PhaseTransition Theory of Instabilities. IV. Critical Points on the Maclaurin Sequence and Nonlinear Fission Processes
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 the final paper of this series, we examine higher than secondharmonic disturbances applied to Maclaurin spheroids, the corresponding bifurcating sequences, and their relation to nonlinear fission processes.
The triangle and ammonite sequences bifurcate from the two thirdharmonic neutral points on the Maclaurin sequence, while the square and onering sequences bifurcate from two of the three known fourthharmonic neutral points. The onering sequence has been analyzed in Christodoulou et al. (1995b). In the other three cases, secular instability does not set in at the corresponding bifurcation points because the sequences stand and terminate at higher energies relative to the Maclaurin sequence. Consequently, an anticipated (numerically unresolved) thirdorder phase transition at the ammonite bifurcation and numerically resolved secondorder phase transitions at the triangle and square bifurcations are strictly forbidden. Furthermore, the ammonite sequence exists at higher rotation frequencies as well and is similar in every respect to the pearshaped sequence that has been analyzed in Christodoulou et al. (1995c).
There is no known bifurcating sequence at the point of thirdharmonic dynamical instability. This point represents a discontinuous λtransition of type 3 that brings a Maclaurin spheroid on a dynamical timescale directly to the binary sequence while the original symmetry and topology are broken in series. The remaining fourthharmonic neutral point also appears to be related to a type3 λtransition which however takes place from the lower turning point of the onering sequence toward the starting point and then on toward the stable branch of the threefluidbody (triple) sequence. A third type3 λtransition, taking place from the onering sequence toward the starting point and then on toward the stable branch of the fourfluidbody (quadruple) sequence, is also discussed.
The tworing sequence bifurcates from the axisymmetric sixthharmonic neutral point on the Maclaurin sequence also toward higher energies initially but eventually turns around and proceeds to lower energies relative to the Maclaurin sequence. The point where the two sequences have equal energies represents a fourth type of λtransition which is not preceded by a firstorder phase transition. This type4 λtransition results in double fission on a secular timescale: a Maclaurin spheroid breaks into two coaxial axisymmetric tori that rotate uniformly and with the same frequency.
Finally, our nonlinear approach easily identifies resonances between the Maclaurin sequence and various multifluidbody sequences that cannot be detected by linear stability analyses. Resonances appear as firstorder phase transitions at points where the energies of the two sequences are nearly equal but the lower energy state belongs to one of the multifluidbody sequences. Three nonlinear resonances leading to the turning points of the binary, triple, and quadruple sequences are described.
 Publication:

The Astrophysical Journal
 Pub Date:
 June 1995
 DOI:
 10.1086/175809
 arXiv:
 arXiv:astroph/9505101
 Bibcode:
 1995ApJ...446..510C
 Keywords:

 GALAXIES: FORMATION;
 STARS: BINARIES: GENERAL;
 HYDRODYNAMICS;
 INSTABILITIES;
 STARS: FORMATION;
 Astrophysics
 EPrint:
 23 pages, postscript, compressed, uuencoded. Figs. (6) available by anonymous ftp from ftp://asta.pa.uky.edu/shlosman/paper4/ , get *.ps.Z). To appear in ApJ