Editing Apps/MaclaurinSpheroidSequence (section)

===Angular Velocity or T/|W| ''vs.'' Angular Momentum===

Figures 5 and 6, respectively, show how the square of the angular velocity and how the energy ratio, &tau;, vary with the square of the angular momentum for models along the Maclaurin spheroid sequence.  In generating these plots, following the lead of {{ EH83a }}, we have normalized the square of the angular velocity by <math>~4\pi G \rho</math> &#8212; a factor of four larger than the normalization used in EFE &#8212; and we have adopted a slightly different angular-momentum-squared normalization, namely,
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<tr>
  <td align="right">
<math>j^2</math>
  </td>
  <td align="center">
<math>\equiv \frac{L^2}{4\pi G M^{10/3} \rho^{-1 / 3}} = </math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{2^8 \pi^4}  \biggr)^{1/3}  \frac{L^2}{(GM^3\bar{a})} \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
Note that in {{ Wong74 }} &#8212; see the NOTE appended to his Table 2 (p. 686) &#8212; the parameter <math>x</math> provides the measure of the configuration's specific angular momentum; specifically,
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  <td align="right">
<math>x_\mathrm{Wong74} </math>
  </td>
  <td align="center">
<math>\equiv \frac{25}{12} \biggl(\frac{4\pi}{3}\biggr)^{1 / 3}\frac{L^2\rho^{1 / 3}}{G M^{10/3} } = </math>
  </td>
  <td align="left">
<math>
\frac{5^2}{2^2} \biggl(\frac{4\pi}{3}\biggr)^{4 / 3}  j^2 \, .
</math>
  </td>
</tr>
</table>

----

Alternatively, as has already been [[#MPT77angmom|highlighted above]], {{ MPT77 }} adopt the dimensionless parameter (see their Eq. 4.1),
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<tr>
  <td align="right">
<math>
L_*^2 \equiv 
\biggl(\frac{4\pi}{3}\biggr)^{1 / 3}\frac{L^2}{G M^{10/3} \rho^{-1 / 3}}
=
3 \biggl(\frac{4\pi}{3}\biggr)^{4 / 3} j^2
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{L^2}{(GM^3\bar{a})} \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

<table border="0" align="center" cellpadding="5"><tr><td align="center">
<table border="0" align="center" cellpadding="3">
<tr>
  <td align="center">'''Figure 5'''</td>
<td align="center" rowspan="3">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
  <td align="center">'''Figure 6'''</td>
</tr>
<tr>
<td align="center">
[[File:EH83Fig3.png|center|350px|Maclaurin Spheroid Sequence]]
</td>
<td align="center">
[[File:EH83Fig4.png|center|350px|Maclaurin Spheroid Sequence]]
</td>
</tr>
<tr>
  <td align="center">
This solid black curve also appears in:
<div align="center">
Fig. 3 (p. 1134) of [https://ui.adsabs.harvard.edu/abs/1983PThPh..69.1131E/abstract Eriguchi &amp; Hachisu (1983)]<br />
Fig. 3 (p. 487) of [https://ui.adsabs.harvard.edu/abs/1986ApJS...61..479H/abstract Hachisu (1986)]<br />
Fig. 4 (p. 4507) of [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract Basillais &amp; Hur&eacute; (2019)]
</div>
  </td>
  <td align="center">
This solid black curve also appears in:
<div align="center">
Fig. 4 (p. 487) of [https://ui.adsabs.harvard.edu/abs/1986ApJS...61..479H/abstract Hachisu (1986)]
</div>
  </td>
</tr>
</table>
</td></tr></table>

<span id="OmegaMax">As above,</span> the small solid-green square marker identifies the location along both sequences where the system with the maximum angular velocity resides:
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  <td align="right">
<math>\biggl[ j^2, \frac{\omega_0^2}{4\pi G \rho}, \tau \biggr]</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ 0.010105, 0.112333, 0.237894 \biggr] \, .</math>
  </td>
</tr>
</table>