Editing AxisymmetricConfigurations/PoissonEq (section)

=====Full Three-Dimensional Generality=====

Following the lead of [http://adsabs.harvard.edu/abs/1975ApJ...199..619B Black &amp; Bodenheimer (1975)], we will insert into this integral relation the Green's function expression for <math>~|\vec{x}^{~'}- \vec{x} |^{-1} </math> as given in terms of ''Spherical Harmonics'', <math>~Y_{\ell m}</math>, which in turn can be written in terms of ''Associated Legendre Functions.''  [[#Ylm|Table 2, below]], provides the primary details.

Written in the context of a spherical coordinate system we have,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-G \int \frac{1}{|\vec{x}^{~'} - \vec{x}|} ~\rho(\vec{x}^{~'}) d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-G \int 
\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{4\pi}{2\ell+1} \biggl[  \frac{r_<^\ell}{r_>^{\ell+1}} \biggr] Y_{\ell m}^*(\theta^', \phi^') Y_{\ell m}(\theta,\phi)
~\rho(r^', \theta^', \phi^') d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-4\pi G  
\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)} \biggl[ \frac{1}{r^{\ell+1}}\int_0^r (r^')^\ell  Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^' 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; <math>~
+ r^\ell \int_r^\infty (r^')^{-(\ell+1)}  Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^' \biggr] \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 66, Eq. (2-122)
  </td>
</tr>
</table>

If the distance from the origin, <math>~r</math>, of a boundary point (''i.e.'', any point lying along the dashed green lines in Figure 1) is greater than the distance from the origin, <math>~r^'</math>, of ''all'' of the (pink) mass elements, the second integral can be completely ignored.  We have, then, an expression that will henceforth be referred to as,
<!-- OLD COMMENT
In making this last step, we have moved the radial distance with its associated exponent, <math>~(r_>)^{-(\ell+1)}</math>, outside of the mass integral.  At the same time, we have left the alternate radial distance along with its associated exponent, <math>~(r_<)^\ell</math>, inside the integral and, accordingly, have labeled it with a "prime" to emphasize its association with the integral.  This has been done under the assumption that ''every'' (pink) mass element (tagged by a "primed" coordinate) lies closer to the coordinate origin than ''every'' point on the boundary (dashed green lines).  The integral must be split into two parts with the locations (inside or outside of the integral) of <math>~r_></math> and <math>~r_<</math> swapped in the second part of the integral if, in any case, the point on the boundary lies closer to the coordinate origin than ''any'' (pink) mass element(s).
-->
<div align="center" id="PotentialA">
<table border="0" align="center">
<tr>
<td align="center" colspan="3">
<font color="#770000">'''Form A of the Boundary Potential'''</font><br />
</td>
</tr>
<tr>
  <td align="right">
<math>~ \Phi_B(r,\theta,\phi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-4\pi G  
\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{+\ell} \frac{Y_{\ell m}(\theta,\phi)}{(2\ell+1)} \biggl[ \frac{1}{r^{\ell+1}}\int_0^r (r^')^\ell  Y_{\ell m}^*(\theta^', \phi^')
~\rho(r^', \theta^', \phi^') d^3x^' \biggr] \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[https://archive.org/details/ClassicalElectrodynamics2nd Jackson (1975)], p. 137, Eq. (4.2)<br />
[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 238, Eqs. (7.53) - (7.54)
  </td>
</tr>
</table>
</div>

Rewriting this expression for <math>~\Phi_B</math> in terms of cylindrical coordinates &#8212; which aligns with our chosen grid coordinate system &#8212; and admitting that in practice our summation over the index, <math>~\ell</math>, cannot extend to infinity, we have,

<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi_B(\varpi, \phi, z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-4\pi G 
\sum_{\ell=0}^{\ell_\mathrm{max}} \sum_{m=-\ell}^{+\ell} \frac{Y_{\ell m}}{(2\ell+1)} \biggl[ \varpi^2 + z^2 \biggr]^{-(\ell+1)/2} \int  Y_{\ell m}^* \biggl[ (\varpi^')^2 + (z^')^2 \biggr]^{\ell/2} 
~\rho(\varpi^', \phi^', z^') d^3x^' \, .
</math>
  </td>
</tr>
</table>

Note that, as a consequence of assuming that our configurations have equatorial-plane symmetry, the weighted integral over the mass distribution necessarily goes to zero anytime the sum of the two indexes, <math>~(\ell + m)</math>, is an odd number.  This is because, in each of these situations &#8212; again, see [[#Ylm|Table 2, below]] for details &#8212; the <math>~Y_{\ell m}</math> includes an overall factor of <math>~\cos\theta</math>, which necessarily switches signs between the two hemispheres.  After setting <math>~\ell_\mathrm{max}=4</math> &#8212; and dropping all terms in the summation for which the index sum, <math>~(\ell + m)</math>, is odd &#8212; this expression becomes precisely the relation that was used to determine the ''boundary'' values of the gravitational potential in our earliest set of simulations; see, for example, [http://adsabs.harvard.edu/abs/1978PhDT.........6T Tohline (1978)], [http://adsabs.harvard.edu/abs/1980ApJ...235..866T Tohline (1980)], and [http://adsabs.harvard.edu/abs/1980ApJ...242..209B Bodenheimer, Tohline, &amp; Black (1980)].