# Polynomial Expressions for Delta T

### Fred Espenak

Using ΔT values derived from the historical record and from direct observations (see: Table 1 and Table 2 ), a series of polynomial expressions have been created to simplify the evaluation of ΔT for any time during the interval -1999 to +3000.

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 both the
*Thousand Year Canon of Solar Eclipses 1501 to 2500 [Espenak]* and the
*Five Millennium Canon of Solar Eclipses: -1999 to +3000 [Espenak and Meeus]*.

Before the year -500, calculate:

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

Between years -500 and +500, we use the data from Table 1, except that for the year -500 we changed the value 17190 to 17203.7 in order to avoid a discontinuity with the previous formula at that epoch. The value for ΔT is given by a polynomial of the 6th degree, which reproduces the values in Table 1 with an error not larger than 4 seconds:

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, we again use the data from Table 1 to derive a polynomial of the 6th degree.

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

The polynomial expressions for ΔT above are from the *Five Millennium Canon of Solar Eclipses: -1999 to +3000*.

Between years 2005 and 2015, calculate:

ΔT = 64.69 + 0.2930 * t where: t = y - 2000

Between years 2015 and 3000, calculate:

ΔT = 67.62 + 0.3645 * t + 0.0039755 * t^2 where: t = y - 2015

The expression covering the period 2005 to 2015 was derived from the most recent values of ΔT.
The last expression covering the years 2015 and beyond is an extrapolation based on recent values of ΔT combined with the long term trend obtained by fitting a quadratic function to the values of ΔT from the historic records. It is an updated expression based on van der Sluys (2010).
These last two polynomial expressions are from the *Thousand Year Canon of Solar Eclipses 1501 to 2500*.

The largest deviation in the value of ΔT between the polynomial expressions above and the *historically derived values* occurs in the period 500 to 1600 and is less than 4 seconds.
This accuracy is acceptable so the polynomial expressions have been used in evaluating ΔT for all eclipses on * EclipseWise.com*.

The uncertainty in ΔT can be used to assess the standard error in values in the distant past or far future.

## References

Morrison, L. and Stephenson, F. R., "Historical Values of the Earth's Clock Error ΔT and the Calculation of Eclipses", J. Hist. Astron., Vol. 35 Part 3, August 2004, No. 120, pp 327-336 (2004).

van der Sluys, M. , *hemel.waarnemen.com/Computing/deltat.html* (2010).

## Acknowledgment

This page is adapted from: