Calculation of Terrestrial and Universal time scale difference, ΔT

$$\Delta$$T is defined as the 'measure of the cumulative effect of the departure of the Earth's rotation period from the fixed-length day of atomic time'.

I tried to calculate $$\Delta$$T corresponding to October 17, 2003 05:30:30 UT. This calculation of $$\Delta$$T is achieved through the following equations.

1.) UTC = UT1 - DUT1.

We have UT1 = October 17, 2003 05:30:30 UT.

From IERS Bulletin D (No. 89), DUT1 from April 3, 2003 0 hour UTC up to further notice is -0.4 seconds. Bulletin D No. 90 was issued on March 24, 2004. Therefore, DUT1 = -0.4 seconds.

UTC = UT1 - (-0.4 seconds) = UT1 + 0.4 seconds. This provides UTC = October 17, 2003 05:30:30.4

2.) TAI = UTC + Delta AT.

'$$\Delta$$AT' from 1999-01-01 to 2005-12-31 was +32.0 seconds. This provides TAI = October 17, 2003 05:31:2.4

3.) TT = TAI + 32.184 seconds.

This provides TT = October 17, 2003 05:31:34.584

4.) Delta T = TT - UT1

This provides $$\Delta$$T = 00:01:4.584 = 64.584 seconds.

Thus, the computed value of '$$\Delta$$T' for October 17, 2003 05:30:30 UT is 64.584 seconds.

This calculated value of '$$\Delta$$T' needs to be compared with the literature.

1.) From 2003-1-1 to 2003-12-1, the website 'http://maia.usno.navy.mil/ser7/deltat.data' provide 64.4734 to 64.5654 seconds.

2.) From 2003-1-1 to 2004-01-01, the website 'https://www.stjarnhimlen.se/comp/time.html' provide +64.47 to +64.57 seconds.

Would someone help to understand the accuracy of the calculated 'Delta T' against the numbers provided on the websites.

• T can be temperature, period, time, kinetic energy ... Always name your symbols. They don't necessarily self-explain. Also put your problem into context what you try to do. Here it obvously is some time (difference). But what is the actual problem you try to solve? Apr 9, 2022 at 10:01
• I have done the edit, and changed Delta T to $\Delta$T. I may take some time to incorporate the same to the equations as well. Thank you. Apr 9, 2022 at 10:28
• @planetmaker It was fairly clear even without the edit that the OP was asking about time, in particular, the difference between Terrestrial Time (TT) and Universal Time (UT1). This time difference is called $\Delta \text T$. Link: en.wikipedia.org/wiki/ΔT_(timekeeping). Apr 9, 2022 at 12:00
• Universal Time comes in a number of flavors. Coordinated Universal Time (UTC; the acronym doesn't work in any language) is TAI plus or minus an integral number of leap seconds. The intent of leap seconds is to keep UTC more or less in sync with the Earth's rotation (UT1). The Earth's rotation is measured by UT0, UT1, the now defunct UT2, and multiple other convoluted measures. Timekeeping is a mess. A rather hot mess. Apr 9, 2022 at 12:18
• @PM2Ring I changed the $\Delta T$ in the title to ΔT (plain text). The reason I did this was we have been advised to not use MathJax in titles. Apr 9, 2022 at 15:59

IERS Bulletin D reports DUT1 to the nearest tenth of a second. Do not use Bulletin D if you want finer values of DUT1. DUT1 is now estimated with an accuracy in the tens of microseconds. Be very careful regarding mixing and matching across various IERS products. The one exception regarding mixing and matching is DUT1 across leap second boundaries.

I recommend you use the EOP 14 C04 (IAU2000A) series and interpolate, taking care when interpolating across leap second boundaries. Also don't go overboard on the interpolating polynomial. A cubic interpolating polynomial is probably good enough and will avoid ringing due to Runge's phenomenon. Even a simple linear interpolation might be good enough, depending on your needs.

Question: "Would someone help to understand the accuracy of the calculated 'Delta T' against the numbers provided on the websites."

The nature of Delta T (ΔT) reflects an irregularly-fluctuating natural phenomenon: it is a quantity that fundamentally has to be observed and measured, rather than calculated.

Evidence suggesting that mean solar time is after all somewhat irregular -- and not uniform as had been long believed -- began to be accumulated from about the mid-19th century onwards. During the 1920s and 1930s the evidence for this irregularity came to be generally accepted as convincing. The first timescale independent of the fluctuations of mean solar time ("Ephemeris Time") was devised in the late 1940s. The Wikipedia article on Ephemeris Time gave (as of late 2015) a reasonably good account of the history. Some of the linked articles are also useful, including the article on Delta T, but the current version of the article on Delta T has unfortunately been amended to include a misleading suggestion that Delta T is a product of calculation -- some older versions, still accessible through the history tab, were more accurate.

Since the fundamental values of Delta T are products of observation, of course these intrinsically belong to the past -- even if sometimes a very recent past. The historical values of Delta T vary in accuracy, from very high accuracy for the past few decades, maybe to 0.0001 second as at the US Naval Observatory website, to much lower accuracies for centuries-old dates, for which Delta T had to be estimated (recently) from a variety of old historical observations. (See especially the historical work of Stephenson, Morrison and Hohenkerk, e.g. at (http://astro.ukho.gov.uk/nao/lvm/)).

Naturally, present-day time-users tend to wish for current quantities rather than history. So, attempts are regularly made to predict Delta T for the near future. The results of these attempts are at best approximate. This can be seen from graphs that compare past predictions with what then turned out to be the actual measured quantities.

An example is the second out of three graphs shown at the web-page for "Delta T : Past, present and Future" on the "Astronomical Almanac Online" website here. This shows "Current values and short term predictions of Delta T" from 2000 to 2022 -- including the odd and unexplained fact that in the interval 2020-2022 the measured values of Delta T stopped trending upwards, and have even slightly diminished. The graph shows clearly how quickly and how much the actual measurements began to deviate from both of two sets of calculated predictions. One set is of IERS predictions made for about 2006 onwards (which turned out lower than actual, about 2 seconds lower by 2022). Another set is a series of USNO predictions (which turned out higher than actual, over 2 seconds higher by 2022).

For a long time, the US Naval Observatory website was a good source of the latest data and predictions about Delta T. But in the last two years and more, its webpages were sometimes inaccessible, and data for latest measured values and most recent predictions seemed to be updated either seldom or not at all. But recently I found that I could again access updates, now via (https://www.cnmoc.usff.navy.mil/Organization/United-States-Naval-Observatory/Earth-Orientation-Department/USNO-Earth-Orientation-Products/Long-term-Delta-T/) and its links to:

"Monthly values of (TT - UT1) since 1973. Updated quarterly" -- here are the latest few entries at the time of writing this:

>    Date      (TT-UT1, secs)
>  2021 10  1     69.2893
>  2021 11  1     69.2880
>  2021 12  1     69.2908
>  2022  1  1     69.2945
>  2022  2  1     69.2913


and to the latest "Predictions of (TT-UT1) from the present through 2027. Updated annually." -- here are a few entries for the period about the time of writing this:

>  Year      TT-UT1 (s)   Uncertainty (s)
> 2021.25    69.358221    0.000004
> 2021.50    69.351422    0.000003
> 2021.75    69.41        0.12
> 2022.00    69.48        0.16
> 2022.25    69.57        0.22
> 2022.50    69.63        0.27
> 2022.75    69.65        0.33
> 2023.00    69.73        0.40


It can be seen, even from these two brief table-quotes, that even the expert predictions, maybe only about a year or two old, have already begun to deviate appreciably from the actuals. The IERS website gives essentially the same data, but it divides Delta T into components given on different web-pages. The USNO combined format -- when available -- seems much the most convenient. I suggest that websites showing values other than these, however they may have been calculated, should be regarded as approximations; but the best of them include some very useful approximations for past Delta T values, for example calculated by formulae that define a collection of spline-curve segments, saving the volume and inconvenience of large lookup tables while losing negligible amounts of accuracy.