Editing 2DStructure/ToroidalCoordinates (section)

==Statement of the Problem==

===Expression for the Axisymmetric Potential===
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl &amp; Tohline (1999; hereafter CT99)] derive an expression for the Newtonian gravitational potential in terms of a ''Compact Cylindrical Green's Function'' expansion.  They show &#8212; see, for example, their equation (31) &#8212; that when expressed in terms of cylindrical coordinates, the potential at any meridional location, <math>\varpi = R_*</math> and <math>~Z = Z_*</math>, due to an axisymmetric mass distribution, <math>~\rho(\varpi, Z)</math>, is
<div align="center">
<math>
\Phi(R_*,Z_*) = - \frac{2Gq_0}{R_*^{1/2}}  ,
</math>
</div>
where,
<div align="center">
<math>
q_0 = \int_\Sigma \varpi^{1/2} Q_{-1/2}(\Chi) \rho(\varpi, Z) d\sigma,
</math>
</div>
<math>~d\sigma = d\varpi dZ</math> is a differential area element in the meridional plane, and the dimensionless argument (the modulus) of the special function, <math>~Q_{-1/2}</math>, is,
<div align="center">
<math>
\Chi \equiv \frac{R_*^2 + \varpi^2 + (Z_* - Z)^2}{2R_* \varpi} .
</math>
</div>
Next, following the lead of CT99, we note that according to the Abramowitz &amp; Stegun (1965),
<div align="center">
<math>Q_{-1/2}(\Chi) = \mu K(\mu) \, ,</math>
</div>
where, the function <math>~K(\mu)</math> is the complete elliptic integral of the first kind and, for our particular problem,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~2(1+\Chi)^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\biggl[ 1+\frac{R_*^2 + \varpi^2 + (Z_* - Z)^2}{2R_* \varpi}\biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{4R_*\varpi}{(R_* + \varpi)^2 + (Z_* - Z)^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Hence, we can write,
<div align="center">
<math>
q_0 = \int\int \varpi^{1/2} \mu K(\mu) \rho(\varpi, Z) d\varpi dZ \, .
</math>
</div>

As has been explained in [[Appendix/Ramblings/ToroidalCoordinates#Confirmation_Provided_by_Trova.2C_Hur.C3.A9_and_Hersant|an accompanying set of notes]], this is precisely the same expression for the gravitational potential that [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T A. Trova, J.-M. Hur&eacute; and F. Hersant (2012; MNRAS, 424, 2635)] used in their study of the potential of self-gravitating, axisymmetric discs.  

Our objective, here, is to examine whether or not it might be advantageous to transform this expression to one in which the double integral is performed on a toroidal, rather than a cylindrical, coordinate system.

===Chosen Test Mass Distribution===

For purposes of illustration, we will follow the lead of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)] &#8212; see the left-hand panel of the following figure ensemble &#8212; and seek to determine the gravitational potential, both inside and outside, of a uniform-density, equatorial-plane torus whose (pink) meridional cross-section is exactly a circle.  More specifically, as illustrated in our Figure 1 &#8212; see the right-hand panel of the following figure ensemble &#8212; at all azimuthal angles, a cross-section through the (pink) torus is prescribed by the familiar algebraic expression for an off-center circle, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\varpi_t - \varpi)^2 + Z^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_t^2 \, .</math>
  </td>
</tr>
</table>
</div>

Everywhere inside this toroidal surface we set <math>~\rho(\varpi, Z) = \rho_0</math>, that is, the density is uniform with the value, <math>~\rho_0</math>.

<div align="center" id="THH12Figure4">
<table border="1" cellpadding="8">

<tr><td align="center">
Figure 4 extracted without modification from p. 2640 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Hur&eacute; &amp; Hersant (2012)]
<br />
<i>"The Potential of Discs from a 'Mean Green Function'"</i><br />
Monthly Notices of the Royal Astronomical Society, vol. 424, pp. 2635-2645 &copy; RAS
</td>
<th align="center"><font size="+1">Our Figure 1</font></th>
</tr>

<tr>
<td align="center">
[[File:Figure4THH2012.png|350px|To be inserted: Fig. 4 from Trova, Hur&eacute; &amp; Hersant (2012)]]
</td>
<td align="center">
[[File:DiagramToroidalCoordinates.png|350px|Diagram of Torus and Toroidal Coordinates]]
</td>
</tr>
</table>
</div>

<span id="OffCenterCircle">Notice that another off-center circle &#8212; this one with a purple perimeter and otherwise white, rather than pink &#8212; appears in our Figure 1 diagram.  In the discussion that follows, it will be used to represent the meridional-plane cross-section of one axisymmetric surface in an MF53 toroidal-coordinate system.  Here we simply point out that this "surface" is also prescribed by an algebraic expression for an off-center circle, namely,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(R_0 - \varpi)^2 + (Z_0 - Z)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_0^2 \, .</math>
  </td>
</tr>
</table>
</div>