Mass distribution

As a mass distribution geoscientists, astronomers, physicists and engineers refer to the spatial distribution of mass within a solid body or well-defined fluid . Through the mathematical or coordinative description of the masses - for example as delimited units, as a distribution of mass points or as a radially symmetrical density function - various parameters of the body can be calculated and physical models of its interior can be set up.

The mass distribution of the body determines u. a. the position of the center of gravity , the moment of inertia and influences the dynamic behavior, such as natural vibrations . Also Functional the gravity potential in the interior and exterior space are influenced by the arrangement of the masses, in particular if the distribution asymmetrical is. Therefore one can deduce the inner structure of planets from orbital disturbances of satellites .

In technology, for example in ships, the position of the volume contained in the hull in relation to the mass distribution is decisive for the buoyancy behavior and thus the stability of the ship in the water.

Density distribution in fluids

In principle, the mass distribution is also important for liquid and gaseous bodies . On earth one can assume a homogeneous distribution of the molecules or atoms for small volumes or incompressible liquids . However, the further the volume under consideration extends (especially vertically), the more precisely the relationship between density and pressure has to be modeled (see gas laws and compression module ). In meteorology and oceanography , the effect of currents must also be taken into account.

The astronomy studies the mass and density distribution also very extensive fluids. Examples are: interstellar gas , dark clouds , gas planets and the interior of stars .

Inhomogeneous mass distributions play an important role u. a. in the formation and development of stars and in the internal structure of planets .

Modeling of inhomogeneous rotating bodies

For the mathematical modeling of celestial bodies and their physical behavior, astronomers and geophysicists are familiar with a number of methods in which the interior of the body is simulated by a large number of mass points and their mutual gravity . In the case of bodies with a high temperature, the thermodynamics must also be taken into account, to which there are also magnetic and other interactions, as well as high- energy radiation and nuclear physical processes inside stars .

Shell-shaped bodies at low temperatures, on the other hand, can be described - mathematically more strictly - by methods of potential theory or by hydrostatic equilibrium figures, the individual shells of which have a homogeneous mass distribution (i.e. constant density ).

With rotating bodies (wheels, shafts , components of machines, earth , other celestial bodies) the mass distribution has strong repercussions on the deformation and stability - see imbalance of a wheel, flattening and pole movement of the earth - and in the case of extremely uneven distribution can result from cracks and uneven centrifugal force lead to breakage .

Equilibrium figures of planets

If the volume and mass of a planet are known, its mean density follows - for the earth, for example, 5.52 g / cm³, for Jupiter 1.30 g / cm³. Because of the increasing pressure towards the center, the density must also increase inwards. With a known rotation time and flattening, this results in the possibility of determining the internal density curve with the help of equilibrium figures.

The simplest model is a homogeneous MacLaurin ellipsoid , for which the parameters of the earth (rotation 1 sidereal day, density 5.52) result in a flattening of 1: 231. In fact, the flattening of the earth is 1: 298.25 (equator radius 6378.13 km, pole radius 6356.74 km), because the earth's core is much denser and contributes less to the centrifugal force than the earth's mantle . With a two-shell ellipsoid ( Wiechert model , density approx. 4 and 12 g / cm) one comes closer to the true conditions, with further density parameters the moment of inertia of the earth can also be represented precisely.

For Jupiter (rotation time about 10 hours) there is an even stronger increase in density, without which its large flattening of 1:16 would not be explainable.

This method works less with Mars because it is small (low centrifugal force) and also asymmetrical (lowlands in the north, crater and highlands in the south). Here, however, the precession movement of its axis of rotation allows the rock density of the core to be estimated.