Eclipse Predictions and Earth's Rotation

Fred Espenak

Introduction

When Sir Isaac Newton first published his revolutionary theory of gravitation in the Principia (1687), it laid the ground work for the prediction of planetary motion throughout the solar system. Edmund Halley played a pivital role in motivating Newton to develop this mathematical description of gravity. In fact, Halley even financed much of the Principia's publication costs.

Halley was quite curious about the orbits of the planets. Using Newton's Principia, Halley calculated orbits for the comets of 1531, 1607, and 1682 and discovered that they must be successive returns of the same object. He correctly predicted that the comet would return in 1758 and it has been known as Halley's Comet ever since. He also devised a method to determine Earth's distance from the Sun using rare transits of Venus across the Sun's disk.

Although not as well known, Halley also made important scientific contributions in his studies of eclipses. He is credited with one of the first eclipse map showing the path of the Moon's shadow across England during the upcoming total eclipse of 1715. He also rediscovered the Saros cycle of 18 years plus 10 or 11 days (depending on the number of intervening leap years) over which eclipses seem to repeat. The Saros was used by Chaldeans and Babylonians (and later, the Greeks) for simple lunar eclipse predictions but it was unknown in Halley's day. Using Newton's Theory of the Moon's Motion (or TMM) and the Saros cycle, Halley made a series of calculations to identify ancient eclipses in the literature. But Halley soon encountered a problem. The eclipse paths he predicted were shifted with respect to the historical records. Either the Moon was accelerating in its orbit or Earth's rotation rate was slowing down (i.e. - length of the day was increasing). Although both are actually true, Halley correctly identified the increasing length of the day as the primary culprit. It took another 300 years to understand why.

Earth's Rotation

The ocean tides are casued by the gravitational pull of the Moon and, to a lesser extent, the Sun. But as the tides are attracted to the Moon, the oceans appear to rise and fall while Earth rotates beneath them. This tidal friction gradually transfers angular momentum from Earth to the Moon. Earth looses energy and slows down while the Moon gains the energy and consequently its orbital period and distance from Earth increase.

The tides are not the only factor changing Earth's rotation rate. The melting of continental ice sheets at the end of the last glacial period result in the rise of land masses that were depressed by the enormous weight of the ice sheets. This "post-glacial rebound" of the land allows the return flow of mantle material back under the de-glaciated landmasses particularly in the polar regions. Through conservation of angular momentum, the resulting shift in mass from the equator to the poles will cause Earth's rotation to slow down. However, it will take many thousands of years for the land to reach an equilibrium level because of the extreme viscosity of the mantle.

The Moon's average distance from Earth is increasing by 3.8 centmeters per year. Such a precise value is possible due to the Apollo laser reflectors which the astronauts left behind during the lunar landing missions. Eventually, the Moon's distance will increase so much that it will be to far away to produce total eclipses of the Sun (See: Extinction of Total Solar Eclipses).

In comparison, the secular change in the rotation rate of Earth currently increases the length of day by 2.3 milliseconds per century. While this amount may seem astonishingly small, its accumulated effects have important consequences. In one century, Earth looses about 40 seconds, while in one millennium, the planet is over one hour "behind schedule." Astronomers use the quantity Delta T to describe this time difference.

Unfortunately, Earth's rotation is not slowing down at a uniform rate. Non-tidal effects of climate (global warming, polar ice caps and ocean depths) and the dynamics of Earth's molten core make it impossible to predict the exact value of Delta T in the remote past or distant future.

Good values of Delta T only exist sometime after the invention of the telescope (1610). Careful analysis of telescopic timings of stellar occultations by the Moon permits the direct measurement of Delta T during this time period. Prior to the 1600's, values of Delta T must rely on historical records of the naked eye observations of eclipses and occultations. Such observations are rare in the literature and of coarse precision.

Stephenson and collaborators have made a number of important contributions concerning Earth's rotation during the past several millennia. In particular, they have identified hundreds of eclipse and occultation observations in early European, Middle Eastern and Chinese annals, manuscripts, canons and records. In spite of their relatively low precision, these data represent our only record of the value of Delta T during the past several millennia.

In particular, Stephenson and Morrison (1984) have fit hundreds of records with simple polynomials to achieve a best fit for describing the value of Delta T from 700 BCE to 1600 CE. An abbreviated table of their results is as follows:

Table 1 - Value of ΔT (1500 BCE to 1500 CE)
Year	Delta T (sec)	Longitude Shift
1500 BCE	39610s (= 11h 00m)	165.0°
1000 BCE	27364s (= 07h 36m)	114.0°
500 BCE	17444s (= 04h 51m)	72.7°
1 BCE	9848s (= 02h 44m)	41.0°
500 CE	4577s (= 01h 16m)	19.1°
1000 CE	1625s (= 00h 27m)	6.8°
1500 CE	275s (= 00h 05m)	1.1°

Note: BCE (Before Common Era) and CE (Common Era) are secular alternatives for the terms BC and AD, respectively. For more information, see Year Dating Conventions.

Take special note of the column labeled "Longitude Shift." This is the amount that an eclipse path must be shifted in order to take into account the cumulative effects of Delta T. The historical eclipse and occultation records for Stephenson and Morrison (1984) only extends back to about 700 BCE. Thus, any values of Delta T before this time must either be 1) a direct extrapolation from known values, or 2) based on theoretical models of purely tidal breaking of Earth's rotation. The best available solution is probably to combine both of the above methods when looking into the distant past (before 1000 BCE), but the uncertainties grow so rapidly that no meaningful results can be obtained earlier than about 2000 BCE.

Stephenson and Houlden [1986] estimate the uncertainties in the adopted values of Delta T as follows:

Table 2 - Uncertainty of ΔT (400 BCE to 1900 CE)
Year	Uncertainty (Time)	Uncertainty (Longitude)
1500 BCE	~900 sec	~4°
400 BCE	~420 sec	~2°
1000 CE	~80 sec	20' (0.33°)
1600 CE	30 sec	7.5' (0.13°)
1700 CE	5 sec	75"
1800 CE	1 sec	15"
1900 CE	0.1 sec	1.5"

The uncertainty in Delta T means that reliable eclipse paths prior to about 1500 BCE are not possible. Similarly, all future values of Delta T are simple extrapolations of current values and trends. Such estimates are prone to growing uncertainty as one extrapolates further and further into the furure. By the year 3000 CE, the value of Delta T could be on the order of one hour with an extrapolated uncertainty of about ten minutes or several degrees in longitude.

In more recent work, Stephenson (1997) has made improvements in his analysis of Delta T using additional historical records.

Table 3 - Value of ΔT (500 BCE to 1500 CE)
Year	Delta T (sec)	Longitude Shift
500 BCE	16800s (= 04h 40m)	70.0°
1 BCE	10600s (= 02h 57m)	44.2°
500 CE	5700s (= 01h 35m)	23.7°
1000 CE	1600s (= 00h 27m)	6.7°
1500 CE	180s (= 00h 03m)	0.8°

If a computer program does not incorporate these values into its calculations, then the resultant eclipse paths will contain longitude shifts inconsistent with the best estimates for Delta T.

References for Rotation and Delta T

Dickey, J.O., "Earth Rotation Variations from Hours to Centuries", in: I. Appenzeller (ed.), Highlights of Astronomy: Vol. 10 (Kluwer Academic Publishers, Dordrecht/Boston/London, 1995), pp. 17-44.

Meeus, J., "The Effect of Delta T on Astronomical Calculations", Journal of the British Astronomical Association, 108 (1998), 154-156.

Morrison, L.V. and Ward, C. G., "An analysis of the transits of Mercury: 1677-1973", Mon. Not. Roy. Astron. Soc., 173, 183-206, 1975.

Spencer Jones, H., "The Rotation of the Earth, and the Secular Accelerations of the Sun, Moon and Planets", Monthly Notices of the Royal Astronomical Society, 99 (1939), 541-558.

Stephenson, F.R. & Morrison, L.V., "Long-Term Changes in the Rotation of the Earth: 700 BC to AD 1980", Philosophical Transactions of the Royal Society of London, Ser. A, 313 (1984), 47-70.

Stephenson F.R and Houlden M.A., Atlas of Historical Eclipse Maps: East Asia 1500 BD - AD 1900, Cambridge Univ.Press., 1986.

Stephenson, F.R. & Morrison, L.V., "Long-Term Fluctuations in the Earth's Rotation: 700 BC to AD 1990", Philosophical Transactions of the Royal Society of London, Ser. A, 351 (1995), 165-202.

Stephenson F.R., Historical Eclipses and Earth's Rotation , Cambridge Univ.Press, 1997.

Links to Additional Eclipse Predictions

Home - home page of EclipseWise with predictions for both solar and lunar eclipses

Solar Eclipses - primary page for solar eclipse predictions
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Lunar Eclipses - primary page for lunar eclipse predictions
Lunar Eclipse Links - detailed directory of links

MrEclipse.com - eclipse resources and tips on photography
Solar Eclipses for Beginners - a primer on solar eclipse basics
Lunar Eclipses for Beginners - a primer on lunar eclipse basics

Acknowledgments

The information presented on this web page is based on material published in Thousand Year Canon of Solar Eclipses 1501 to 2500 and Thousand Year Canon of Lunar Eclipses 1501 to 2500.

For permission to reproduce this data or figures, please contact Fred Espenak. Any reproduction must be accompanied by an acknowledgment:

"Eclipse Predictions by Fred Espenak, www.EclipseWise.com"