Gravitational field of the homogeneous rotational ellipsoidal body: a simple derivation and applications

Abstract

We calculate the gravitational intensity and potential of a homogeneous body
with the shape of the rotational ellipsoid. The calculation is performed in ellipsoidal coordinates
and uses the properties of harmonic functions expressed as ellipsoidal harmonics.
The resulting formulae for the internal and external fields are expressed in ellipsoidal coordinates
and (in the case of external field) also in spherical coordinates. The results are
used in the calculation of the gravitational field of a layered body whose layer boundaries
are rotational ellipsoids with common centre and rotational axis; the density in each layer
is constant. The equilibrium of such a layered rotating body is examined: it is found that
there is no equilibrium for such a body except the case that the body is homogeneous
(thus proving once more the important, but rarely mentioned, fact).