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Exact solutions for the gravitational potential of a family of heterogeneous spheroids


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An infinite family of heterogeneous spheroids has been found for which exact or closed-form solutions for the Newtonian gravitational potential can be given. The family includes both axisymmetric spheroids and spheroids where the matter density varies harmonically with the azimuthal angle. For the axisymmetric family of spheroids, which have no azimuthal dependence of the density, the potential external to the spheroid is of the same form as the potential exterior to a spheroidal homoeoid. It is therefore constant on the surface of the spheroid and on all external spheroidal surfaces confocal with it. The potential is also constant on all internal confocal spheroidal surfaces, with the value on each confocal surface dependent on the density distribution chosen. However, the density is not constant on either concentric or confocal spheroids. These solutions can be considered to be generalizations of analogous spherical solutions given in a companion paper by Conway. For the classical solutions for homogeneous spheroids, the surface is not equipotential, and these are not included within the new family of solutions, except in the special case of a homogeneous sphere.

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