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

**(Dynamical Time) or**

**TD****(Terrestrial Time).**

**TT**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**

on the site of NASA: https://eclipse.gsfc.nasa.gov/LEcat5/time.html*Time*and**Espenak, Fred**:**Meeus, Jean**http://eclipse.gsfc.nasa.gov/Secat5/deltatpoly.html*Delta T: Terrestrial Time, Universal Time and Algorithms for Historical Periods*- An overview on the site of
:http://www.staff.science.uu.nl/~gent0113/deltat/deltat.htm**Robert van Gent**