Editing ThreeDimensionalConfigurations/RiemannStype (section)

===Summary===

<span id="Fig1">It is often useful to discuss</span> the properties of Riemann S-type ellipsoids in the context of what we will refer to as the traditional "EFE Diagram" &#8212; a two-dimensional parameter space defined by the axis ratio ranges, 0 &le; b/a &le; 1 and 0 &le; c/a &le; 1.  It is useful to appreciate at the outset, for example, that Riemann S-type ellipsoids only populate a subset of the EFE Diagram's entire parameter space.  More specifically, they all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram that is shown here, on the right.  Keeping this in mind, we summarize here a sequence of steps that should be taken in order to construct and thereby quantitatively detail all of the physical properties that are associated with any Riemann S-type ellipsoid that lies in this allowed region of the EFE Diagram.

<table border="0" align="right" cellpadding="5">
<tr><td align="center">'''Figure 1'''</td></tr>
<tr><td align="center">
[[File:EFEdiagram02.png|right|350px|EFE Diagram]]
</td></tr>
<tr><td align="center">Caption:  See [[#Fig2|Figure 2, below]]</td></tr>
</table>
<ol>
<li>Specify numerical values for any two of the three key parameters:  <math>~b/a, c/a, f \equiv \zeta/\Omega_f</math>.  The value of the third (unspecified) parameter can then be found by determining the root(s) of the [[#Based_on_Virial_Equilibrium|virial-equilibrium-based expression]],
 <table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{a^2 b^2}{(a^2 + b^2)^2} \biggr] f^2 
+ \biggl[ \frac{2a^2 b^2 B_{12}}{c^2 A_3 - a^2 b^2 A_{12}} \biggr]\frac{f}{a^2 + b^2} + 1 \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;48, Eq. (35)</font> ]</td></tr>
</table>
<ol type="A">
<li>If you specified the values of <math>~b/a</math> and <math>~c/a</math>, then values of the three parameters, <math>~A_1, A_2, A_3</math> &#8212; as well as the related parameters, <math>~A_{12}, B_{12}</math> &#8212; can be immediately determined from the [[#General_Coefficient_Expressions|above general coefficient expressions]] and [[#A12B12|related relations]] as long as you have an algorithm that can be used to evaluate incomplete elliptic integrals of the first and second kind. The governing virial-equilibrium-based expression then becomes a quadratic equation whose pair of roots give two physically viable values of the parameter, <math>~f</math>; we will refer to them as <math>~f_+</math> and <math>~f_-</math>
<br />NOTE:  If the chosen pair of axis ratios places your configuration ''above'' the Jacobi/Dedekind sequence in the familiar "EFE Diagram," then the parameter, <math>~f</math>, will invariably be negative; if it is ''below'' the Jacobi/Dedekind sequence, <math>~f</math> will invariably be positive. <br />NOTE as well:  This is the method that we have used, below, in order to replicate various equilibrium configurations that have been [[#Models_Examined_by_Ou_.282006.29|studied by Ou (2006)]].
</li>
<li>If, instead, you specified the value of <math>~f</math> and (only) one of the ellipsoid's axis ratios, then an iterative numerical scheme &#8212; such as a [https://brilliant.org/wiki/newton-raphson-method/ Newton Raphson method] &#8212; will need to be used in order to determine a physically viable (real) root of this nonlinear, virial-equilibrium-based expression.  This root will provide the value of the equilibrium ellipsoid's other axis ratio.  
<br />NOTE:  The [[ThreeDimensionalConfigurations/JacobiEllipsoids#Jacobi_Ellipsoids|Jacobi/Dedelind sequence]] is determined in this manner by setting <math>~f = 0</math>, then determining what value of the c/a axis ratio is consistent with various selected values of 0 < b/a &le; 1.
</li>
<li>If, as in step 1.B, only one value of the parameter, <math>~f</math>, is known, the other relevant value may be obtained from the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_+ \cdot f_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{a^2 + b^2}{ab}\biggr]^2 \, .
</math>
  </td>
</tr>
</table>
In either case, if <math>~|f_\pm | < 1</math> the model will be referred to as being a ''Jacobi-like'' &#8212; or, ''Direct'' &#8212; configuration because the magnitude of the configuration's spin frequency, <math>~|\Omega_f|</math>, is larger than the magnitude of the frequency, <math>~|\zeta|</math>, that characterizes internal motions (vorticity).  On the other hand, if <math>~|f_\pm | > 1</math> the model will be referred to as being a ''Dedekind-like'' &#8212; or, ''Adjoint'' &#8212; configuration because the internal motions dominate.
<br />NOTE:  A so-called ''self-adjoint'' model sequence will arise when <math>~f_+ = f_-</math> for all values of the axis ratio, 0 < b/a &le; 1.  There are two such sequences, namely, when <math>~f_+ = f_- = (a^2 + b^2)/(ab)</math> &#8212; this is the curve labeled, "X =+1" in the EFE Diagram shown here on the right &#8212; or when <math>~f_+ = f_- = -(a^2 + b^2)/(ab)</math> &#8212; this is the curve labeled, "X = - 1.  In the familiar EFE diagram, these curves intersect the Maclaurin sequence (where, b/a = 1) when, respectively, <math>~f_+ = +2</math> and <math>~f_+ = -2</math>.  
</li>
</ol>
</li>
<li>
Once a consistently specified set of parameters, <math>~b/a, c/a</math> and <math>~f</math>, is known, the configuration's spin frequency may be straightforwardly obtained from another [[#Based_on_Virial_Equilibrium|virial-equilibrium-based expression]], namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\Omega_f^2}{\pi G \rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2B_{12} \biggl[ 1 + \frac{a^2 b^2 \cdot f^2}{(a^2 + b^2)^2}  \biggr]^{-1} \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;48, Eq. (33)</font> ]</td></tr>
</table>
</li>
<li>
Once a consistently specified pair of parameters, <math>~\Omega_f</math> and <math>~f</math>, is known, the configuration's vorticity can immediately be determined via the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta = \boldsymbol{\hat{k}} \zeta </math>
  </td>
  <td align="center">
&nbsp; &nbsp; where, &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\zeta =  (f \Omega_f) \, .</math>  </td>
</tr>
</table>
</li>
<li>
At every location inside a Riemann S-type ellipsoid, the fluid vorticity must be related to the underlying velocity field via the expression, <math>~\vec\zeta = \nabla \times {\vec{v}}_\mathrm{rot}</math>.  In order for the vorticity to be uniform throughout the configuration &#8212; everywhere being represented by the vector, <math>~\vec\zeta = \boldsymbol{\hat{k}} \zeta</math> &#8212; we realize that the velocity field is properly described by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\vec{v}}_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lambda \biggl[ \boldsymbol{\hat{\imath}} \biggl(\frac{a}{b}\biggr)y - \boldsymbol{\hat{\jmath}} \biggl(\frac{b}{a}\biggr)x \biggr]  \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\lambda</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{ab}{a^2 + b^2} \biggr] \zeta \, .</math>
  </td>
</tr>
</table>

{{ LL96 }}, p. 700, &sect;2, Eq. (1)<br />
{{ LL96b }}, p. 930, &sect;3, Eqs. (3.1) &amp; (3.2)<br />
{{ Ou2006 }}, p. 550, &sect;2, Eqs. (3) &amp; (17)
</div>
</li>
</ol>