Besselian elements

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Outline of umbra and penumbra (green) during total solar eclipse on Earth's surface and in fundamental plane (red)

The Besselian elements are schedular values to calculate and predict the local circumstances of occultations for an observer on Earth. This method is particularly used for solar eclipses but also applied for occultations of stars or planets by the Moon and transits of Venus or Mercury. In addition for lunar eclipses a similar method is used, at which the shadow is cast on the Moon instead of the Earth.[1]

For solar eclipses for instance it is possible to calculate the duration of totality for a distinct location on Earth based on the Besselian elements, and the path of the umbra on the Earth's surface can be calculated. This method was developed in the 1820s by Friedrich Wilhelm Bessel, a German mathematician and astronomer, and afterwards improved by William Chauvenet.

The basic concept is that Besselian elements describe the movement of the shadow cast by the occulting body – for solar eclipses this is the shadow of the Moon – for a fittingly chosen plane, called the fundamental plane. Comparatively few values are needed to accurately describe the movement of the shadow in this plane. Based on this, the next step is to project the shadow cone onto Earth's surface, only now Earth's rotation, the flattening of Earth and latitude, longitude and elevation of the observer have to be taken into account.[2]

The fundamental plane is the geocentric, normal plane of the shadow axis. In other words, the plane through the Earth's center perpendicular to the shadow axis, which is the line through the centers of the occulting and the occulted body.[3] One advantage, among others, of choosing this plane is that the outline of the shadow is always a circle, and there is no perspective distortion.


  1. ^ P. Kenneth Seidelmann: Explanatory Supplement to the Astronomical Almanac. page 421f, see further reading
  2. ^ P. Kenneth Seidelmann: Explanatory Supplement to the Astronomical Almanac. page 435f, see further reading
  3. ^ Hermann Mucke, Jean Meeus: Canon of solar eclipses: -2003 to +2526. Astronomisches Büro, Wien 1992, Seite XXXIII–LI

Further reading[edit]

  • P. Kenneth Seidelmann (Hrsg.): Explanatory Supplement to the Astronomical Almanac. University Science Books, Sausalito 2006, ISBN 1-891389-45-9
  • Robin M. Green: Spherical astronomy. Cambridge University Press, Cambridge 1985, ISBN 0-521-23988-5
  • William Chauvenet: A manual of spherical and practical astronomy. J. B. Lippincott & Co, Philadelphia 1863 (online)