Shapes and gravitational fields of rotating two-layer Maclaurin ellipsoids: Application to planets and satellites

TL;DR

This paper compares exact and approximate solutions for the shape and gravitational field of rotating two-layer ellipsoids, deriving new formulas and applying them to planetary models, highlighting limitations of classical theories at high rotation rates.

Contribution

It introduces an explicit formula for the external gravitational coefficient and a numerical approach to the theory of figures, improving modeling of planetary shapes and fields.

Findings

Radau-Darwin formula fails for high rotational parameters

Exact solutions provide more accurate planetary shape modeling

Two-layer models approximate terrestrial and icy planets effectively

Abstract

The exact solution for the shape and gravitational field of a rotating two-layer Maclaurin ellipsoid of revolution is compared with predictions of the theory of figures up to third order in the small rotational parameter of the theory of figures. An explicit formula is derived for the external gravitational coefficient $J_2$ of the exact solution. A new approach to the evaluation of the theory of figures based on numerical integration of ordinary differential equations is presented. The classical Radau-Darwin formula is found not to be valid for the rotational parameter \epsilon_2 = \Omega^2/(2\pi G\rho_2) >= 0.17 since the formula then predicts a surface eccentricity that is smaller than the eccentricity of the core-envelope boundary. Interface eccentricity must be smaller than surface eccentricity. In the formula for $\epsilon_2$, $\Omega$ is the angular velocity of the two-layer body, $\rho_2$ is the density of the outer layer, and G is the gravitational constant. For an envelope density of 3000 kg m^(-3) the failure of the Radau-Darwin formula corresponds to a rotation period of about 3 hr. Application of the exact solution and the theory of figures is made to models of Earth, Mars, Uranus, and Neptune. The two-layer model with constant densities in the layers can provide realistic approximations to terrestrial planets and icy outer planet satellites. The two-layer model needs to be generalized to allow for a continuous envelope (outer layer) radial density profile in order to realistically model a gas or ice giant planet.