PhaseTransition Theory of Instabilities. I. SecondHarmonic Instability and Bifurcation Points
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 wellknown sequences of rotating, selfgravitating fluid and stellar systems such as the Maclaurin spheroids, the Jacobi, Dedekind, and Riemann ellipsoids, and the fluid/stellar disks. Our approach stems from the GinzburgLandau theory of phase transitions. In this paper, we focus on the Maclaurin sequence of oblate spheroidal equilibria and on the effects of nonaxisymmetric, secondharmonic disturbances. The freeenergy approach has been pioneered in astrophysics by Bertin & Radicati (1976) who showed that the secular instability beyond the MaclaurinJacobi bifurcation can be interpreted as a secondorder phase transition. We show that secondorder phase transitions appear on the Maclaurin sequence also at the points of dynamical instability (bifurcation of the x = +1 selfadjoint Riemann sequence) and of bifurcation of the Dedekind sequence. The distinguishing characteristic of each secondorder phase transition is the conservation/nonconservation of an integral of motion (a "conserved/nonconserved current") which, in effect, determines uniquely whether the transition appears or not. The secular instability beyond the Jacobi bifurcation appears only if circulation is not conserved. The secular instability at the Dedekind bifurcation appears only if angular momentum is not conserved. We show by an explicit calculation that, in the presence of dissipation agents that violate one or the other conservation law, the global minimum of the freeenergy function beyond the onset of secular instability belongs to the Jacobi and to the Dedekind sequence, respectively. In the case of a "perfect" fluid which conserves both circulation and angular momentum, the "secular" phase transitions are no longer realized and the Jacobi/Dedekind bifurcation point becomes irrelevant. The Maclaurin spheroid remains at the global minimum of the freeenergy function up to the bifurcation point of the x = +1 Riemann sequence. The x = +1 equilibria have lower free energy than the corresponding Maclaurin spheroids for the same values of angular momentum and circulation. Thus, a "dynamical" second order phase transition is allowed to take place beyond this bifurcation point. This phase transition brings the spheroid, now sitting at a saddle point of the freeenergy function, to the new global minimum on the x = +1 Riemann sequence. Circulation is not conserved in stellar systems because the stresstensor gradient terms that appear in the Jeans equations of motion include "viscositylike" offdiagonal terms of the same order of magnitude as the conventional "pressure" gradient terms. For this reason, globally unstable axisymmetric stellar systems evolve toward the "stellar" Jacobi sequence on dynamical timescales. This explains why the Jacobi bifurcation is a point of dynamical instability in stellar systems but only a point of secular instability in viscous fluids. The secondorder phase transitions on the Maclaurin sequence are discussed in relation to the dynamical instability of stellar systems, the λtransition of liquid ^{4}He, the secondorder phase transition in superconductivity, and the mechanism of spontaneous symmetry breaking.
 Publication:

The Astrophysical Journal
 Pub Date:
 June 1995
 DOI:
 10.1086/175806
 arXiv:
 arXiv:astroph/9409039
 Bibcode:
 1995ApJ...446..472C
 Keywords:

 GALAXIES: FORMATION;
 HYDRODYNAMICS;
 INSTABILITIES;
 STARS: BINARIES: GENERAL;
 STARS: FORMATION;
 Astrophysics
 EPrint:
 Postscript, compressed, uuencoded. Submitted to ApJ