PhaseTransition Theory of Instabilities. III. The ThirdHarmonic Bifurcation on the Jacobi Sequence and the Fission Problem
Abstract
In Christodoulou et al. (1995a, b, hereafter Papers I and II), we used a freeenergy minimization approach that stems from the GinzburgLandau 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, selfgravitating fluid systems. Based on the physical picture that emerged from this method, we investigate here the secular and dynamical thirdharmonic instabilities that are presumed to appear first and at the same point on the Jacobi sequence of incompressible zerovorticity ellipsoids.
Poincaré (1885) found a bifurcation point on the Jacobi sequence where a thirdharmonic mode of oscillation becomes neutral. A sequence of pearshaped equilibria branches off at this point, but this result does not necessarily imply secular instability. The total energies of the pearshaped objects must also be lower than those of the corresponding Jacobi ellipsoids with the same angular momentum. This condition is not met if the pearshaped objects are assumed to rotate uniformly. Near the bifurcation point, such uniformly rotating pearshaped objects stand at higher energies relative to the Jacobi sequence (e.g., Jeans 1929). This result implies secular instability in pearshaped 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 thirdharmonic bifurcation point.
Cartan (1924) found that dynamical thirdharmonic instability also sets in at the Jacobipear 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 pearshaped 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).
The fourthharmonic 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 λtransition (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 zerovorticity Jacobi ellipsoid to abandon the sequence.
A pearshaped sequence has been obtained numerically by Eriguchi, Hachisu, & 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 pearshaped 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 λtransition associated with the pearshaped sequence.
In this case, the first λtransition on the Jacobi sequence is of type 2 and appears past the higher turning point of the dumbbellbinary sequence. This transition has been described in Paper II. The Jacobi ellipsoid undergoes fission on a secular timescale and a shortperiod binary is produced. The classical fission hypothesis of binary star formation of Poincaré and Darwin is thus feasible. In all stages, evolution proceeds quasi statically, and thus the resulting fission is retarded just as was anticipated by Tassoul (1978). Modern approaches to the fission problem (Lebovitz 1972; Ostriker & 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 pearshaped sequence and by hydrodynamical simulations of viscous Jacobi ellipsoids prior to and past the λpoint of the dumbbellbinary sequence.
 Publication:

The Astrophysical Journal
 Pub Date:
 June 1995
 DOI:
 10.1086/175808
 arXiv:
 arXiv:astroph/9505008
 Bibcode:
 1995ApJ...446..500C
 Keywords:

 GALAXIES: FORMATION;
 STARS: BINARIES: GENERAL;
 HYDRODYNAMICS;
 INSTABILITIES;
 STARS: FORMATION;
 Astrophysics
 EPrint:
 22 pages (including 2 figures), postscript, compressed, uuencoded. Also available by anonymous ftp at ftp://asta.pa.uky.edu/shlosman/paper3 , mget *.ps.Z). To appear in ApJ