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

摘要

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 equilibria. Physically, this instability is not realized in viscous Jacobi ellipsoids because the vortical modes disappear in the presence of any amount of viscosity (cf. Narayan, Goldreich, & Goodman 1987).