Editing ThreeDimensionalConfigurations/CAREs

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=New Thoughts Regarding the Formation of Binary Stars=
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It is February, 2020 and I have begun to develop some new ideas regarding the manner by which binary stars might form from initially equilibrium configurations.  These new thoughts have been sparked by my recent experience playing with &#8212; and, in particular, employing COLLADA to help build visually insightful models of &#8212; Type I Riemann ellipsoids. 
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==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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==Andalib's Work==
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Can we develop an SCF technique that let's us generate a wide range of CAREs?  This may be impossible, given that the Cazes models exhibit a violin Mach surface and, therefore, do not actually represent steady-state structures. 
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What should the 3D velocity field look like?  The Cazes CAREs appear to exhibit an angular-velocity profile that is independent of the vertical (z) coordianate.
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Using an approach analogous to that used by [[Apps/Korycansky_Papaloizou_1996#Korycansky_and_Papaloizou_.281996.29|Korycansy &amp; Papaloizou (1996)]], Andalib was able to generate two-dimensional compressible equilibrium structures that have a wide variety of nonaxisymmetric (bar-like ''and'' binary like) shapes and internal velocity distributions; they all have uniform vortensity.  Some look quite similar to the flows found in Cazes quasi-steady-state CAREs, including the common-envelope binary flow.  How do we extend this work to 3D structures?  I developed some on-line notes that are helpful, but incomplete; this was done following a discussion in the spring of 2010 that I had with David and Eric at BYU.  I tried, for example, enforcing no motion in the vertical (z) direction, but the remaining constraint equations were still pretty daunting.
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==New Idea==

Perhaps we should move away from Riemann S-type ellipsoids and attempt instead to develop an SCF technique that can be used to construct a variety of Type I Riemann ellipsoids.  This might be useful because:
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  <li>
The velocity flow-field is not constrained to be independent of z.  (Although a related constraint appears to be in place.)  This would lead to significant redesign of my above-mentioned "Incomplete" on-line notes.  This state of affairs might demand that each system evolve on a viscous time scale toward a flow field that has no z-component.
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In the Type I Riemann ellipsoids, the steady-state flow-field already shows a pair of off-axis circulations &#8212; analogous to what is seen in a binary system &#8212; even though the underlying density distribution is uniform.  This flow would presumably get "locked in" as the configuration cooled and encourage/necessarily imply the development of off-axis density maxima develop.
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Given that the spin-axis of the interflow is not aligned with the spin-axis of the ellipsoidal figure, one can imagine that a binary system formed from such a configuration would have a wide range of interesting properties; the spin and orbital angular momentum axes would be different from one another, for example.  This reminds me of work that Peter Bodenheimer did in the late 70s &#8212; see [https://ui.adsabs.harvard.edu/abs/1978ApJ...224..488B/abstract P. Bodenheimer (1978, ApJ, 224, 448)] &#8212; when he used fragmentation of "rings' to estimate how multiple multiple star systems might divide up the angular momentum.
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=See Also=
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! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/Challenges|<b>Construction<br />Challenges</b>]]<br />(Pt. 1)
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! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/ChallengesPt2|<b>Construction<br />Challenges</b>]]<br />(Pt. 2)
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! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/ChallengesPt3|<b>Construction<br />Challenges</b>]]<br />(Pt. 3)
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! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/ChallengesPt4|<b>Construction<br />Challenges</b>]]<br />(Pt. 4)
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! style="height: 150px; width: 150px; background-color:white; border-right: 2px dashed black; " |[[ThreeDimensionalConfigurations/ChallengesPt5|<b>Construction<br />Challenges</b>]]<br />(Pt. 5)
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! style="height: 150px; width: 150px; background-color:white; " |[[ThreeDimensionalConfigurations/ChallengesPt6|<b>Construction<br />Challenges</b>]]<br />(Pt. 6)
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