Editing AxisymmetricConfigurations/PoissonEq (section)

===Stahler (1983)===

<table border="0" cellpadding="5" width="100%" align="center"><tr><td align="left">
<table border="1" cellpadding="5" align="left"><tr>
<td align="center" bgcolor="red">&nbsp; 2D &nbsp;</td>
<td align="center" bgcolor="yellow">&nbsp; Tor &nbsp;</td>
</tr></table>
</td></tr></table>

[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] used a [[AxisymmetricConfigurations/HSCF#Introduction|self-consistent-field]] technique to construct equilibrium sequences of rotationally flattened, isothermal gas clouds.  At each iteration step, the method that he adopted to evaluate the gravitational potential along the outer boundary of his computational mesh was essentially the same as the method used by [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] &#8212; see our [[#Deupree_.281974.29|above description]] &#8212; to model the time-dependent behavior of pulsating stars.  It appears as though Stahler was unaware of Deupree's earlier development of this method to evaluate the boundary potential, as Deupree's (1974) paper is not among Stahler's list of references.  

<!--
In the chapter of this H_Book that focuses on a discussion of [[Apps/DysonWongTori|Dyson-Wong tori]], we have included the expression for the [[Apps/DysonWongTori#RingPotential|gravitational potential of a thin ring]] of mass, <math>~M</math>, that passes through the meridional plane at coordinate location, <math>~(\varpi^', z^') = (a, 0)</math>, as derived, for example, by [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)] and by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential W. D. MacMillan (1958; originally, 1930)], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\varpi, z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{2GMc}{\pi\rho_1}\biggr] K(k)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{2GM}{\pi } \biggr]\frac{1}{\sqrt{(\varpi+a)^2 + z^2}} \times K(k) \, ,
</math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~k</math>
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{4a\varpi }{(\varpi+a)^2 + z^2} \biggr]^{1 / 2}  \, .
</math>
  </td>
</tr>
</table>
</div>
-->
In describing [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler's (1983a)] method, we will first draw upon 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.)  Stahler has argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' &#8212; <math>~\delta M(\varpi^', z^')</math>  being the appropriate differential mass contributed by each ring element &#8212; positioned at various meridional coordinate locations throughout the mass distribution.  According to his independent derivation, the differential contribution to the potential, <math>~\delta\Phi_g(\varpi, z)</math>, due to each differential mass element is (see his equation 11 and the explanatory text that follows it):
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta\Phi_g(\varpi,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[\frac{2G}{\pi \varpi^'}\biggr] \frac{\delta M}{[(\alpha + 1)^2 + \beta^2]^{1 / 2}}
\times K\biggl\{ \biggl[ \frac{4\alpha}{(\alpha+1)^2 + \beta^2} \biggr]^{1 / 2} \biggr\}
</math>
  </td>
</tr>
<!--
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[\frac{2G}{\pi }\biggr] \frac{\delta M}{[(\varpi^' \alpha + \varpi^')^2 + (\varpi^' \beta)^2]^{1 / 2}}
\times K\biggl\{ \biggl[ \frac{4\alpha (\varpi^')^2}{(\varpi^' \alpha+\varpi^')^2 + (\varpi^' \beta)^2} \biggr]^{1 / 2} \biggr\}
</math>
  </td>
</tr>
-->
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[\frac{2G}{\pi }\biggr] \frac{\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}
\times K\biggl\{ \biggl[ \frac{4\varpi^' \varpi}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Stahler's expression for each ''thin ring'' contribution is clearly the same as our "key equation" expression for <math>~\Phi_\mathrm{TR}</math> if the individual ring being considered cuts through the meridional plane at <math>~(\varpi^', z^') = (a, 0)</math>.

In the context of our [[#For_Axisymmetric_Systems|above discussion of a Green's function expression written in terms of toroidal functions]], we have shown that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\varpi, z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2G}{\varpi^{1 / 2}} \int\int (\varpi^')^{1 / 2} \mu K(\mu) \rho(\varpi^', z^') d\varpi^' dz^' \, ,</math>
  </td>
</tr>
</table>
</div>
where, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu^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] \, .
</math>
  </td>
</tr>
</table>
</div>
Recognizing that, for axisymmetric structures, the differential mass element is, <math>~dM^' = 2\pi \rho(\varpi^', z^') \varpi^' d\varpi^' dz^'</math>, this integral expression may be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\varpi, z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2G}{\varpi^{1 / 2}} \int\int (\varpi^')^{1 / 2} \mu K(\mu) \biggl[ \frac{dM^'}{2\pi \varpi^'} \biggr]  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{G}{\pi} \int\int \biggl[ \frac{1}{\varpi^'\varpi}\biggr]^{1 / 2} \biggl[\frac{4\varpi^' \varpi}{(\varpi + \varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} K(\mu) dM^'  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2G}{\pi} \int\int  \biggl[\frac{1}{(\varpi + \varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} K(\mu) dM^'  \, .</math>
  </td>
</tr>
</table>
</div>
We see that our expression for the differential contribution to the potential exactly matches [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler's (1983a)].  It is therefore fair to say that Stahler's expression for the gravitational potential is derivable from a 3D Green's function that is written in terms of toroidal functions.  We note that, in his study of axisymmetric systems, Stahler made the decision to evaluate the gravitational potential both inside as well as outside of the mass distribution using the same integral expression.