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A 2016 paper (Stephenson FR, Morrison LV, Hohenkerk CY., "Measurement of
the Earth’s rotation: 720 BC to AD 2015," Proc. R. Soc. A 472: 20160404,
http://dx.doi.org/10.1098/rspa.2016.0404) provided a set of polynomial
approximations for ∆T, the time scale difference TT-UT1.

http://rspa.royalsocietypublishing.org/content/472/2196/20160404
http://astro.ukho.gov.uk/nao/lvm/

The polynomials are valid from -720.0 to 2016.0. Outside that span we
can use the parabola that best fits the whole data set: if t = (year -
1825.0) / 100, then ∆T = -320 + 32.5 * t^2.

Although a constant deceleration of Earth's rotation implies a parabolic
graph of ∆T, a parabola may be a poor fit in the short term. For
example, figure 10 of the paper (a graph of ∆T from 1550 to the present)
doesn't show the parabola because it's off the graph. At 2018.0 the
parabolic formula gives -200 s, whereas the true value is +69.

We can get more reasonable values by starting at the last polynomial,
which ends at 2016.0, when ∆T = 68.041. Then extrapolate from that point
by integrating the length of day expression in the paper. That's how the
values on the UKHO page ("ΔT & lod from −2000 to +2500") were obtained.

It doesn't give a formula for the integral, but I have derived one,
where t = year - 1825.0. To extrapolate into the future from 2016.0:

∆T = -293.600 + .00325 * t^2 + 349 * cos(.00419 * t)

Note t is years, not centuries. The input to the cosine function is
radians, not degrees.

To extrapolate into the past from -720.0, change the constant from
-293.600 to -385.979. With those constants there is less than one
millisecond discontinuity at the junctions with the polynomials, and the
extrapolations back to -2000 and forward to +2500 are practically
identical (within the estimated error) to the values on the UKHO page.

   

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