[say something on the determination of ΔT from historically dated eclipses]
Until recently, the most complete analysis of ΔT from all available observations of solar eclipses, timed lunar eclipses and other data was published by Stephenson & Morrison (1995) and in Stephenson (1997).
Year | ΔT | Year | ΔT | Year | ΔT | Year | ΔT | |||
–500 | +16800 | 50 | +10100 | 600 | +4700 | 1150 | +900 | |||
–450 | +16000 | 100 | +9600 | 650 | +4300 | 1200 | +750 | |||
–400 | +15300 | 150 | +9100 | 700 | +3800 | 1250 | +600 | |||
–350 | +14600 | 200 | +8600 | 750 | +3400 | 1300 | +470 | |||
–300 | +14000 | 250 | +8200 | 800 | +3000 | 1350 | +380 | |||
–250 | +13400 | 300 | +7700 | 850 | +2600 | 1400 | +300 | |||
–200 | +12800 | 350 | +7200 | 900 | +2200 | 1450 | +230 | |||
–150 | +12200 | 400 | +6700 | 950 | +1900 | 1500 | +180 | |||
–100 | +11600 | 450 | +6200 | 1000 | +1600 | 1550 | +140 | |||
–50 | +11100 | 500 | +5700 | 1050 | +1350 | 1600 | +110 | |||
0 | +10600 | 550 | +5200 | 1100 | +1100 |
In this analysis the tidal acceleration parameter was assumed to be –26.0 "/cy/cy.
These values are used in Shinobu Takesako’s Windows 95 program EmapWin for plotting the circumstances of solar eclipses from 3000 BC to AD 3000 and in Kerry Shetline’s online planetarium program Sky View Café.
An analysis of Stephenson et al. (1997) of the solar eclipse of 9 April 1567 observed by Christopher Clavius in Rome suggests that the 16th-century values for ΔT may be somewhat higher. This would also remove the apparent jump in ΔT around 1600 when compared with the modern data.
ΔT(s) = –745 + 16.18 u + 28.863 u^{2} |
with u = (year – 2000)/100.
A reanalysis of the available data by Morrison & Stephenson (2004, 2005) resulted in the following recommended values:
Year | ΔT | σ | Year | ΔT | σ | Year | ΔT | σ | ||
–1000 | +25400* | 640 | 0 | +10580 | 260 | 1000 | +1570 | 55 | ||
–900 | +23700* | 590 | 100 | +9600 | 240 | 1100 | +1090 | 40 | ||
–800 | +22000* | 550 | 200 | +8640 | 210 | 1200 | +740 | 30 | ||
–700 | +20400 | 500 | 300 | +7680 | 180 | 1300 | +490 | 20 | ||
–600 | +18800 | 460 | 400 | +6700 | 160 | 1400 | +320 | 20 | ||
–500 | +17190 | 430 | 500 | +5710 | 140 | 1500 | +200 | 20 | ||
–400 | +15530 | 390 | 600 | +4740 | 120 | 1600 | +120 | 20 | ||
–300 | +14080 | 360 | 700 | +3810 | 100 | 1700 | +9 | 5 | ||
–200 | +12790 | 330 | 800 | +2960 | 80 | |||||
–100 | +11640 | 290 | 900 | +2200 | 70 |
Values before –700 (labelled *) were obtained from the empirical relation ΔT(s) = –20 + 32 u^{2}, with u = (year – 1820)/100. The tidal acceleration parameter was assumed to be –26.0 "/cy/cy.
Between –1000 and 1200 the standard error (σ) in the value of ΔT was estimated from the relation: σ(ΔT) = 0.8 u^{2}.
During the past decades several semi-empirical analytical relations have been suggested in the literature as an aid for predicting past and future values for ΔT. When the tidal acceleration parameter is assumed to be constant in time, this results in a parabolic relation for ΔT as function of time (u), or:
ΔT = a + b u + c u^{2} |
where a, b and c are constants that can be obtained from historical observations of solar and lunar eclipse timings and other data. The origin of u is often chosen in such a way that the linear term vanishes (b = 0).
In September 1952, the eighth General Assembly of the International Astronomical Union in Rome adopted the following formula:
ΔT (s) = 24.349 + 72.318 u + 29.950 u^{2} + small fluctuations |
with u = (year – 1900)/100, or the time measured in centuries since 1900.
This formula was based on a study of the post-1650 observations of the Sun, the Moon and the planets by Spencer Jones (1939).
This single-parabolic relation (the influence of the “small fluctuations” was assumed to be negligible in the historical past) was used by Meeus in his Astronomical Formulæ for Calculators (1979, 1982) and in the lunar and solar eclipse tables of Mucke & Meeus (198?) and Meeus & Mucke (198?). It was also adopted in the PC program SunTracker Pro (Zephyr Services, 19??).
In 1960, a slightly modified version of the above relation was adopted in the Astronomical Ephemeris:
ΔT (s) = 24.349 + 72.3165 u + 29.949 u^{2} + small fluctuations |
with u = (year – 1900)/100, or the time measured in centuries since 1900.
The tables of Tuckerman (1962, 1964) list the positions of the Sun, the Moon and the planets at 5- and 10-day intervals from 601 BCE to 1649 CE. The listed positions are for 19h 00m (mean) local time at Babylon/Baghdad (i.e. near sunset) or 16h 00m GMT. From the difference in the adopted solar theory (Leverrier, 1857) with that of Newcomb (1895), Stephenson & Houlden (1981) and Houlden & Stephenson (1986) derived the following ΔT relation that is implicitly used in the Tuckerman tables:
ΔT (s) = 4.87 + 35.06 u + 36.79 u^{2} |
with u = (year – 1900)/100, or the time measured in centuries since 1900.
The same relation was also implicitly adopted in the syzygy tables of Goldstine (1973).
Single-parabolic fit by P.M. Muller & F.R. Stephenson (1975).
ΔT (s) = +66.0 + 120.38 u + 45.78 u^{2} |
with u = (year – 1900)/100, or the time measured in centuries since 1900.
Based on n = –37.5 "/cy/cy. Cf. also Stephenson & Clark (1978), chapter 2.
Single-parabolic fit by F.R. Stephenson (1978).
ΔT (s) = +20 + 114 u + 38.30 u^{2} |
with u = (year – 1900)/100, or the time measured in centuries since 1900.
Based on n = –30.0 "/cy/cy.
Single-parabolic fit by L.V. Morrison & F.R. Stephenson.
ΔT (s) = –15 + 32.5 u^{2} |
with u = (year – 1810)/100, or the time measured in centuries since 1810.
This relation was adopted in Bretagnon & Simon’s Planetary Programs and Tables from –4000 to +2800 (1986) and in the PC planetarium program RedShift (Maris Multimedia, 1994-2000).
Double-parabolic fit by F.R. Stephenson & L.V. Morrison.
ΔT (s) = 1360 + 320 u + 44.3 u^{2} | (–391 < year < 948 CE) |
and
ΔT (s) = 25.5 u^{2} | (948 CE < year < 1600 CE) |
with u = (year – 1800)/100, or the time measured in centuries since 1800.
Double-parabolic fit by F.R. Stephenson & M.A. Houlden,
ΔT (s) = 1830 – 405 u + 46.5 u^{2} | (year < 948 CE) |
with u = (year – 948)/100, or the time measured in centuries since 948 CE, and
ΔT (s) = 22.5 u^{2} | (948 CE < year < 1600 CE) |
with u = (year – 1850)/100, or the time measured in centuries since 1850.
These relations are used in the PC planetarium program Guide 7 (Project Pluto, 1999).
The following single-parabolic relation closely approximates the ΔT values given by Fred Espenak in his Fifty Year Canon of Solar Eclipses: 1986 – 2035 (1987) and in his Fifty Year Canon of Lunar Eclipses: 1986 – 2035 (1989).
ΔT (s) = 67 + 61 u + 64.3 u^{2} |
with u = (year – 2000)/100, or the time measured in centuries since 2000.
This relation should not be used before around 1950 or after around 2100 (Espenak, pers. comm.).
The following single-parabolic fit was obtained by K.M. Borkowski from an analysis of 31 solar eclipse records dating between 2137 BCE to 1715 CE:
ΔT (s) = 40 + 35.0 u^{2} |
with u = (year – 1625)/100, or the time measured in centuries since 1625.
The solar eclipse records were compared with the ELP 2000-85 lunar theory of Chapront-Touzé & Chapront (1988) which adopts a tidal acceleration parameter of –23.8946 "/cy/cy.
Note that Borkowski’s ΔT relation is strongly biased by the inclusion of speculative ΔT values inferred from two very early but highly doubtful solar eclipse records: the so-called Ugarit eclipse (dated to 3 May 1375 BCE by Borkowski) and the legendary Chinese eclipse of Xi-Ho (22 October 2137 BCE) mentioned in the Shu Jing.
The following double-parabolic fit was adopted by Michelle Chapront-Touzé & Jean Chapront in the shortened version of the ELP 2000-85 lunar theory in their Lunar Tables and Programs from 4000 B.C. to A.D. 8000 (1991):
ΔT (s) = 2177 + 495 u + 42.4 u^{2} | (–391 < year < 948 CE) |
and
ΔT (s) = 102 + 100 u + 23.6 u^{2} | (+948 < year < +1600) |
with u = (year – 2000)/100, or the time measured in centuries since 2000.
The relations are based on those of Stephenson & Morrison (1984), but slightly modified to make them compatible with the tidal acceleration parameter of –23.8946 "/cy/cy adopted in the ELP 2000-85 lunar theory.
Six years later, Jean Chapront, Michelle Chapront-Touzé & G. Francou published an improved set of orbital constants for the ELP 2000-85 lunar theory in which they adopted a revised lunar acceleration parameter of –25.7376 "/cy/cy to obtain a close fit the JPL DE403 theory of the planets.
ΔT (s) = 2177 + 497 u + 44.1 u^{2} | (year < +948) |
and
ΔT (s) = 102 + 102 u + 25.3 u^{2} | (+948 < year < +1600 or year > 2000) |
with u = (year – 2000)/100, or the time measured in centuries since 2000.
These relations are also recommended by Jean Meeus in the second edition of his Astronomical Algorithms (1998), but in order to avoid a discontinuity of about 37 seconds with the observed values around the year 2000, he suggests adding the linear term to the latter relation:
+0.37 (year – 2100) | (+2000 < year < +2100) |
These relations are used in Shinobu Takesako’s EmapWin program for plotting the circumstances of solar eclipses from 3000 B.C. to A.D. 3000 and in Kerry Shetline’s interactive planetarium Sky View Café.
The JPL Solar System Dynamics Group of the NASA Jet Propulsion Laboratory (California Institute of Technology) supports an interactive website JPL Horizons for calculating high-precision positions of the solar system bodies from the most recent and accurate algorithms. For dates before 1620, the JPL Horizons website uses the following ΔT relations:
ΔT (s) = 31.0 u^{2} | (–2999 < year < +948) |
with u = (year – 1820)/100, or the time measured in centuries since 1820, and
ΔT (s) = 50.6 + 67.5 u +22.5 u^{2} | (+948 < year < +1620) |
with u = (year – 2000)/100, or the time measured in centuries since 2000. The source of the pre-948 relation is unclear, the post-948 relation was taken from Stephenson & Houlden (1986).
Note that their polynomial relations imply a 526.6-second jump in ΔT around 948 CE.
In 2006, Fred Espenak and Jean Meeus recommended the following polynomial relations.
Period | ΔT (s) | u = |
before –500 | –20 + 32 u^{2} | (year – 1820)/100 |
–500 to 500 | +10583.6 – 1014.41 u + 33.78311 u^{2}
– 5.952053 u^{3}
– 0.1798452 u^{4} + 0.022174192 u^{5} + 0.0090316521 u^{6} |
year/100 |
500 to 1600 | +1574.2 – 556.01 u + 71.23472 u^{2}
+ 0.319781 u^{3}
– 0.8503463 u^{4} – 0.005050998 u^{5} + 0.0083572073 u^{6} |
(year – 1000)/100 |
1600 to 1700 | +120 – 98.08 u – 153.2 u^{2} + u^{3}/0.007129 | (year – 1600)/100 |
1700 to 1800 | +8.83 + 16.03 u – 59.285 u^{2} + 133.36 u^{3} – u^{4} /0.01174 | (year – 1700)/100 |
1800 to 1860 | +13.72 – 33.2447 u + 68.612 u^{2}
+ 4111.6 u^{3}
– 37436 u^{4}
+ 121272 u^{5} – 169900 u^{6} + 87500 u^{7} |
(year – 1800)/100 |
1860 to 1900 | +7.62 + 57.37 u – 2517.54 u^{2}
+ 16806.68 u^{3}
– 44736.24 u^{4}
+ u^{5}/0.0000233174 |
(year – 1860)/100 |
1900 to 1920 | –2.79 + 149.4119 u – 598.939 u^{2} + 6196.6 u^{3} – 19700 u^{4} | (year – 1900)/100 |
1920 to 1941 | +21.20 + 84.493 u – 761.00 u^{2} + 2093.6 u^{3} | (year – 1920)/100 |
1941 to 1961 | +29.07 + 40.7 u – u^{2}/0.0233 + u^{3}/0.002547 | (year – 1950)/100 |
1961 to 1986 | +45.45 + 106.7 u – u^{2}/0.026 – u^{3}/0.000718 | (year – 1975)/100 |
1986 to 2005 | +63.86 + 33.45 u – 603.74 u^{2}
+ 1727.5 u^{3} + 65181.4 u^{4}
+ 237359.9 u^{5} |
(year – 2000)/100 |
2005 to 2050 | +62.92 + 32.217 u + 55.89 u^{2} | (year – 2000)/100 |
2050 to 2150 | –205.72 + 56.28 u + 32 u^{2} | (year – 1820)/100 |
after 2150 | –20 + 32 u^{2} | (year – 1820)/100 |
These polynomial relations were developed for the following publications:
These relations are also adopted in the solar, lunar and planetary ephemeris program SOLEX, developed and maintained by Aldo Vitagliano.