# Eclipse Predictions and Earth's Rotation

## 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 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 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. Ê

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:

```
Year        delta-T             Longitude
(sec)                Shift

1500 BCE     39610 = 11h 00m      165.0°
1000 BCE     27364 = 07h 36m      114.0°
500 BCE     17444 = 04h 51m       72.7°
1 BCE      9848 = 02h 44m       41.0°
500 CE       4577 = 01h 16m       19.1°
1000 CE       1625 = 00h 27m        6.8°
1500 CE        275 = 00h 05m        1.1°

```

Note: BCE (Before Common Era) and CE (Common Era) are secular alternatives for the terms BC and AD, respectively.

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:

```                     Year          Uncertainty       Uncertainty
(Time)          (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.

```                     Year        delta-T             Longitude
(sec)                Shift

500 BCE     16800 = 04h 40m       70.0°
1 BCE     10600 = 02h 57m       44.2°
500 CE       5700 = 01h 35m       23.7°
1000 CE       1600 = 00h 27m        6.7°
1500 CE        180 = 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.