## The Effects on Ancient Observations

The cumulative slowing of the Earth's rotation means that events in the far past actually occurred later than we would calculate them to have occurred on the basis of uniform time (see Figure 4.13 to understand how this happens). Although the computation of Earth's deceleration is determined from timed observations of lunar and planetary phenomena, observation of an event with a recorded local solar time provides an invaluable and necessary check. Indeed, there are other causes for rotation variation and the correction of mean solar to Ephemeris (now Terrestrial) Time depends on these empirical determinations.

Ancient eclipses are among the most useful events for checking the deceleration rate. The ancient lunar and solar eclipses used by several authors include Arabic, Greek, Babylonian, and Chinese sources; these are discussed in §5 in the context of eclipses; details of the eclipses are discussed in subsequent chapters in the contexts of the cultures in which the observational records were taken. This work has important consequences for historical studies: Chronologies are sometimes based on eclipse records, at least in part, so it is important to sort out what can be determined convincingly without using the very events that are to be calibrated! At this stage, we summarize the determinations of the deceleration.

The time difference, AT, between a uniform measure of time, effectively the "Terrestrial Time" (TT), and a measure of the moment of observation as given by the hour angle of the Sun at a specified location on Earth, the Universal Time (UT), is, in the notation26 of Stephenson and Clark (1978),

where b and c are constants and e is a quantity called the deceleration parameter.27 The quantity t is in Julian centuries measured from 1900 a.d., and AT is in seconds. Muller and Stephenson (1975) present the solutions of these data in which b and c are determined as well as e. Their results (summarized also in Stephenson and Clark 1978) included the calculation of the lunar acceleration (the accelerations in the mean longitudes of the Moon, Sun, and Mercury were observed prior to the identification of the Earth's varying rotation), and in their notation are n = -37.5 ± 5arc x sec/century2, b = +66.0, c = 120.38s/century, e = -91.6 ± 10 s/century2 so that = V2 e « 40.8 s/cy2. Uncertainties are not given for b and c, but the combined result of short term variation of all terms aside from that due to tidal friction does not exceed ±100s (however, see below for evidence suggesting larger variation). In terms of relative change in angular velocity, where w is the current angular rotation speed of the Earth, w/w = 3.17 x 10-10e = -29.0 ± 3 x 10-"century-1. (4.27)

This corresponds to a rate of increase in the length of the day of 2.5(± 0.3) x 10-3 s/century or 1 second in 40,000 years (recall our earlier example in which we assumed a rate of slowing of 1s/100,000y). More recent work for the interval 700 b.c. to the present by Stephenson and Morrison (1995) suggests, however, that the slowing of the Earth's rotation has measurable variation, as evidenced by residuals to spline fittings, amounting to as much as 1000 s or more for specific, timed events (Stephenson and Morrison 1995, Fig. 6). They have revised estimates of the underlying deceleration. In particular, they (and others) have argued that action of tidal friction due to the Moon is, at present, partly offset by a negative (deceleration) term, the latter implying a speedup. With an adopted value28 for the lunar acceleration, n = -26

26 The formula originally used Ephemeris Time (ET) rather than Terrestrial Time. The difference between ET and atomic time (TAI) was determined to be 32!184 in 1984;the difference between the predecessor of TT (TDT) and TAI was also set at this value. Since then AT has increased (to ~85s in 2004). See 4.1.1.2 for the relation between time and time intervals and Stephenson (1997) for a thorough discussion of the history of these different definitions of time.

27 It should be noted that other works use different notation as well as different algebraic expressions. As a consequence, the coefficient of T 2 is sometimes called "c" with values of ~-e. Note that, in this context, e is neither an eccentricity nor an ellipticity.

28 A value borne out by more recent determinations, such as that of Dickey et al. (1994), -25.88 ± 0.5arcsec/century2. A lunar acceleration of -26 arcsec/cy2 results in a recession of the Moon from the Earth of 3.86m/century, and an increase in the length of the month of 0.038s/cy. See Stephenson (1997) for a much more complete discussion of the methods used to get these values.

rotation direction

Figure 4.13. The slowing of the Earth's rotation means that events in the distance past occurred later than calculated, and eclipses are seen at more easterly sites, under the assumption of uniform time (DT = 0). Here, we depict the longitude shift of January 14,484 a.d., as seen from space at the onset of the eclipse (at sunrise) in two ways: (a) a Polar sketch (drawing by E.F. Milone) and (b) Merca-tor view sketch (map from Liu and Fiala 1992 software; modified by E.F. Milone).

± 0.5"/cy2 [based on Williams, Newhall, and Dickey, 1992 (25.9 ± 0.5"/cy2); Christodoulidis et al. 1988 (-25.27"/cy2)], Stephenson and Morrison (1995, p. 170) found a slowing of the Earth due to tidal friction alone of 2.3 ± 0.1 millisec/century (ms/cy), resulting in an accumulation

ATtldal = TT - UT = qTt2 = +42(±2) x tSeconds, (4.28)

where t is in centuries measured positively from 1800.

From Babylonian and other data, Stephenson and Morrison (1995, pp. 188-193), revise their preferred value for the observed AT to

which implies a rate of change of the length of the day of +1.70 ± 0.05 ms/cy. Because +2.3 ms/cy is the expected rate from tidal friction, the difference, -0.6 ms/cy, suggests a source of speedup. It is probably not due to variation in tidal friction in the "shallow seas" and deep ocean basins, where the friction supposedly has its greatest effects, because the average ocean depth over the past 2700 years has not varied by more than an estimated 1-2 m, and the areas of both deep oceans and shallow seas have remained approximately steady over that time. One possible source is a change in the Earth's oblateness as the Earth changes shape in response to the "post-glacial rebound" of melting ice sheets built during the last ice age. Studies of the orbital perturbations of artificial satellites suggest a change in the J2 zonal harmonics term that describes the Earth's shape; this in turn implies a deceleration of -0.44 ± 0.05 ms/cy (the negative sign indicating a speedup). Further residuals in their fittings of Babylonian, Chinese, Arab, and European data lead Stephenson and Morrison (1995, p. 199) to conclude that there are periodic fluctuations of ±4ms over a period of ~1500y. These fluctuations provide a limitation on the usefulness of ancient records. We will continue the discussion of the importance of the determinations of AT in the context of eclipses in §5.2.1.3.

The periodicities of the diurnal motions of the stars and other objects and of the "wandering" motions of the Sun, Moon, and planets have been our main concern until now. Knowledge of the motions of both Sun and Moon leads to an understanding of eclipses, a subject to which we turn in Chapter 5. In §5.8, we describe variations in brightness among the stars, random as well as complex periodic effects, and the impacts these phenomena have had on the ancient world, and thus on ours.

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