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    Delta T long term formula
    From: Paul Hirose
    Date: 2018 Feb 3, 09:38 -0800

    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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