Phase-Transition Theory of Instabilities. III. The Third-Harmonic Bifurcation on the Jacobi Sequence and the Fission Problem
In Papers I and II, we have used a free-energy minimization approach that stems from the Landau-Ginzburg theory of phase transitions to describe in simple and clear physical terms the secular and dynamical instabilities as well as the bifurcations along equilibrium sequences of rotating, self-gravitating fluid systems. Here we investigate the secular and dynamical 3rd-harmonic instabilities that appear first on the Jacobi sequence of incompressible zero- vorticity ellipsoids. Poincare found a bifurcation point on the Jacobi sequence where a 3rd-harmonic mode becomes neutral. A sequence of pear-shaped equilibria branches off at this point but stands at higher energies. Therefore, the Jacobi ellipsoids remain secularly and dynamically stable. Cartan found that dynamical 3rd-harmonic instability also sets in at the Jacobi-pear bifurcation. We find that Cartan's instability leads to differentially rotating objects and not to uniformly rotating pear-shaped equilibria. We demonstrate that the pear-shaped sequence exists at higher energies and at higher rotation relative to the Jacobi sequence. The Jacobi ellipsoid undergoes fission on a secular time scale and a short-period binary is produced. The classical fission hypothesis of binary-star formation of Poincare and Darwin is thus feasible.