Besselian Elements
Encyclopedia
The Besselian Elements are schedular values to calculate and predict the local circumstances of occultation
Occultation
An occultation is an event that occurs when one object is hidden by another object that passes between it and the observer. The word is used in astronomy . It can also refer to any situation wherein an object in the foreground blocks from view an object in the background...

s for an observer on Earth. This method is particularly used for solar eclipses but also applied for occultation
Occultation
An occultation is an event that occurs when one object is hidden by another object that passes between it and the observer. The word is used in astronomy . It can also refer to any situation wherein an object in the foreground blocks from view an object in the background...

s of star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s or planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

s by the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

 and transit
Astronomical transit
The term transit or astronomical transit has three meanings in astronomy:* A transit is the astronomical event that occurs when one celestial body appears to move across the face of another celestial body, hiding a small part of it, as seen by an observer at some particular vantage point...

s of Venus
Transit of Venus
A transit of Venus across the Sun takes place when the planet Venus passes directly between the Sun and Earth, becoming visible against the solar disk. During a transit, Venus can be seen from Earth as a small black disk moving across the face of the Sun...

 or Mercury
Transit of Mercury
A transit of Mercury across the Sun takes place when the planet Mercury comes between the Sun and the Earth, and Mercury is seen as a small black dot moving across the face of the Sun....

. In addition for lunar eclipse
Lunar eclipse
A lunar eclipse occurs when the Moon passes behind the Earth so that the Earth blocks the Sun's rays from striking the Moon. This can occur only when the Sun, Earth, and Moon are aligned exactly, or very closely so, with the Earth in the middle. Hence, a lunar eclipse can only occur the night of a...

s a similar method is used, at which the shadow is casted on the Moon instead of the Earth.

For solar eclipses for instance it's possible to calculate the duration of totality for a distinct location on Earth based on the Besselian Elements, also the path of the umbra on Earth surface can be calculated. This method was developed in the 1820th by Friedrich Wilhelm Bessel, a German mathematician and astronomer, and afterwards improved by William Chauvenet
William Chauvenet
William Chauvenet was an early American educator. A professor of mathematics, astronomy, navigation, and surveying, he was always known and well liked among students and faculty....

.

The basic concept is that Besselian Elements describe the movement of the shadow casted of the occulting body – for solar eclipses this is the shadow of the Moon – for a fittingly chosen plane, called fundamental plane. Comparatively few values are sufficient to describe the movement of the shadow in this plane – with adequate accuracy. Based on this the next step is to project
Graphical projection
Graphical projection is a protocol by which an image of a three-dimensional object is projected onto a planar surface without the aid of mathematical calculation, used in technical drawing.- Overview :...

 the shadow cone onto Earth's surface, only now Earth's rotation, the flattening
Flattening
The flattening, ellipticity, or oblateness of an oblate spheroid is a measure of the "squashing" of the spheroid's pole, towards its equator...

 of Earth and latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

, longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

 and elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

 of the observer have to be taken into account.

The fundamental plane is the geocentric normal plane of the shadow axis, in other words, the plane through the Earth's center perpendiculer to the shadow axis, that is the line through the centers of occulting and occulted body. To choose this plane has – besides others – the advantage that the outline of the shadow is always a circle and there' no perspective distortion
Perspective projection distortion
Perspective projection distortion is the inevitable misrepresentation of three-dimensional space when drawn or "projected" onto a two-dimensional surface....

.

Further reading

  • 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)
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