## Delta T

In order to calculate for a given place on earth the precise circumstances of an eclipse, it is necessary to know the difference between "Terrestrial Time" (TT), formerly "Ephemeris Time" (ET), and "Universal Time" (UT).

This difference TT - UT = DeltaT.

Especially in the case of solar eclipses a precise knowledge of deltaT is indispensable, since deltaT determines a.o. the times of contact and the magnitude of the eclipse at a given place.

For the calculation of solar eclipses from the 17th century on to the present time precise results can be obtained since for that period deltaT is known sufficiently accurately from telescopic observations.

For investigations on older eclipses things are quite different.

Up till around 1985, deltaT was calculated using the following formula:

(1)      deltaT(seconds) = 24.349 + 72.318 T + 29.950 T^2 + small fluctuations
(T = centuries since 1900.0)

This formula was derived from observations since AD 1650 by Spencer JONES (1939) and Gerald M.CLEMENCE (1948), officially accepted by the International Astronomical Union in 1952 and slightly corrected in 1960.

Since 1980 new formulae were proposed a.o. by STEPHENSON & MORRISON (1984) and STEPHENSON & HOULDEN (1986). These were the result of additional research on observations of old solar eclipses in China and the Arab world.

The following formulae were proposed:

(2)      period up till AD 948 (STEPHENSON & HOULDEN, 1986)

deltaT(seconds) = 1830 - 405 E + 46. 5 E^2
(E = centuries since 948 AD)

deltaT(seconds) = 22. 5 t^2
(t = centuries since 1850 AD)

In his recent book "Historical Eclipses and Earth's Rotation" (1997) STEPHENSON presents a new analysis of most if not all known solar and lunar eclipses that occurred during the period -700 to +1600. As a result he presents the following "new" values for deltaT (given here per century):

Table (4)
 year dT(sec) -500 16800 -400 15300 -300 14000 -200 12800 -100 11600 0 10600 +100 9600 +200 8600 +300 7700 +400 6700 +500 5700 +600 4700 +700 3800 +800 3000 +900 2200 +1000 1600 +1100 1100 +1200 750 +1300 470 +1400 300 +1500 180 +1600 110

If we calculate the value for deltaT using the 3 given formulas and compare the results with those presented by Stephenson (1997), we obtain the following table for the period -2000 to +1700:

Year Form(1)
Jones
Form(2)
St&H(86)
Form(3)
St&H(86)
Table(4)
Steph(97)
-2000 42757 54181 - -
-190040524 51081- -
-1800 38350 48073- -
-170036236 45159- -
-1600 34181 42338 - -
-1500 32187 39610 - -
-1400 30253 36975 - -
-130028378 34433 - -
-1200 26564 31984- -
-1100 24809 29627 - -
-1000 23115 27364 - -
-900 21480 25194 - -
-800 19905 23117 - -
-700 18390 21133 - -
-600 16935 19242 - -
-500 15539 17444 - 16800
-400 14204 15738- 15300
-300 12929 14126 - 14000
-200 11713 12607 - 12800
-100 10557 11181 - 11600
09462 9848 -10600
100 8426 8608 - 9600
200 74507461 - 8600
300 6534 6406 -7700
400 5678 5445 -6700
500 4882 4577 - 5700
600 4145 3802 -4700
700 3469 3120 - 3800
800 2852 2531 - 3000
900 2296 2035 -2200
1000 1799-1625 1600
1100 1362 -1265 1100
1200 985 - 950 750
1300 668 - 680 470
1400 411- 455 300
1500 214 - 275 180
1600 76 - 140110
1700 -1-- -
1800 -19 -- -
1900 24- - -
2000 126 - - -

For the period AD 1700 to AD 2000 the observed values are:
+1700     +9 seconds
+1800     +13.7 sec
+1900     -2.7 sec
+2000     +64 sec

It is important to pay attention to the differences in deltaT, depending on the author especially when studying old eclipses, say before AD 1000.
In AD 1500 there is a difference of 95 seconds between the values given by Stephenson & Houlden in 1986 and those by Stephenson in 1997;
In AD 1000 the difference is only 25 seconds between the values given by Stephenson & Houlden and Stephenson, but there is a difference of almost 200 seconds with the Jones' values;
In AD 500 the difference between the Stephensons & Houlden (1986) and Stephenson (1997) values amount to 1123 seconds, while the difference with Jones' values reach 818 seconds;
For the earlier centuries the differences are rapidly increasing to exceed 11400 seconds (this is more than 3 hours !!) around 2000 BC.

When studying ancient eclipses it is important to keep these uncertainties in mind.

Errors of a few minutes will probable hardly affect our ideas of the perception of ancient observers, except perhaps in the case of total or near total solar eclipses that might become near total or total respectively, or in the case of eclipses starting or ending at dawn or sundown.
For the more remote past, when uncertainties in delta T become increasingly important, precise calculations for a given place tend to become very problematic indeed.

Felix Verbelen

Bibliography
• DANJON A. - Astronomie Générale (Blanchart, 1980)
• STEPHENSON F.R and HOULDEN M.A. - Atlas of Historical Eclipse Maps - Cambridge Univ.Press. (1986)
• Explanatory Supplement to the Astronomical Almanac - University Science Books (1992)
• STEPHENSON F.R - Historical Eclipses and Earth's Rotation - Cambridge Univ.Press. (1997)
Homepage