Editing Appendix/Ramblings/ToroidalCoordinates (section)

===Summary of Structure===
As derived [[Apps/PapaloizouPringleTori|elsewhere]], the accretion tori constructed by Papaloizou &amp; Pringle (1984; hereafter PP84) have the following surface properties.  For a given choice of the dimensionless Bernoulli constant, <math>C_\mathrm{B}^'</math>,

<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>
~\chi_\mathrm{inner}
</math>
  </td>
  <td align="center">
<math>~ =</math> 
  </td>
  <td align="left"><math>
~\frac{1}{1 + \sqrt{1-2C_\mathrm{B}^'}} \, ;
  </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\chi_\mathrm{outer} 
</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left"><math>
~\frac{1}{1 - \sqrt{1-2C_\mathrm{B}^'}} \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\chi_0 
</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left"><math>
~\frac{1}{2} (\chi_\mathrm{outer} + \chi_\mathrm{inner}) = \frac{1}{2C_\mathrm{B}^'} \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\delta 
</math>
  </td>
  <td align="center">
<math>~\equiv</math> 
  </td>
  <td algin="left">
<math>
~\frac{\chi_\mathrm{outer} - \chi_\mathrm{inner}}{\chi_0} = 2\sqrt{1-2C_\mathrm{B}^'} \, .
</math>
  </td>
</tr>
</table>
</div>

So if I want to construct PP84 tori that are approximately the same size/shape as the MF53 tori illustrated above, I should choose values of the dimensionless Bernoulli constant as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
~{\chi_0}\biggr|_\mathrm{PP84} 
</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left"><math>
~{\chi_0}\biggr|_\mathrm{MF53}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\Rightarrow~~~~ \frac{1}{2C_\mathrm{B}^'}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\xi_1}{(\xi_1^2-1)^{1/2}}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\Rightarrow~~~~ C_\mathrm{B}^'
</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left">
<math>~\frac{(\xi_1^2-1)^{1/2}}{2\xi_1} \, .
</math>
  </td>
</tr>
</table>
</div>

In the accompanying figure labeled "Papaloizou-Pringle Tori," we've drawn three different <math>~C_\mathrm{B}^' = \mathrm{constant}</math> meridional contours for the PP84 tori where the values of the dimensionless Bernoulli constants have been chosen to produce values of <math>~\chi_0</math> that are identical to the values displayed by the three MF53 tori shown above.  The following table details properties of these three PP84 tori that have been constructed in an effort to facilitate comparison with the table shown above for MF53 tori.

<table align="center" border="1" cellpadding="8">
<tr>
  <th align="center" colspan="6">
<font color="maroon">
Properties of <math>C_\mathrm{B}^' = \mathrm{constant}</math> PP84 Toroidal Surfaces
</font>
  </th>
</tr>
<tr>
  <td align="center">
Curve in<br />Figure
  </td>
  <td align="center">
<math>C_\mathrm{B}^'</math>
  </td>
  <td align="center">
<math>\chi_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>\chi_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>\chi_0</math>
  </td>
  <td align="center">
<math>\delta</math>
  </td>
</tr>

<tr>
  <td align="center">
Red
  </td>
  <td align="center">
0.208
  </td>
  <td align="center">
0.567
  </td>
  <td align="center">
4.234
  </td>
  <td align="center">
2.400
  </td>
  <td align="center">
1.528
  </td>
</tr>

<tr>
  <td align="center">
Blue
  </td>
  <td align="center">
0.276
  </td>
  <td align="center">
0.599
  </td>
  <td align="center">
3.019
  </td>
  <td align="center">
1.809
  </td>
  <td align="center">
1.338
  </td>
</tr>

<tr>
  <td align="center">
Gold
  </td>
  <td align="center">
0.373
  </td>
  <td align="center">
0.665
  </td>
  <td align="center">
2.019
  </td>
  <td align="center">
1.342
  </td>
  <td align="center">
1.009
  </td>
</tr>
</table>