Editing ThreeDimensionalConfigurations/ChallengesPt4 (section)

====Summary====
In a [[ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body.  As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = constant and dz'/dt = 0, and the planar orbit is defined by the expression for an,
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<td align="center" colspan="3"><font color="maroon">'''Off-Center Ellipse'''</font></td>
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  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{x'}{x_\mathrm{max}} \biggr]^2 + \biggl[\frac{y' - y_c(z')}{y_\mathrm{max}} \biggr]^2 \, .</math>
  </td>
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''Tipped Orbit Frame'' (yellow, primed) <br />
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  <td align="center">[[File:TippedAxes03.png|350px|Tipped Orbital Planes]]</td>
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Given that b/a = 1.25 and c/a = 0.4703 for our chosen [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model|Example Type I Ellipsoid]], we find that, <math>~\theta = - 1.18122 ~\mathrm{rad} = -67.68^\circ</math>.
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Notice that the offset, <math>~y_c</math>, is a function of the tipped plane's vertical coordinate, <math>~z'</math>.  As a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,
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  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_\mathrm{max}\cos(\dot\varphi t)</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~y' - y_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math>
  </td>
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  <td align="right">
<math>~\dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_c - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\dot{y}' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>
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We have determined that (numerical value given for our [[ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model|chosen example Type I ellipsoid]]),
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  <td align="right">
<math>~\tan\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{b^2 \beta \Omega_2}{c^2 \gamma \Omega_3}  
=
-2.43573\, ,
</math>
  </td>
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<tr><td align="center" colspan="3"><b><font color="red">This definition of tan(&theta;) is inconsistent with all others!</font></b></td>
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where, as has also been specified [[ThreeDimensionalConfigurations/ChallengesPt3#betagamma|in an accompanying discussion]],
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  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2}
=
+1.13451
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} 
=
+1.80518\, .
</math>
  </td>
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We also have determined that,
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<math>~\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^4</math>
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  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^4 (c^4 \gamma^2 \Omega_3^2 + b^4 \beta^2 \Omega_2^2)}{b^4 c^4(\gamma^2\Omega_3^2 + \beta^2\Omega_2^2)}
~~~\Rightarrow ~~~\frac{x_\mathrm{max}}{y_\mathrm{max}}  = 1.26218  \, ,
</math>
  </td>
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  <td align="right">
<math>~{\dot\varphi}^4 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^4}{b^4 c^4} \biggl(\gamma^2\Omega_3^2 + \beta^2\Omega_2^2 \biggr) (c^4 \gamma^2 \Omega_3^2 + b^4 \beta^2 \Omega_2^2)
~~~\Rightarrow ~~~ \dot\varphi = 1.59862\, ,
</math>
  </td>
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  <td align="right">
<math>~\frac{y_c}{z_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\sin\theta
~~~\Rightarrow~~~~ \frac{y_c}{z_0} = -0.92507
\, .</math>
  </td>
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</table>