PHASE-TRANSITION THEORY OF INSTABILITIES .2. 4TH-HARMONIC BIFURCATIONS AND LAMBDA-TRANSITIONS

We use a free-energy minimization approach to describe in simple and c lear physical terms the secular and dynamical instabilities as well as the bifurcations along equilibrium sequences of rotating, self-gravit ating fluid systems. Our approach is fully nonlinear and stems from th e Ginzburg-Landau theory of phase transitions. In this paper, we exami ne fourth-harmonic axisymmetric disturbances in Maclaurin spheroids an d fourth-harmonic nonaxisymmetric disturbances in Jacobi ellipsoids. T hese two cases are very similar in the framework of phase transitions. It has been conjectured (Hachisu and Eriguchi 1983) that third-order phase transitions, manifested as smooth bifurcations in the angular mo mentum-rotation frequency plane, may occur on the Maclaurin sequence a t the bifurcation point of the axisymmetric one-ring sequence and on t he Jacobi sequence at the bifurcation point of the dumbbell-binary seq uence. We show that these transitions are forbidden when viscosity mai ntains uniform rotation. The uniformly rotating one-ring/dumbbell equi libria close to each bifurcation point and their neighboring uniformly rotating nonequilibrium states have higher free energies than the Mac laurin/Jacobi equilibria of the same mass and angular momentum. These high-energy states act as free-energy barriers preventing the transiti on of spheroids/ellipsoids from their local minima to the free-energy minima that exist on the low rotation frequency branch of the one-ring /binary sequence. At a critical point, the two minima of the free-ener gy function are equal, signaling the appearance of a first-order phase transition. This transition can take place beyond the critical point only nonlinearly if the applied perturbations contribute enough energy to send the system over the top of the barrier (and if, in addition, viscosity maintains uniform rotation). In the angular momentum-rotatio n frequency plane, the one-ring and dumbbell-binary sequences have the shape of an ''inverted S'' and two corresponding turning points each. Because of this shape, the free-energy barrier disappears suddenly pa st the higher turning point, leaving the spheroid/ellipsoid on a saddl e point but also causing a ''catastrophe'' by permitting a ''secular'' transition toward a one-ring/binary minimum energy state. This transi tion appears as a typical second-order phase transition, although ther e is no associated sequence bifurcating at the transition point (cf. C hristodoulou et al. 1995a). Irrespective of whether a nonlinear first- order phase transition occurs between the critical point and the highe r turning point or an apparent second-order phase transition occurs be yond the higher turning point, the result is fission (i.e., ''spontane ous breaking'' of the topology) of the original object on a secular ti mescale: the Maclaurin spheroid becomes a uniformly rotating axisymmet ric torus, and the Jacobi ellipsoid becomes a binary. The presence of viscosity is crucial since angular momentum needs to be redistributed for uniform rotation to be maintained. We strongly suspect that the '' secular catastrophe'' is the dynamical analog of the notorious A-trans ition of liquid He-4 because it appears as a ''second-order'' phase tr ansition with infinite ''specific heat'' at the point where the free-e nergy barrier disappears suddenly. This transition is not an elementar y catastrophe. In contrast to this case, a ''dynamical catastrophe'' t akes place from the bifurcation point to the lower branch of the Macla urin toroid sequence because all conservation laws are automatically s atisfied between the two equilibrium states. Furthermore, the free-ene rgy barrier disappears gradually, and this transition is part of the e lementary cusp catastrophe. This type of ''lambda-transition'' is the dynamical analog of the Bose-Einstein condensation of an ideal Bose ga s. The phase transitions of the dynamical systems are briefly discusse d in relation to previous numerical simulations of the formation and e volution of protostellar systems. Some technical discussions concernin g related results obtained from linear stability analyses, the breakin g of topology, and the nonlinear theories of structural stability and catastrophic morphogenesis are included in an appendix.