Editing Apps/MaclaurinToroid (section)

==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>