PhaseTransition Theory of Instabilities. II. FourthHarmonic Bifurcations and lambda Transitions
Abstract
We use a freeenergy minimization approach 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. Our approach is fully nonlinear and stems from the GinzburgLandau theory of phase transitions. In this paper, we examine fourthharmonic axisymmetric disturbances in Maclaurin spheroids and fourthharmonic nonaxisymmetric disturbances in Jacobi ellipsoids. These two cases are very similar in the framework of phase transitions.
It has been conjectured (Hachisu & Eriguchi 1983) that thirdorder phase transitions, manifested as smooth bifurcations in the angular momentumrotation frequency plane, may occur on the Maclaurin sequence at the bifurcation point of the axisymmetric onering sequence and on the Jacobi sequence at the bifurcation point of the dumbbellbinary sequence. We show that these transitions are forbidden when viscosity maintains uniform rotation. The uniformly rotating onering/dumbbell equilibria close to each bifurcation point and their neighboring uniformly rotating nonequilibrium states have higher free energies than the Maclaurin/Jacobi equilibria of the same mass and angular momentum. These highenergy states act as freeenergy barriers preventing the transition of spheroids/ellipsoids from their local minima to the freeenergy minima that exist on the low rotation frequency branch of the onering/binary sequence. At a critical point, the two minima of the freeenergy function are equal, signaling the appearance of a firstorder 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 momentumrotation frequency plane, the onering and dumbbellbinary sequences have the shape of an "inverted S" and two corresponding turning points each. Because of this shape, the freeenergy barrier disappears suddenly past the higher turning point, leaving the spheroid/ellipsoid on a saddle point but also causing a "catastrophe" by permitting a "secular" transition toward a onering/binary minimum energy state. This transition appears as a typical secondorder phase transition, although there is no associated sequence bifurcating at the transition point (cf. Christodoulou et al. 1995a).
Irrespective of whether a nonlinear firstorder phase transition occurs between the critical point and the higher turning point or an apparent secondorder phase transition occurs beyond the higher turning point, the result is fission (i.e., "spontaneous breaking" of the topology) of the original object on a secular timescale: the Maclaurin spheroid becomes a uniformly rotating axisymmetric 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 λtransition of liquid ^{4}He because it appears as a "secondorder" phase transition with infinite "specific heat" at the point where the freeenergy barrier disappears suddenly. This transition is not an elementary catastrophe. In contrast to this case, a "dynamical catastrophe" takes place from the bifurcation point to the lower branch of the Maclaurin toroid sequence because all conservation laws are automatically satisfied between the two equilibrium states. Furthermore, the freeenergy barrier disappears gradually, and this transition is part of the elementary cusp catastrophe. This type of "λtransition" is the dynamical analog of the BoseEinstein condensation of an ideal Bose gas.
The phase transitions of the dynamical systems are briefly discussed in relation to previous numerical simulations of the formation and evolution of protostellar systems. Some technical discussions concerning related results obtained from linear stability analyses, the breaking of topology, and the nonlinear theories of structural stability and catastrophic morphogenesis are included in an appendix.
 Publication:

The Astrophysical Journal
 Pub Date:
 June 1995
 DOI:
 10.1086/175807
 arXiv:
 arXiv:astroph/9504062
 Bibcode:
 1995ApJ...446..485C
 Keywords:

 GALAXIES: KINEMATICS AND DYNAMICS;
 GALAXIES: STRUCTURE;
 INSTABILITIES;
 STARS: FORMATION;
 STARS: ROTATION;
 Astrophysics
 EPrint:
 34 pages, postscript, compressed,uuencoded. 7 figures available in postscript, compressed form by anonymous ftp from asta.pa.uky.edu (cd /shlosman/paper2 mget *.ps.Z). To appear in ApJ