Phase-Transition Theory of Instabilities. II. Fourth-Harmonic Bifurcations and Lambda-Transitions

We use a free-energy minimization approach to describe 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 this paper, we examine fourth-harmonic axisymmetric disturbances in Maclaurin spheroids and fourth-harmonic nonaxisymmetric disturbances in Jacobi ellipsoids. These two cases are very similar in the framework of phase transitions. Irrespective of whether a nonlinear first-order phase transition occurs between the critical point and the higher turning point or an apparent second-order 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 time scale: 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. The phase transitions of the dynamical systems are briefly discussed in relation to previous numerical simulations of the formation and evolution of protostellar systems.Comment: 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 Ap

Gravitational instabilities in protostellar disks

The nonaxisymmetric stability of self-gravitating, geometrically thick accretion disks has been studied for protostellar systems having a wide range of disk-to-central object mass ratios. Global eigenmodes with four distinctly different characters were identified using numerical, nonlinear hydrodynamic techniques. The mode that appears most likely to arise in normal star formation settings, however, resembles the 'eccentric instability' that was identified earlier in thin, nearly Keplerian disks: It presents an open, one-armed spiral pattern that sweeps continuously in a trailing direction through more than 2-pi radians, smoothly connecting the inner and outer edges of the disk, and requires cooperative motion of the point mass for effective amplification. This particular instability promotes the development of a single, self-gravitating clump of material in orbit about the point mass, so its routine appearance in our simulations supports the conjecture that the eccentric instability provides a primary route to the formation of short-period binaries in protostellar systems

Nonlinear Development of the Secular Bar-mode Instability in Rotating Neutron Stars

We have modelled the nonlinear development of the secular bar-mode instability that is driven by gravitational radiation-reaction (GRR) forces in rotating neutron stars. In the absence of any competing viscous effects, an initially uniformly rotating, axisymmetric $n=1/2$ polytropic star with a ratio of rotational to gravitational potential energy $T/|W| = 0.181$ is driven by GRR forces to a bar-like structure, as predicted by linear theory. The pattern frequency of the bar slows to nearly zero, that is, the bar becomes almost stationary as viewed from an inertial frame of reference as GRR removes energy and angular momentum from the star. In this ``Dedekind-like'' state, rotational energy is stored as motion of the fluid in highly noncircular orbits inside the bar. However, in less than 10 dynamical times after its formation, the bar loses its initially coherent structure as the ordered flow inside the bar is disrupted by what appears to be a purely hydrodynamical, short-wavelength, ``shearing'' type instability. The gravitational waveforms generated by such an event are determined, and an estimate of the detectability of these waves is presented.Comment: 25 pages, 9 figures, accepted for publication in ApJ, refereed version, updated, for quicktime movie, see http://www.phys.lsu.edu/~ou/movie/fmode/new/fmode.b181.om4.2e5.mo

A new criterion for Bar-Forming Instability in Rapidly Rotating Gaseous and Stellar Systems. II. Nonaxisymmetric Form

We have previously introduced the parameter `alpha' as an indicator of stability to m=2 nonaxisymmetric modes in rotating, self-gravitating, axisymmetric, gaseous and stellar systems. This parameter can be written as a function of the total rotational kinetic energy, the total gravitational potential energy, and as a function of the topology/connectedness and the geometric shape of a system. Here we extend the stability criterion to nonaxisymmetric equilibrium systems, such as ellipsoids and elliptical disks and cylinders. We test the validity of this extension by considering predictions for previously published, gaseous and stellar, nonaxisymmetric models. The above formulation and critical values account accurately for the stability properties of m=2 modes in gaseous Riemann S-type ellipsoids (including the Jacobi and Dedekind ellipsoids) and elliptical Riemann disks as well as in stellar elliptical Freeman disks and cylinders: all these systems are dynamically stable except for the stellar elliptical Freeman disks that exhibit a relatively small region of m=2 dynamical instability.Comment: 17 pages, postscript, compressed, uuencoded. 10 figures available by anonymous ftp from ftp://asta.pa.uky.edu/shlosman/bar2/ (mget *.ps.Z). To appear in Ap.J

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

We use a free-energy minimization approach to describe 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 Landau-Ginzburg theory of phase transitions. Here we examine higher than 2nd-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 3rd-harmonic neutral points on the Maclaurin sequence while the square and one-ring sequences bifurcate from two of the three known 4th harmonic neutral points. 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. There is no known bifurcating sequence at the point of 3rd-harmonic dynamical instability. 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.Comment: 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 Ap