Editing 3Dconfigurations/DescriptionOfRiemannTypeI (section)

===Frequency Ratio===

In the context of [[ThreeDimensionalConfigurations/RiemannStype#Equilibrium_Conditions_for_Riemann_S-type_Ellipsoids|Riemann S-type ellipsoids]], we have found it useful to examine model sequences along which the frequency ratio,
<table border="0" align="center" cellpadding="8">
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  <td align="right"><math>f</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\frac{\zeta_3}{\Omega_3}\, ,</math></td>
</tr>
</table>
is constant.  Below, we will examine how such sequences behave across the domain of Type I Riemann Ellipsoids.  In anticipation of this discussion, here we examine how <math>f</math> varies along the ''limiting'' Maclaurin spheroid sequence.  

Adopting the parameter,
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  <td align="right"><math>\mathcal{H} \equiv \frac{16B_{13}}{\Omega^2_\mathrm{Mc}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{8}{e^2}
</math>
  </td>
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</table>
we have the relation,
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  <td align="right">
<math>\frac{2\Omega_3}{\Omega_\mathrm{Mc}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \pm  (1 + \mathcal{H} )^{1 / 2} \, .</math>
  </td>
</tr>
</table>
But we also see that,
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<math>f_\mathrm{Mc} = \biggl(\frac{\zeta_3}{\Omega_3}\biggr)_\mathrm{Mc}</math>
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  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\biggl[\frac{\Omega_\mathrm{Mc}}{\Omega_3} - 1 \biggr] \, .</math>
  </td>
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Combining these last two expressions gives,
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  <td align="right">
<math>\frac{4}{f_\mathrm{Mc} +2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \pm  (1 + \mathcal{H} )^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 \pm  \biggl[ 1 + \frac{8}{e^2} \biggr]^{1 / 2}</math>
  </td>
</tr>

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  <td align="right">
<math>\Rightarrow ~~~ \frac{4e}{f_\mathrm{Mc} +2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>e \pm  ( 8 + e^2 )^{1 / 2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ f_\mathrm{Mc}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4e\biggl[ e \pm  ( 8 + e^2 )^{1 / 2} \biggr]^{-1} - 2 \, .</math>
  </td>
</tr>
</table>

In an [[Apps/MaclaurinSpheroidSequence#Maclaurin_Spheroid_Sequence|accompanying discussion of the Maclaurin spheroid sequence]], a number of different plots have been used to display how various physical parameters vary along the sequence.  The solid curve that appears in Figure 1 of that discussion has been redrawn as a black-dotted curve in the left-hand panel of Figure 1 of ''this'' chapter (immediately below); it shows how <math>\Omega^2_\mathrm{Mc}</math> varies with the spheroid's eccentricity, <math>e</math>.  The small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides:
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<math>~\biggl[ e, \frac{\omega_0^2}{\pi G \rho} \biggr]</math>
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<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ 0.92995, 0.449331 \biggr] \, .</math>
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<tr>
  <td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;32, p. 80, Eqs. (9) &amp; (10)</td>
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</table>

 
<table border="1" align="center" cellpadding="8" width="80%">
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  <td align="center" colspan="2">
<b>Figure 1: &nbsp; Parameter Variations Along the Maclaurin Spheroid Sequence</b>
  </td>
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<td align="center">
[[File:JacobiSequenceTooA.png|400px|center|JacobiSequenceToo]]
</td>
<td align="center">
[[File:f_McA.png|400px|center|FrequencyRatio]]
</td>
</tr>
<tr>
  <td align="left" width="50%">Analogous to Figure 5 from &sect;32, p. 79 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>]; shows how the square of the normalized rotation frequency varies with eccentricity, <math>e = (1 - a_3/a_1)^{1 / 2},</math> along the (black-dotted) Maclaurin sequence and along the Jacobi sequence (series of purple circular markers).</td>
  <td align="left">&nbsp;</td>
</tr>
</table>


<table border="1" cellpadding="10" align="center" colspan="10">
<tr>
  <td align="center" rowspan="3">Note:</td>
  <td align="center" rowspan="3"><math>e</math></td>
  <td align="center" rowspan="3"><math>\Omega_\mathrm{Mc}^2</math></td>
<td align="center" rowspan="7" bgcolor="gray" width="1%">&nbsp;</td>
  <td align="center" colspan="4">Limiting Riemann S-type Ellipsoids</td>
<td align="center" rowspan="7" bgcolor="gray" width="1%">&nbsp;</td>
  <td align="center" colspan="4" bgcolor="pink">Limiting Type I Riemann Ellipsoids</td>
</tr>
<tr>
  <td align="center" colspan="2">Direct</td>
  <td align="center" colspan="2">Adjoint</td>
  <td align="center" colspan="2">Direct</td>
  <td align="center" colspan="2">Adjoint</td>
</tr>
<tr>
  <td align="center" colspan="1"><math>\Omega_3^2</math></td>
  <td align="center" colspan="1"><math>f = \frac{\zeta_3}{\Omega_3}</math></td>
  <td align="center" colspan="1"><math>(\Omega_3^\dagger)^2</math></td>
  <td align="center" colspan="1"><math>f^\dagger = \frac{\zeta_3^\dagger}{\Omega_3^\dagger}</math></td>
  <td align="center" colspan="1"><math>\Omega_3^2</math></td>
  <td align="center" colspan="1"><math>f = \frac{\zeta_3}{\Omega_3}</math></td>
  <td align="center" colspan="1"><math>(\Omega_3^\dagger)^2</math></td>
  <td align="center" colspan="1"><math>f^\dagger = \frac{\zeta_3^\dagger}{\Omega_3^\dagger}</math></td>
</tr>

<tr>
  <td align="center">(a)</td>
  <td align="right">0.00000</td>
  <td align="right">0.00000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
</tr>

<tr>
  <td align="center">(b)</td>
  <td align="right">0.81267</td>
  <td align="right">0.37423</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
</tr>

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  <td align="center">(c)</td>
  <td align="right">0.92995</td>
  <td align="right">0.44933</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
</tr>

<tr>
  <td align="center">(d)</td>
  <td align="right">0.95289</td>
  <td align="right">0.44022</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
  <td align="right">000</td>
</tr>

<tr>
  <td align="left" colspan="13">
Notes:
<ol type="a">
<li>Nonrotating sphere, <math>c/a \rightarrow 1</math>; also, self-adjoint Riemann S-type ellipsoid</li>
<li>Bifurcation to Jacobi (''direct'') and Dedekind (''adjoint'') sequences</li>
<li>Configuration with maximum <math>\Omega^2_\mathrm{Mc}</math></li>
<li>Onset of dynamical instability; also, self-adjoint Riemann S-type ellipsoid
<li>Infinitesimally thin disk, <math>c/a \rightarrow 0</math>
</ol>
  </td>
</tr>
</table>