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 \(n\rightarrow \infty \). If T does not coincide with a Maclaurin ellipsoid, then \(|I_n|\sim B\varepsilon ^n/n\) 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.