Editing 2DStructure/ToroidalCoordinates (section)

===Associated Analytic Expressions===
We define the following terms that are functions only of the four principal model parameters, <math>~(a, Z_0,  \varpi_t, r_t)</math>, and therefore can be treated as constants while carrying out the pair of nested integrals that determine <math>~q_0</math>:

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

<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="4">&nbsp;</td>
  <td align="center">Parameters evaluated for Figure 2</td>
</tr>
<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>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
\frac{5^2}{2^4\cdot 3^2} \approx 0.17361111
</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>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~\frac{17 \cdot 1409}{5^4} \approx 38.3248 
</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>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="left">
<math>~
-~\frac{5^2}{2^6\cdot 3}\biggl[ 1\mp \sqrt{ \frac{17\cdot 1409}{3^2\cdot 5^4}} \biggr] 
</math>
  </td>
</tr>
</table>

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

Then,
<div align="center">
<math>\xi_1|_\mathrm{max} = 
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2  \biggr]^{-1/2}
\approx 1.1927843</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\xi_1|_\mathrm{min} = 
\biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2  \biggr]^{-1/2}
\approx 1.0449467\, ,</math>
</div>

which implies,
<div align="center">
<math>x_\mathrm{max} = \tanh^{-1}\biggl( \frac{a}{\varpi_t-\beta_+} \biggr) 
\approx 0.61137548 </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>x_\mathrm{min} = \tanh^{-1}\biggl( \frac{a}{\varpi_t-\beta_-} \biggr) 
\approx 0.29871048 \, .</math>
</div>

----


Also let,
<div align="center">
<math>~\xi_1 \rightarrow \cosh x \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; in which case,
&nbsp; &nbsp; &nbsp; &nbsp; 
<math>~(\xi_1^2 - 1)^{1/2} \rightarrow \sinh x</math>
&nbsp; &nbsp; and,
&nbsp; &nbsp;  
<math>~(1-\xi_1^{-2})^{-1/2} = \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \rightarrow \coth x\, ,</math>
</div>
and define,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr><td align="center">
<table border="0" cellpadding="5" align="center">
<tr><td align="center" colspan="4">&nbsp;</td><td align="center" colspan="3">Evaluated for</td></tr>
<tr>
  <td align="center" colspan="4">&nbsp;</td>
  <td align="center"><math>~x_\mathrm{max} ~(i.e.~\xi_1|_\mathrm{max})</math><p></p>
----
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="center"><math>~x_\mathrm{min} ~(i.e.~\xi_1|_\mathrm{min})</math><p></p>
----
  </td>
</tr>
<tr>
  <td align="right">
<math>~A(\xi_1)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{Z_0}{a} \biggr)^2 + \biggl[ \coth x - \frac{\varpi_t}{a} \biggr]^2 
</math> 
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="center">
<math>~ \approx 5.23510376 </math> 
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="center">
<math>~ \approx 6.49460545</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B(\xi_1)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\coth x - \frac{\varpi_t}{a} + \biggl( \frac{2\varpi_t Z_0^2}{a\kappa} \biggr) </math>
  </td>
<td align="center">&nbsp;  <math>\rightarrow</math>  &nbsp;</td>
  <td align="center">
<math>~\approx 14.1645439</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="center">
<math>~\approx 15.77670608</math>
  </td>
</tr>
</table>

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

<table border="1" cellpadding="8" align="right">
<tr><th align="center"><font size="+1">Figure 3</font></th></tr>
<tr>
<td align="center">
[[File:ConstantXi2.png|300px|Diagram of Torus and xi_2-constant Toroidal Coordinate curve]]
</td>
</tr>
</table>
Then, for each value of the radial coordinate, <math>~\xi_1 = \cosh x</math>, within the range,
<div align="center">
<math>~x_\mathrm{max} \geq x \geq x_\mathrm{min} \, ,</math>
</div>
the limiting values <math>~(\pm)</math> of the angular coordinate are,

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

<tr>
  <td align="right">
<math>~\xi_2\biggr|_\pm</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \cosh x - \sinh x \biggl( \frac{2a^2}{\kappa}\cdot \frac{A}{B} \biggr)
\biggl[1 \pm \sqrt{1-\frac{AC}{B^2}}\biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>

This provides the desired analytic expression for the two limits of integration over the angular coordinate that must be carried out in order to evaluate the key function, <math>~q_0</math>, in our determination of the gravitational potential at the cylindrical-coordinate location, <math>~(R_*, Z_*) = (a, Z_0)</math>.  

We should note that at both limits &#8212; that is, ''at'' the two values of <math>~x</math> for which the functions <math>~A</math> and <math>~B</math> have been evaluated, immediately above &#8212; the ratio under the square-root,
<div align="center">
<math>~\frac{AC}{B^2} = 1 \, .</math>
</div>
Hence, at both radial-coordinate limits, the two angular-coordinate limits, <math>~\xi_2|_\pm</math>, are degenerate.  Furthermore, we have discovered that the value of this degeneracy angle is exactly the same at both radial-coordinate limits; specifically, we find that, for our chosen test-mass distribution,
<div align="center">
<math>~\xi_2\biggr|_\mathrm{deg} \approx 0.885198 \, .</math>
</div>
This means that exactly the same <math>\xi_2</math>-constant, toroidal-coordinate "line" intersects the surface of the (pink) test-mass torus at both the point of first contact <math>~(\xi_1|_\mathrm{max})</math> and the point of final contact <math>~(\xi_1|_\mathrm{min})</math>.  In order to illustrate this, this particular <math>\xi_2</math>-constant "line" has been traced by a sequence of red dots in Figure 3.  As just described, it passes smoothly through the points on the surface of the (pink) torus where the <math>\xi_1</math>-circles make first (small black-dotted circle) and final (large black-dotted circle) contact.