Secular Acceleration of the Moon used in
Five Millennium Canon of Solar Eclipses

NASA Technical Publication TP-2006-214141
by Fred Espenak and Jean Meeus

Ocean tides are caused by the gravitational pull of the Moon (and, to a lesser extent, the Sun). The resulting tidal bulge in Earth's oceans is dragged ahead of the Moon in its orbit due to the daily rotation of Earth. As a consequence, the ocean mass offset from the Earth-Moon line exerts a pull on the Moon and accelerates it in its orbit. Conversely, the Moon's gravitational tug on this mass exerts a torque that decelerates the rotation of Earth. The length of the day gradually increases as energy is transferred from Earth to the Moon, causing the lunar orbit and period of revolution about Earth to increase.

This secular acceleration of the Moon is small but it has a cumulative effect on the Moon's position when extrapolated over many centuries. Direct measurements of the acceleration have been only been possible since 1969 using the Apollo retro-reflectors left on the Moon. The results from Lunar Laser Ranging show that the Moon's mean distance from Earth is increasing by 3.8 cm per year (Dickey, et al., 1994). The corresponding acceleration in the Moon's ecliptic longitude is -25.858 "/cy^2 (Chapront, Chapront-Touze, and Francou, 2002). This is the value we have adopted in our lunar ephemeris calculations.

There is a close correlation between the Moon's secular acceleration and changes in the length of the day. However, the relationship is not exact because the lunar orbit is inclined anywhere from about 18.5° to 28.5° to Earth's equator. The parameter ΔT is a measure of the accumulated difference in time between an ideal clock and one based on Earth's rotation as it gradually slows down. Published determinations of ΔT from historical eclipse records have been done assuming a secular acceleration of -26 "/cy^2 (Morrison and Stephenson, 2004). Since we have adopted a slightly different value for the secular acceleration, we must make a small correction "c" to the published values of ΔT as follows:

	       c = - 0.91072 * ( -25.858 + 26.0 ) *  t^2
		     where: t = (year-1955)/100


	       ΔT (corrected) = ΔT + c

Evaluation of the correction from -2000 to +3000 yields the following.

Corrections to ΔT Due to Secular Acceleration
Year Correction
-2000 -202
-1000 -133
0000 -49
+1000 -12
+2000 0
+3000 -14

Clearly the correction is only important for negative years although it is significantly smaller than the actual uncertainty in ΔT itself.

The secular acceleration of the Moon is very poorly known and may not be constant. Careful records for its derivation only go back about a century. Before then, spurious and often incomplete eclipse and occultation observations from medieval and ancient manuscripts comprise the data base. In any case, the current value implies an increase in the length of day [LOD] of about 2.3 milliseconds per century. Such a small amount may seem insignificant, but it has very measurable cumulative effects. At this rate, time as measured through Earth's rotation is losing about 84 seconds per century^2 when compared to atomic time.

Five Millennium Canon of Solar Eclipses

Five Millennium Canon of Lunar Eclipses