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Besselian elements

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.

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. 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.