Homeoid

3-D homeoid

2-D homeoid

A homeoid is a shell in three dimensions that is bordered by two concentric , similar ellipsoids . In two dimensions, a homeoid is an elliptical ring that is bordered by corresponding ellipses .

Mathematical definition

Becomes the outer boundary by an implicitly given ellipsoid

{\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} + {\ frac {z ^ {2} } {c ^ {2}}} = 1}

with the semiaxes described, so is through for the inner boundary ${\ displaystyle \! \ a, b, c}$ ${\ displaystyle 0 \ leq m \ leq 1}$

{\ displaystyle {\ frac {x ^ {2}} {a ^ {2}}} + {\ frac {y ^ {2}} {b ^ {2}}} + {\ frac {z ^ {2} } {c ^ {2}}} = m ^ {2}}

given.

In the limiting case of one speaks of thin , in the other case of thick homeoids. ${\ displaystyle \! \ m \ to 1}$

Physical meaning I

The physical meaning of homeoids in potential theory lies in the fact that no force is exerted on a test mass or charge within a homeoid that is homogeneously filled with mass or charge. The corresponding potential is therefore constant, see also the principle of defoliation . This does not apply to other elliptical shells, e.g. B. Focaloid .

The potential in the exterior of a thin homeoid is constant on ellipsoids that are confocal to this homeoid. These remarkable properties have already been demonstrated by Isaac Newton .

In astronomy and geophysics, the theory of homeoids can be used to calculate equilibrium figures. Since the density of all larger celestial bodies increases inwards, they can be modeled like onion skin through thin layers of the same density .

Definition of homeoidal distribution

One speaks of a homeoidal density distribution when the layers of constant density of a mass or charge distribution are given by concentric, similar ellipsoids.

Lines of constant density of a homeoidal distribution

Physical meaning II

Within a homeoidal density distribution, only those layers contribute to the force effect on a body, which are located within the similar ellipsoid which is concentric to the boundary and which runs through the body.

literature

S. Chandrasekhar : Ellipsoidal Figures of Equilibrium (= The Silliman Foundation Lectures 42). Yale University Press, New Haven CT et al. 1969, ISBN 0-300-01116-4 .
Edward John Routh : A Treatise on Analytical Statics. Volume II. Cambridge University Press, Cambridge 1882.