Editing Appendix/Ramblings/Bordeaux (section)

===The Hur&eacute;, ''et al'' (2020) Presentation===

{{ SGFworkInProgress }}

====Notation====

In [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], the major and minor radii of the torus surface ("shell") are labeled, respectively, R<sub>c</sub> and b, and their ratio is denoted, 
<div align="center">
<math>~e \equiv \frac{b}{R_c} \, .</math>

[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eq. (1)
</div>
The authors work in cylindrical coordinates, <math>~(R, Z)</math>, whereas we refer to this same coordinate-pair as, <math>~(\varpi_W, z_W)</math>.  The quantity,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
[R + (R_c + b\cos\theta_H)]^2 + [Z - b\sin\theta_H]^2 \, .
</math>
  </td>
</tr>
</table>
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eqs. (5) &amp; (7)
</div>

We have affixed the subscript "H" to their meridional-plane angle, &theta;, to clarify that it has a different coordinate-base definition from the meridional-plane angle, &theta;, that appears in our above discussion of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] work. In their paper, the subscript "0" is used in the case of an infinitesimally thin hoop <math>~(b \rightarrow 0)</math>, that is to say,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta_0^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[R + R_c]^2 + Z^2 \, .
</math>
  </td>
</tr>
</table>
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;3, p. 5827, Eq. (13)
</div>

Generally, the argument (modulus) of the complete elliptic integral functions is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k_H</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2}{\Delta}\biggl[ R (R_c + b\cos\theta_H) \biggr]^{1 / 2}  \, ,
</math>
  </td>
</tr>
</table>
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eq. (4)
</div>
and, as stated in the first sentence of their &sect;3, reference may also be made to the ''complementary modulus'',
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k'_H</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[1 - k_H^2]^{1 / 2} \, .</math>
  </td>
</tr>
</table>

(Again, we have affixed the subscript "H" in order to differentiate from the modulus employed by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].)  And in the case of an infinitesimally thin hoop <math>~(b\rightarrow 0)</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[k^2_H]_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4R R_c}{\Delta_0^2}  \, .
</math>
  </td>
</tr>
</table>
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;3, p. 5827, Eq. (12)
</div>

====Key Finding====
On an initial reading, it appears as though the most relevant section of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] paper is &sect;8 titled, ''The Solid Torus.''  They write the gravitational potential in terms of the series expansion,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Psi_\mathrm{grav}(\vec{r})</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\Psi_0 + \sum\limits_{n=1}^N \Psi_n \, ,
</math>
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;7, p. 5831, Eq. (42)
</div>
where, after setting <math>~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c </math> and acknowledging that <math>~V_{0,0} = 1 \, ,</math> we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Psi_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}([k_H]_0) \biggr]
</math>
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eqs. (52) &amp; (53)
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{G\pi \rho_0 R_c b^2}{4 (k'_H)^2 \Delta_0^3} \biggl\{
[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H)
\biggr\} \, .
</math>
  </td>
</tr>
</table>

[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eq. (54)
</div>

Rewriting this last expression in a form that can more readily be compared with Wong's work, we obtain,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{2^3\pi}{e^2} \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{ 1 }{(k'_H)^2 \Delta_0^3}\biggl\{
[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0} +  \frac{\boldsymbol{K}(k_H)}{\Delta_0} \, .
</math>
  </td>
</tr>
</table>

<span id="Step01">Hence, also,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{ \Psi_0 }{GM}  + \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2}{\pi}\biggl\{
\frac{\boldsymbol{K}([k_H]_0)}{\Delta_0}
\biggr\} + 
\frac{e^2}{2^3\pi}\biggl\{
\frac{\boldsymbol{K}(k_H)}{\Delta_0}
- \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2}{\pi \Delta_0}\biggl\{
\boldsymbol{K}([k_H]_0)
- 
\frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) \biggr\}
- \frac{2}{\pi\Delta_0} \cdot \frac{e^2}{2^4}\biggl\{\biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] 
\biggr\} \boldsymbol{E}(k_H) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{K}([k_H]_0)
- 
\frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) 
+  \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr]  \boldsymbol{E}(k_H) \, .
</math>
  </td>
</tr>
</table>