Editing Apps/MaclaurinToroid (section)

=Maclaurin Toroid (EH85)=

{{ EH85full }} &#8212; hereafter, {{ EH85hereafter }} &#8212; have constructed a set of uniform-density, axisymmetric configurations that show how the Maclaurin toroid sequence is connected to the Maclaurin spheroid sequence.  The following table displays the structural characteristics of these configurations; the numbers in the first four columns have been drawn directly from Table 1 of {{ EH85hereafter }}.  The quantity, <math>E_\mathrm{EH85}</math>, that has been used to normalize the total energy in, for example, the fourth column of this table, is given by the expression,
<table border="0" align="center" cellpadding="3">
<tr>
  <td align="right"><math>E_\mathrm{EH85}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>(4\pi G)^2 M^5/J^2 \, .</math></td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ EH85 }}, &sect;2.2, p. 291, Eq. (7)
  </td>
</tr>
</table>
For purposes of comparison between the separate published works of {{ MPT77hereafter }} and {{ EH85hereafter }}, here we desire to shift back to the normalization adopted by {{ MPT77hereafter }}, namely,
<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> 
\frac{3}{5}E_\mathrm{T78} \, .
</math>
  </td>
</tr>
</table>

Building on our [[Apps/MaclaurinSpheroidSequence#EnergyNorm|separate discussion of energy normalizations]] where we showed that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ \frac{E_\mathrm{EH85}}{E_\mathrm{T78}} \biggr]^3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{3(4\pi)^2}{j^6} \, ,
</math>
  </td>
</tr>
</table>
we recognize immediately that,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{E_\mathrm{EH85}}{E_0} \biggr] = \biggl[ \frac{E_\mathrm{EH85}}{E_\mathrm{T78}} \biggr] \cdot \frac{E_\mathrm{T78}}{E_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\frac{5}{3} \biggl[ \frac{3(4\pi)^2}{j^6} \biggr]^{1 / 3} 
=
\frac{5(4\pi/3)^{2 / 3}}{j^2} 
\, .
</math>
  </td>
</tr>
</table>

We have evaluated this conversion factor and the consequential normalized total energy for each of the {{ EH85hereafter }} equilibrium configurations and have presented the results in columns six and seven, respectively, of the following table. 


<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center" colspan="4">
Data extracted from Table 1 (p. 290) of &hellip;<br />{{ EH85figure }}
  </td>
  <td align="center" colspan="3">
Our Determination
  </td>
</tr>
<tr>
  <td align="center"><math>h_0^2</math></td>
  <td align="center"><math>j^2</math></td>
  <td align="center"><math>\frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math></td>
  <td align="center"><math>\frac{T_\mathrm{rot} + W_\mathrm{grav}}{E_\mathrm{EH85}}</math></td>
  <td align="center"><math>L_* \equiv (4\pi/3)^{2 / 3} (3j^2)^{1 / 2}</math></td>
  <td align="center"><math>\frac{E_\mathrm{EH85}}{E_0} = \frac{5(4\pi/3)^{2 / 3}}{j^2}</math></td>
  <td align="center"><math>\frac{T_\mathrm{rot} + W_\mathrm{grav}}{E_0}</math></td>
 
</tr>

<tr>
  <td align="center"><math>5.802\times 10^{-2}</math></td>
  <td align="center"><math>3.964\times 10^{-2}</math></td>
  <td align="center"><math>0.445</math></td>
  <td align="center"><math>- 1.03\times 10^{-3}</math></td>
  <td align="center"><math>0.8961</math></td>
  <td align="center"><math>327.76</math></td>
  <td align="center"><math>-0.3376</math></td>
</tr>

<tr>
  <td align="center"><math>5.834\times 10^{-2}</math></td>
  <td align="center"><math>3.816\times 10^{-2}</math></td>
  <td align="center"><math>0.439</math></td>
  <td align="center"><math>- 1.02\times 10^{-3}</math></td>
  <td align="center"><math>0.8792</math></td>
  <td align="center"><math>340.48</math></td>
  <td align="center"><math>-0.3473</math></td>
</tr>

<tr>
  <td align="center"><math>5.916\times 10^{-2}</math></td>
  <td align="center"><math>3.752\times 10^{-2}</math></td>
  <td align="center"><math>0.437</math></td>
  <td align="center"><math>- 1.02\times 10^{-3}</math></td>
  <td align="center"><math>0.8718</math></td>
  <td align="center"><math>346.28</math></td>
  <td align="center"><math>-0.3532</math></td>
</tr>

<tr>
  <td align="center"><math>6.075\times 10^{-2}</math></td>
  <td align="center"><math>3.718\times 10^{-2}</math></td>
  <td align="center"><math>0.436</math></td>
  <td align="center"><math>- 1.02\times 10^{-3}</math></td>
  <td align="center"><math>0.8678</math></td>
  <td align="center"><math>349.45</math></td>
  <td align="center"><math>-0.3564</math></td>
</tr>

<tr>
  <td align="center"><math>6.416\times 10^{-2}</math></td>
  <td align="center"><math>3.209\times 10^{-2}</math></td>
  <td align="center"><math>0.412</math></td>
  <td align="center"><math>- 9.73\times 10^{-4}</math></td>
  <td align="center"><math>0.8063</math></td>
  <td align="center"><math>404.89</math></td>
  <td align="center"><math>-0.3939</math></td>
</tr>

<tr>
  <td align="center"><math>6.766\times 10^{-2}</math></td>
  <td align="center"><math>3.090\times 10^{-2}</math></td>
  <td align="center"><math>0.403</math></td>
  <td align="center"><math>- 9.61\times 10^{-4}</math></td>
  <td align="center"><math>0.7912</math></td>
  <td align="center"><math>420.47</math></td>
  <td align="center"><math>-0.4041</math></td>
</tr>

<tr>
  <td align="center"><math>7.070\times 10^{-2}</math></td>
  <td align="center"><math>3.016\times 10^{-2}</math></td>
  <td align="center"><math>0.389</math></td>
  <td align="center"><math>- 9.53\times 10^{-4}</math></td>
  <td align="center"><math>0.7816</math></td>
  <td align="center"><math>430.79</math></td>
  <td align="center"><math>-0.4105</math></td>
</tr>

<tr>
  <td align="center"><math>6.739\times 10^{-2}</math></td>
  <td align="center"><math>3.192\times 10^{-2}</math></td>
  <td align="center"><math>0.376</math></td>
  <td align="center"><math>- 9.74\times 10^{-4}</math></td>
  <td align="center"><math>0.8041</math></td>
  <td align="center"><math>407.04</math></td>
  <td align="center"><math>-0.3965</math></td>
</tr>

<tr>
  <td align="center"><math>5.376\times 10^{-2}</math></td>
  <td align="center"><math>3.751\times 10^{-2}</math></td>
  <td align="center"><math>0.368</math></td>
  <td align="center"><math>- 1.04\times 10^{-3}</math></td>
  <td align="center"><math>0.8717</math></td>
  <td align="center"><math>346.38</math></td>
  <td align="center"><math>-0.3602</math></td>
</tr>

<tr>
  <td align="center"><math>4.007\times 10^{-2}</math></td>
  <td align="center"><math>4.502\times 10^{-2}</math></td>
  <td align="center"><math>0.365</math></td>
  <td align="center"><math>- 1.12\times 10^{-3}</math></td>
  <td align="center"><math>0.9550</math></td>
  <td align="center"><math>288.60</math></td>
  <td align="center"><math>-0.3232</math></td>
</tr>

<tr>
  <td align="center"><math>3.202\times 10^{-2}</math></td>
  <td align="center"><math>5.100\times 10^{-2}</math></td>
  <td align="center"><math>0.366</math></td>
  <td align="center"><math>- 1.18\times 10^{-3}</math></td>
  <td align="center"><math>1.0164</math></td>
  <td align="center"><math>254.76</math></td>
  <td align="center"><math>-0.3006</math></td>
</tr>

<tr>
  <td align="center"><math>2.525\times 10^{-2}</math></td>
  <td align="center"><math>5.975\times 10^{-2}</math></td>
  <td align="center"><math>0.369</math></td>
  <td align="center"><math>- 1.26\times 10^{-3}</math></td>
  <td align="center"><math>1.1002</math></td>
  <td align="center"><math>217.45</math></td>
  <td align="center"><math>-0.2740</math></td>
</tr>

<tr>
  <td align="center"><math>1.811\times 10^{-2}</math></td>
  <td align="center"><math>7.364\times 10^{-2}</math></td>
  <td align="center"><math>0.378</math></td>
  <td align="center"><math>- 1.37\times 10^{-3}</math></td>
  <td align="center"><math>1.2214</math></td>
  <td align="center"><math>176.43</math></td>
  <td align="center"><math>-0.2417</math></td>
</tr>

<tr>
  <td align="center"><math>1.127\times 10^{-2}</math></td>
  <td align="center"><math>9.914\times 10^{-2}</math></td>
  <td align="center"><math>0.396</math></td>
  <td align="center"><math>- 1.53\times 10^{-3}</math></td>
  <td align="center"><math>1.4171</math></td>
  <td align="center"><math>131.05</math></td>
  <td align="center"><math>-0.2005</math></td>
</tr>

<tr>
  <td align="center"><math>4.812\times 10^{-3}</math></td>
  <td align="center"><math>1.618\times 10^{-1}</math></td>
  <td align="center"><math>0.430</math></td>
  <td align="center"><math>- 1.76\times 10^{-3}</math></td>
  <td align="center"><math>1.8104</math></td>
  <td align="center"><math>80.300</math></td>
  <td align="center"><math>-0.1413</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 5</b><br />
[[File:EH85nine.png|400px|EH85nine]]
</td>
<td align="center">
&nbsp;<br /><b>Figure 6</b><br />
[[File:EH85ten.png|400px|EH85ten]]
</td>
</tr>
</table>

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