Editing 2DStructure/ToroidalCoordinates (section)

==Special Case==

In the context of the (pink torus) test-mass distribution being discussed here, a review of the [[#Properties|properties of a toroidal coordinate system]] highlights one particularly interesting (cylindrical) coordinate location at which the gravitational potential should be evaluated:
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<math>~\{ R_*, Z_* \} = \biggl\{r_t \biggl[\biggl( \frac{\varpi_t}{r_t} \biggr)^2 - 1  \biggr]^{1/2}, 0  \biggr\} \, .</math>
</div> 
By placing the origin of the toroidal coordinate system at this and ''only'' this location &#8212; that is, by setting,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~a</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_t \biggl[\biggl( \frac{\varpi_t}{r_t} \biggr)^2 - 1  \biggr]^{1/2} = (\varpi_t^2 - r_t^2)^{1/2} \, ,</math>
  </td>
</tr>
</table>
</div>

&#8212; the limits of integration can be specified very simply: There is only one <math>\xi_1</math>-circle that intersects the surface of the torus and it does so by, not simply intersecting, but by perfectly aligning with the surface.  This perfect overlap with the surface of the torus happens for the coordinate circle associated with,
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<math>~\xi_1 = \xi_s \equiv \frac{\varpi_t}{r_t}  \, .</math>
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All <math>\xi_1</math>-circles larger than this one &#8212; that is, all circles corresponding to values of 
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<math>~\xi_1 < \xi_s </math>
</div>
&#8212; lie entirely outside of the (pink) torus and therefore need not be included in the integration to determine the gravitational potential; while all <math>\xi_1</math>-circles smaller than this one &#8212; that is, all circles corresponding to values of  
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<math>~\xi_1 > \xi_s </math>
</div>
&#8212; lie entirely inside of the (pink) torus.  Hence, in this special case the ''radial''-coordinate integration limits are,
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<math>~\xi_1|_\mathrm{min} = \frac{\varpi_t}{r_t} </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; <math>~~\xi_1|_\mathrm{max} = \infty  \, .</math>
</div>
Furthermore, because the corresponding <math>\xi_1</math>-circles fall entirely inside (or on the surface of) the torus for all values of <math>~\xi_1</math> in this latter range, the integral over the "angular" toroidal coordinate, <math>~\theta \equiv \cos^{-1}\xi_2</math>, will have the simple limits,
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<math>~\theta_\mathrm{min} = - \pi </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~\theta_\mathrm{max} = + \pi \, .</math>
</div>
(More practically, we will integrate from zero to <math>~\pi</math>, and double the result.)  Hence, the inner integral over our toroidal system's angular coordinate gives,

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

<tr>
  <td align="right">
<math>~
\int\limits_{-\pi}^{\pi} \biggl[ \frac{d\theta}{(\xi_1 - \cos\theta)^{5/2}} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\int\limits_{0}^{\pi} \biggl[ \frac{d\theta}{(\xi_1 - \cos\theta)^{5/2}} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{4(\xi_1+1)^{1/2} }{3(\xi_1^2 - 1)^2 } \biggr[ \frac{
\sin \theta(5\xi_1^2 - 4\xi_1 \cos \theta - 1)}{(\xi_1+1)^{1/2} (\xi_1 - \cos \theta)^{3/2}} 
- 4\xi_1 E\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) 
+ (\xi_1-1) F\biggl( \frac{\pi-\theta}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]_{0}^{\pi} 
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{4(\xi_1+1)^{1/2} }{3(\xi_1^2 - 1)^2 } \biggr[ 
4\xi_1 E\biggl( \frac{\pi}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) 
- (\xi_1-1) F\biggl( \frac{\pi}{2} \, , \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{4(\xi_1+1)^{1/2} }{3(\xi_1^2 - 1)^2 } \biggr[ 4\xi_1 E\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) - (\xi_1-1) K\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]
\, .
</math>
  </td>
</tr>
</table>
</div>

So, we have,

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

<tr>
  <td align="right">
<math>~q_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1/2} a^{5/2} \rho_0
\int\limits_{\varpi_t/r_t}^\infty [ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{-1/2} K(\mu) 
\biggl\{ \frac{4(\xi_1+1)^{1/2} }{3(\xi_1^2 - 1)^2 } \biggr[ 4\xi_1 E\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) - (\xi_1-1) K\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]
\biggr\}
d\xi_1   \, ,
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2^{5/2} a^{5/2} \rho_0}{3} 
\int\limits_{\varpi_t/r_t}^\infty \frac{(\xi_1+1)^{1/2}K(\mu) }{(\xi_1^2 - 1)^2[ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{1/2} }
\biggr[ 4\xi_1 E\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) - (\xi_1-1) K\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]
d\xi_1   
</math>
  </td>
</tr>

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  <td align="right">
<math>~\Rightarrow ~~~~\Phi(\sqrt{\varpi_t^2 - r_t^2}, 0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{2^{7/2} (\varpi_t^2 - r_t^2) G\rho_0}{3} 
\int\limits_{\varpi_t/r_t}^\infty \frac{(\xi_1+1)^{1/2}K(\mu) }{(\xi_1^2 - 1)^2[ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{1/2} }
\biggr[ 4\xi_1 E\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) - (\xi_1-1) K\biggl( \sqrt{\frac{2}{\xi_1 + 1}} \biggr) \biggr]
d\xi_1   \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2(\xi_1^2 - 1)^{1/2}}{(\xi_1^2 - 1)^{1/2}+\xi_1} \biggr]^{1/2}  
\, .</math>
  </td>
</tr>
</table>
</div>

<table border="1" align="center" width="80%" cellpadding="10"><tr><td align="left">
TO BE DONE:


As we have detailed in an [[Apps/DysonWongTori#The_Coulomb_Potential|accompanying discussion]], using toroidal coordinates [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] derived an expression for the potential at any point inside or outside of a uniform-density torus that has a circular cross-section &#8212; as illustrated by the pink torus, above.  At some point, we should compare the expression that Wong derived to our independent derivation, as presented in this chapter; in particular, our "special case" should be compared with Wong's equation (2.57), which gives an expression for the potential after integration over both of the angular toroidal coordinates, <math>~(\theta, \psi)</math>, but before the radial integration has been completed.  [[Apps/DysonWongTori#CompareWithCohl|His expression is]],

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

<tr>
  <td align="right">
<math>~\Phi_\mathrm{Wong}(\eta^',\theta^') </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-GU(\eta^',\theta^')
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{2^{9 / 2}G\rho_0 a^2}{3} (\cosh \eta^' - \cos \theta^')^{1 / 2} \sum\limits_n \epsilon_n \cos(n\theta^') \int_{\eta_0}^\infty d\eta 
\biggl[ \frac{Q^2_{n-1 / 2}(\cosh\eta)}{\sinh\eta} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') 
\, .
</math>
  </td>
</tr>
</table>
</div>
This expression should be evaluated at <math>~\eta^' = +\infty</math>, in which case it appears as though it should match our "special case" result for all values of the polar angle, <math>~\theta^'</math>.
</td></tr></table>

Let's see if the expression simplifies somewhat by adopting a variable, 
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<math>~\epsilon \equiv \frac{1}{\xi_1}</math>
&nbsp; &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math>&nbsp; &nbsp; &nbsp; &nbsp; 
<math>~d\xi_1 ~\rightarrow ~ d(\epsilon^{-1}) = - \epsilon^{-2} d\epsilon \, .</math>
</div>
Making this substitution, and defining,
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<math>\Phi_\mathrm{norm} \equiv \frac{2^{3} (\varpi_t^2 - r_t^2) G\rho_0}{3} \, ,</math>
</div>

we have,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_\mathrm{norm}^{-1} \Phi(\sqrt{\varpi_t^2 - r_t^2}, 0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~   
\int\limits_{r_t/\varpi_t}^0 \frac{2^{1/2}\epsilon(1+\epsilon)^{1/2}K(\mu) }{(1 - \epsilon^2)^2[ (1-\epsilon^2)^{1/2}+1 ]^{1/2} }
\biggr[ 4E\biggl( \sqrt{\frac{2\epsilon}{1 + \epsilon}} \biggr) - (1-\epsilon) K\biggl( \sqrt{\frac{2\epsilon}{1 + \epsilon}} \biggr) \biggr]
d\epsilon  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~   
\int\limits_{r_t/\varpi_t}^0 
\frac{\epsilon \mu  K(\mu)}{(1 - \epsilon^2)^2} \biggl[ \frac{1+\epsilon}{1 - \epsilon} \biggr]^{1/4}
\biggr[ 4E\biggl( \sqrt{\frac{2\epsilon}{1 + \epsilon}} \biggr) - (1-\epsilon) K\biggl( \sqrt{\frac{2\epsilon}{1 + \epsilon}} \biggr) \biggr]
d\epsilon  \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mu</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^{1/2} [ 1 + (1- \epsilon^2)^{-1/2} ]^{-1/2} \, .</math>
  </td>
</tr>
</table>
</div>