Editing 2DStructure/ToroidalCoordinates (section)

==Code Layout==
Here are some suggestions related to the development of a computer program that can perform the one-dimensional integral over <math>~\xi_1</math>.
<pre>
STEP 1:  Specify numerical values for torus parameters &hellip; varpi_t &amp; r_t

STEP 2:  Associate the parameters "a" and "Z0" with the cylindrical coordinate 
         location (R*,Z*) at which gravitational potential is to be evaluated.

STEP 3:  Given the numerical values for these four parameters, calculate "kappa", 
         "C", "beta_plus", and "beta_minus" as prescribed by the following algebraic 
         relations &hellip;
</pre>

<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\kappa</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
Z_0^2 + a^2  - (\varpi_t^2 - r_t^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1 + \biggl( \frac{2Z_0}{\kappa}\biggr)^2 ( \varpi_t^2 - r_t^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
- \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] 
</math>
  </td>
</tr>
</table>

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

<pre>
STEP 4:  The (numerical values of the) limits of integration are set by the following 
         two expressions &hellip;
</pre>
<div align="center">
<p><math>\xi_1|_\mathrm{max} = 
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2} </math></p>

<p>and</p>

<p><math>\xi_1|_\mathrm{min} = 
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}\, .</math></p>
</div>

<pre>
STEP 4 (cont.):  Establish a 1D grid with, say, 100 zones, assigning values of xi_1 
         that are equally spaced between these two limiting values; let "delta" be the 
         grid spacing.  (Perhaps use logarithmic spacing.)  The mid-point value of 
         the coordinate location of the accompanying 99 grid cells should also be 
         determined.

STEP 5:  For each of the 99 grid-cell coordinate values, "xi1", calculate the following 
         parameter, or aggregate-term, values:
</pre>

<div align="center" id="Parameters">
<table border="1" cellpadding="5" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{2^{5/2} G \rho_0 a^{2}}{3} \, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mu</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[
\frac{2(\xi_1^2-1)^{1/2}}{(\xi_1^2-1)^{1/2} + \xi_1} \biggr]^{1/2} \, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{coef}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{(\xi_1+1)^{1/2}K(\mu) }{(\xi_1^2 - 1)^2 [ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{1/2} } \, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{tempbeta}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\varpi_t}{a} - \frac{\xi_1}{(\xi_1^2-1)^{1/2}} \, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{Z_0}{a}\biggr)^2 + (\mathrm{tempbeta})^2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\varpi_t Z_0^2}{a\kappa}\biggr) - (\mathrm{tempbeta}) \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl(\frac{\varpi_i}{a}\biggr)_\pm</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\kappa}{2a^2}\cdot \frac{\mathrm{B}}{\mathrm{A}} 
\biggl[1 \pm \sqrt{1-\frac{AC}{B^2}}  \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi_2\biggr|_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi_1 - \frac{ (\xi_1^2-1)^{1/2}}{(\varpi_i/a)_\pm}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\theta_\mathrm{max}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cos^{-1}(\xi_2|_+)
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\theta_\mathrm{min}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cos^{-1}(\xi_2|_-)
\, ;
</math>
  </td>
</tr>
</table>

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

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

<tr>
  <td align="right">
<math>~T(\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\sin \theta(5\xi_1^2 - 4\xi_1 \cos \theta - 1)}{(\xi_1+1)^{1/2} (\xi_1 - \cos \theta)^{3/2}} 
- 4\xi_1 E\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) 
+ (\xi_1-1) F\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Phi(a,Z_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\Phi_0 \sum_{n=1}^{99} \mathrm{delta}\cdot \mathrm{coef} \cdot [T(\theta_\mathrm{max}) - T(\theta_\mathrm{min})] \, .
</math>
  </td>
</tr>
</table>
</div>