Editing Apps/DysonWongTori (section)

====Uniform-Density, Circular Torus====

[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)] tackles this same problem in &sect;5.2 of his article.  The following table identifies the parameter mapping between the notation adopted by [http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima (2016)] and our adopted conventions; note that, when Fukushima attaches an asterisk to a coordinate or variable, it appears as a superscript whereas when we have used an asterisk, it appears as a subscript.

<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="2">Parameter Mapping</th>
</tr>
<tr>
  <td align="center">[http://adsabs.harvard.edu/abs/2016AJ....152...35F Fukushima's (2016)]</td>
  <td align="center">[[2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential|Our Analysis]]</td>
</tr>
<tr>
  <td align="center"><math>~(R, z)</math></td>
  <td align="center"><math>~(R_*, Z_*)</math></td>
</tr>
<tr>
  <td align="center"><math>~(R^*, z^*)</math></td>
  <td align="center"><math>~(\varpi, Z)</math></td>
</tr>
<tr>
  <td align="center"><math>~R_\mathrm{CU}</math></td>
  <td align="center"><math>~\varpi_t</math></td>
</tr>
<tr>
  <td align="center"><math>~R_\mathrm{RU}</math></td>
  <td align="center"><math>~r_t</math></td>
</tr>
<tr>
  <td align="center"><math>~R_\mathrm{1U}</math></td>
  <td align="center"><math>~\varpi_t - r_t</math></td>
</tr>
<tr>
  <td align="center"><math>~R_\mathrm{2U}</math></td>
  <td align="center"><math>~\varpi_t + r_t</math></td>
</tr>
</table>
Fukushima's expression for the ''external'' gravitational potential is (see his equations 138 - 142),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_U(R,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4G\rho_\mathrm{0U} 
\int_{R_\mathrm{1U}}^{R_\mathrm{2U}} R^* ~dR^* 
\int_{-z_\mathrm{U}}^{z_\mathrm{U}}\biggl[ \frac{K(m^*)}{p^*}\biggr] dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4G\rho_\mathrm{0U} 
\int_{R_\mathrm{1U}}^{R_\mathrm{2U}} R^* ~dR^* 
\int_{-z_\mathrm{U}}^{z_\mathrm{U}}\biggl[ \frac{K(m^*)}{\sqrt{(R + R^*)^2 + (z - z^*)^2 }}\biggr] dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \Phi_U(R_*,Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4G\rho_\mathrm{0U} 
\int_{\varpi_t - r_t}^{\varpi_t + r_t} \varpi ~d\varpi 
\int_{-Z_*(\varpi)}^{Z_*(\varpi)} \biggl[ \frac{K(\mu^2)}{\sqrt{(R_* + \varpi)^2 + (Z_* - Z)^2 }}\biggr] dZ
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4G\rho_\mathrm{0U} 
\int_{\varpi_t - r_t}^{\varpi_t + r_t}  \varpi ~d\varpi 
\int_{-Z_*(\varpi)}^{Z_*(\varpi)} \biggl[ \frac{\mu^2}{4R_*\varpi} \biggr]^{1 / 2} K(\mu^2) dZ
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{2G\rho_\mathrm{0U} }{R_*^{1 / 2}}
\int_{\varpi_t - r_t}^{\varpi_t + r_t}  
\int_{-Z_*(\varpi)}^{Z_*(\varpi)} \biggl[ \mu^2 \biggr]^{1 / 2} K(\mu^2) \varpi^{1 / 2} ~d\varpi dZ
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m^* = \mu^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{4R^*\varpi}{(R_*+\varpi)^2 + (Z_* - Z)^2} \, ,
</math>
  </td>
</tr>
</table>
</div>
and the integration limit,
<div align="center">
<math>~Z_*(\varpi) = \sqrt{r_t^2 - (\varpi - \varpi_t)^2 } \, .</math>
</div>
This matches the [[#Our_Integral_Expressions|integral expression that we have derived and repeated, above]].