In 1824, the Prussian astronomer and mathematician Friedrich Bessel introduced a new method for the prediction of solar eclipses. It was so successful that it remains today as the most powerful technique, even with the application of the digital computer. The key to Bessel's method is the expression of the ephemerides of the Sun and Moon in terms of the Moon's shadow with respect to Earth's center. This change in the frame of reference greatly simplifies the mathematics and geometry without any sacrifice in accuracy.
To define the Besselian elements of an eclipse, a plane is passed through the center of Earth which is fixed perpendicular to axis of the lunar shadow. This is called the fundamental plane and on it is constructed an X-Y rectangular coordinate system with its origin at the geocenter. The axes of this system are oriented with north in the positive Y direction and east in the positive X direction. The Z axis is perpendicular to the fundamental plane and parallel to the shadow axis. The X-Y coordinates of the shadow axis can now be expressed in units of the equatorial radius of Earth. The radii of the penumbral and umbral shadows on the fundamental plane are also tabulated as L1 and L2, respectively. The direction of the shadow axis on the celestial sphere is defined by its declination 'd' and ephemeris hour angle 'm'. Finally, the angles which the penumbral and umbral shadow cones make with the shadow axis are expressed as f1 and f2, respectively. These eight parameters, often tabulated at hourly intervals serve as the only input needed to characterize an eclipse. The details of actual eclipse calculations can be found in the references listed below.
Chauvenet, W., Manual of Spherical and Practical Astronomy, Vol.1, 1891 (Dover edition 1961).
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.
