Editing Appendix/Ramblings/ForCohlHoward (section)

==Introductory Remarks==

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On 4 January 2023, Shawn Walker reintroduced the following strategy:

"<font color="darkgreen">But if I remember correctly, I think the idea was to start with some baby steps.  Perhaps have a fixed domain (the star) and solve the Euler equations in this with perhaps a fixed “fake” gravitational potential.  Just to get something working.  Then add the Poisson equation for the gravity, which would couple with the fluid.</font>"
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Joel's response:  I am completely with you regarding the idea of starting with baby steps.  But if we totally focus on Riemann S-Type ellipsoids, even the first few baby steps can be very realistic.  Immediately below is a table that itemizes my recommendation regarding which ellipsoids should be modeled fist, second, third, etc. and why.  What follows is a short preview.

Every S-Type configuration is a uniform-density ellipsoid (semi-axes: a, b, c) whose gravitational potential -- inside, outside, and on the surface of the ellipsoid -- is known analytically in terms of incomplete elliptic integrals of the 1st and 2nd kind (or simpler, trigonometric functions).  When the chosen ellipsoid is viewed from a frame that is rotating (about its c-axis) with a characteristic frequency, <math>\Omega</math>, the ellipsoidal configuration appears stationary.  As a result, the (analytically specified) gravitational potential does not vary with time.  For our purposes, this can naturally serve as the "fixed fake" potential to which you refer.  

As viewed from the proper (<math>\Omega</math>) rotating frame, we also know the exact solution to the Euler equations throughout every S-Type ellipsoid.  It is a steady-state flow in which all Lagrangian fluid elements move along closed elliptical orbits that &hellip;
<ul>
  <li>lie in planes parallel to the system's (''a-b'') equatorial plane;</li>
  <li>have identical ellipticities (that is, identical axis ratios, <math>b/a</math>);</li>
  <li>have identical orbital frequencies.</li>
</ul>  

So, as we try to build a code that can solve the Euler equations, we can (A) build a coordinate grid that is rotating with the desired (<math>\Omega</math>) frequency; (B) "guess" a velocity flow-field that corresponds to what we know to be the correct, steady-state solution; (C) hold fixed the analytically known gravitational potential; (D) then use the code to integrate the Euler equations forward in time and see how well the configuration maintains a steady-state flow.  We will have successfully debugged the "Euler equations" code if steady-state has been achieved for a variety of different Riemann S-Type ellipsoids.

Once the "Euler equations" code has been properly debugged, we can then add a Poisson solver; this will allow us to examine non-steady-state flows that result from natural instabilities among the set of Riemann S-Type ellipsoids.

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Other useful attributes of Riemann S-Type ellipsoids &hellip;
<ul>
<li>Each is a (incompressible) uniform-density ellipsoid with semi-axes, <math>(a, b, c)</math>; ''usually'' without loss of generality we are able to set <math>a = 1</math>.</li>
<li>The steady-state velocity flow-field that is associated with each equilibrium configuration can be characterized by the values of two physical parameters: a frequency, <math>\Omega</math>, that is associated with the rate of the ellipsoidal figure's spin about its <math>c</math>-axis; and, a vorticity, <math>\zeta</math>, associated with the elliptical orbital motion of Lagrangian fluid elements ''inside and on the surface of'' the ellipsoid, when viewed from a frame that is rotating with the frequency, <math>\Omega</math>.</li>
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Details &hellip;
<ul>
<li>A pair of equilibrium configurations exists if the specified set of axis ratios <math>(b/a, c/a)</math> falls between the "upper-self-adjoint" (USA; <math>x = -1</math>) and "lower-self-adjoint" (LSA; <math>x=+1</math>) sequences; this "horn-shaped" sub-domain is identified in both figure panels (1a) and (1b), immediately below. </li>
<li>At each point in this <math>(b/a, c/a)</math> sub-domain, the pair of equilibrium models are distinguished from one other by the behavior of the fluid flow;</li>
<li>Each is steady-state when viewed from ... </li>
</ul>
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  <td align="center" colspan="3">Interesting Domains and/or Equilibrium Sequences in the <math>(b/a, c/a)</math> Diagram</td>
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Figure 2 extracted from &hellip;<br />
{{ Chandrasekhar65_XXVfigure }}
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Boundary defined by USA and LSA sequences
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Jacobi Sequence:<br />(blue) Points defined by data in Table IV of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 6, &sect;39 (p. 103)</font>; <br />(red) points generated here from [[#Roots_of_the_Governing_Relation|above-defined roots of the governing relation]].
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<b>(1a)</b><br />
[[File:ChandrasekharFig2annotated.png|400px|Chandra Diagram]]
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<b>(1b)</b><br />
[[File:SelfAdjointPlot.png|400px|USA and LSA Plot]]
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<b>(1c)</b><br />
[[File:EFEdiagram02.png|400px|Jacobi/Dedekind Sequence]]
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