Editing ThreeDimensionalConfigurations/CAREs (section)

==Learning from Classical Fission Hypothesis==

According to the classical fission hypothesis as described in particular by Lebovitz, relying especially on equilibrium models of incompressible fluids, an initially axisymmetric configuration (Maclaurin spheroid) can spontaneously deform into an ellipsoidal configuration when the Jacobi sequence bifurcates from the Maclaurin sequence.  (Models along the Jacobi sequence are examples of Riemann S-type ellipsoids.)  Upon further contraction/cooling along the Jacobi sequence, the equilibrium configuration eventually encounters an m = 3 instability; or, alternatively, it may have an opportunity to deform gradually into a peanut-shaped, "contact" binary system.

Using self-consistent-field techniques, numerous groups have been able to numerically construct compressible analogs of Maclaurin spheroids.  Most have employed polytropic equations of state having a variety of different index values; generally speaking, highly flattened configurations can be constructed only if they are differentially rotating.

Over the years, especially through collaborations with Dick Durisen, Harold Williams, and John Cazes, nonlinear hydrodynamic techniques have been used to model the spontaneous development of nonaxisymmetric structure &#8212; almost exclusively, m = 2 bar-like distortions &#8212; in equilibrium models that are compressible analogs of Maclaurin spheroids.  Mostly we have examined models that are ''dynamically'' unstable, which means that we have chosen models that have a T/|W| that is significantly higher than what is necessary to encounter the Jacobi-sequence bifurcation.  One key exception is the modeling performed by Shangli Ou in collaboration with Lee Lindblom; we introduced a post-Newtonian radiation-reaction term into the simulation and were able to watch lower T/|W| models deform and evolve to the Dedekind sequence, which is adjoint sequence to the Jacobi sequence.

While evolving high T/|W| models that are dynamically unstable to the bar-mode, John Cazes demonstrated that the initially axisymmetric configuration evolves to a quasi-steady-state ellipsoidal configuration that, in many respects, can be described as a compressible analog of a Riemann S-type ellipsoid (CARE).  [Note:  He obtained qualitatively similar evolutionary results whether his initial Maclaurin-like model had an n' = 3/2 angular velocity profile, or had uniform vortensity.]  This relatively long-lived bar-like configuration is differentially rotating (see relevant movie) and exhibits a "violin Mach surface" where the fluid transitions from supersonic to subsonic regions.  The mild shock fronts associated with the violin Mach surface necessarily introduce dissipation, and therefore each CARE is, strictly speaking, cannot be steady-state.  Two especially relevant elements of Cazes' simulations are the following:

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In Chandrasekhar's EFE, there is a chapter/subsection that discusses the nonlinear evolution of models from the Maclaurin sequence to Riemann S-type configurations.  Evidently in the mid-to-late 60s, one researcher published a paper in which he "numerically integrated" a few such evolutions to illustrate how evolution toward a specific Riemann model might occur.  It appears as though, using energy minima arguments, Christodoulou has presented a similar evolution; he did this, in part, in an effort to perhaps quantitatively understand Cazes' simulation results.
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Cazes slowly "cooled" one of his CARE models and discovered that the bar became even more elongated over time and, eventually, encountered a "radial" oscillation which sloshed the fluid back and forth between a centrally condensed configuration to a configuration having a pair of off-axis density maxima &#8212; presenting the appearance of a common-envelope binary.  The flow-field (see movie) showed some common-envelope flow, but also displayed circulation about the off-axis density maxima.  This is the closest we have come to actually ''seeing'' an event that loosely can be described as binary formation.
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