ET = UT + ΔT

UT = ET - ΔT

Polynomial Expressions for Delta T (ΔT)

Adapted from "Five Millennium Canon of Solar Eclipses" [Espenak and Meeus]

Using the ΔT values derived from the historical record and from direct observations.

We define the decimal year "y" as follows:

y = year + (month - 0.5)/12

This gives "y" for the middle of the month, which is accurate enough given the precision in the known values of ΔT. The following polynomial expressions can be used calculate the value of ΔT (in seconds) over the time period covered by of the Five Millennium Canon of Solar Eclipses: -1999 to +3000.

Before the year -500, calculate:

ΔT = -20 + 32 * u^2
where: u = (year-1820)/100

Between years -500 and +500 calculate:

ΔT = 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
where: u = y/100

Between years +500 and +1600 calculate:

ΔT = 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
where: u = (y-1000)/100

Between years +1600 and +1700, calculate:

ΔT = 120 - 0.9808 * t - 0.01532 * t^2 + t^3 / 7129
where: t = y - 1600

Between years +1700 and +1800, calculate:

ΔT = 8.83 + 0.1603 * t - 0.0059285 * t^2 + 0.00013336 * t^3 - t^4 / 1174000
where: t = y - 1700

Between years +1800 and +1860, calculate:

ΔT = 13.72 - 0.332447 * t + 0.0068612 * t^2 + 0.0041116 * t^3 - 0.00037436 * t^4 + 0.0000121272 * t^5 - 0.0000001699 * t^6 + 0.000000000875 * t^7
where: t = y - 1800

Between years 1860 and 1900, calculate:

ΔT = 7.62 + 0.5737 * t - 0.251754 * t^2 + 0.01680668 * t^3 -0.0004473624 * t^4 + t^5 / 233174
where: t = y - 1860

Between years 1900 and 1920, calculate:

ΔT = -2.79 + 1.494119 * t - 0.0598939 * t^2 + 0.0061966 * t^3 - 0.000197 * t^4
where: t = y - 1900

Between years 1920 and 1941, calculate:

ΔT = 21.20 + 0.84493*t - 0.076100 * t^2 + 0.0020936 * t^3
where: t = y - 1920

Between years 1941 and 1961, calculate:

ΔT = 29.07 + 0.407*t - t^2/233 + t^3 / 2547
where: t = y - 1950

Between years 1961 and 1986, calculate:

ΔT = 45.45 + 1.067*t - t^2/260 - t^3 / 718
where: t = y - 1975

Between years 1986 and 2005, calculate:

ΔT = 63.86 + 0.3345 * t - 0.060374 * t^2 + 0.0017275 * t^3 + 0.000651814 * t^4 + 0.00002373599 * t^5
where: t = y - 2000

Between years 2005 and 2050, calculate:

ΔT = 62.92 + 0.32217 * t + 0.005589 * t^2
where: t = y - 2000

This expression is derived from estimated values of ΔT in the years 2010 and 2050. The value for 2010 (66.9 seconds) is based on a linearly extrapolation from 2005 using 0.39 seconds/year (average from 1995 to 2005). The value for 2050 (93 seconds) is linearly extrapolated from 2010 using 0.66 seconds/year (average rate from 1901 to 2000).

Between years 2050 and 2150, calculate:

ΔT = -20 + 32 * ((y-1820)/100)^2 - 0.5628 * (2150 - y)

The last term is introduced to eliminate the discontinuity at 2050.

After 2150, calculate:

ΔT = -20 + 32 * u^2 where: u = (year-1820)/100

All values of ΔT based on Morrison and Stephenson [2004] assume a value for the Moon's secular acceleration of -26 arcsec/cy^2. However, the ELP-2000/82 lunar ephemeris employed in the Canon uses a slightly different value of -25.858 arcsec/cy^2. Thus, a small correction "c" must be added to the values derived from the polynomial expressions for ΔT before they can be used in the Canon

c = -0.000012932 * (y - 1955)^2

Since the values of ΔT for the interval 1955 to 2005 were derived independent of any lunar ephemeris, no correction is needed for this period.

The uncertainty in ΔT over this period can be estimated from scatter in the measurements.