The coordinates of the Sun used in these eclipse predictions have been calculated on the basis of the VSOP87 theory constructed by P. Bretagnon and G. Francou  at the Bureau des Longitudes, Paris. This theory gives the ecliptic longitude and latitude of the planets, and their radius vector, as sums of periodic terms. In our calculations, we used the complete set of periodic terms of version D of VSOP87 (this version provides the positions referred to the mean equinox of the date).
For the Moon, use has been made of the theory ELP-2000/82 of M. Chapront-Touze and J. Chapront , again of the Bureau des Longitudes. This theory contains a total of 37862 periodic terms, namely 20560 for the Moon's longitude, 7684 for the latitude, and 9618 for the distance to Earth. But many of these terms are very small: some have an amplitude of only 0.00001 arcsecond for the longitude or the latitude, and of 2 centimeters for the distance. In our computer program, we neglected all periodic terms with coefficients smaller than 0.0005 arcsecond in longitude and latitude, and smaller than 1 meter in distance. Due to neglecting the very small periodic terms, the Moon's positions calculated in our program have a mean error (as compared to the full ELP theory) of about 0.0006 second of time in right ascension, and about 0.006 arcsecond in declination. The corresponding error in the calculated times of the phases of a solar eclipse is of the order of 1/40 second, which is much smaller than the uncertainties in predicted values of ΔT, and also much smaller than the error due to neglecting the irregularities (mountains and valleys) at the lunar limb.
Improved expressions for the mean arguments L', D, M, M' and F have been taken from Chapront, Chapront-Touze, and Francou . A major consequence of this work is to bring the secular acceleration of the Moon's longitude (-25.858 "/cy^2) into good agreement with Lunar Laser Ranging (LLR) observations from 1972 to 2001 (see: Secular Acceleration of the Moon).
The center of figure of the Moon does not coincide exactly with its center of mass. To compensate for this property in their eclipse predictions, many of the national institutes employ an empirical correction to the center of mass position of the Moon. This correction is typically +0.50" in longitude and -0.25" in latitude. Unfortunately, the large variation in lunar libration from one eclipse to the next minimizes the effectiveness of the empirical correction. We choose to ignore this convention and have performed all calculations using the Moon's center of mass position. In any case, it has no practical impact on the present work.
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Bretagnon, P., and G. Francou, "Planetary theories in rectangular and spherical variables: VSOP87 solution," Astron. Astrophys., 202(309), (1988).
Chapront-Touze, M., and J. Chapront, "The Lunar Ephemeris ELP 2000," Astron. Astrophys., 124(1), pp.50-62 (1983).
Chapront-Touze, M., and J. Chapront, Lunar Tables and Programs from 4000 B.C. to A.D. 8000, Willmann-Bell, Richmond, Virginia (1991).
Chapront, J., M. Chapront-Touze, and G. Francou, "A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR measurements," Astron. Astrophys., 387, pp. 700-709 (2002).
Espenak, F, and Meeus, J. Five Millennium Canon of Lunar Eclipses: -1999 to +3000 (2000 BCE to 3000 CE), NASA Tech. Pub. 2008-214172, NASA Goddard Space Flight Center, Greenbelt, Maryland (2009).
Espenak, F, and Meeus, J. Five Millennium Catalog of Lunar Eclipses: -1999 to +3000 (2000 BCE to 3000 CE), NASA Tech. Pub. 2008-214173, NASA Goddard Space Flight Center, Greenbelt, Maryland (2009).