Editing Apps/Wong1973Potential (section)

====Equatorial Plane Behavior====
=====Interior Solution (n = 0)=====

Let's examine in more detail how the potential varies with radial position in the equatorial plane, focusing first on the ''interior'' region &#8212; that is, <math>~\infty \ge \eta \ge \eta_0</math> &#8212; and (cylindrical-coordinate-based) radial locations, <math>~\varpi < a</math>, in which case the corresponding value of the (toroidal-coordinate-based) polar angle is, <math>~\theta = \pi</math>.  In this region, the zeroth-order contribution to the potential is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Interior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{
Q_{-\frac{1}{2}}(\cosh\eta) [B_0(\cosh\eta_0)]
- Q^2_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
where, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(\cosh\eta_0) \cdot Q_{- \frac{1}{2}}^2(\cosh\eta_0) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(\cosh\eta_0) \cdot Q^2_{+ \frac{1}{2}}(\cosh\eta_0) \, .
</math>
  </td>
</tr>
</table>

Now if <math>~\cosh\eta_0 = 3</math>, as in the above example illustration, then, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sinh\eta_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[3^2-1]^{1 / 2} = 2^{3/2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ D_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2^{3/2}}{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] 
= \frac{2^{3/2}}{3\pi^2} \biggl[ \frac{2^{9/2}}{3}\biggr]
=\biggl[\frac{2^3}{3\pi}\biggr]^{2} = 0.720506195 \, ;
</math>
  </td>
</tr>
</table>
and, drawing from the derivations and example double-precision evaluations of selected toroidal functions provided in an [[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|accompanying appendix]],

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

<tr>
  <td align="right">
<math>~B_0(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(3) \cdot Q^2_{+ \frac{1}{2}}(3) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl[ 1.597386605 \cdot 1.104816977 \biggr]
+ \frac{3}{2} \biggl\{   \frac{0.8346268417 }{\sqrt{3^2-1}} \biggr[ 
-\frac{3}{2} Q^1_{+\frac{1}{2}}(3) - \frac{3}{2} Q^1_{-\frac{1}{2}}(3)
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.882409920
+ \frac{3^2}{2^2} \biggl\{   \frac{0.8346268417 }{\sqrt{3^2-1}} \biggr[ 
0.1718911443 + 0.6753219405
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.882409920
+ \frac{3^2}{2^2} \biggl\{   
\frac{1}{2^2}\biggr\} = 1.444909920 \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~0.720506195 
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{ 1.444909920
Q_{-\frac{1}{2}}(\cosh\eta) - Q^2_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, .
</math>
  </td>
</tr>
</table>

=====Exterior Solution (n = 0)=====
In the region ''exterior'' to the torus, the zeroth-order contribution to the potential is,

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

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-D_0
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{
P_{-\frac{1}{2}}(\cosh\eta) [C_0(\cosh\eta_0)]
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
where, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2}~Q_{+\frac{1}{2}}(\cosh \eta_0) Q_{- \frac{1}{2}}^2(\cosh \eta_0) 
+ \frac{3}{2}~ Q_{- \frac{1}{2}}(\cosh \eta_0)~Q^2_{+ \frac{1}{2}}(\cosh \eta_0) \, .
</math>
  </td>
</tr>
</table>

Drawing again from the derivations and example double-precision evaluations of selected toroidal functions provided in an [[Appendix/SpecialFunctions#Toroidal_Function_Evaluations|accompanying appendix]], this means that, for <math>~\cosh\eta_0 = 3</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~C_0(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2}~Q_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2}~ Q_{- \frac{1}{2}}(3)\cdot Q^2_{+ \frac{1}{2}}(3) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} \biggl[ 0.1128885424 \cdot 1.104816977 \biggr]
+ \frac{3}{2} \biggl\{   \frac{1.311028777 }{\sqrt{3^2-1}} \biggr[ 
-\frac{3}{2} Q^1_{+\frac{1}{2}}(3) - \frac{3}{2} Q^1_{-\frac{1}{2}}(3)
\biggr] \biggr\}
</math>
  </td>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.0623605885
+ \frac{3^2}{2^2} \biggl\{   \frac{1.311028777 }{\sqrt{3^2-1}} \biggr[ 
0.1718911443 + 0.6753219405
\biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0.0623605885
+ \frac{3^2}{2^2} \biggl\{  0.3926990811 \biggr\} = 0.945933521 \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\biggl(\frac{a}{GM} \biggr) \Phi_{\mathrm{W}0} (\eta,\theta) \biggr|_\mathrm{Exterior} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~0.720506195~
[\cosh\eta - \cos\theta]^{1 / 2} ~\biggl\{0.945933521~
P_{-\frac{1}{2}}(\cosh\eta)
\biggr\} \, .
</math>
  </td>
</tr>
</table>



<!--
=====Second Attempt=====

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM}\biggr)\Phi_\mathrm{W}(\eta,\theta)\bigr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~D_0  (\cosh\eta - \cos\theta)^{1 / 2} 
\sum\limits^\infty_{n=0} \epsilon_n \cos(n\theta) 
~B_n(\cosh\eta_0)~Q_{n-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_n(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ (n+\tfrac{1}{2})P_{n+\frac{1}{2}}(\cosh{\eta_0}) Q_{n - \frac{1}{2}}^2(\cosh{\eta_0}) 
- (n - \tfrac{3}{2}) ~ P_{n - \frac{1}{2}}(\cosh{\eta_0})~Q^2_{n + \frac{1}{2}}(\cosh{\eta_0}) \, ,
</math>
  </td>
</tr>
</table>

For the model illustrated above, <math>~\cosh\eta_0 = 3</math>, in which case,
<div align="center">
<math>~D_0
= \frac{2^{3/2} }{3\pi^2} \cdot  \frac{(\sinh^2\eta_0)^{3/2}}{\cosh\eta_0} 
= \frac{2^{3/2} }{3\pi^2} \cdot  \frac{(3^2-1)^{3/2}}{3} 
= \biggl(\frac{2^3 }{3\pi} \biggr)^2 = 0.720506194 \, .
</math>
</div>
And, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~P_{+\frac{1}{2}}(3) \cdot Q_{- \frac{1}{2}}^2(3) 
+ \frac{3}{2} ~ P_{- \frac{1}{2}}(3) \cdot Q^2_{+ \frac{1}{2}}(3) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2} ~[1.597386605] \cdot [1.104816977 ]
+ \frac{3}{2} ~ [0.8346268417] \cdot [Q^2_{+ \frac{1}{2}}(3) ]
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~Q_{n - \frac{1}{2}}^{2}(z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(z^2-1)^{-\frac{1}{2}} \{ (n-\tfrac{3}{2}) z Q^1_{n - \frac{1}{2}}(z) - (n+\tfrac{1}{2})Q^1_{n - \frac{3}{2}}(z)\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ Q_{+\frac{1}{2}}^{2}(3)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(3^2-1)^{-\frac{1}{2}} \biggl[ -\frac{3}{2}  Q^1_{+\frac{1}{2}}(3) - \frac{3}{2}Q^1_{- \frac{1}{2}}(3) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{3}{2^{5/2}}  \biggr)
\biggl[  Q^1_{+\frac{1}{2}}(3) + Q^1_{- \frac{1}{2}}(3) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl( \frac{3}{2^{5/2}}  \biggr)
\biggl[  -0.1718911443 - 0.6753219405 \biggr] = 0.4493025877 \, .
</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B_0(\cosh\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~ \frac{1}{2} ~[1.597386605] \cdot [1.104816977 ]
+ \frac{3}{2} ~ [0.8346268417] \cdot [0.4493025877 ]
=
1.444909919 \, .
</math>
  </td>
</tr>
</table>

As a result, including only the zeroth-order term in the series summation gives,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{a}{GM}\biggr)\Phi_\mathrm{W0}(\eta,\theta)\bigr|_\mathrm{interior}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~D_0  (\cosh\eta - \cos\theta)^{1 / 2}  
~B_0(\cosh\eta_0)~Q_{-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~1.041066547  (\cosh\eta - \cos\theta)^{1 / 2}  ~Q_{-1 / 2}(\cosh\eta)  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{ 2\pi} \biggl[ \frac{ \sinh\eta }{(\cosh\eta - \cos\theta) } \biggr]^2
~ - ~1.041066547  ~\sqrt{\frac{\cosh\eta - \cos\theta}{\cosh\eta +1}} ~K\biggl( \sqrt{\frac{1}{\cosh\eta +1}} \biggr) \, .
</math>
  </td>
</tr>
</table>

-->