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The exact solution for the shape and gravitational field of a rotatingtwo-layer Maclaurin ellipsoid of revolution is compared with predictions of thetheory of figures up to third order in the small rotational parameter of thetheory of figures. An explicit formula is derived for the externalgravitational coefficient $J_2$ of the exact solution. A new approach to theevaluation of the theory of figures based on numerical integration of ordinarydifferential equations is presented. The classical Radau-Darwin formula isfound not to be valid for the rotational parameter \epsilon_2 = \Omega^2/(2\piG\rho_2) >= 0.17 since the formula then predicts a surface eccentricity that issmaller than the eccentricity of the core-envelope boundary. Interfaceeccentricity 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 anenvelope density of 3000 kg m^(-3) the failure of the Radau-Darwin formulacorresponds to a rotation period of about 3 hr. Application of the exactsolution 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 canprovide realistic approximations to terrestrial planets and icy outer planetsatellites. The two-layer model needs to be generalized to allow for acontinuous envelope (outer layer) radial density profile in order torealistically model a gas or ice giant planet.

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