Equilibrium figure of a rotating body composed of two-layered incompressible fluids with application to the theory of lunar origin

Recent geochemical research on the bulk composition of the Moon implies that the lunar material comes from the terrestrial magma ocean. Rotational fission seems to be a favorable candidate for the mechanism to tear off the magma ocean although it has several physical difficulties. In most of the previous studies on rotational fission, the proto-Earth was treated as a single-layered body. In this study, we modeled the proto-Earth as a superposition of two layers of incompressible fluids with different densities to take into account the effect of inhomogeneity of the proto-Earth. We first looked for the gravitational equilibrium figure of such rotating two-layered axisymmetric masses numerically. We found that, when the rotation is slow, both the body surface and the boundary between the two layers are well approximated by spheroids. As the body rotates faster, the body surface begins to deviate from a spheroid while the boundary between the two layers stays spheroidal. Furthermore, when the rotational angular momentum beyonds a certain critical value, the outward centrifugal force overcomes the inward gravitational force and the body can no longer keep a single figure. We next investigated the stability of the equilibrium figures against small perturbations using the criterion derived by Chandrasekhar & Lebovitz (1962). For the cases where the values of the parameters are suitable for the proto-Earth, the condition of secular instability is almost the same as that for the Maclaurin spheroid. Moreover, the body reaches the critical angular momentum before dynamical instability occurs. However, the critical angular momentum is almost the same as that needed to trigger dynamical instability. Therefore, even if the inhomogeneity of the proto-Earth is taken account, rotational fission is an unlikely mechanism for lunar origin.