Editing Apps/MaclaurinToroid (section)

=Maclaurin Toroid (MPT77)=
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! style="height: 125px; width: 125px;" |[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like|<b>Maclaurin<br />Toroid Sequence</b>]]<br />{{ MPT77hereafter }}
|}

In a [[#/Apps/DysonPotential|separate chapter]], we focused on the pioneering work of {{ Dyson1893full }}, {{ Dyson1893Part2full }} and, more recently, {{ Wong74full }}, who determined the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. We will refer to these ''uniformly rotating'' configurations as "Dyson-Wong tori."

Here, we summarize the work of {{ MPT77full }} &#8212; hereafter, {{ MPT77hereafter }} &#8212; who constructed a sequence of toroidal-shaped, self-gravitating, incompressible configurations that are not uniformly rotating but, rather, have a distribution of angular momentum that is identical to the distribution found in a uniformly rotating, uniform-density sphere.  As we have pointed out in our [[AxisymmetricConfigurations/SolutionStrategies#Uniform-Density_Initially_(n'_=_0)|associated overview of "simple rotation curves"]], this chosen (cylindrical) radial distribution of specific angular momentum is given by the expression,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\dot\varphi\varpi^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5L}{2M}\biggl\{ 1 - [1 - m(\varpi) ]^{2 / 3} \biggr\} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Stoeckly65 }}, &sect;II.c, Eq. (12)<br />
{{ OM68 }}, Eq. (45)<br />
{{ BO70 }}, Eq. (12)<br />
{{ BO73 }}, Eq. (3)<br />
{{ EH85 }}, Eq. (1)<br />
{{ HTE87 }}, Eq. (6)
  </td>
</tr>
</table>
where, <math>L</math> is the total angular momentum, <math>M</math> is the total mass, the mass fraction,
<div align="center">
<math>m(\varpi) \equiv \frac{M_\varpi(\varpi)}{M} \, ,</math>
</div>
and <math>M_\varpi(\varpi)</math> is the mass enclosed within a cylinder of radius, <math>\varpi</math>. Such equilibrium models are often referred to as <math>n' = 0</math> configurations, although {{ MPT77hereafter }} do not use this terminology.  Following the lead of {{ MPT77hereafter }}, we will refer to each of their equilibrium configurations as a "Maclaurin Toroid."

==Maclaurin Spheroid Reminder==

<span id="L*">As has been demonstrated</span> in our [[Apps/MaclaurinSpheroidSequence#Corresponding_Total_Angular_Momentum|accompanying discussion of the Maclaurin spheroid sequence]], the (square of the) normalized angular momentum that is associated with a spheroid of eccentricity, <math>e \equiv (1 - c^2/a^2)^{1 / 2}</math>, is,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>L_*^2 \equiv \frac{L^2}{(GM^3\bar{a})}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ MPT77 }}, &sect;IVa, p. 591, Eq. (4.2)
  </td>
</tr>
</table>
In that [[Apps/MaclaurinSpheroidSequence#tau|same discussion]], we have demonstrated that the corresponding ratio of rotational to gravitational potential energy is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tau \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2e^2\sin^{-1} e}\biggl[ (3-2e^2)\sin^{-1} e - 3e(1-e^2)^{1 / 2}\biggr]  
\, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ MPT77 }}, &sect;IVc, p. 594, Eq. (4.4)
  </td>
</tr>
</table>

[[Apps/MaclaurinSpheroidSequence#Figs3and4|Figure 4 from this accompanying discussion]] shows how <math>L_*</math> varies with <math>\tau</math> along the Maclaurin Spheroid sequence.  In an effort to conform to {{ MPT77hereafter }}'s presentation, our Figure 1 (immediately below) displays the same information as displayed in Figure 4 of this separate chapter, but the axes have been swapped and the maximum displayed value of <math>L_*</math> has been extended from 1 to 1.5.

<table border="1" align="center"><tr><td align="center">
<table border="0" align="center" cellpadding="0">
<tr>
<td align="center" rowspan="2">
<b>EFE Diagram</b><br />
[[File:OurEFEannotated.png|300px|OurEFE]]
</td>
<td align="center">
&nbsp;<br /><b>Figure 1</b><br />
[[File:MPT77fiveModified.png|300px|MPT77five]]
</td>
<td align="center">
&nbsp;<br /><b>Figure 2</b><br />
[[File:MPT77sixModified.png|300px|MPT77six]]
</td>
</tr>
<tr>
  <td align="center" colspan="2">
The multicolor curve that appears here in Figures 1 and 2 also appears as a solid black curve in, respectively,
<br />Fig. 5 (p. 594) and Fig. 4 (p. 593) of {{ MPT77 }}
  </td>
</tr>
</table>

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

{{ MPT77hereafter }} also evaluate the normalized total energy, <math>E_\mathrm{tot}/|E_0|</math>, of each of their constructed equilibrium configurations, where
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>E_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
T_\mathrm{rot} + W_\mathrm{grav} \, ,
</math>
  </td>
</tr>
</table>
and, according to the caption of their Figure 4, <math>E_0</math> is <font color="darkgreen">"&hellip; the energy of a nonrotating sphere of equal mass and volume."</font>  Drawing from our [[Apps/MaclaurinSpheroidSequence#EnergyNorm|separate discussion of the Maclaurin spheroid sequence]], it would be reasonable to assume that the energy normalization adopted by {{ MPT77hereafter }} is the same as the normalization used by [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], namely,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>E_\mathrm{T78}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{4\pi}{3}\biggr)^{1 / 3}G (M^5 \rho)^{1 / 3} \, .
</math>
  </td>
</tr>
</table>
For models along the Maclaurin spheroid sequence, this normalization leads to expressions for the two key energy terms of the form,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{T78}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{T78}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{3}{5}(1-e^2)^{1 / 6} \cdot \frac{\sin^{-1}e }{e}  \, ;</math>
  </td>
</tr>
</table>
in which case, in the limit of a nonrotating sphere, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\lim_{e\rightarrow 0}\biggl[ \frac{T_\mathrm{rot} + W_\mathrm{grav}}{E_\mathrm{T78}}\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
- \frac{3}{5} \, .
</math>
  </td>
</tr>
</table>
But in Figure 4 of {{ MPT77hereafter }}, the point along the Maclaurin spheroid sequence &#8212; the solid, black curve &#8212; that represents a nonrotating <math>(L_* = 0)</math> sphere has a normalized energy, <math>(E_\mathrm{tot}/E_0) = -1.</math>  We conclude, therefore, that the normalization adopted by {{ MPT77hereafter }} is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>E_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\tfrac{3}{5}E_\mathrm{T78} \, .
</math>
  </td>
</tr>
</table>
Our Figure 2 (immediately above) attempts to quantitatively replicate the behavior of the Maclaurin spheroid sequence that is shown in Figure 4 (p. 213) of {{ MPT77hereafter }}; the ordinate depicts, on a base-10 logarithmic scale, how the total energy varies with the spheroid's angular momentum over the range, <math>0 \le L_* \le 1.50</math>.  More specifically, for eccentricities over the range, <math>0 \le e \le 0.99998967881</math>, the corresponding value of the spheroid's normalized angular momentum is obtained from the [[#L*|above expression for]] <math>L_*</math>, and the normalized energy is given by the relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{E_\mathrm{tot}}{E_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{5}{3}\biggl[ \frac{T_\mathrm{rot}}{E_\mathrm{T78}} + \frac{W_\mathrm{grav}}{E_\mathrm{T78}}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{1}{2} 
\biggl[ (3 - 2e^2)\frac{\sin^{-1}e}{e} - 3(1-e^2)^{1 / 2} \biggr] \frac{(1-e^2)^{1 / 6}}{e^2}
- 
(1-e^2)^{1 / 6} \cdot \frac{\sin^{-1}e }{e} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{(1-e^2)^{1 / 6}}{2e^2}\biggl\{
\biggl[ (3 - 2e^2)\frac{\sin^{-1}e}{e} - 3(1-e^2)^{1 / 2} \biggr] 
- 
\frac{2e^2\sin^{-1}e }{e} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{(1-e^2)^{1 / 6}}{2e^2}
\biggl[ (3 - 4e^2)\frac{\sin^{-1}e}{e} 
- 3(1-e^2)^{1 / 2} \biggr] 
\, .
</math>
  </td>
</tr>
</table>

==Constructed Maclaurin Toroid Models==

{{ MPT77 }} did not create a tabulated description of the models that they constructed along their so-called "Maclaurin Toroid" sequence.  Throughout their paper, however, they highlighted some properties of a selected group of equilibrium models.  Column (2) of Table 1, immediately below, provides a list of the values of the normalized angular momentum, <math>L_*</math>, that corresponds to the Maclaurin Toroid models that have been explicitly referenced in their discussion.

<table border="1" align="center" cellpadding="5" width="75%">
<tr>
  <td align="center" colspan="6">'''Table 1'''</td>
</tr>

<tr>
  <td align="center" rowspan="2">Model</td>
  <td align="center" rowspan="2"><math>L_*</math></td>
  <td align="center" rowspan="1" colspan="3">Spheroid Equivalent</td>
  <td align="left" rowspan="3">Notes &hellip;</td>
</tr>

<tr>
  <td align="center" rowspan="1" colspan="1"><math>e</math></td>
  <td align="center" rowspan="1" colspan="1"><math>\tau</math></td>
  <td align="center" rowspan="1" colspan="1"><math>\frac{E_\mathrm{tot}}{E_0} = \tfrac{5}{3}(T+W)/E_\mathrm{T78}</math></td>
</tr>

<tr>
  <td align="center" rowspan="1" colspan="1">(1)</td>
  <td align="center" rowspan="1" colspan="1">(2)</td>
  <td align="center" rowspan="1" colspan="1">(3)</td>
  <td align="center" rowspan="1" colspan="1">(4)</td>
  <td align="center" rowspan="1" colspan="1">(5)</td>
</tr>

<tr>
  <td align="center"><math>L^-</math></td>
  <td align="center"><math>\sim 0.775</math></td>
  <td align="center"><math>\sim 0.99409</math></td>
  <td align="center"><math>\sim 0.40585</math></td>
  <td align="center"><math>-0.41685</math></td>
  <td align="center">(a)</td>
</tr>

<tr>
  <td align="center"><math>L_0</math></td>
  <td align="center"><math>0.792 \pm 0.002</math></td>
  <td align="center"><math>0.9949 \pm 0.0001</math></td>
  <td align="center"><math>0.41195</math></td>
  <td align="center"><math>-0.40439</math></td>
  <td align="center">(b)</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>0.8732</math></td>
  <td align="center"><math>0.9975</math></td>
  <td align="center"><math>0.4367</math></td>
  <td align="center"><math>-0.35023</math></td>
  <td align="center">(c)</td>
</tr>

<tr>
  <td align="center"><math>L^+</math></td>
  <td align="center"><math>0.965175</math></td>
  <td align="center"><math>0.99892</math></td>
  <td align="center"><math>0.45747</math></td>
  <td align="center"><math>-0.29762</math></td>
  <td align="center">(d)</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>0.9852</math></td>
  <td align="center"><math>0.99910</math></td>
  <td align="center"><math>0.46104</math></td>
  <td align="center"><math>-0.28754</math></td>
  <td align="center" rowspan="4">(e)</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>1.0731</math></td>
  <td align="center"><math>0.99960</math></td>
  <td align="center"><math>0.47369</math></td>
  <td align="center"><math>-0.24745</math></td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>1.1489</math></td>
  <td align="center"><math>0.99980</math></td>
  <td align="center"><math>0.48125</math></td>
  <td align="center"><math>-0.21841</math></td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>1.2262</math></td>
  <td align="center"><math>0.9998999</math></td>
  <td align="center"><math>0.486665</math></td>
  <td align="center"><math>-0.19329</math></td>
</tr>

<tr>
  <td align="left" colspan="6">
'''Notes:'''
<ol type="a">
<li>Toroid does not exist.</li>
<li>Total energy of toroid is same as the total energy of Maclaurin spheroid with same <math>L_*</math>.</li>
<li>Marginally stable Maclaurin spheroid and associated toroid; see {{ MPT77hereafter}}'s Figure 2 (p. 592).  Also, one (of five) meridional cross-sections displayed in {{ MPT77hereafter}}'s Figure 3 (p. 593).</li>
<li>''Analytically known'' (!) onset of dynamical instability along Maclaurin spheroid sequence; see &sect; 33 of EFE and the last row of Table B.1 from {{ Bardeen71 }}.</li>
<li>Four (of five) meridional cross-sections displayed in {{ MPT77hereafter}}'s Figure 3 (p. 593).</li>
</ol>
  </td>
</tr>
</table>


Using a simple iterative technique, we have determined the value of <math>e</math> that corresponds to each tabulated <math>L_*</math> if we assume that the referenced model is a ''Maclaurin spheroid'', not a toroid; the value that corresponds to each spheroid's eccentricity is listed in column (3) of the table.  In turn, columns (4) and (5) list the values of <math>\tau</math> and <math>E_\mathrm{tot}/E_0</math> that is associated with a Maclaurin spheroid that has the stated eccentricity.  If &#8212; using the coordinate pair, <math>(L_*, \tau)</math> &#8212; we were to position any one of these models into our Figure 1, above, the model would fall on the solid black portion of the "Maclaurin spheroid sequence." Similarly, if &#8212; using the coordinate pair, <math>(L_*, E_\mathrm{tot}/E_0)</math> &#8212; we were to position any one of these models into our Figure 2, above, the model would fall on the solid black portion of the "Maclaurin spheroid sequence."


<table border="1" align="center" cellpadding="5">
<tr>
 <td align="center" colspan="6">'''Table 2'''<br /><hr />Data extracted via pencil &amp; ruler measurement from Figs. 4 and 5 of &hellip;<br />{{ MPT77figure }}

 </td>
</tr>

<tr>
  <td align="center" rowspan="2">Model</td>
  <td align="center" rowspan="2"><math>L_*</math></td>
  <td align="center" rowspan="1" colspan="2">"Measured" <font color="red">Spheroid</font> Properties</td>
  <td align="center" rowspan="1" colspan="2">"Measured" <font color="red">Toroid</font> Properties</td>
</tr>

<tr>
  <td align="center" rowspan="1" colspan="1"><math>\tau</math></td>
  <td align="center" rowspan="1" colspan="1"><math>\frac{E_\mathrm{tot}}{E_0}</math></td>
  <td align="center" rowspan="1" colspan="1"><math>\tau</math></td>
  <td align="center" rowspan="1" colspan="1"><math>\frac{E_\mathrm{tot}}{E_0}</math></td>
</tr>

<tr>
  <td align="center" rowspan="1" colspan="1">(1)</td>
  <td align="center" rowspan="1" colspan="1">(2)</td>
  <td align="center" rowspan="1" colspan="1">(3)</td>
  <td align="center" rowspan="1" colspan="1">(4)</td>
  <td align="center" rowspan="1" colspan="1">(5)</td>
  <td align="center" rowspan="1" colspan="1">(6)</td>
</tr>

<tr>
  <td align="center"><math>L^-</math></td>
  <td align="center"><math>\sim 0.775</math></td>
  <td align="center"><math>\sim 0.4098</math></td>
  <td align="center">---</td>
  <td align="center"><math>0.3784</math></td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center"><math>L_0</math></td>
  <td align="center"><math>0.792 \pm 0.002</math></td>
  <td align="center"><math>0.4196</math></td>
  <td align="center"><math>-0.4464</math></td>
  <td align="center"><math>0.3745</math></td>
  <td align="center"><math>- 0.4464</math></td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>0.8732</math></td>
  <td align="center"><math>0.4431</math></td>
  <td align="center"><math>-0.3670</math></td>
  <td align="center"><math>0.3667</math></td>
  <td align="center"><math>-0.3819</math></td>
</tr>

<tr>
  <td align="center"><math>L^+</math></td>
  <td align="center"><math>0.965175</math></td>
  <td align="center"><math>0.4620</math></td>
  <td align="center"><math>-0.3069</math></td>
  <td align="center"><math>0.3627</math></td>
  <td align="center"><math>-0.3441</math></td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>0.9852</math></td>
  <td align="center">---</td>
  <td align="center"><math>-0.2944</math></td>
  <td align="center"><math>0.3623</math></td>
  <td align="center"><math>-0.3388</math></td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>1.0731</math></td>
  <td align="center"><math>0.4725</math></td>
  <td align="center"><math>-0.2565</math></td>
  <td align="center"><math>0.3608</math></td>
  <td align="center"><math>-0.3061</math></td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>1.1489</math></td>
  <td align="center"><math>0.4804</math></td>
  <td align="center"><math>-0.2287</math></td>
  <td align="center"><math>0.3588</math></td>
  <td align="center"><math>-0.2795</math></td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>1.2262</math></td>
  <td align="center"><math>0.4882</math></td>
  <td align="center"><math>-0.2019</math></td>
  <td align="center"><math>0.3573</math></td>
  <td align="center"><math>-0.2585</math></td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center"><math>1.5</math></td>
  <td align="center">---</td>
  <td align="center"><math>-0.1366</math></td>
  <td align="center"><math>0.3529</math></td>
  <td align="center"><math>-0.1983</math></td>
</tr>
</table>


<table border="1" align="center"><tr><td align="center">
<table border="0" align="center" cellpadding="0">
<tr>
<td align="center">
&nbsp;<br /><b>Figure 3</b><br />
[[File:MPT77seven.png|400px|MPT77seven]]
</td>
<td align="center">
&nbsp;<br /><b>Figure 4</b><br />
[[File:MPT77eight.png|400px|MPT77eight]]
</td>
</tr>
<tr>
  <td align="center">
Compare to:<br />Fig. 5 (p. 594) of {{ MPT77 }}
  </td>
  <td align="center">
Compare to:<br />Fig. 4 (p. 593) of {{ MPT77 }}
  </td>
</tr>
</table>

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