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 [1988] 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 these calculations, we use 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 [1983], 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) in the lunar limb profile.
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.
The revised value used for the Moon's secular acceleration (n-dot) is -26 arc-sec/cy*cy, as deduced by Morrison and Ward [1975] from 250 years of Mercury transit observations. The value for delta-T (delta-T = DT - UT) is based on the work of Morrison and Stephenson [2004]
The predictions use a smaller value for the Moon's radius k (=0.272281) than the one adopted by the 1982 IAU General Assembly (k=0.2725076). The smaller k is a better approximation of the Moon's minimum diameter and results in a slightly shorter total or longer annular eclipse when compared with calculations using the IAU value. No correction has been made between the Moon's center of mass and center of figure. The difference is small and of consequence only where careful timings are needed or for observations near the northern or southern path limits. In such cases, a more detailed predictions are required which include the effects of the Moon's limb profile.
The Besselian elements computed for solar eclipses using the VSOP87 and ELP2000/82 ephemerides were originally prepared for the Five Millennium Canon of Solar Eclipses: -1999 to +3000 [Espenak and Meeus].
References
Bretagnon, P. and Francou, G., "Planetary Solutions VSOP87", Astronomy and Astrophysics, vol. 202, 309B, 1988.
Chapront-Touze, M. and Chapront, J., "The lunar ephemeris ELP 2000", Astronomy and Astrophysics, vol. 190, no. 1-2, Jan. 1988, p. 342-352.
Chapront-Touze, M. and Chapront, J., "ELP 2000-85 - A semi-analytical lunar ephemeris adequate for historical times", Astronomy and Astrophysics, vol. 190, no. 1-2, Jan. 1988, p. 342-352.
Espenak, F., and J. Meeus., Five Millennium Canon of Solar Eclipses: -1999 to +3000, NASA TP–2006-214141, NASA Goddard Space Flight Center, Greenbelt, Maryland, 648 pp, 2006.
Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac, Her Majesty's Nautical Almanac Office, London, 1974.
Meeus, J., Elements of Solar Eclipses: 1951 - 2200, Willmann-Bell, Inc., Richmond, 1989.
Morrison, L. and Stephenson, F. R., "Historical Values of the Earth's Clock Error ΔT and the Calculation of Eclipses", J. Hist. Astron., Vol. 35 Part 3, August 2004, No. 120, pp 327-336 (2004).
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.
Newcomb, S. "Tables of the Motion of the Earth on its Axis Around the Sun",Astron. Papers Amer. Eph., Vol. 6, Part I, 1895.
Stephenson F.R., Historical Eclipses and Earth's Rotation , Cambridge Univ.Press, 1997.
Stephenson F.R and Houlden M.A., Atlas of Historical Eclipse Maps, Cambridge Univ.Press., Cambridge, 1986.