Editing Apps/MaclaurinSpheroidSequence (section)

===Equilibrium Angular Velocity===

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  <td align="center">'''Figure 1'''</td>
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[[File:EFE Omega2vsECCwithThomsonTait2.png|center|350px|Maclaurin Spheroid Sequence]]
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  <td align="center">
The dark blue circular markers locate 15 of the 18 individual models identified in Table 1. The solid black curve derives from our evaluation of the function, <math>~\omega_0^2(e)\, ;</math> this curve also may be found in:
<div align="center">
Fig. 5 (p. 79) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>];<br />
Fig. 7.2 (p. 173) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]<br />
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The essential structural elements of each Maclaurin spheroid model are uniquely determined once we specify the system's axis ratio, <math>~c/a</math>, or the system's meridional-plane eccentricity, <math>~e</math>, where

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  <td align="right">
<math>~e</math>
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  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[1 - \biggl(\frac{c}{a}\biggr)^2\biggr]^{1 / 2} \, ,</math>
  </td>
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which varies from ''e = 0'' (spherical structure) to ''e = 1'' (infinitesimally thin disk).  According to our [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|accompanying derivation]], for a given choice of <math>~e</math>, the square of the system's equilibrium angular velocity is,

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  <td align="right">
<math>
~ \omega_0^2
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
2\pi G \rho \biggl[ A_1 - A_3 (1-e^2) \biggr] \, ,
</math>
  </td>
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  <td align="center" colspan="3">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;32, p. 77, Eq. (4)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;4.5, p. 86, Eq. (52)<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], &sect;7.3, p. 172, Eq. (7.3.18)
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where,

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  <td align="right">
<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>
</tr>

<tr>
  <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>
  </td>
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  <td align="center" colspan="3">
{{ TT1867 }}, &sect;522, p. 392, Eqs. (9) &amp; (7)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;17, p. 43, Eq. (36)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;4.5, p. 85, Eqs. (48) &amp; (49)<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], &sect;7.3, p. 170, Eq. (7.3.8)
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</table>

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'''Table 1'''<br />Data copied from<br />{{ TT1867 }}, &sect;772, p. 614
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  <td align="center" width="50%"><math>~e</math></td>
  <td align="center"><math>~\frac{\omega_0^2}{2\pi G \rho}</math></td>
  <td align="center" bgcolor="lightgrey" rowspan="10">&nbsp; &nbsp;</td>
  <td align="center" width="50%"><math>~e</math></td>
  <td align="center"><math>~\frac{\omega_0^2}{2\pi G \rho}</math></td>
</tr>
<tr>
  <td align="center">0.10</td>
  <td align="center">0.0027</td>
  <td align="center">0.91</td>
  <td align="center">0.2225</td>
</tr>
<tr>
  <td align="center">0.20</td>
  <td align="center">0.0107</td>
  <td align="center">0.92</td>
  <td align="center">0.2241</td>
</tr>
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  <td align="center">0.30</td>
  <td align="center">0.0243</td>
  <td align="center">0.93</td>
  <td align="center">0.2247</td>
</tr>
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  <td align="center">0.40</td>
  <td align="center">0.0436</td>
  <td align="center">0.94</td>
  <td align="center">0.2239</td>
</tr>
<tr>
  <td align="center">0.50</td>
  <td align="center">0.0690</td>
  <td align="center">0.95</td>
  <td align="center">0.2213</td>
</tr>
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  <td align="center">0.60</td>
  <td align="center">0.1007</td>
  <td align="center">0.96</td>
  <td align="center">0.2160</td>
</tr>
<tr>
  <td align="center">0.70</td>
  <td align="center">0.1387</td>
  <td align="center">0.97</td>
  <td align="center">0.2063</td>
</tr>
<tr>
  <td align="center">0.80</td>
  <td align="center">0.1816</td>
  <td align="center">0.98</td>
  <td align="center">0.1890</td>
</tr>
<tr>
  <td align="center">0.90</td>
  <td align="center">0.2203</td>
  <td align="center">0.99</td>
  <td align="center">0.1551</td>
</tr>
</table>
</td></tr></table>
<span id="MaclaurinFrequency">In other words,</span>

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  <td align="right">
<math>
~ \frac{\omega_0^2}{2\pi G \rho }
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
(3-2e^2)(1-e^2)^{1 / 2} \cdot \frac{\sin^{-1}e}{e^3} - \frac{3(1-e^2)}{e^2}
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ TT1867 }}, &sect;771, p. 613, Eq. (1)<br />
[<b>[[Appendix/References#Lamb32|<font color="red">Lamb32</font>]]</b>], 6<sup>th</sup> Ed. (1932), Ch. XII, &sect;374, p. 701, Eq. (6)  &#8212; set <math>~\zeta^2 = (1-e^2)/e^2</math><br />
[https://ui.adsabs.harvard.edu/abs/1886RSPS...41..319D/abstract G. H. Darwin (1886)], p.322, Eq. (14) &#8212; set <math>~\gamma = \sin^{-1}e</math><br />
[https://ui.adsabs.harvard.edu/abs/1928asco.book.....J/abstract J. H. Jeans (1928)], &sect;192, p. 202, Eq. (192.4)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;32, p. 78, Eq. (6)<br />
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], &sect;7.3, p. 172, Eq. (7.3.18)
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Figure 1 shows how the square of the angular velocity varies with eccentricity along the Maclaurin spheroid sequence; given the chosen normalization unit,  <math>~\pi G\rho</math>, it is understood that the density of the configuration is held fixed as the eccentricity is varied.  


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Examining the Maclaurin spheroid sequence "<font color="orange">&hellip; we see that the value of <math>~\omega_0^2</math> increases gradually from zero to a maximum as the eccentricity <math>~e</math> rises from zero to about 0.93, and then (more quickly) falls to zero as the eccentricity rises from 0.93 to unity.</font>" &hellip; "<font color="orange">If the angular velocity exceed the value</font> associated with this maximum, "<font color="orange">&hellip; equilibrium is impossible in the form of an ellipsoid of revolution.  If the angular velocity fall short of this limit there are always two ellipsoids of revolution which satisfy the conditions of equilibrium.  In one of these the eccentricity is greater than 0.93, in the other less.</font>"</td></tr>
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--- {{ TT1867 }}, &sect;772, p. 614.
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The extremum of the curve occurs where <math>d\omega_0^2/de = 0</math>; that is, it occurs where,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\frac{\sin^{-1}e}{e}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>(1 - e^2)^{1 / 2} \biggl[\frac{9 - 2e^2}{9 - 8e^2}\biggr] \, .</math></td>
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</table>

In our Figure 1, the small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides:
<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\biggl[ e, \frac{\omega_0^2}{\pi G \rho} \biggr]</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ 0.92995, 0.449331 \biggr] \, .</math>
  </td>
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  <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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