# Maclaurin ellipsoid

In geophysics and planetology , the **Maclaurin ellipsoid** is a **homogeneous ****ellipsoid** that is used to calculate theoretical planetary and earth models .
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McLaurin ellipsoids have the exact shape of an ellipsoid of revolution and a constant density ("homogeneous"). Each of these possible models can be assigned a period of rotation during which it is in hydrostatic equilibrium . The free surface of the ellipsoid thus represents a level surface .

The Physical Geodesy uses the Maclaurin ellipsoids to the theoretical development of equilibrium figures . An ellipsoid with the dimensions of the earth , its mean density of 5.52 g / cm³ and the duration of rotation of 86.164 seconds would have a flattening of 1: 230, whereas the actual flattening of the earth is 1: 298.2. From this it was deduced by Colin Maclaurin as early as the 18th century that the density of the earth must increase sharply inward.

## Further equilibrium models

In the middle of the 20th century, Karl Ledersteger calculated various model series in order to verify the dynamic flattening of the earth derived from the moon's orbit . For this he approached the Earth's mantle and core of the Earth by *double-skin* on models of different density, in which, however flattening had slightly different - see Wiechert model . These two-shell models, first developed by the geophysicist Wiechert, can be linked to the geophysical data on the depth and density of the earth's core.

Even further approximation is achieved if, instead of constant density, an inwardly increasing density function is chosen, i.e. H. when a transition is made from McLaurin ellipsoids to *one-parameter equilibrium* figures (density as a parameter of depth). However, their treatment is mathematically difficult and their combination to two-shell models is still unsolved.

The counter-model of this almost linear increase in density is the *spheroid of the greatest mass concentration* - a model that roughly corresponds to a balloon envelope with a massive, punctiform center. With the dimensions of the earth it would have a flattening of about 1: 400, which would make it much more spherical than the homogeneous ellipsoid. The earth is therefore much closer to the homogeneous ellipsoid - which would correspond to a celestial body made of incompressible liquid - than a mass concentration in the center of the earth .

## See also

## Sources and web links

- K. Ledersteger:
*Astronomical and Physical Geodesy*, Stuttgart 1968 - https://link.springer.com/article/10.1007%2FBF02001091