Editing Apps/DysonWongTori (section)

===Bannikova et al. (2011) ===

In &sect;2 of their paper, [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al. (2011)] develop an integral expression &#8212; ultimately, their equation (8) &#8212;  for the gravitational potential both inside and outside of a uniform-density, axisymmetric, thick torus that has a circular cross-section.  They accomplish this by imagining that the (axisymmetric) torus is composed of a set of infinitely thin, axisymmetric rings with their planes being parallel to the torus symmetry plane.  [Note that, 28 years earlier, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983)] adopted precisely this same strategy to evaluate the gravitational potential of axisymmetric, inhomogeneous configurations; see our [[Apps/ReviewStahler83#Stahler.27s_.281983.29_Rotationally_Flattened_Isothermal_Configurations|separate discussion of Stahler's work in the context of rotationally flattened, isothermal structures]].] In recording, here, the expression derived by Bannikova et al. (2011), we will substitute parameter names as detailed in the following "mapping" table.

<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/2011MNRAS.411..557B Bannikova et al. (2011)]</td>
  <td align="center">[[2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential|Our Analysis]]</td>
</tr>
<tr>
  <td align="center"><math>~(x, z)</math></td>
  <td align="center"><math>~(R_*, Z_*)</math></td>
</tr>
<tr>
  <td align="center"><math>~(x^', z^')</math></td>
  <td align="center"><math>~(\varpi-\varpi_t, Z)</math></td>
</tr>
<tr>
  <td align="center"><math>~R</math></td>
  <td align="center"><math>~\varpi_t</math></td>
</tr>
<tr>
  <td align="center"><math>~R_0</math></td>
  <td align="center"><math>~r_t</math></td>
</tr>
<tr>
  <td align="center"><math>~r_0 \equiv \frac{R_0}{R}</math></td>
  <td align="center"><math>~\frac{r_t}{\varpi_t}</math></td>
</tr>
<tr>
  <td align="center"><math>~\eta^'</math></td>
  <td align="center"><math>~\frac{\varpi - \varpi_t}{\varpi_t}</math></td>
</tr>
<tr>
  <td align="center"><math>~\zeta^'</math></td>
  <td align="center"><math>~\frac{Z}{\varpi_t}</math></td>
</tr>
<tr>
  <td align="center"><math>~\rho</math></td>
  <td align="center"><math>~\frac{R_*}{\varpi_t}</math></td>
</tr>
<tr>
  <td align="center"><math>~\zeta</math></td>
  <td align="center"><math>~\frac{Z_*}{\varpi_t}</math></td>
</tr>
</table>

Integrating [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B Bannikova et al.'s (2011)] equation (7) over the volume of the torus &#8212; after inserting the expression for <math>~\phi_r</math> provided by their equation (5), and making the stipulated substitution, <math>~M_c \rightarrow dM</math> &#8212; we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(R_*,Z_*)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{G}{\pi \varpi_t} \iint\limits_\mathrm{torus} \phi_r dM = \frac{G}{\pi \varpi_t} \biggl[ \frac{M}{\pi r_0^2} \biggr] \iint\limits_\mathrm{torus} \phi_r d\eta^' d\zeta^'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{GM}{\pi^2 \varpi_t r_0^2} \iint\limits_\mathrm{torus} \biggl[ \frac{(1 + \eta^')m_r}{\rho} \biggr]^{1 / 2} K(m_r) d\eta^' d\zeta^'
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{GM}{\pi^2 \varpi_t} \biggl( \frac{\varpi_t}{r_t}\biggr)^2 \iint\limits_\mathrm{torus} 
\biggl[ \frac{(\varpi/\varpi_t)m_r}{R_*/\varpi_t} \biggr]^{1 / 2} K(m_r) \biggl[ \frac{d\varpi dZ}{\varpi_t^2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2G}{R_*^{1 / 2}} \biggl[ \frac{M}{2\pi^2 \varpi_t r_t^2}\biggr]\iint\limits_\mathrm{torus} 
\varpi^{1 / 2} m_r^{1 / 2}  K(m_r) ~d\varpi dZ \, ,
</math>
  </td>
</tr>
</table>
</div>
where, according to their equation (6),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m_r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4\rho(1 + \eta^')}{(1 + \eta^' + \rho)^2 + (\zeta - \zeta^')^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4 R_* \varpi}{(\varpi + R_*)^2 + (Z_* - Z)^2} \, .
</math>
  </td>
</tr>
</table>
</div>
Recognizing that the density of material in a uniform-density torus is the total mass divided by the [[2DStructure/ToroidalCoordinates#Total_Mass|volume of the torus]], ''i.e.,'' it is <math>~M/(2\pi^2\varpi_t r_t^2)</math>, we see that this integral expression is identical to the one [[#General.2C_Two-Dimensional_Integral|we have professed, above]], is the correct one.