Phase-Transition Theory of Instabilities. IV. Critical Points on the Maclaurin Sequence and Nonlinear Fission Processes

Abstract

view Abstract Citations (9) References (34) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Phase-Transition Theory of Instabilities. IV. Critical Points on the Maclaurin Sequence and Nonlinear Fission Processes Christodoulou, Dimitris M. ; Kazanas, Demosthenes ; Shlosman, Isaac ; Tohline, Joel E. Abstract We use a free-energy 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, self-gravitating fluid systems. Our approach is fully nonlinear and stems from the Ginzburg-Landau theory of phase transitions. In the final paper of this series, we examine higher than second-harmonic 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 third-harmonic neutral points on the Maclaurin sequence, while the square and one-ring sequences bifurcate from two of the three known fourth-harmonic neutral points. The one-ring 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) third-order phase transition at the ammonite bifurcation and numerically resolved second-order 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 pear-shaped sequence that has been analyzed in Christodoulou et al. (1995c). There is no known bifurcating sequence at the point of third-harmonic dynamical instability. This point represents a discontinuous λ-transition 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 fourth-harmonic neutral point also appears to be related to a type-3 λ-transition which however takes place from the lower turning point of the one-ring sequence toward the starting point and then on toward the stable branch of the three-fluid-body (triple) sequence. A third type-3 λ-transition, taking place from the one-ring sequence toward the starting point and then on toward the stable branch of the four-fluid-body (quadruple) sequence, is also discussed. The two-ring sequence bifurcates from the axisymmetric sixth-harmonic 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 λ-transition which is not preceded by a first-order phase transition. This type-4 λ-transition 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 multi-fluid-body sequences that cannot be detected by linear stability analyses. Resonances appear as first-order phase transitions at points where the energies of the two sequences are nearly equal but the lower energy state belongs to one of the multi-fluid-body 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:astro-ph/9505101 Bibcode: 1995ApJ...446..510C Keywords: GALAXIES: FORMATION; STARS: BINARIES: GENERAL; HYDRODYNAMICS; INSTABILITIES; STARS: FORMATION; Astrophysics E-Print: 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 full text sources arXiv | ADS | Related Materials (3) Part 1: 1995ApJ...446..472C Part 2: 1995ApJ...446..485C Part 3: 1995ApJ...446..500C