Laplace series of ellipsoidal figures of revolution

Abstract

Theory of the figures of equilibrium was developed actively during XIX century when causes making the form of observable massive celestial bodies (the Sun, planets, moons) almost ellipsoidal were discovered. The existence of exactly ellipsoidal figures was established. The gravitational potential of such figures can be presented by Laplace series. Its coefficients (Stokes constants In) are defined by a certain integral operator. The general term of the series was found under condition that equidensits (surfaces of equal density) are homothetic to the outer surface of the ellipsoid of revolution. First few terms were found for several other mass distributions. Here we have found the general term of the series under condition that equidensits are ellipsoids of revolution with an oblateness increasing from the centre to the surface. Simple estimates and asymptotics of In are also found. It turned out that asymptotics depends on the mean density, the density on the surface of the outer ellipsoid, and its oblateness only. Refs 12. Fig. 1.