Editing Apps/Wong1973Potential (section)

==Technical Background Review==

Drawing principally from the published works of {{ Wong73full }} and {{ CT99full }}, in an [[2DStructure/ToroidalGreenFunction#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|accompanying discussion]] we have detailed how the gravitational potential of any finite mass distribution can be determined using toroidal coordinates, rather than, say, cartesian, spherical or cylindrical coordinates.  Specifically for [[2DStructure/ToroidalGreenFunction#Axisymmetric_Systems|axisymmetric systems]], the relevant expression derived by {{ Wong73 }} &#8212; involving a pair of integrals over the meridional-plane coordinates, <math>~(\eta,\theta)</math>, and a summation over all ''polar angle'' modes &#8212; is,

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

<tr>
  <td align="right">
<math>\Phi(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 2Ga^2 (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n 
\int d\eta^' ~\sinh\eta^'~P^0_{n-1 / 2}(\cosh\eta_<) ~Q^0_{n-1 / 2}(\cosh\eta_>)
\int d\theta^' \biggl\{ \frac{ ~\cos[n(\theta - \theta^')]}{(\cosh\eta^' - \cos\theta^')^{5/2}} \biggr\}
\rho(\eta^',\theta^') \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Wong73 }}, p. 293, Eq. (2.55)
  </td>
</tr>
</table>
while the expression derived by {{ CT99 }} &#8212; involving different integrand expressions and no summation &#8212; is,

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

<tr>
  <td align="right">
<math>\Phi(\eta,\theta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> - 2Ga^2 \biggl[ \frac{(\cosh\eta - \cos\theta)}{\sinh\eta } \biggr]^{1 / 2} 
\int d\eta^' \int d\theta^' \biggl[ \frac{\sinh\eta^'}{(\cosh\eta^' - \cos\theta^')^5} \biggr]^{1 / 2} Q_{- 1 / 2}(\Chi) \rho(\eta^',\theta^') \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ CT99 }}, p. 88, Eqs. (31) &amp; (32a)
  </td>
</tr>
</table>
where, <math>a</math> is the radius of the toroidal-coordinate system's ''anchor'' ring, <math>P^0_{n-\frac{1}{2}}</math> and <math>Q^0_{n-\frac{1}{2}}</math> are zero-order half-integer-degree associated Legendre functions of, respectively, the first and second kind, and
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\cosh\eta \cdot \cosh\eta^' - \cos(\theta^' - \theta) }{ \sinh\eta \cdot \sinh\eta^'} \, .
</math>
  </td>
</tr>
</table>

In the subsection of our [[2DStructure/ToroidalGreenFunction#Rearranging_Terms_and_Incorporating_Special-Function_Relations|accompanying discussion]] titled, "Rearranging Terms and Incorporating Special-Function Relations," we have explicitly demonstrated that these two expressions lead to identical evaluations of the gravitational potential.

Here we build upon this technical foundation and detail how {{ Wong73 }} was able to complete both integrals to derive an analytic expression for the potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>R/d</math>.  This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, is underappreciated.