Explanation of Besselian Elements for Lunar Eclipses

by Fred Espenak

Under Construction

This page is bases on solar eclipses and must be updated for lunar eclipses

In order to predict the general characteristics as well as the local circumstances for a lunar eclipse, high accuracy ephemerides for both the Sun and the Moon are required. Conventional ephemerides tabulate the positions and distances of these bodies with respect to Earth's center. However, the lunar eclipse calculator is primarily interested in the position, dimensions and velocity of Earth's shadow with respect to the Moon.

In 1824, the Prussian astronomer and mathematician Friedrich Bessel introduced a method for the prediction of solar eclipses in which the size and position of the lunar shadow is first calculated with respect to Earth. A similar technique can be used in the prediction of lunar eclipses

To define the Besselian elements of a lunar eclipse, a plane is passed through Earth's shadow at the distance of the Moon and oriented perpendicular to the shadow axis. This is called the fundamental plane on which an X-Y rectangular coordinate system is constructed with its origin centered on the shadow. 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 identical to the shadow axis.

The X-Y coordinates of the Moon in the fundamental plane 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 identified 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 'µ'. 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 and their variations over time serve as the only input needed to characterize a lunar eclipse. They may be tabulated at hourly intervals, or expressed as a series of third order polynomials over the several hour period of the eclipse.

The eight Besselian elements needed to characterize a solar eclipse can be summarized as follows:

  • x, y - Cartesian coordinates of the lunar shadow axis in the (in units or Earth's equatorial radius)
  • L1, L2 - Radii of the Moon's penumbral and umbral/antumbral shadows in the (in units or Earth's equatorial radius)
  • d - Declination of the Moon's shadow axis on the celestial sphere
  • µ - Hour angle of the Moon's shadow axis on the celestial sphere
  • f1, f2 - Angles of the penumbral and umbral/antumbral shadow cones with respect to the axis of the lunar shadow

The details for actual eclipse calculations using the Besselian elements 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.