Editing AxisymmetricConfigurations/PoissonEq (section)

====Comparison With Other Related Derivations====
Now, given that Deupree chose to construct and evolve his models using a spherical coordinate system, he would have specified the relevant lengths, <math>~p</math> and <math>~q</math>, and each ring's differential cross-section, <math>~\delta A</math>, in terms of spherical coordinates.  In an effort to more clearly illustrate the connection between Deupree's expression for the (differential contribution to the) boundary potential and the expression for the boundary potential that we have [[#For_Axisymmetric_Systems|derived above for axisymmetric systems]], we will insert expressions for these terms that apply, instead, to a cylindrical-coordinate mesh.

Following the same line of reasoning as has been presented in our [[Apps/DysonWongTori#CylindricalLocation|accompanying discussion of MacMillan's work]], if the meridional-plane locations of the infinitesimally thin ring and the desired point on the boundary are, respectively, <math>~(\varpi^',z^')</math> and <math>~(\varpi,z)</math>, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k = \sqrt{1 - c^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[\frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2 }\biggr]^{1 / 2} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{c}{p}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[(\varpi + \varpi^')^2 + (z - z^')^2  \biggr]^{- 1 / 2} = \frac{k}{\sqrt{4\varpi \varpi^'}} \, .</math>
  </td>
</tr>
</table>
</div>
Hence &#8212; after acknowledging that, in cylindrical coordinates, the radius of each "infinitesimally thin ring" is, <math>~a = \varpi^'</math>, and the differential cross-section of each ring is, <math>~\delta A = \delta\varpi^' \delta z^'</math> &#8212; Deupree's expression for the (differential contribution to the) potential may be rewritten as,

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

<tr>
  <td align="right">
<math>~\delta\Phi_B(\varpi,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr]  K(k) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2G (\delta M) }{\pi } \cdot \frac{K(k) }{ \sqrt{(\varpi + a)^2 + (z - z^')^2 }}  \, ,  </math>
  </td>
</tr>
</table>
or it may be rewritten as,

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

<tr>
  <td align="right">
<math>~\delta\Phi_B(\varpi,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr]  K(k) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2G }{\sqrt{\varpi}} \biggl[ \delta\varpi^' \delta z^' \rho(\varpi^', z^') \sqrt{\varpi^'} k K(k) \biggr] \, .</math>
  </td>
</tr>
</table>

Notice that the first of these two rewritten expressions aligns perfectly with our "[[Appendix/EquationTemplates#Other_Equations_with_Assigned_Templates|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,
<table border="0" align="center" cellpadding="10"><tr><td align="center">
{{ Math/EQ TRApproximation }}
</td>
<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png|225px|link=Apps/DysonWongTori#ThinRingContours]]</td>
</tr></table>
(See our [[Apps/DysonWongTori#ThinRingContours|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.)  Next, referring back to the expression that was [[#For_Axisymmetric_Systems|derived above for axisymmetric systems from a toroidal-function-based Green's function]], namely,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(\varpi,z)\biggr|_{2D}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2G}{\sqrt{\varpi}} \int\limits_{\varpi^'} \int\limits_{z^'} d\varpi^' dz^' \rho(\varpi^',z^') \sqrt{\varpi^'}  
\mu K(\mu) \, , 
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[http://adsabs.harvard.edu/abs/1999ApJ...527...86C H. S. Cohl &amp; J. E. Tohline (1999)], p. 89, Eqs. (31) &amp; (32)
  </td>
</tr>
</table>
where,
<table border="0" align="center">
<tr>
  <td align="right">
<math>~\mu </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{4\varpi \varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2}
\, ,
</math>
  </td>
</tr>
</table>
we see that the second of these rewritten expressions for Deupree's <math>~\delta\Phi_B</math> aligns perfectly with our derived expression for the differential contribution to the potential of any axisymmetric mass distribution.  It is therefore fair to say that the expression that Deupree used to determine the gravitational potential along the boundary of his modeled configurations is derivable from a 3D Green's function that is written in terms of ''toroidal functions''.