Editing 3Dconfigurations/DescriptionOfRiemannTypeI (section)

===Basic Relations===

In Chapter 7, &sect;51(c) (pp. 165 - 166) of [[Appendix/References#EFE|[<font color="red">EFE</font>] ]], Chandrasekhar shows that <font color="orange">"&hellip; the ''entire'' [[Apps/MaclaurinSpheroidSequence|Maclaurin sequence]] can be considered as limiting <math>(a_2/a_1 \rightarrow 1)</math> forms of the Riemann Ellipsoids of type I."</font>  First, we [[Apps/MaclaurinSpheroidSequence#Equilibrium_Angular_Velocity|recall]] that, as viewed from the inertial frame, each Maclaurin spheroid of eccentricity, <math>e = (1 - a_3^2/a_1^2)^{1 / 2}</math>, rotates uniformly with angular velocity,
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  <td align="right"><math>\Omega_\mathrm{Mc}^2 \equiv \frac{\omega_0^2}{\pi G \rho} = 2e^2 B_{13} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2(3-2e^2)(1 - e^2)^{1 / 2} \cdot \frac{\sin^{-1} e}{e^3} - \frac{6(1-e^2)}{e^2} \, .</math>
  </td>
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  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;32, pp. 77-78, Eqs. (4) &amp; (6)
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Given the specified value of the semi-axis ratio, <math>a_3/a_1</math>, the properties of the limiting Riemann Type I ellipsoid are given by the expressions,
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  <td align="right">
<math>2\Omega_3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\Omega_\mathrm{Mc} \pm \biggl[16 B_{13} + \Omega^2_\mathrm{Mc}\biggr]^{1 / 2} \, ,</math>
  </td>
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<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 7, &sect;51(c), p. 166, Eq. (215)</font> </td></tr>
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and,
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  <td align="right">
<math>\zeta_3</math>
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  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2(\Omega_\mathrm{Mc} - \Omega_3) \, ,</math>
  </td>
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<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 7, &sect;51(c), p. 165, Eq. (212)</font> </td></tr>
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where,
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  <td align="right">
<math>B_{13}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{A_1 a_1^2 - A_3 a_3^2}{a_1^2 - a_3^2} \biggr] \, ;</math>
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<tr><td align="center" colspan="3">[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], <font color="#00CC00">Chapter 3, &sect;21, Eqs. (105) &amp; (107)</font></td></tr>
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and,

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<math>
~A_1
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
\frac{1}{e^2} \biggl[\frac{\sin^{-1}e}{e} - (1-e^2)^{1/2}  \biggr](1-e^2)^{1/2} \, ,
</math>
  </td>
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  <td align="right">
<math>
~A_3
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
\frac{2}{e^2} \biggl[(1-e^2)^{-1/2} -\frac{\sin^{-1}e}{e} \biggr](1-e^2)^{1/2} \, .
</math>
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  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;17, p. 43, Eq. (36)
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<font color="red">REMINDER:</font>&nbsp; For a given choice of the eccentricity, there are two viable solutions &hellip; the ''direct'' configuration and its ''adjoint.''  In the context of [[ThreeDimensionalConfigurations/RiemannStype#Expressions_Supplied_by_EFE|Riemann S-type ellipsoids]], this pair of solutions arises from the choice of the sign <math>(\pm)</math> in the  expression for <math>\Omega_3</math>; in the context of Type I Riemann ellipsoids (<i>i.e.</i>, here) the pair arises from the choice of the sign <math>(\mp)</math> in the <font color="red">STEP #3</font> determination of <math>\beta</math> and <math>\gamma</math>.  In both physical contexts, the ''direct'' (Jacobi-like) solution results from selecting the ''inferior'' sign while the ''adjoint'' (Dedekind-like) solution results from selecting the ''superior'' sign.