Factor t and delta T

If you know the Julian Day (see:  Julian day and Julian day number) it is easy to calculate factor t. This is a representation of a century related to the date of January 1, 2000. We simplify this century to 36525 days. This does not affect accuracy as all calculations are based on this value.

For the calculation of observations at the earth, the standard value for t will suffice. It will do for houses, positions and the horizon etcetera.

If you calculate phaenomena outside the earth you will need to make a correction for factor t.

The earth is not rotating with a constant speed. The speed diminishes slowly. The change is irregularly and therefore difficult to predict. UT takes this into account because it is based on the rotation of the earth. For the calculation of houses you will use the standard UT.

But for the calculation of planetary positions you will need a more constant time. This time is called  TDT (Terrestrial Dynamical Time), also just TD (Dynamical Time) or TT (Terrestrial Time).

The difference between TD and UT is delta T.

The correction for delta T is small: currently about 1 minute. But for historical periods the value for delta T can easily add up to more than an hour.

You will need TD, and therefore delta T, only for the calculation of planetary positions and in principle for the calculation of the obliquity of the earth (the angle between ecliptic and equator). But in the last case the difference will be not noticable and you can base your calculations on factor t as calculated for the standard UT.

Formulas

Calculation of factor t :

Calculate the Julian Day JD (see Julian day and Julian day number) and then use the formula

t = (JD – 2451545) / 36525

Delta T : Only if you want to calculate factor t based on TD you will have to apply a correction to UT first.

TD = UT + delta T

Now calculate the Julian Day but based on date and time in TD, then calculate t.

Delta T is difficult to predict. We de have formulas but these are only valid for a limited period. In most cases you will not need these values. If you calculate planetary positions based on the Swiss Ephemeris you can just use UT and the Swiss Ephemeris will apply any correction for delta T automatically. And for the calculation of the obliquity you can ignore the difference as it is much too small.

The following formulas are based on Meeus and Espenak:

Attention: in the formulas for periods from 1600 you will find the variable t again, this is also part of the century but you do have to use the value for t as defined in the formula hereafter, and not t as described earlier.

In the formulas ‘^’ means ‘to the power’, so u^2 is the square of u .

Calculate the year y and round this to a full month.

y = year + (month – 0.5)/12

Before -500:

u = (y-1820)/100

deltaT = -20 + 32 * u^2

From -500 until +500:

u = y/100

deltaT = 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

From +500 until + 1600:

u = (y-1000)/100

deltaT = 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

From +1600 until +1700:

t = y – 1600

deltaT = 120 – 0.9808 * t – 0.01532 * t^2 + t^3 / 7129

From +1700 until +1800:

t = y – 1700

deltaT = 8.83 + 0.1603 * t – 0.0059285 * t^2 + 0.00013336 * t^3 – t^4 / 1174000

From +1800 until +1860:

t = y – 1800

deltaT = 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

From +1860 until +1900:

t = y – 1860

deltaT = 7.62 + 0.5737 * t – 0.251754 * t^2 + 0.01680668 * t^3 -0.0004473624 * t^4 + t^5 / 233174

From +1900 until +1920:

t = y – 1900

deltaT = -2.79 + 1.494119 * t – 0.0598939 * t^2 + 0.0061966 * t^3 – 0.000197 * t^4

From +1920 until +1941:

t = y – 1920

deltaT = 21.20 + 0.84493*t – 0.076100 * t^2 + 0.0020936 * t^3

From +1941 until +1961:

t = y – 1950

deltaT = 29.07 + 0.407*t – t^2/233 + t^3 / 2547

From +1961 until +1986:

t = y – 1975

deltaT = 45.45 + 1.067*t – t^2/260 – t^3 / 718

From +1986 until +2005:

t = y – 1975

deltaT = 63.86 + 0.3345 * t – 0.060374 * t^2 + 0.0017275 * t^3 + 0.000651814 * t^4 + 0.00002373599 * t^5

From +2005 until +2050:

t = y – 2000

deltaT = 62.92 + 0.32217 * t + 0.005589 * t^2

Van +2050 tot +2150:

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

After +2150:

u = (y-1820)/100

deltaT = -20 + 32 * u^2

Example calculations

Calculation of factor t.

As example we will use the date of November 2, 2016 (Gregorian Calendar) 21:17:30 UT. Calculate the Julian Day, you can check the calculation at the page about JD.

The calculated JD is  2457695.387152778

Use this value in the formula for factor t

t = JD – 2451545 / 36525

t = (2457695.387152778 – 2451545) / 36525

t = 6150.387152778 / 36525

t = 0.168388423

Calculation of Delta T for November 2, 2016:

2016 is beween 2005 and 2050, the formula will be:

t = y – 2000

deltaT = 62.92 + 0.32217 * t + 0.005589 * t^2

t = 2016 – 2000 = 16

deltaT = 62.92 + 0.32217 * t + 0.005589 * t^2

deltaT = 62.92 + 0.32217 * 16 + 0.005589 * 16^2

deltaT = 62.92 + 5.15472 + 1.430784

deltaT = 69.505504 seconds

Calculate TD :

TD = UT + deltaT

so TD = 21:17:30 + 69.5 seconds = 21:18:39.5

References