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. 2016 Dec;472(2196):20160404.
doi: 10.1098/rspa.2016.0404.

Measurement of the Earth's rotation: 720 BC to AD 2015

F R Stephenson  1 , L V Morrison  2 , C Y Hohenkerk  3
Affiliations

Measurement of the Earth's rotation: 720 BC to AD 2015

F R Stephenson et al. Proc Math Phys Eng Sci. 2016 Dec.

Abstract

New compilations of records of ancient and medieval eclipses in the period 720 BC to AD 1600, and of lunar occultations of stars in AD 1600-2015, are analysed to investigate variations in the Earth's rate of rotation. It is found that the rate of rotation departs from uniformity, such that the change in the length of the mean solar day (lod) increases at an average rate of +1.8 ms per century. This is significantly less than the rate predicted on the basis of tidal friction, which is +2.3 ms per century. Besides this linear change in the lod, there are fluctuations about this trend on time scales of decades to centuries. A power spectral density analysis of fluctuations in the range 2-30 years follows a power law with exponent -1.3, and there is evidence of increased power at a period of 6 years. There is some indication of an oscillation in the lod with a period of roughly 1500 years. Our measurements of the Earth's rotation for the period 720 BC to AD 2015 set firm boundaries for future work on post-glacial rebound and core-mantle coupling which are invoked to explain the departures from tidal friction.

Keywords: core–mantle coupling; eclipses; length of day; occultations; sea level; tidal friction.

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Figures

Figure 1.
Figure 1.
Difference in longitude between JPL lunar ephemeris DE430 and the analytical ephemeris designated j=2 incorporating a value of −26′′ cy−2 for the Moon’s tidal acceleration. The dashed line is the average difference of about 10′′ in longitude, which produces a systematic difference of about 20s in ΔT.
Figure 2.
Figure 2.
Values of ΔT for all the Babylonian-timed observations −720 to −9 listed in the electronic supplementary material, tables S1–S4. (The conversion factor for deriving the sidereal rotational displacement angle of the Greenwich meridian, Δθ, measured in radians, from ΔT, measured in seconds of mean solar time, is −7.29×10−5 ΔT.) The observation in parenthesis was not used in fitting curves to the data (see electronic supplementary material section S4, −666).
Figure 3.
Figure 3.
Babylonian observations −720 to −65 with weights greater than 2 after −560, which is indicated by a vertical dotted line. The observations before −560 are intrinsically less accurate (see §3b(i)). The two observations in brackets around −200 were treated as outliers: an observation at −666 is intrinsically doubtful.
Figure 4.
Figure 4.
Values of ΔT for Chinese lunar eclipse timings 434–1280 listed in the electronic supplementary material, table S5. The observations are less accurate before 900 (indicated by a vertical dotted line). Observations in brackets were treated as outliers.
Figure 5.
Figure 5.
Values of ΔT for Chinese solar eclipse timings 586–1277 listed in the electronic supplementary material, table S6. The observations are less accurate before 900 (indicated by a vertical dotted line). The observation in brackets was treated as an outlier.
Figure 6.
Figure 6.
Values of ΔT for Greek lunar and solar eclipse timings −200 to 364, listed in the electronic supplementary material, table S7. The observation in brackets was treated as an outlier.
Figure 7.
Figure 7.
Values of ΔT for Arab solar eclipse timings 829–1004 listed in the electronic supplementary material, table S8. The observation in brackets was treated as an outlier.
Figure 8.
Figure 8.
Values of ΔT for Arab lunar eclipse timings 854–1019 listed in the electronic supplementary material, table S9.
Figure 9.
Figure 9.
Results for ΔT for collected timed observations −720 to 1280 and the untimed total solar eclipse of 1567. The dotted red curve is the parabola given by equation (4.1). The black curve is the spline curve described in §4b.The grey curve is the parabola (equation (1.5)), predicted on the basis of tidal friction. The observations in brackets were treated as outliers, apart from a Babylonian observation in −666 which is intrinsically doubtful.
Figure 10.
Figure 10.
Results for ΔT for timed data 1623–2015: lunar occultations of stars (478 843 observations); lunar occultation of Jupiter;fourth contacts of solar eclipses (1623–1670). The untimed total solar eclipse of 1567 is also plotted. A sample of up to 10 observations in any one year are plotted to avoid saturation, which otherwise would give a false impression of the scatter of the data. The black curve is the spline curve described in §4b.
Figure 11.
Figure 11.
Solution space for ΔT: total/annular eclipses −708 to 454. The dotted red curve is the parabola given by equation (4.1).The black curve is the spline curve described in §4b. The grey curve is the parabola (equation (1.5)), predicted on the basis of tidal friction.
Figure 12.
Figure 12.
Solution space for ΔT: total/annular eclipses 454–1567. The dotted red curve is the parabola given by equation (4.1). The black curve is the spline curve described in §4b.The grey curve is the parabola (equation (1.5)), predicted on the basis of tidal friction.
Figure 13.
Figure 13.
Solution space for ΔT: −719 to 360, large partial solar eclipses; solar and lunar eclipses rose or set eclipsed; estimates of degree of obscuration of lunar and solar eclipses at rising or setting. The dotted red curve is the parabola given by equation (4.1). The black curve is the spline curve described in §4b.The grey curve is the parabola (equation (1.5)), predicted on the basis of tidal friction.
Figure 14.
Figure 14.
Solution space for ΔT: the dotted red curve is the parabola given by equation (4.1). The black curve is the spline curve described in §4b.The grey curve is the parabola (equation (1.5)), predicted on the basis of tidal friction.
Figure 15.
Figure 15.
Plot of the residuals δΔT with respect to the parabola (4.1) represented as a straight line. The key to the symbols is as in figures 9–14. The black curve is the spline fit discussed in §4b.
Figure 16.
Figure 16.
lod 1960–2015 derived from lunar occultations: black curve from cubic splines with knots at 3 year intervals; green curve from fitting by loess with smoothing parameter q=0.08. Lod 1962–2015 from IERS data smoothed by 2 year moving average. (The conversion factor for deriving the change in the rotational velocity Δω in rad s−1 from the lod in ms is −0.843×10−12 lod.)
Figure 17.
Figure 17.
lod 1985–2015 with atmosphere angular momentum subtracted.
Figure 18.
Figure 18.
lod −2000 to 2500. The dotted red line is the average measured rate of change in the lod, +1.78±0.03 ms cy−1, which is equivalent to an acceleration of −4.7±0.1×10−22 rad s−2. The shaded grey area shows the change expected on the basis of tidal friction, +2.3±0.1 ms cy−1, equivalent to −6.2±0.4× 10−22 rad s−2. The black curve is the slope on the spline fit shown in figures 9 and 10. The green-dashed curve is the extrapolation of the oscillation (equation (5.1)).
Figure 19.
Figure 19.
lod 1700–2015 derived from lunar occultations. The black curve is the slope on the spline curve displayed in figure 10. The green curve is the half-yearly values derived from the loess smoothing after 1800 as described in §5b. The red curve is the IERS data as displayed in figure 16.
Figure 20.
Figure 20.
Power spectrum density (PSD) of half-yearly lod values in the years 1800–2015 obtained from the slope on local polynomial regression of occultation ΔT values.
See this image and copyright information in PMC

References

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