Dynamic fission instability of dissipative protoplanets

All theories of fission require a catastrophic, dynamic phase in order to produce two separate bodies. We have used nonlinear numerical and linear analytical calculations to show that the dynamic fission instability probably does not occur in dissipative protoplanets. The numerical calculations were performed with a three-spatial-dimension hydrodynamical code, with the proto-planet represented by a fluid with a Murnaghan equation of state. The kinetic energy in the protoplanet (other than rigid body rotation) is dissipated throughout the evolution in order to simulate the effects of viscous dissipation. Protoplanets rotating above the limit for dynamic instability were given initial asymmetric density perturbations; in each case the asymmetry did not grow during a time on the order of the rotational period. This dynamical stability has been verified by including the dissipative terms in the tensor-virial equation analysis for the stability of a Maclaurin spheroid: the dynamic instability vanishes when the dissipative terms are included, while the secular instability (with a growth time much larger than the rotational period) remains. The result applies to bodies of radius R with a kinematic viscosity ν? 4 × 10¹³ (R/6400 km)²cm²sec^?1, and hence may be applicable to any terrestrial protoplanet which is not totally molten. Current thermal histories for the Earth predict a partially molten mantle with a viscosity greater than this critical value. Depending on the detailed rheology of the early Earth, our results appear to rule out the possibility of forming the Earth-Moon system through a dynamic fission instability.