# Balance figure

Under equilibrium figures , more precisely hydrostatic equilibrium figures , geophysics and astronomy understand a fluid, stable physical structure of celestial bodies in which all inner boundary and level surfaces as well as the free surface are in complete equilibrium of all forces . In nature, however , this ideal state is nowhere 100 percent given.

## An idealized earth

The mean sea ​​level of the earth - which one continues to think of as a geoid under the continents and generally used as a reference to altitude - would not be an exact equilibrium area even if the terrain was leveled. Only when the water completely immobile covered the whole planet and had the same depth everywhere would the sea level be the surface of a hydrostatic equilibrium figure. The force of gravity on such a (and on any natural) level surface is not the same everywhere, but it always acts perpendicular to the surface, so that no water drop would move from the point. However, all internal level surfaces and rock boundaries would also have to meet an analogous condition - which is obviously not entirely the case. The earth's crust is only about 90% in equilibrium, the liquid interior almost 100%.

In order to be able to calculate the ideal shape of the earth or a planet , the model has to be simplified: at least the irregularly structured crust would have to be leveled and all of its rock layers too. To calculate an equilibrium of forces, at least 5 quantities are relevant for each inner layer of the earth:

partly also the course of

when thermodynamics are taken into account in the latest models .

Even with a completely "regularized" earth crust (according to K.Ledersteger ), residual deviations remained in the earth's interior. However, the more homogeneous the magma of the earth's mantle and the material in the earth's core , the more precisely it is possible to infer their interfaces, density and possibly temperature. All of these phenomena have an effect on the outer shape of the earth and are therefore of fundamental importance for geodesy . In the case of equilibrium, Clairaut's theorem also establishes a simple relationship between the flattening of the earth and the gravitational field .

## MacLaurin and Wiechert models

The simplest model is a completely homogeneous, liquid ellipsoid , which has exactly the mass, size and rotation of the earth ( MacLaurin ellipsoid ). The theoretical solution lies in a conditional equation for these three quantities, which Colin Maclaurin published in Edinburgh in 1742 ( A Treatise of Fluxions ). However, the result is an earth flattening of 1: 231 instead of actually f = (ab) / a = 1: 298.25 (equator half-axis a = 6378.13 km and polar half-axis b = 6356.74 km).

The earth's core must therefore be much denser than the average of the entire earth (5.52 g / cm³). The next better model is a two-shell model with a liquid-homogeneous shell and core. Its theoretical solution is already much more complicated (in Ref. 2 at least 10 pages), because the pressure in the “core of the earth” also depends on the weight of the mantle. It comes from Emil Wiechert ( About the mass distribution in the interior of the earth , Nachr. Kgl. Ges. Wiss. Göttingen 1897 ), which is why the model is called the Wiechert model by geoscientists . The question of where the core-mantle boundary is to be applied depends on the result: the model can only be realistic if the two density values ​​are plausible and result in the above-mentioned flattening value  f (plus an additional "shape parameter" from the satellite geodesy ). Of the infinite possibilities, K. Ledersteger (1966, Zeitschrift für Geophysik) considers the density values ​​4.17 and 12.44 g / cm³ as well as a core depth of 2900 km for those that come closest to the seismically explored structure of the earth .

In fact, however, the crust density is on average 2.7 g / cm³, while the density of the earth's mantle increases inward from 3.3 to over 5 g / cm³, and that of the earth's core is probably 10 to 14 g / cm³. A third model is therefore suitable, the density of which increases linearly towards the inside - with a factor that has yet to be determined. Karl Ledersteger and László Egyed (see item 3) call this series of models " one-parameter equilibrium figures ".

The ideal would now be to approximate the earth “ defoliated ” around the earth's crust by a combination of the Wiechert and the one-parameter model. The theoretical solution is of course extremely complicated and is bypassed today by numerically rather than analytically integrated is. For a small to fit body Earth "exploded" (eg., By homoeoid to formulate the equilibrium conditions -shells), but is a separate problem.

## Balance figures in other areas

In astrophysics models play for the interior of the sun and stars an increasing role. Their main features can no longer be shown here, because the prevailing temperatures (a few thousand to many million degrees) bring other physical processes to the fore. The energy balance is just as important as the hydrostatic equilibrium inside such a hot gas ball : only those models can be plausible in which each layer of the sun's interior transmits as much energy as it receives from below. In the center, the nuclear fusion of hydrogen in helium generates intense gamma radiation that changes upwards into X-rays and UV rays and only finally into light . In between there must be a distinctive layer from which the energy is transported by convection . Only in the last few years has it been possible to simulate such complicated models on large computers .

The problem of equilibrium is also known in the construction industry , for example when finding the shape of planar structures . A possible model approach is that all network nodes are in equilibrium of forces, while the edge nodes of the surface are fixed in space (i.e. given by external conditions).

## Individual evidence

1. Without wind and tides, this would also be the case with the actual geoid , although it has undulations of up to 100 m due to the continents and sea depths .
2. Gert Eilbracht: Workshop "Light surface structures - building with membranes" . In: Rudolfinum . 2007, p. 75–77 ( PDF on ZOBODAT [accessed February 5, 2018]).

## literature

• Heinrich Bruns , The Figure of the Earth , Berlin 1878 .
• Karl Ledersteger, Volume V (JEK), Astronomical and Physical Geodesy , 871 p., Verlag JBMetzler, Stuttgart 1969.
• Laszlo Egyed, Solid Earth Physics , 370 p., Akadémiai Kiadó, Budapest 1969.
• Kiepenheuer / Hanslmeier, Introduction to Solar Physics?