Editing Apps/MaclaurinSpheroidSequence (section)

===Energy Ratio, T/|W|===

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'''Table 2:''' &nbsp;[[Appendix/Ramblings/PowerSeriesExpressions#Maclaurin_Spheroid_Index_Symbols|Limiting Values]]
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&nbsp;
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<b><math>e \rightarrow 0</math></b>
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<b><math>\frac{c}{a} \rightarrow 0</math></b>
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<tr>
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<b><math>A_1</math></b>
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<math>\frac{2}{3}\biggl[1 - \frac{e^2}{5} - \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
  </td>
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<math>\frac{\pi}{2} \biggl( \frac{c}{a}\biggr) - 2\biggl(\frac{c}{a}\biggr)^2+ \mathcal{O}\biggl(\frac{c^3}{a^3}\biggr)</math>
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<tr>
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<b><math>A_3</math></b>
  </td>
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<math>\frac{2}{3}\biggl[1 + \frac{2e^2}{5} + \mathcal{O}\biggl(e^4\biggr)\biggr]</math>
  </td>
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<math> 2 - \pi \biggl( \frac{c}{a}\biggr) + 4\biggl(\frac{c}{a}\biggr)^2
- \mathcal{O}\biggl(\frac{c^3}{a^3}\biggr)</math>
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<b><math>~\frac{\sin^{-1}e}{e}</math></b>
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<math>~1 + \frac{e^2}{6} + \mathcal{O}\biggl(e^4\biggr)</math>
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<math>~\frac{\pi}{2} - \biggl(\frac{c}{a}\biggr) +\frac{\pi}{4}\biggl(\frac{c}{a}\biggr)^2
- \mathcal{O}\biggl(\frac{c^3}{a^3}\biggr)</math>
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<b><math>~\tau \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math></b>
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<math>~0</math>
  </td>
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<math>~\frac{1}{2}</math>
  </td>
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</table>
The rotational kinetic energy of each uniformly rotating Maclaurin spheroid is given by the expression,
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<math>~T_\mathrm{rot}</math>
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<math>~=</math>
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<math>~
\frac{1}{2}I \omega_0^2
=\frac{Ma^2}{5} \cdot 2\pi G\rho
\biggl[ A_1 - (1-e^2)A_3\biggr]
</math>
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</tr>

<tr>
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&nbsp;
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<math>~=</math>
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<math>~
\frac{2^3 \pi^2}{3\cdot 5} \cdot G\rho^2 a^4 c 
\biggl[ A_1 - (1-e^2)A_3\biggr]
</math>
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</tr>

<tr>
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&nbsp;
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<math>~=</math>
  </td>
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<math>~
\frac{2^3\pi^2}{3\cdot 5} \cdot  G\rho^2 a^5  
\biggl[ \frac{(1-e^2)}{e^3} ~(3 - 2e^2)\sin^{-1}e - \frac{3(1-e^2)^{3 / 2}}{e^2} \biggr] 
\, ;
</math>
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</table>
and the gravitational potential energy of each configuration is,
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<math>~W_\mathrm{grav}</math>
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<math>~=</math>
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<math>~
- \frac{3}{5} \cdot \frac{GM^2}{c} \biggl[ A_1 + \frac{1}{2}(1-e^2)A_3 \biggr] 
=
- \frac{3}{2\cdot 5} \cdot \frac{G}{c} \biggl[ \frac{2^2\pi \rho a^2 c}{3} \biggr]^2 \biggl[ 2A_1 + (1-e^2)A_3 \biggr] 
</math>
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<tr>
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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~
- \frac{2^3\pi^2}{3\cdot 5}\cdot G\rho^2 a^4 c \biggl[ 2A_1 + (1-e^2)A_3 \biggr] 
</math>
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</tr>

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&nbsp;
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<math>~=</math>
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<math>~
- \frac{2^4 \pi^2}{3\cdot 5} \cdot G\rho^2 a^5 (1-e^2) \cdot \frac{\sin^{-1}e }{e}  \, .</math>
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<span id="EnergyNorm">&nbsp;</span>
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<div align="center">'''Energy Normalization'''</div>
In his tabulation of the properties of Maclaurin Spheroids &#8212; see Appendix D (p. 483) of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] &#8212; Tassoul adopted the following energy normalization:
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<math>E_\mathrm{T78}</math>
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<math>=</math>
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<math>
(4/3)\pi G \rho M {\bar{a}}^2 \, ,
</math>
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</table>
where, as [[#Corresponding_Total_Angular_Momentum|above]],
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<math>\bar{a}</math>
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<math>\equiv</math>
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<math>
(a^2 c)^{1 / 3}  
= a \biggl(\frac{c}{a}\biggr)^{1 / 3}
= a (1 - e^2)^{1 / 6}
\, .
</math>
  </td>
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</table>
Given that, <math>M = (4/3)\pi \rho a^2c = (4/3)\pi \rho a^3 (1 - e^2)^{1 / 2}\, ,</math> we can write instead,
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<math>E_\mathrm{T78}</math>
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<math>=</math>
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<math>
(4/3)\pi G [\rho (4/3)\pi \rho a^3 (1 - e^2)^{1 / 2}] a^2 (1-e^2)^{1 / 3} 
</math>
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</tr>

<tr>
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&nbsp;
  </td>
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<math>=</math>
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<math>
(2^4 \pi^2/3^2)G \rho^2 a^5 (1 - e^2)^{5 / 6} 
</math>
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</tr>

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&nbsp;
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<math>=</math>
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<math>
\biggl(\frac{4\pi}{3}\biggr)^{1 / 3} G (M^5\rho)^{1 / 3} \, .
</math>
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</table>
After normalization, then, we have,
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<math>\frac{T_\mathrm{rot}}{E_\mathrm{T78}}</math>
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<math>=</math>
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<math> 
\frac{3}{2\cdot 5} 
\biggl[ (3 - 2e^2)\frac{\sin^{-1}e}{e} - 3(1-e^2)^{1 / 2} \biggr] \frac{(1-e^2)^{1 / 6}}{e^2}
\, ;
</math>
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</table>
and,
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<math>\frac{W_\mathrm{grav}}{E_\mathrm{T78}}</math>
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<math>=</math>
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<math>
- \frac{3}{5}(1-e^2)^{1 / 6} \cdot \frac{\sin^{-1}e }{e}  \, .</math>
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</table>
<b><font color="darkblue">Example</font></b>  &hellip; to be checked against the relevant line of data from Tables D.1 and D.2 of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]:  &nbsp;If we set <math>e = 0.965646</math>, we find, <math>T_\mathrm{rot}/E_\mathrm{T78} =  0.155578\, ,</math> and <math>W_\mathrm{grav}/E_\mathrm{T78} = -0.518594\, ,</math> which implies that, <math>(T_\mathrm{rot} + W_\mathrm{grav})/E_\mathrm{T78} = -0.363016\, ,</math> and <math>\tau \equiv T_\mathrm{rot}/|W_\mathrm{grav}| = 0.300000\, .</math>

----

Note that {{ Wong74 }} &#8212; see the NOTE appended to his Table 2 (p. 686) &#8212; adopts the normalization,
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<math>E_\mathrm{Wong74}</math>
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<math>=</math>
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<math>
\frac{3}{5}\biggl(\frac{4\pi}{3}\biggr)^{1 / 2} G (M^5\rho)^{1 / 3} 
</math>
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<math>\Rightarrow ~~~ \frac{E_\mathrm{T78} }{E_\mathrm{Wong74}}</math>
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<math>=</math>
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<math>
\frac{5}{3}\biggl(\frac{3}{4\pi}\biggr)^{1 / 6}  \, .
</math>
  </td>
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</table>

----

Alternatively, in {{ EH85 }} &#8212; see Eq. 7 (p. 291) &#8212; and in {{ CKST95d }}  &#8212; see Eq. 1.3 (p. 511)  &#8212; the energy normalization is,
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<math>E_\mathrm{EH85} = E_\mathrm{CKST95d}</math>
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<math>=</math>
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<math>
(4\pi G)^2 M^5 L^{-2} 
</math>
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<math>\Rightarrow ~~~ \biggl[\frac{E_\mathrm{T78}}{E_\mathrm{EH85}} \biggr]^3 = \biggl[\frac{E_\mathrm{T78}}{E_\mathrm{CKST95d}} \biggr]^3</math>
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<math>=</math>
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<math>
\frac{j^6}{3(4\pi)^2} \, . 
</math>
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</table>
</td></tr> 
</table>


<span id="tau">Hence, the energy ratio,</span>
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<math>~\tau \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math>
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<math>~=</math>
  </td>
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<math>~
\frac{ A_1 - (1-e^2)A_3 }{  2A_1 + (1-e^2)A_3 }
</math>
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</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;4.5, p. 86, Eq. (53)
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&nbsp;
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<math>~=</math>
  </td>
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<math>~ 
\biggl[ \frac{(1-e^2)}{e^3} ~(3 - 2e^2)\sin^{-1}e - \frac{3(1-e^2)^{3 / 2}}{e^2} \biggr] 
\biggl[ 2(1-e^2) \cdot \frac{\sin^{-1}e }{e} \biggr]^{-1}
</math>
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</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>~ \frac{3}{2e^2}\biggl[ 1 - \frac{e(1-e^2)^{1 / 2}}{\sin^{-1} e}\biggr] - 1 </math>
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<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>], &sect;7.3, p. 172, Eq. (7.3.24)<br />
[<b>[[Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. I, &sect;10.3, p. 489, Eq. (10.54)
  </td>
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&nbsp;
  </td>
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<math>=</math>
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<math>
\frac{1}{2e^2\sin^{-1} e}\biggl[ (3-2e^2)\sin^{-1} e - 3e(1-e^2)^{1 / 2}\biggr]  
\, .
</math>
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  <td align="center" colspan="3">
{{ MPT77 }}, &sect;IVc, p. 594, Eq. (4.4)
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</table>

Building on an [[Apps/MaclaurinSpheroids#Gravitational_Potential|accompanying discussion of the structure of Maclaurin spheroids]], Table 2 &#8212; shown just above, on the right &#8212; lists the limiting values of several key functions.  Note, in particular, that as the eccentricity varies smoothly from zero (spherical configuration) to unity (infinitesimally thin disk), the energy ratio, <math>~\tau</math>, varies smoothly from zero to one-half.  In his examination of the Maclaurin spheroid sequence, [[Appendix/References#T78|Tassoul (1978)]] chose to use this energy ratio as the ''order parameter'', <span id="Figs3and4">rather than the eccentricity</span>.  

<table border="1" align="center"><tr><td align="center">
<table border="0" align="center" cellpadding="3">
<tr>
  <td align="center">'''Figure 3'''</td>
<td align="center" rowspan="3">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
  <td align="center">'''Figure 4'''</td>
</tr>
<tr>
<td align="center">
[[File:T78Fig4.2omega2.png|center|350px|Maclaurin Spheroid Sequence]]
</td>
<td align="center">
[[File:T78Fig4.2angmom.png|center|350px|Maclaurin Spheroid Sequence]]
</td>
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Solid black curve also may be found in:
<div align="center">
Fig. 4.2 (p. 88) &amp; Fig. 10.1 (p. 236) of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]
</div>
  </td>
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This solid black curve also appears in:
<div align="center">
Fig. 4.2 (p. 88) &amp; Fig. 10.12 (p. 237) of [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>]
</div>
  </td>
</tr>
</table>

</td></tr></table>

Following Tassoul, our Figure 3 shows how the square of the angular velocity varies with <math>~\tau</math>, and our Figure 4 shows how the system angular momentum varies with <math>~\tau</math>.  In these plots, respectively, the square of the angular velocity has been normalized by <math>~2\pi G \rho</math> &#8212; that is, by a quantity that is a factor of two larger than the normalization adopted in EFE &#8212; while the angular momentum has been normalized to the same quantity used in EFE.  As above, the small solid-green square marker identifies the location along the sequence where the system with the maximum angular velocity resides.