PHASE-TRANSITION THEORY OF INSTABILITIES .3. THE 3RD-HARMONIC BIFURCATION ON THE JACOBI SEQUENCE AND THE FISSION PROBLEM

In Christodoulou et al. (1995a, b, hereafter Papers I and II), we used a free-energy minimization approach that stems from the Ginzburg-Landau 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. Based on the physical picture that emerged from this method, we investigate here the secular and dynamical third-harmonic instabilities that are presumed to appear first and at the same point on the Jacobi sequence of incompressible zero-vorticity ellipsoids. Poincare (1885) found a bifurcation point on the Jacobi sequence where a third-harmonic mode of oscillation becomes neutral. A sequence of pear-shaped equilibria branches off at this point, but this result does not necessarily imply secular instability. The total energies of the pear-shaped objects must also be lower than those of the corresponding Jacobi ellipsoids with the same angular momentum. This condition is not met if the pear-shaped objects are assumed to rotate uniformly. Near the bifurcation point, such uniformly rotating pear-shaped objects stand at higher energies relative to the Jacobi sequence. (e.g., Jeans 1929). This result implies secular instability in pear-shaped objects and a return to the ellipsoidal form. Therefore, assuming that uniform rotation is maintained by viscosity, the Jacobi ellipsoids continue to remain secularly stable (and thus dynamically stable as well) past the third-harmonic bifurcation point. Cartan (1924) found that dynamical third-harmonic instability also sets in at the Jacobi-pear bifurcation. This result is irrelevant in the case of uniform rotation because the perturbations used in Cartan's analysis carry vorticity and, by Kelvin's theorem of irrotational motion, cannot cause instability. Such vortical perturbations cause differential rotation that cannot be damped since viscosity has been assumed absent from Cartan's equations. Thus, Cartan's instability leads to differentially rotating objects and not to uniformly rotating pear-shaped vortical modes disappear in the presence of any amount of viscosity (cf. Narayan, Goldreich, and Goodman 1987). The fourth-harmonic bifurcation on the Jacobi sequence leads to the dumbbell equilibria that also have initially higher total energies (Paper II). From these considerations, we deduce that a Jacobi ellipsoid can evolve away from the sequence only via a discontinuous lambda-transition (Paper II), provided there exists a branch of lower energy and broken topology in any of the known bifurcating sequences. The breaking of topology circumvents Kelvin's theorem and allows a zero-vorticity Jacobi ellipsoid to abandon the sequence. A pear-shaped sequence has been obtained numerically by Eriguchi, Hachisu, and Sugimoto (1982). Using their results, we demonstrate that the entire sequence exists at higher energies and at higher rotation frequencies relative to the Jacobi sequence. These results were expected since they were predicted by the classical analytical calculations of Jeans (1929). Furthermore, the computed pear-shaped sequence terminates prematurely above the Jacobi sequence due to equatorial mass shedding and thus has no lower energy branch of broken topology. Therefore, there exists no lambda-transition associated with the pear-shaped sequence. In this case, the first lambda-transition on the Jacobi sequence is of type 2 and appears past the higher turning point of the dumbbell-binary sequence. This transition has been described in Paper II. The Jacobi ellipsoid undergoes fission on a secular timescale arid a short-period binary is produced. The classical fission hypothesis of binary star formation of Poincare and Darwin is thus feasible. In all stages, evolution proceeds quasistatically, and thus the resulting fission is retarded just as was anticipated by Tassoul (1978). Modern approaches to the fission problem (Lebovitz 1972; Ostriker and Bodenheimer 1973), involving perfect fluid masses, are also discussed briefly in the context of phase transitions. The above conclusions can be strengthened by a more accurate computation of the pear-shaped sequence and by hydrodynamical simulations of viscous Jacobi ellipsoids prior to and past the lambda-point of the dumbbell-binary sequence.