Editing ThreeDimensionalConfigurations/RiemannStype (section)

===His Tabulated Model Parameters===
Table 1 (see below) lists a subset of the Riemann S-type ellipsoids that were studied by {{ Ou2006 }}; properties of various so-called ''Direct'' configurations can be found in Ou's Table 1, while properties of various ''Adjoint'' configurations can be found in his Table 5.  Each row of ''our'' Table 1 was constructed as follows:
<ul>
  <li>The pair of axis ratios <math>~(\tfrac{b}{a}, \tfrac{c}{a} )</math> associated with one of Ou's (2006) uniform-density, incompressible <math>~(n=0)</math> ellipsoid models (columns 1 and 2 from Ou's Table 1) has been copied into columns 1 and 2 of ''our'' table.</li>
  <li>Properties of ''Direct Configurations'' &hellip;</li>
  <ul>
    <li>The pair of parameter values <math>~(\omega_\mathrm{analytic}, \lambda_\mathrm{analytic})</math> that is required in order for this to be an <b>equilibrium</b> configuration &#8212; as specified by the above set of analytical expressions from EFE &#8212; is copied from, respectively, columns 11 and 13 of Ou's Table 1 into columns 3 and 4 of ''our'' table; in our table, the "analytic" subscript has been dropped from the column headings.</li>
    <li>The value of the equilibrium configuration's vorticity, <math>~\zeta</math> &#8212; see column 5 of our table &#8212; has been determined from the expression,<br /><table border="0" align="center"><tr><td align="center"><math>~\zeta = - \biggl[ \frac{1 + (b/a)^2}{b/a} \biggr] \lambda \, .</math></td></tr></table></li>
    <li>Column 6 of our table lists the value of the frequency ratio, <math>~f \equiv \zeta/\omega</math>.
  </ul>
  <li>Properties of ''Adjoint Configurations'' [in order to distinguish from ''Direct'' configuration properties, a superscript &dagger; has been attached to each parameter name] &hellip;</li>
  <ul>
    <li>As listed in column 7 of our Table, the "spin" angular velocity of the ''adjoint'' equilibrium configuration has been determined from the vorticity of the ''direct'' configuration via the relation, <br /><table border="0" align="center"><tr><td align="center"><math>~\omega^\dagger = \zeta \biggl[\frac{b/a}{1 + (b/a)^2}\biggr] \, .</math></td></tr></table></li>
    <li>As listed in column 10 of our Table, the ratio <math>~(f^\dagger)</math> of the vorticity to the angular velocity in the ''adjoint'' equilibrium configuration has been determined from the same ratio <math>~(f)</math> in the ''direct'' configuration via the relation, <br /><table border="0" align="center"><tr><td align="center"><math>~f^\dagger = \frac{1}{f} \biggl\{ \frac{[1 + (b/a)^2]^2}{(b/a)^2} \biggr\} \, .</math></td></tr></table></li>
  <li>As indicated, the value of the vorticity in the ''adjoint'' equilibrium configuration (column 9 of our table) has been determined from a product of <math>~\omega^\dagger</math> and <math>~f^\dagger</math>.</li>
    <li>As listed in column 8 of our table, the value of the parameter, <math>~\lambda^\dagger</math>, has been determined from the vorticity in the ''adjoint'' equilibrium configuration via the relation, <br /><table border="0" align="center"><tr><td align="center"><math>~\lambda^\dagger = -~ \zeta^\dagger \biggl[ \frac{b}{a} + \frac{a}{b}\biggr]^{-1} \, .</math></td></tr></table></li>
  </ul>
</ul>

<table border="1" align="center" cellpadding="8" width="90%">
<tr>
  <td align="center" colspan="10">
<b>Table 1: &nbsp; Example Riemann S-type Ellipsoids</b><br />
[Cells with a pink background contain numbers copied directly from Table 1 of {{ Ou2006 }}]<br />
[Cells with a yellow background contain numbers drawn from Table IV (p. 103) of EFE]
  </td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~\frac{b}{a}</math></td>
  <td align="center" rowspan="2"><math>~\frac{c}{a}</math></td>
  <td align="center" rowspan="1" colspan="4">
Properties of<br /><b>''Direct'' Configurations</b>
  </td>
  <td align="center" rowspan="1" colspan="4">
Properties of<br /><b>''Adjoint'' Configurations</b>
  </td>
</tr>
<tr>
  <td align="center" rowspan="1"><math>~\omega = \frac{\Omega}{\sqrt{G \rho}}</math></td>
  <td align="center" rowspan="1"><math>~\lambda</math></td>
  <td align="center" rowspan="1"><math>~\zeta </math></td>
  <td align="center" rowspan="1"><math>~f \equiv \frac{\zeta}{\omega}</math></td>
  <td align="center" rowspan="1"><math>~\omega^\dagger </math></td>
  <td align="center" rowspan="1"><math>~\lambda^\dagger </math></td>
  <td align="center" rowspan="1"><math>~\zeta^\dagger = \omega^\dagger f^\dagger</math></td>
  <td align="center" rowspan="1"><math>~f^\dagger </math></td>
</tr>
<tr>
  <td align="center">(1)</td>
  <td align="center">(2)</td>
  <td align="center">(3)</td>
  <td align="center">(4)</td>
  <td align="center">(5)</td>
  <td align="center">(6)</td>
  <td align="center">(7)</td>
  <td align="center">(8)</td>
  <td align="center">(9)</td>
  <td align="center">(10)</td>
</tr>
<tr>
  <td align="center" rowspan="7" bgcolor="white">0.90</td>
  <td align="center" bgcolor="pink">0.795</td>
  <td align="center" bgcolor="pink">1.14704</td>
  <td align="center" bgcolor="pink">0.43181</td>
  <td align="center">-0.86842</td>
  <td align="center">-0.75709</td>
  <td align="center">-0.43181</td>
  <td align="center">-1.14704</td>
  <td align="center">+2.30682</td>
  <td align="center">-5.3422</td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.641</td>
  <td align="center" bgcolor="pink">1.13137</td>
  <td align="center" bgcolor="pink">0.15077</td>
  <td align="center"> - 0.30322</td>
  <td align="center">- 0.26801</td>
  <td align="center">- 0.15077</td>
  <td align="center">-1.13137</td>
  <td align="center">2.27531</td>
  <td align="center">- 15.0913</td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.590</td>
  <td align="center" bgcolor="pink">1.10661</td>
  <td align="center" bgcolor="pink">0.06406</td>
  <td align="center">-0.12883</td>
  <td align="center">-0.11642</td>
  <td align="center">-0.06406</td>
  <td align="center">-1.10661</td>
  <td align="center">+2.22552</td>
  <td align="center">-34.7411</td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.564</td>
  <td align="center" bgcolor="pink">1.09034</td>
  <td align="center" bgcolor="pink">0.02033</td>
  <td align="center">-0.04089</td>
  <td align="center">-0.03750</td>
  <td align="center">-0.02033</td>
  <td align="center">-1.09034</td>
  <td align="center">+2.19279</td>
  <td align="center">-107.86</td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.538</td>
  <td align="center" bgcolor="pink">1.07148</td>
  <td align="center" bgcolor="pink">- 0.02324</td>
  <td align="center">+0.04674</td>
  <td align="center">+0.04362</td>
  <td align="center">+0.02324</td>
  <td align="center">- 1.07148</td>
  <td align="center">+2.15487</td>
  <td align="center">+92.722</td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.487</td>
  <td align="center" bgcolor="pink">1.02639</td>
  <td align="center" bgcolor="pink">- 0.10880</td>
  <td align="center">+0.21881</td>
  <td align="center">+0.21318</td>
  <td align="center">+0.10880</td>
  <td align="center">-1.02639</td>
  <td align="center">+2.06418</td>
  <td align="center">+18.972</td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.333</td>
  <td align="center" bgcolor="pink">0.79257</td>
  <td align="center" bgcolor="pink">- 0.39224</td>
  <td align="center">+0.78884</td>
  <td align="center">+0.99529</td>
  <td align="center">+0.39224</td>
  <td align="center">-0.79257</td>
  <td align="center">+1.59395</td>
  <td align="center">+4.06370</td>
</tr>
<tr>
  <td align="center" rowspan="4" bgcolor="white">0.28</td>
  <td align="center" bgcolor="pink">0.256</td>
  <td align="center" bgcolor="pink">0.80944</td>
  <td align="center" bgcolor="pink">0.03668</td>
  <td align="center">-0.14127</td>
  <td align="center">-0.17453</td>
  <td align="center">-0.03668</td>
  <td align="center">-0.80944</td>
  <td align="center">+3.11750</td>
  <td align="center">-84.992</td>
</tr>
<tr>
  <td align="center" bgcolor="yellow">0.245083</td>
  <td align="center" bgcolor="yellow">0.796512<sup>a</sup></td>
  <td align="center" bgcolor="yellow">0.0</td>
  <td align="center">0.0</td>
  <td align="center">0.0</td>
  <td align="center">0.0</td>
  <td align="center">&hellip;</td>
  <td align="center">&hellip;</td>
  <td align="center"><math>~\infty</math></td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.231</td>
  <td align="center" bgcolor="pink">0.77651</td>
  <td align="center" bgcolor="pink">- 0.04714</td>
  <td align="center">+0.18156</td>
  <td align="center">+0.23381</td>
  <td align="center">+0.04714</td>
  <td align="center">-0.77651</td>
  <td align="center">+2.99067</td>
  <td align="center">+63.442</td>
</tr>
<tr>
  <td align="center" bgcolor="pink">0.205</td>
  <td align="center" bgcolor="pink">0.72853</td>
  <td align="center" bgcolor="pink">- 0.13511</td>
  <td align="center">+0.52037</td>
  <td align="center">+0.71427</td>
  <td align="center">+0.13511</td>
  <td align="center">-0.72853</td>
  <td align="center">+2.80588</td>
  <td align="center">+20.7674</td>
</tr>
<tr>
  <td align="left" colspan="10">
<sup>a</sup>According to Table IV (p. 103) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], the square of the angular velocity of this Jacobi ellipsoid is, <math>~\Omega^2/(\pi G\rho) = 0.201946</math>; from this value, we find that, <math>~\omega = \sqrt{\pi} \cdot \sqrt{0.201946} = 0.796512</math>.
  </td>
</tr>
</table>