Historical Values of the Earth's Clock Error Δ and the Calculation of Eclipses:

TLDR: In this article, the effect of variations in the Earth's rate of rotation (LOD) or, equivalently, the length of the day (LOD), on the position of the Moon and Sun in the historical past is discussed.

Abstract: Numerous observations of the Moon, Sun and planets are recorded in ancient and medieval history. These observations which include many eclipses and lunar and planetary conjunctions frequently attract the interest of historians of astronomy. If the positions of the Moon and Sun (and to a lesser extent the planets) in the historical past are to be computed with high precision, it is usually necessary to make satisfactory allowance for the effect of variations in the Earth's rate of rotation, or, equivalently, the length of the day (LOD). Long-term variations in the LOD are mainly produced by lunar and solar tides, but other causes such as the continuing rise of land that was glaciated during the last ice-age are also significant. Although actual changes in the LOD amount to only a few hundredths of a second over several thousand years, the cumulative effect (known as ~T) of these minute changes can be very large. For instance, the estimated value of ~T at the epoch 1000 B.C. is as much as 7 hours. During this interval, the Moon can change position by nearly 4°. It is therefore a matter of concern that at present there appears to be a degree of confusion and misapprehension among historians of astronomy over the choice of values of ~T that should be used in making retrospective computations of lunar and solar positions. Accurate knowledge of the value of ~T is often crucial in assessing the local circumstances of solar eclipses. Neglect of variations in the Earth's spin rate would materially affect the calculated positions of where these phenomena could be seen on the Earth's surface. In this paper we shall try to elucidate the necessary procedures and in particular draw attention to several important points regarding ~T in the calculation of solar eclipses. We shall place special emphasis on three specific issues: the adopted time-scale, the importance of tidal friction in the ephemeris of the Moon, and the enumeration of ~T at various epochs in the past.