:50 13:56:57 7:04:47 4:31:42 2:33:57 1:11:32 0:24:22 0:03:12 0:01:20 0:01:42 7 24:18:58 14:40:57 7:29:17 4:48:34 2:44:37 1:17:24 0:26:52 0:03:45 0:01:38 0:02:01 8 19:49:03 12:27:15 6:46:59 4:37:04 2:55:07 1:34:08 0:26:18 0:02:55 0:02:02 0:02:04 M dif. 5:15:14 2:42:40 1:36:12 1:15:09 0:56:23 0:40:33 0:21:02 0:05:58 0:03:09 0:00:22
The last row of the table shows the maximum difference, M dif., between the values in each column. Since the mean motion of the moon in longitude is about 33'/h and the mean motion of elongation from the sun about 30'/h, dividing M dif. by 2 gives the uncertainty due to delta T in the longitude or elongation of the moon in degrees, minutes, and seconds of arc: in -2999 about 2°30', in 0 about 30', in 1600 about 1' 30", and in 1620 about 11", all of these probably optimistic.
The transition from the formulas to tabular interpolation on 1 Jan 1620 produces a discontinuity that cannot possibly be correct but is a necessary consequence of the use of the formulas and the tabular interpolation. The following table shows the transition from 31 Dec 1619 to 1 Jan 1620, the differences in minutes and seconds, and the resulting discontinuity in the mean motion of the moon in seconds of arc with the mean motion taken as 33'/h:
Source 1619 1620 Diff. Moon 1 3:15 2:04 -1:11 -39" 2 1:42 2:04 +0:22 +12 3 1:23 2:04 +0:41 +23 4 1:59 2:04 +0:05 +3 5 0:40 2:04 +1:24 +46 6 1:03 2:04 +1:01 +34 7 1:20 2:04 +0:44 +24 8 1:57 2:04 +0:07 +4
The following graph shows delta T in seconds computed from the formulas and tabular interpolation from 1500 to 2050:
Notice that for all formulas except Tuckerman and Goldstine, delta T increases in 1620, after doing nothing but decrease at earlier dates, and then immediately decreases again.
It should be clear from these examples that delta T must be considered with great caution, and consequently at earlier epochs longitudes and elongations of the moon, and thus the circumstances of lunar and, more so, solar eclipses, must also be considered with great caution.
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