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 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.
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:
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.
Links
- Delta T
- NASA Eclipse Home Page
- Delta T
- Felix Verbelen (
Belgium
)
- Delta T
- Robert van Gent (
The Netherlands
)
- Delta-T
- IERS Rapid Service/Prediction Center