Editing Apps/MaclaurinSpheroidSequence (section)

===Corresponding Total Angular Momentum===

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  <td align="center">'''Figure 2'''</td>
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[[File:EFE_AngMomVsEcc.png|center|350px|Maclaurin Spheroid Sequence]]
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Solid black curve also may be found as:
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Fig. 6 (p. 79) of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>];<br />
Fig. 7.3 (p. 174) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>]
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The total angular momentum of each uniformly rotating Maclaurin spheroid is given by the expression,
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<math>~L</math>
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<math>~=</math>
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<math>~I \omega_0 \, ,</math>
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where, the moment of inertia <math>~(I)</math> and the total mass <math>~(M)</math> of a uniform-density spheroid are, respectively,
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<math>~I</math>
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<math>~=</math>
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<math>~\biggl(\frac{2}{5}\biggr) M a^2 \, ,</math>
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<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
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<math>~M</math>
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<math>~=</math>
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<math>~\biggl( \frac{4\pi}{3} \biggr) \rho a^2c \, .</math>
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Adopting the shorthand notation, <math>\bar{a} \equiv  (a^2 c)^{1 / 3}</math>, we have,
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<math>~L^2</math>
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<math>~=</math>
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<math>~ \frac{2^2 M^2 a^4}{5^2} \biggl[  A_1 - A_3 (1-e^2) \biggr] 2\pi G \biggl[ \frac{3}{2^2\pi} \cdot \frac{M}{a^2c} \biggr]</math>
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&nbsp;
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<math>~=</math>
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<math>~ \frac{6GM^3 {\bar{a}}}{5^2} \biggl[  A_1 - A_3 (1-e^2) \biggr]\biggl(\frac{a}{c}\biggr)^{4/3} </math>
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<math>~\Rightarrow ~~~ \frac{L}{(GM^3\bar{a})^{1 / 2}}</math>
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<math>~=</math>
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<math>~ \frac{6^{1 / 2}}{5} \biggl[  A_1 - A_3 (1-e^2) \biggr]^{1 / 2}(1 - e^2)^{-1 / 3} \, .</math>
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[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;32, p. 78, Eq. (7)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;4.5, p. 86, Eq. (54)
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<span id="MPT77angmom">This also means,</span>
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<math>L_*^2 \equiv \frac{L^2}{(GM^3\bar{a})}</math>
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<math>=</math>
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<math>\frac{6}{5^2} \biggl[ (3-2e^2)(1-e^2)^{1 / 2} \cdot \frac{\sin^{-1}e}{e^3} - \frac{3(1-e^2)}{e^2}\biggr](1 - e^2)^{-2 / 3}  \, .</math>
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{{ MPT77 }}, &sect;IVa, p. 591, Eq. (4.2)
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Figure 2 shows how the system's normalized angular momentum, <math>L_*</math>, varies with eccentricity along the Maclaurin spheroid sequence; given the chosen normalization unit,  <math>~(GM^3\bar{a})^{1 / 2}</math>, it is understood that the mass and the volume &#8212; hence, also the density &#8212; of the configuration are held fixed as the eccentricity is varied.  Strictly speaking, along this sequence the angular momentum asymptotically approaches infinity as <math>~e \rightarrow 1</math>; by limiting the ordinate to a maximum value of 1.2, the plot masks this asymptotic behavior.  The small solid-green square marker identifies the location along this sequence where the system with the maximum angular velocity resides (see Figure 1); this system is not associated with a turning point along this angular-momentum versus eccentricity sequence.