Laplace series for the level ellipsoid of revolution

The outer gravitational potential V of the level ellipsoid of revolution T is uniquely determined by two quantities: the eccentricity (varepsilon ) of the ellipsoid and Clairaut parameter q, proportional to the angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities (varepsilon ) and q are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients (I_n) of Laplace series representing the outer potential of T are uniquely determined by (varepsilon ) and q. In this paper, we have found explicit expressions for Stokes coefficients via (varepsilon ) and q, as well as their asymptotics when (nrightarrow infty ). If T does not coincide with a Maclaurin ellipsoid, then (|I_n|sim Bvarepsilon ^n/n) with a certain constant B. Let us compare this asymptotics with one of (I_n) for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: (|I_n|sim bar{B}varepsilon ^n/(n^2)). Hence, level ellipsoids with ellipsoidal equidensites do not exist. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.