Editing ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI (section)

==Analytic Determination of Equilibrium Model Parameters==
Drawing heavily from &sect;47 (pp. 129 - 132) of [[Appendix/References#EFE|[<font color="red">EFE</font>] ]], in a separate chapter we show how the steady-state 2<sup>nd</sup>-order tensor virial equations can be used to derive the equilibrium structure of [[VE/RiemannEllipsoids#Riemann_Ellipsoids_of_Types_I,_II,_&_III|Riemann Ellipsoids of Type I, II, &amp; III]].  By definition, for these types of Riemann Ellipsoids, the two vectors <math>\vec{\Omega}</math> and <math>~\vec\zeta</math> are not parallel to any of the principal axes of the ellipsoid, and they are not aligned with each other, but they both lie in the <math>y-z</math> plane &#8212; that is to say, <math>~(\Omega_1, \zeta_1) = (0, 0)</math>.  For a given specified density <math>(\rho)</math> and choice of the three semi-axes <math>(a_1, a_2, a_3) \leftrightarrow (a, b, c)</math>, all five of the expressions displayed in that chapter's [[VE/RiemannEllipsoids#SummaryTable|''Summary Table'']] must be used in order to determine the equilibrium configuration's associated values of the five unknowns:  <math>\Pi, (\Omega_2, \zeta_2), (\Omega_3, \zeta_3)</math>.  

In an effort to simplify the constraint-equation expressions &#8212; and following the notation found in [[Appendix/References#EFE|[<font color="red">EFE</font>] ]] &#8212; we adopt the intermediary parameters:
<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\beta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{4a^2}\biggl\{ (4a^2 -b^2 + c^2  ) \mp \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, ;
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (16)</font> ]</td></tr>

<tr>
  <td align="right">
<math>
\gamma 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{4a^2}\biggl\{ (4a^2  + b^2 -c^2)  \mp \biggl[ (4a^2 + c^2 - b^2  )^2 - 16a^2 c^2 \biggr]^{1 / 2} \biggr\} \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (17)</font> ]</td></tr>
</table>

As is emphasized in [[Appendix/References#EFE|EFE]] (Chapter 7, &sect;47, p. 131) "<font color="darkgreen">&hellip; the signs in front of the radicals, in the two expressions, go together.</font>  Furthermore, "<font color="darkgreen">the two roots &hellip; correspond to the fact that, consistent with Dedekind's theorem, two states of internal motions are compatible with the same external figure.</font>"
</td></tr></table>


The five relevant constraint equations are,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>2\biggl[ \frac{3\cdot 5}{2^2\pi abc\rho} \biggr]\Pi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi G \rho c^2 \biggl\{ A_3  
+~
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  } \biggr] 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>0
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
\Omega_2^2\beta \biggl[ \frac{c^2 - b^2}{c^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  }{2a^2 b^2 }
\biggr]
~+~2\pi G\rho 
\biggl[ \frac{a^2(3b^2-4a^2 + c^2)B_{23} + b^2(a^2A_1 - c^2 A_3)  }{a^2b^2} \biggr] \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (170)</font> ]</td></tr>
</table>

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\zeta_2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_2 \beta \biggl[ \frac{a^2 + c^2}{c^2} \biggr] \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (12)</font> ]</td></tr>
</table>

<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>~
-~\Omega_3^2 \gamma \biggl[ \frac{c^2 - b^2}{b^2} \biggr]
\biggl[ \frac{ 4a^4 - a^2 (b^2 + c^2) + b^2 c^2  }{2a^2 c^2 }
\biggr]
~+~2\pi G \rho 
\biggl[ \frac{ a^2( b^2 + 3c^2 - 4a^2 ) B_{23} +c^2(a^2A_1 - b^2 A_2)  }{a^2 c^2 }
\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;51, Eq. (171)</font> ]</td></tr>
</table>

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\zeta_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>~
-~\Omega_3 \gamma \biggl[ \frac{a^2 + b^2}{b^2} \biggr] \, ,
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 7, &sect;47, Eq. (12)</font> ]</td></tr>
</table>

where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>B_{23}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{A_2 b^2 - A_3 c^2}{b^2 - c^2} \biggr] \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[ [[Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;21, Eqs. (105) &amp; (107)</font> ]</td></tr>
</table>

Throughout the [[Appendix/References#EFE|EFE]] discussion  of Type I Riemann Ellipsoids, the assumption is made that <math>b \ge a \ge c</math>.  Presumably then, when evaluating the above set of constraints, we must [[VE/RiemannEllipsoids#Specific_Case_of_a2_%3E_a1_%3E_a3|adopt the associations]], <math>(A_1, a_1) \leftrightarrow (A_m, a_m)</math>, <math>(A_2, a_2) \leftrightarrow (A_\ell, a_\ell)</math>, and <math>(A_3, a_3) \leftrightarrow (A_s, a_s)</math>.  This means that the coefficients, <math>A_1</math>, <math>A_2</math>, and <math>A_3</math> are defined by the expressions,
<div align="center">
<table align="center" border=0 cellpadding="3">
<tr>
  <td align="right">
<math>
A_2
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>2 \biggl( \frac{a_1}{a_2} \biggr)\biggl( \frac{a_3}{a_2} \biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_3
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
2\biggl( \frac{a_1}{a_2}\biggr) \biggl[  \frac{(a_1/a_2) \sin\theta - (a_3/a_2)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr]  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
A_1 = 2 - (A_2 + A_3)
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{ 2a_1 a_3}{a_2^2 }
\biggl[ \frac{
E(\theta, k) 
-~(1-k^2)
F(\theta, k)
-~(a_3/a_1)k^2\sin\theta}{k^2 (1-k^2)\sin^3\theta}
\biggr] \, ,
</math>
  </td>
</tr>

</table>
</div>
where, the arguments of the incomplete elliptic integrals are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta = \cos^{-1} \biggl(\frac{a_3}{a_2} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>k = \biggl[\frac{1 - (a_1/a_2)^2}{1 - (a_3/a_2)^2} \biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>