PHASE-TRANSITION THEORY OF INSTABILITIES .4. CRITICAL-POINTS ON THE MACLAURIN SEQUENCE AND NONLINEAR FISSION PROCESSES

We use a free-energy minimization approach to describe in simple and c lear physical terms the secular and dynamical instabilities as well as the bifurcations along equilibrium sequences of rotating, self-gravit ating fluid systems. Our approach is fully nonlinear and stems from th e Ginzburg-Landau theory of phase transitions. In the final paper of t his series, we examine higher than second-harmonic disturbances applie d to Maclaurin spheroids, the corresponding bifurcating sequences, and their relation to nonlinear fission processes. The triangle and ammon ite sequences bifurcate from the two third-harmonic neutral points on the Maclaurin sequence, while the square and one-ring sequences bifurc ate from two of the three known fourth-harmonic neutral points. The on e-ring sequence has been analyzed in Christodoulou et al. (1995b). In the other three cases, secular instability does not set in at the corr esponding 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 a t the ammonite bifurcation and numerically resolved second-order phase transitions at the triangle and square bifurcations are strictly forb idden. Furthermore, the ammonite sequence exists at higher rotation fr equencies as well and is similar in every respect to the pear-shaped s equence that has been analyzed in Christodoulou et al. (1995c). There is no known bifurcating sequence at the point of third-harmonic dynami cal instability. This point represents a discontinuous lambda-transiti on of type 3 that brings a Maclaurin spheroid on a dynamical timescale directly to the binary sequence while the original symmetry and topol ogy are broken in series. The remaining fourth-harmonic neutral point also appears to be related to a type-3 lambda-transition which however takes place from the lower turning point of the one-ring sequence tow ard the starting point and then on toward the stable branch of the thr ee-fluid-body (triple) sequence. A third type-3 lambda-transition, tak ing place from the one-ring sequence toward the starting point and the n on toward the stable branch of the four-fluid-body (quadruple) seque nce, is also discussed. The two-ring sequence bifurcates from the axis ymmetric sixth-harmonic neutral point on the Maclaurin sequence also t oward higher energies initially but eventually turns around and procee ds to lower energies relative to the Maclaurin sequence. The point whe re the two sequences have equal energies represents a fourth type of l ambda-transition which is not preceded by a first-order phase transiti on. This type-4 lambda-transition results in double fission on a secul ar timescale: a Maclaurin spheroid breaks into two coaxial axisymmetri c 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 detect ed by linear stability analyses. Resonances appear as first-order phas e transitions at points where the energies of the two sequences are ne arly equal but the lower energy state belongs to one of the multi-flui d-body sequences. Three nonlinear resonances leading to the turning po ints of the binary, triple, and quadruple sequences are described.