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=Universit&eacute; de Bordeaux=
+
=Universit&eacute; de Bordeaux (Part 1)=
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Through a research collaboration at the [https://www.u-bordeaux.com Universit&eacute; de Bordeaux], [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract B. Basillais &amp; J. -M. Hur&eacute; (2019), MNRAS, 487, 4504-4509] have published a paper titled, ''Rigidly Rotating, Incompressible Spheroid-Ring Systems:  New Bifurcations, Critical Rotations, and Degenerate States.''
Through a research collaboration at the [https://www.u-bordeaux.com Universit&eacute; de Bordeaux], [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract B. Basillais &amp; J. -M. Hur&eacute; (2019), MNRAS, 487, 4504-4509] have published a paper titled, ''Rigidly Rotating, Incompressible Spheroid-Ring Systems:  New Bifurcations, Critical Rotations, and Degenerate States.''
 +
We discuss this topic in a [[User:Tohline/Appendix/Ramblings/BordeauxSequences#Spheroid-Ring_Systems|separate, accompanying chapter]].
==Exterior Gravitational Potential of Toroids==
==Exterior Gravitational Potential of Toroids==
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract J. -M. Hur&eacute;, B. Basillais, V. Karas, A. Trova, &amp; O. Semer&aacute;k (2020), MNRAS, 494, 5825-5838] have published a paper titled, ''The Exterior Gravitational Potential of Toroids.''  Here we examine how their work relates to the published work by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)], which we have separately [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|discussed in detail]].
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract J. -M. Hur&eacute;, B. Basillais, V. Karas, A. Trova, &amp; O. Semer&aacute;k (2020), MNRAS, 494, 5825-5838] have published a paper titled, ''The Exterior Gravitational Potential of Toroids.''  Here we examine how their work relates to the published work by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)], which we have separately [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|discussed in detail]].
-
===Their Presentation===
+
===Our Presentation of Wong's (1973) Result===
-
On an initial reading, it appears as though the most relevant section of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] paper is &sect;8 titled, ''The Solid Torus.''  They write the gravitational potential in terms of the series expansion,
+
-
<div align="center">
+
-
<table border="0" cellpadding="5" align="center">
+
 +
<table border="1" cellpadding="8" align="center" width="80%">
 +
<tr><td align="center">'''Summary:'''&nbsp; First three terms in [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] expression for the gravitational potential at any point, P(&#x03D6;, z), outside of a uniform-density torus.</td></tr>
 +
<tr><td align="left">
 +
 +
[[File:WongTorusIllustration02.png|500px|center|Wong diagram]]
 +
 +
----
 +
 +
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~\Psi_\mathrm{grav}(\vec{r})</math>
+
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
-
<math>~\approx</math>
+
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-
\Psi_0 + \sum\limits_{n=1}^N \Psi_n \, ,
+
- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
 +
\Upsilon_{W0}(\eta_0) \biggl\{
 +
\frac{a}{ r_1 } \cdot  \boldsymbol{K}(k) \biggr\}\, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>
-
</table>
 
-
 
-
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;7, p. 5831, Eq. (42)
 
-
</div>
 
-
where, after setting <math>~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c </math> and acknowledging that <math>~V_{0,0} = 1 \, ,</math> we can write,
 
-
<div align="center">
 
-
<table border="0" cellpadding="5" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~\Psi_0 </math>
+
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 47: Line 49:
   <td align="left">
   <td align="left">
<math>~
<math>~
-
- \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}(k_0) \biggr]
+
- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
 +
\Upsilon_{W1}(\eta_0) \times \cos\theta
 +
\biggl\{ \frac{a}{r_2} \cdot
 +
\boldsymbol{E}(k) \biggr\} \, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>
-
</table>
 
-
 
-
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eqs. (52) &amp; (53)
 
-
</div>
 
-
and,
 
-
<div align="center">
 
-
<table border="0" cellpadding="5" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]</math>
+
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\varpi, z)\biggr|_\mathrm{exterior}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 68: Line 66:
   <td align="left">
   <td align="left">
<math>~
<math>~
-
- \frac{G\pi \rho_0 R_c b^2}{4 (k')^2 \Delta_0^3} \biggl\{
+
- \biggl( \frac{2^{3} }{3\pi^3} \biggr)\Upsilon_{W2}(\eta_0)
-
[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k) - (k')^2 \Delta_0^2 \boldsymbol{K}(k)
+
\times \cos(2\theta)
-
\biggr\} \, .
+
\biggl\{  
 +
\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k)
 +
-  
 +
\frac{a}{r_1}  \cdot \boldsymbol{K}(k)  
 +
\biggr\} \, ,
</math>
</math>
   </td>
   </td>
Line 76: Line 78:
</table>
</table>
-
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eq. (54)
+
where, once the major ( R ) and minor ( d ) radii of the torus &#8212; as well as the vertical location of its equatorial plane (Z<sub>0</sub>) &#8212; have been specified, we have,
-
</div>
+
-
Note that the argument of the elliptic integral functions is,
+
-
<div align="center">
+
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~k</math>
+
<math>~a^2 </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 91: Line 90:
   <td align="left">
   <td align="left">
<math>~
<math>~
-
\frac{2\sqrt{\varpi R}}{\Delta}
+
R^2 - d^2</math>
 +
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
 +
<math>~\cosh\eta_0 \equiv \frac{R}{d} ~~~~\Rightarrow ~~~\sinh\eta_0 = \frac{a}{d}
 +
\, ,
</math>
</math>
   </td>
   </td>
-
<td align="center">&nbsp; &nbsp; where, &nbsp; &nbsp;</td>
+
</tr>
 +
 
 +
<tr>
   <td align="right">
   <td align="right">
-
<math>~\Delta</math>
+
<math>~r_1^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~(\varpi + a)^2 + (z - Z_0)^2 \, ,</math>
 +
  </td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
<math>~r_2^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math>
 +
  </td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
<math>~\cos\theta</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \, ,</math>
 +
  </td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
<math>~k</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 103: Line 143:
   <td align="left">
   <td align="left">
<math>~
<math>~
-
\biggl[ (R + \varpi)^2 + (Z-z)^2 \biggr]^{1 / 2} \, .
+
\biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2}
 +
= \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2}
 +
= \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2}  
 +
\, .
</math>
</math>
   </td>
   </td>
Line 109: Line 152:
</table>
</table>
-
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eqs. (4) &amp; (5)
+
----
-
</div>
+
-
===Our Presentation of Wong's (1973) Result===
+
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="left" colspan="2">&nbsp;</td>
 +
  <td align="left" colspan="1">Leading Coefficient Expressions &hellip;</td>
 +
  <td align="right" colspan="1" width="30%">&hellip; evaluated for:&nbsp; &nbsp;</td>
 +
  <td align="center" colspan="1"><math>~\frac{R}{d} = \cosh\eta_0 = 3</math>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
<math>~\Upsilon_{W0}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left" colspan="2">
 +
<math>~
 +
\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]  \biggl\{
 +
K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ]
 +
+ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ]
 +
- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
 +
\biggr\}  \, ,
 +
</math>
 +
  </td>
 +
  <td align="center"><font color="red">7.134677</font></td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
<math>~\Upsilon_{W1}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left" colspan="2">
 +
<math>~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] 
 +
\biggl\{
 +
K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)]
 +
+~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)]
 +
-~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)]
 +
\biggr\}
 +
\, ,
 +
</math>
 +
  </td>
 +
  <td align="center"><font color="red">0.130324</font></td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
<math>~\Upsilon_{W2}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left" colspan="2">
 +
<math>~
 +
\frac{2^{3 / 2}}{3^2}  \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{
 +
K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13]
 +
+  2  K ( k_0 ) \cdot E(k_0)  [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ]
 +
</math>
 +
  </td>
 +
  <td align="center">&nbsp;</td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left" colspan="2">
 +
<math>~
 +
-~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ]
 +
\biggr\} \, ,
 +
</math>
 +
  </td>
 +
  <td align="center"><font color="red">0.003153</font></td>
 +
</tr>
 +
<tr><td align="left" colspan="5">where,</td></tr>
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left" colspan="2">
 +
<math>~
 +
\biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, .
 +
</math>
 +
  </td>
 +
  <td align="center"><font color="red">0.707106781</font></td>
 +
</tr>
 +
</table>
 +
NOTE:  In evaluating these "leading coefficient expressions" for the case, <math>~R/d = 3</math>, we have used the complete elliptic integral evaluations, '''K'''(k<sub>0</sub>) = <font color="red">1.854074677</font> and  '''E'''(k<sub>0</sub>) = <font color="red">1.350643881</font>.
 +
</td></tr>
 +
</table>
====Setup====
====Setup====
From our [[User:Tohline/Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression,
From our [[User:Tohline/Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression,
 +
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 134: Line 274:
</tr>
</tr>
</table>
</table>
-
 
+
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], &sect;II.D, p. 294, Eqs. (2.59) &amp; (2.61)
 +
</div>
where,
where,
 +
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 168: Line 310:
</tr>
</tr>
</table>
</table>
 +
[http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], &sect;II.D, p. 294, Eq. (2.63)
 +
</div>
and where, in terms of the major ( R ) and minor ( d ) radii of the torus &#8212; or their ratio, &epsilon; &equiv; d/R,  
and where, in terms of the major ( R ) and minor ( d ) radii of the torus &#8212; or their ratio, &epsilon; &equiv; d/R,  
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 1,148: Line 1,292:
</table>
</table>
-
 
+
<span id="Qrecurrence">&nbsp;</span>
<table border="1" align="center" width="80%" cellpadding="10">
<table border="1" align="center" width="80%" cellpadding="10">
<tr><td align="left">
<tr><td align="left">
Line 1,252: Line 1,396:
   <td align="left">
   <td align="left">
<math>~
<math>~
-
-~\frac{1}{2^2}\biggl\{ z k_0~K ( k_0 )   
+
-~\frac{1}{2^2}
 +
\biggl\{ z k_0~K ( k_0 )   
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\}  
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\}  
</math>
</math>
Line 1,294: Line 1,439:
   <td align="left">
   <td align="left">
<math>~
<math>~
-
5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0)  \, ,
+
5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0)   
 +
</math>
 +
  </td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
5 \biggl\{ [2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)] \biggr\}
 +
+ z \biggl\{ z k_0~K ( k_0 ) 
 +
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} \biggl\{  [5 z]
 +
~-~z (z^2+3) \biggr\} E(k_0)
 +
+ \biggl\{ z^2 k_0~ 
 +
- [(z-1)(z^2-1)]^{-1 / 2} [ 2^{-3 / 2} \cdot 5 (z-1)] \biggr\}K(k_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^{-3 / 2}(z+1)^{-1 / 2} [4 z^2  -  5  ]K(k_0)
 +
-~2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} (z^2 - 2)z  E(k_0)
</math>
</math>
   </td>
   </td>
Line 1,576: Line 1,769:
   <td align="left">
   <td align="left">
<math>~
<math>~
 +
\biggl[ \frac{2(3z^2 - 1)}{(z^2-1)}      \biggr]K(k_0)\cdot E(k_0)
-~\biggl[ \frac{z}{(z+1)} \biggr] K(k_0)\cdot K(k_0)  
-~\biggl[ \frac{z}{(z+1)} \biggr] K(k_0)\cdot K(k_0)  
-~\biggl[ \frac{ 5z}{(z-1)}  \biggr] E(k_0)\cdot E(k_0)  
-~\biggl[ \frac{ 5z}{(z-1)}  \biggr] E(k_0)\cdot E(k_0)  
-
+~\biggl[ \frac{2(3z^2 - 1)}{(z^2-1)}      \biggr]K(k_0)\cdot E(k_0)
 
</math>
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~(z_0^2-1)C_1(z_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2(3z^2 - 1) K(k_0)\cdot E(k_0)
 +
-~z_0(z_0-1) K(k_0)\cdot K(k_0)
 +
-~5z_0(z_0+1) E(k_0)\cdot E(k_0) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
Hence, we have,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\eta,\theta)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{2^{3} a}{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]  C_1(\cosh\eta_0) \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2^2} \biggr]
 +
\boldsymbol{E}(k) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
====Third (n = 2) Term====
 +
 +
=====Part A=====
 +
The third (n = 2) term in [[User:Tohline/Apps/Wong1973Potential#D0andCn|Wong's (1973) expression for the exterior potential]] is,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-D_0
 +
(\cosh\eta - \cos\theta)^{1 / 2} \cdot 2 \cos(2\theta) \cdot  C_2(\cosh\eta_0)P_{+\frac{3}{2}}(\cosh\eta) \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
where, <math>~D_0</math> is the same as [[#Setup|above]], and,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~C_2(\cosh\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\tfrac{5}{2}Q_{+\frac{5}{2}}(\cosh \eta_0) Q_{+\frac{3}{2}}^2(\cosh \eta_0)
 +
- \tfrac{1}{2} Q_{+\frac{3}{2}}(\cosh \eta_0)~Q^2_{+ \frac{5}{2}}(\cosh \eta_0) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<table border="1" align="center" width="80%" cellpadding="10">
 +
<tr><td align="left">
 +
In order to evaluate <math>~C_2(z)</math>, we will need the following pair of expressions in addition to the ones already used:
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~Q^0_{m-\tfrac{1}{2}}</math> [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|recurrence]] with m = 3, gives:  &nbsp; &nbsp;
 +
<math>~Q_{+\tfrac{5}{2}}(z_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{8}{5} z~Q_{+\tfrac{3}{2}}(z_0) - \frac{3}{5} Q_{+\tfrac{1}{2}}(z_0)</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow~~~15Q_{+\tfrac{5}{2}}(z_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~8 z~\biggl[ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0) \biggr]
 +
-
 +
9 \biggl[ z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0)  \biggr]</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
z~k_0 K(k_0) \biggl[ 8(4z^2 - 1 ) - 9 \biggr]
 +
+
 +
[2(z+1)]^{1 / 2} E(k_0) \biggl[-32z^2 + 9  \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
z~k_0 K(k_0) [ 32z^2 - 17 ]
 +
+
 +
[2(z+1)]^{1 / 2} E(k_0) [9 -32z^2 ] \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
Hence, &nbsp; &nbsp; <math>~Q_{+\frac{5}{2}}(3)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~0.002080867 \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
And, setting m = 2 in the [[#Qrecurrence|above recurrence relation for]] <math>~Q^2_{m+\frac{1}{2}}(z)</math> gives,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl[ (m - \tfrac{3}{2})Q^{2}_{m+\tfrac{1}{2}} (z) \biggr]_{m=2}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ (2m)z Q_{m-\tfrac{1}{2}}^{2}(z) - (m + \tfrac{3}{2})Q^{2}_{m - \tfrac{3}{2}}(z) \biggr]_{m=2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~  Q^{2}_{+\tfrac{5}{2}} (z) </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
8z Q_{+\tfrac{3}{2}}^{2}(z) - 7 Q^{2}_{+ \tfrac{1}{2}}(z)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
8z \biggl[ 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) \biggr] - 7 Q^{2}_{+ \tfrac{1}{2}}(z)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
40z Q^{2}_{- \tfrac{1}{2}}(z_0)
 +
- [32z^2 +7]Q_{+\tfrac{1}{2}}^{2}(z_0) 
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
40z \biggl\{
 +
[2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)]
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ \frac{[32z^2 +7]}{4} \biggl\{
 +
z k_0~K ( k_0 ) 
 +
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)
 +
\biggr\} 
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~  4Q^{2}_{+\tfrac{5}{2}} (z) </math>  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^5\cdot 5z \biggl\{ 2^{1 / 2}
 +
[(z-1)(z^2-1)]^{-1 / 2} [zE(k_0) ]
 +
-
 +
2^{-3 / 2}[(z-1)(z^2-1)]^{-1 / 2} [(z-1)K(k_0)]
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ [32z^2 +7] \biggl\{
 +
z k_0~K ( k_0 ) 
 +
~-~2^{1 / 2}(z^2+3) [ (z-1)(z^2-1) ]^{-1 / 2} E(k_0)
 +
\biggr\} 
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
2^{11 / 2}\cdot 5  [z^2 ] - 2^{1 / 2} [32z^2 +7]  (z^2+3)
 +
\biggr\}[(z-1)(z^2-1)]^{-1 / 2} E(k_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~2^{7 / 2}\cdot 5 \biggl\{ [(z-1)(z^2-1)]^{-1 / 2} (z-1) \biggr\} z K(k_0)
 +
+ [32z^2 +7] \biggl\{ 2^{1 / 2}[z + 1]^{-1 / 2} \biggr\}  z K ( k_0 )
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^{1 / 2}\biggl\{ 32z^2 - 33 \biggr\}  z [z + 1]^{-1 / 2}  K ( k_0 )
 +
-~2^{1 / 2}
 +
\biggl\{
 +
32z^4 - 57  z^2  + 21
 +
\biggr\}[(z-1)(z^2-1)]^{-1 / 2} E(k_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^{1 / 2}[z + 1]^{-1 / 2} [z-1]^{-1 }\biggl[ (32z^2 - 33)  z (z-1)  K ( k_0 )
 +
-~(32z^4 - 57  z^2  + 21)E(k_0) \biggr]
 +
\, .
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
Hence, &nbsp; &nbsp; <math>~Q^2_{+\frac{5}{2}}(3)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~0.03377378 \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
</td></tr></table>
 +
 +
=====Part B=====
 +
 +
Let's evaluate <math>~C_2(z)</math> specifically for the case where <math>~z = \cosh\eta_0 = 3</math>, using the already separately evaluated values of the four relevant toroidal functions.  We find,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~2C_2(3)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~5Q_{+\frac{5}{2}}(3) Q_{+\frac{3}{2}}^2(3)
 +
- Q_{+\frac{3}{2}}(3)~Q^2_{+ \frac{5}{2}}(3)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
5\cdot ( 0.002080867 ) \times ( 0.132453829 ) -  ( 0.014544576 ) \times (0.03377378 )
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
8.868687\times 10^{-4} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
Next, let's develop a consolidated expression for <math>~C_2(z_0)</math> that replaces all the toroidal functions with complete elliptic integrals of the first and second kind.
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~2C_2(z_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~5Q_{+\frac{5}{2}}(z_0) Q_{+\frac{3}{2}}^2(z_0)
 +
- Q_{+\frac{3}{2}}(z_0)~Q^2_{+ \frac{5}{2}}(z_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{1}{3}\biggl\{
 +
z~k_0 K(k_0) [ 32z^2 - 17 ]
 +
+
 +
[2(z+1)]^{1 / 2} E(k_0) [9 -32z^2 ]
 +
\biggr\}
 +
\times \biggl\{
 +
2^{-3 / 2}(z+1)^{-1 / 2} [4 z^2  -  5  ]K(k_0)
 +
-~2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} (z^2 - 2)z  E(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{1}{2^2\cdot 3}
 +
\biggl\{ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0)  \biggr\}
 +
\times \biggl\{
 +
2^{1 / 2}[z + 1]^{-1 / 2} [z-1]^{-1 }\biggl[ (32z^2 - 33)  z (z-1)  K ( k_0 )
 +
-~(32z^4 - 57  z^2  + 21)E(k_0) \biggr]
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ 2^{2} \cdot 3 (z^2-1) C_2(z_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
K(k_0) z[ 32z^2 - 17 ]
 +
+
 +
(z+1) E(k_0) [9 -32z^2 ]
 +
\biggr\}
 +
\times \biggl\{
 +
(z-1) [4 z^2  -  5  ]K(k_0)
 +
-~4 (z^2 - 2)z  E(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- ~
 +
\biggl\{ (4z^2 - 1 )  K(k_0) - 4 z(z+1) E(k_0)  \biggr\}
 +
\times \biggl\{
 +
(32z^2 - 33)  z (z-1)  K ( k_0 )
 +
-~(32z^4 - 57  z^2  + 21)E(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
(z-1)[ 32z^2 - 17 ] [4 z^2  -  5  ]z K(k_0) \cdot K(k_0)
 +
-~4 (z^2 - 2)z^2 [ 32z^2 - 17 ]  K(k_0) \cdot E(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+  \biggl\{
 +
(z-1) (z+1) [9 -32z^2 ] [4 z^2  -  5  ]K(k_0) \cdot E(k_0)
 +
-~4 (z^2 - 2)z (z+1) [9 -32z^2 ] E(k_0) \cdot E(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ ~   
 +
\biggl\{
 +
(32z^4 - 57  z^2  + 21)(4z^2 - 1 ) K(k_0) \cdot E(k_0)
 +
-~(32z^2 - 33)  z (z-1)(4z^2 - 1 )  K ( k_0 ) \cdot K(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ ~ 
 +
\biggl\{
 +
4 z(z+1)(32z^2 - 33)  z (z-1)  K ( k_0 ) \cdot E(k_0)
 +
-~4 z(z+1)(32z^4 - 57  z^2  + 21)E(k_0) \cdot E(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~(z-1)\biggl\{
 +
\biggl[
 +
( 32z^2 - 17 ) (4 z^2  -  5  )z \biggr]
 +
-~\biggl[ (32z^2 - 33)  z (4z^2 - 1 ) \biggr] \biggr\} K ( k_0 ) \cdot K(k_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+  \biggl\{ \biggl[
 +
(z-1) (z+1) (9 -32z^2 ) (4 z^2  -  5  )\biggr] 
 +
-~\biggl[ 4 (z^2 - 2)z^2 ( 32z^2 - 17 ) \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ ~   
 +
\biggl[
 +
(32z^4 - 57  z^2  + 21)(4z^2 - 1 ) \biggr]
 +
+ ~  \biggl[ 4 z(z+1)(32z^2 - 33)  z (z-1)\biggr]\biggr\}  K ( k_0 ) \cdot E(k_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~2z(z+1) \biggl\{ \biggl[ 2 (32z^4 - 57  z^2  + 21) \biggr]
 +
+~2\biggl[  (z^2 - 2) (9 -32z^2 )\biggr] \biggr\} E(k_0) \cdot E(k_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
z(z-1)\biggl\{[ 52 - 64z^2 ] \biggr\} K ( k_0 ) \cdot K(k_0)
 +
-~4z(z+1) \biggl\{ [ 3 +16z^2 ]\biggr\} E(k_0) \cdot E(k_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+  \biggl\{
 +
\biggl[ 5(32z^4 - 41z^2 + 9 ) \biggr] 
 +
- \biggl[ (32z^4 - 57  z^2  + 21)\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ ~   
 +
4z^2\biggl[ (32z^4 - 57  z^2  + 21)
 +
+  (32z^4 - 65z^2 + 33)  + (-32z^4 + 41z^2 -9 )  +~( -32z^4 + 81z^2 - 34 )
 +
\biggr]\biggr\}  K ( k_0 ) \cdot E(k_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
4z(z-1)\biggl\{ 13 - 16z^2  \biggr\} K ( k_0 ) \cdot K(k_0)
 +
-~4z(z+1) \biggl\{ [ 3 +16z^2 ]\biggr\} E(k_0) \cdot E(k_0)
 +
+  8\biggl\{
 +
16z^4 -13z^2 + 3 \biggr\}  K ( k_0 ) \cdot E(k_0) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
Finally, let's evaluate this consolidated expression for the specific case of <math>~z_0 = \cosh\eta_0 = 3</math>, remembering that in this specific case <math>~k_0  = 2^{-1 / 2}</math>, <math>~K(k_0) = 1.854074677</math>, and <math>~E(k_0) = 1.350643881</math>.  We find,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~2C_2(z_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
[2 \cdot 3 (z^2-1) ]^{-1} \biggl\{
 +
4z(z-1) [ 13 - 16z^2 ] K ( k_0 ) \cdot K(k_0)
 +
-~4z(z+1) [ 3 +16z^2 ] E(k_0) \cdot E(k_0)
 +
+  8[
 +
16z^4 -13z^2 + 3 ]  K ( k_0 ) \cdot E(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
[48 ]^{-1} \biggl\{
 +
-24[ 131 ] K ( k_0 ) \cdot K(k_0)
 +
-~48 [ 147] E(k_0) \cdot E(k_0)
 +
+  8[ 1182 ]  K ( k_0 ) \cdot E(k_0)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
8.8708 \times 10^{-4} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<font color="red">This matches the numerically evaluated expression, from above (6/30/2020)</font>.  There is a tremendous amount of cancellation between the three key terms in this expression, so the match  is only to three significant digits.</tr>
 +
 +
=====Part C=====
 +
 +
Next &hellip;
 +
 +
<table border="1" cellpadding="8" align="center" width="60%">
 +
<tr><td align="left">
 +
<div align="center">'''Useful Relations from Above'''</div>
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\cosh\eta</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{r_1^2 + r_2^2}{2r_1 r_2} \, ;</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\sinh\eta</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{r_1^2 - r_2^2}{2r_1 r_2} \, ;</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\varpi</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{r_1^2 - r_2^2}{2a} \, ;</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\cosh\eta - \cos\theta</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{2a^2}{r_1 r_2} \, ;</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~ \cos\theta</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{r_1^2 + r_2^2-4a^2}{2r_1 r_2} \, ;</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2}{\coth\eta + 1}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{4a\varpi}{r_1^2} \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
</td></tr>
 +
</table>
 +
 +
 +
Now, from our tabulation of [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|example recurrence relations]], we see that,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~ P_{+\frac{3}{2}}(\cosh\eta)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{4}{3} \cdot \cosh\eta ~P_{+\frac{1}{2}}(\cosh\eta) - \frac{1}{3} ~ P_{-\frac{1}{2}}(\cosh\eta) </math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{4}{3} \cdot \cosh\eta \biggl[ \frac{\sqrt{2}}{\pi} (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) \biggr]
 +
-
 +
\frac{1}{3} ~ \biggl[ \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)\biggr]</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{2^{1 / 2}}{3\pi}
 +
\biggl[ 4 \cosh\eta (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) 
 +
-
 +
(\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)\biggr]
 +
\, ,</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
where, as above,
 +
<div align="center">
 +
<math>~
 +
k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} 
 +
=
 +
\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2}
 +
=
 +
\biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} \, .
 +
</math>
 +
</div>
 +
So we have,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-\frac{2^{5/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]
 +
C_2(\cosh\eta_0)\cos(2\theta)
 +
\biggl\{ (\cosh\eta - \cos\theta)^{1 / 2}P_{+\frac{3}{2}}(\cosh\eta) \biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-\frac{2^{3} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]
 +
C_2(\cosh\eta_0)\cos(2\theta)\biggl( \frac{2a^2}{r_1 r_2}\biggr)^{1 / 2}
 +
\biggl\{
 +
4 \cosh\eta (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) 
 +
-
 +
(\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-\frac{2^{3} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]
 +
C_2(\cosh\eta_0)\cos(2\theta)\biggl( \frac{2a^2}{r_1 r_2}\biggr)^{1 / 2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\times
 +
\biggl\{
 +
4 \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \biggl[\frac{r_1^2 - r_2^2}{2r_1 r_2} \biggr]^{+1 / 2} \biggl[ \frac{4a}{r_1^2} \biggl( \frac{r_1^2 - r_2^2}{2a} \biggr)\biggr]^{-1 / 2}  \boldsymbol{E}(k) 
 +
-
 +
\biggl[ \frac{r_1^2 - r_2^2}{2r_1 r_2}  \biggr]^{-1 / 2} ~\biggl[ \frac{4a}{r_1^2}\biggl( \frac{r_1^2 - r_2^2}{2a} \biggr) \biggr]^{1 / 2} \boldsymbol{K}(k)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-\frac{2^{9/2} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]
 +
C_2(\cosh\eta_0)\cos(2\theta)
 +
\times
 +
\biggl\{
 +
\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k) 
 +
-
 +
\frac{a}{r_1}  \cdot \boldsymbol{K}(k)
 +
\biggr\} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
Finally, inserting the expression for <math>~\sinh^2\eta_0~ C_2(\cosh\eta_0) = (z_0^2-1)C_2(z_0)</math> that we have derived, above, gives,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-\frac{2^{9/2} }{3^3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
 +
\times \cos(2\theta)
 +
\biggl\{
 +
\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k) 
 +
-
 +
\frac{a}{r_1}  \cdot \boldsymbol{K}(k)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\times \biggl\{
 +
z(z-1) [ 13 - 16z^2 ] K ( k_0 ) \cdot K(k_0)
 +
-~z(z+1) [ 3 +16z^2 ] E(k_0) \cdot E(k_0)
 +
+  2 [ 16z^4 -13z^2 + 3 ]  K ( k_0 ) \cdot E(k_0)
 +
\biggr\} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
====Summary====
 +
 +
Once the major ( R ) and minor ( d ) radii of the torus have been specified, two key model parameters can be immediately determined, namely,
 +
<div align="center">
 +
<math>~a^2 \equiv R^2 - d^2\, ,</math> &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; <math>~\cosh\eta_0 \equiv \frac{R}{d} \, ,</math>
 +
</div>
 +
in which case also, <math>~\sinh\eta_0 = a/d \, .</math>  Once the mass-density ( &rho;<sub>0</sub> ) of the torus has been specified, the torus mass is given by the expression,
 +
<div align="center">
 +
<math>~M = 2\pi^2 \rho_0 d^2 R \, .</math>
 +
</div>
 +
In addition to the principal pair of meridional-plane coordinates, <math>~(\varpi, z)</math>, it is useful to define the pair of distances,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~r_1^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~(\varpi + a)^2 + (z - Z_0)^2 \, ,</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~r_2^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
where, the equatorial plane of the torus is located at <math>~z = Z_0</math>.  As we have shown above, the leading (n = 0) term in Wong's (1973) series-expression for the gravitational potential anywhere outside the torus is,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{2^{3} }{3\pi^3}
 +
\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
 +
\frac{a}{ r_1 } \cdot  \boldsymbol{K}(k)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\times \biggl\{
 +
K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ]
 +
+ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ]
 +
- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
 +
\biggr\}  \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
where, the two distinctly different arguments &#8212; one with, and one without a zero subscript &#8212; of the complete elliptic-integral functions are,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} 
 +
=
 +
\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2}
 +
= \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2}
 +
= \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2}  \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
As we also have shown above, the second (n = 1) term in Wong's (1973) series-expression for the exterior gravitational potential is,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]  \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot
 +
\boldsymbol{E}(k)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~\times
 +
\biggl\{
 +
K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)]
 +
+~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)]
 +
-~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)]
 +
\biggr\}
 +
\, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
Note that a transformation from the <math>~(r_1, r_2)</math> coordinate pair to the toroidal-coordinate pair <math>~(\eta, \theta)</math> includes the expression,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\cos\theta</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
So this (n = 1) term's explicit dependence on "cos(n&theta;)" is clear.  Finally,  the third (n = 2) term in Wong's (1973) series-expression for the exterior gravitational potential is,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-\frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]
 +
\times \cos(2\theta)
 +
\biggl\{
 +
\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k) 
 +
-
 +
\frac{a}{r_1}  \cdot \boldsymbol{K}(k)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\times \frac{2^{3 / 2}}{3^2}\biggl\{
 +
K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13]
 +
+  2  K ( k_0 ) \cdot E(k_0)  [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ]
 +
\biggr\} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
===The Hur&eacute;, ''et al'' (2020) Presentation===
 +
 +
{{LSU_WorkInProgress}}
 +
 +
====Notation====
 +
 +
In [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], the major and minor radii of the torus surface ("shell") are labeled, respectively, R<sub>c</sub> and b, and their ratio is denoted,
 +
<div align="center">
 +
<math>~e \equiv \frac{b}{R_c} \, .</math>
 +
 +
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eq. (1)
 +
</div>
 +
The authors work in cylindrical coordinates, <math>~(R, Z)</math>, whereas we refer to this same coordinate-pair as, <math>~(\varpi_W, z_W)</math>.  The quantity,
 +
<div align="center">
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Delta^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
[R + (R_c + b\cos\theta_H)]^2 + [Z - b\sin\theta_H]^2 \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eqs. (5) &amp; (7)
 +
</div>
 +
 +
We have affixed the subscript "H" to their meridional-plane angle, &theta;, to clarify that it has a different coordinate-base definition from the meridional-plane angle, &theta;, that appears in our above discussion of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] work. In their paper, the subscript "0" is used in the case of an infinitesimally thin hoop <math>~(b \rightarrow 0)</math>, that is to say,
 +
<div align="center">
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Delta_0^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
[R + R_c]^2 + Z^2 \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;3, p. 5827, Eq. (13)
 +
</div>
 +
 +
Generally, the argument (modulus) of the complete elliptic integral functions is,
 +
<div align="center">
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_H</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{2}{\Delta}\biggl[ R (R_c + b\cos\theta_H) \biggr]^{1 / 2}  \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;2, p. 5826, Eq. (4)
 +
</div>
 +
and, as stated in the first sentence of their &sect;3, reference may also be made to the ''complementary modulus'',
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k'_H</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~[1 - k_H^2]^{1 / 2} \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
(Again, we have affixed the subscript "H" in order to differentiate from the modulus employed by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].)  And in the case of an infinitesimally thin hoop <math>~(b\rightarrow 0)</math>,
 +
<div align="center">
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~[k^2_H]_0</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{4R R_c}{\Delta_0^2}  \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;3, p. 5827, Eq. (12)
 +
</div>
 +
 +
====Key Finding====
 +
On an initial reading, it appears as though the most relevant section of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] paper is &sect;8 titled, ''The Solid Torus.''  They write the gravitational potential in terms of the series expansion,
 +
<div align="center">
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Psi_\mathrm{grav}(\vec{r})</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\Psi_0 + \sum\limits_{n=1}^N \Psi_n \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;7, p. 5831, Eq. (42)
 +
</div>
 +
where, after setting <math>~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c </math> and acknowledging that <math>~V_{0,0} = 1 \, ,</math> we can write,
 +
<div align="center">
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Psi_0 </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}([k_H]_0) \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eqs. (52) &amp; (53)
 +
</div>
 +
and,
 +
<div align="center">
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{G\pi \rho_0 R_c b^2}{4 (k'_H)^2 \Delta_0^3} \biggl\{
 +
[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H)
 +
\biggr\} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)], &sect;8, p. 5832, Eq. (54)
 +
</div>
 +
 +
Rewriting this last expression in a form that can more readily be compared with Wong's work, we obtain,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^3\pi}{e^2} \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{ 1 }{(k'_H)^2 \Delta_0^3}\biggl\{
 +
[\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0} +  \frac{\boldsymbol{K}(k_H)}{\Delta_0} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<span id="Step01">Hence, also,</span>
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~ \frac{ \Psi_0 }{GM}  + \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~- \frac{2}{\pi}\biggl\{
 +
\frac{\boldsymbol{K}([k_H]_0)}{\Delta_0}
 +
\biggr\} +
 +
\frac{e^2}{2^3\pi}\biggl\{
 +
\frac{\boldsymbol{K}(k_H)}{\Delta_0}
 +
- \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0}
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~- \frac{2}{\pi \Delta_0}\biggl\{
 +
\boldsymbol{K}([k_H]_0)
 +
-
 +
\frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) \biggr\}
 +
- \frac{2}{\pi\Delta_0} \cdot \frac{e^2}{2^4}\biggl\{\biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr]
 +
\biggr\} \boldsymbol{E}(k_H)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\boldsymbol{K}([k_H]_0)
 +
-
 +
\frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H)
 +
+  \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr]  \boldsymbol{E}(k_H) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
===Compare First Terms===
 +
 +
Rewriting the first term in the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] series expression for the potential, we have,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{\Psi_0}{GM} </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{2}{\pi} \biggl\{ \frac{\boldsymbol{K}([k_H]_0) }{[ (\varpi_W + R_c)^2 + z_W^2]^{1 / 2}} \biggr\} \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
where,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~[k_H]_0</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{4\varpi_W R_c}{\Delta_0^2}  \biggr]^{1 / 2}
 +
=
 +
\biggl\{ \frac{4\varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2} \biggr\}^{1 / 2} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
For comparison, the first term in Wong's expression is,
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{\Phi_\mathrm{W0}}{GM} </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
 +
\Upsilon_{W0}(\eta_0) \biggl\{
 +
\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
where,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~a^2 </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
R^2 - d^2 ~~~\Rightarrow ~~~ a = R_c(1 - e^2)^{1 / 2} \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~r_1^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[ \varpi + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + \biggl[z - Z_0 \biggr]^2 \, ,</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{ \frac{4\varpi R_c(1-e^2)^{1 / 2}}{[\varpi + R_c(1-e^2)^{1 / 2}]^2 + [z - Z_0]^2} \biggr\}^{1 / 2}
 +
\, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Upsilon_{W0}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{\sinh\eta_0}{\cosh\eta_0}\biggl\{
 +
K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ]
 +
+ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ]
 +
- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
 +
\biggr\}  \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{(1-e^2)^{1 / 2}}{e^2} \biggl\{
 +
- K(k_0)\cdot K(k_0) (1-e)
 +
+ 2K(k_0)\cdot E(k_0) (1+e^2)
 +
- E(k_0)\cdot E(k_0) (1+e)
 +
\biggr\}  \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{2}{1+1/e} \biggr]^{1 / 2}
 +
=
 +
\biggl[ \frac{2e}{1+e} \biggr]^{1 / 2} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
This expression is correct for any value of the aspect ratio, <math>~e</math>.  But let's set <math>~Z_0 = 0</math> &#8212; as Hur&eacute;, et al. (2020) have done &#8212; then see how the expression simplifies for an infinitesimally thin hoop, that is, if we let <math>~e \rightarrow 0</math>.  First we note that,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k\biggr|_{e\rightarrow 0}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{ \frac{4\varpi R_c}{[\varpi + R_c]^2 + z^2} \biggr\}^{1 / 2}
 +
\, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
so in this limit the modulus of the complete elliptic integral of the first kind becomes identical to the modulus used by Hur&eacute;, et al. (2020), <math>~[k_H]_0</math>.  Next, we note that,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~r_1\biggr|_{e\rightarrow 0}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~[( \varpi + R_c )^2 + z^2]^{1 / 2} \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
As a result, we can write,
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{\Phi_\mathrm{W0}}{GM} \biggr|_{e\rightarrow 0}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{\Psi_0}{GM}  \cdot
 +
\biggl[
 +
\biggl( \frac{2^{2} }{3\pi^2} \biggr)
 +
\Upsilon_{W0}(\eta_0)
 +
\biggr]_{e\rightarrow 0} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
Now let's evaluate the coefficient, <math>~\Upsilon_{W0}</math>, in the limit of <math>~e \rightarrow 0</math>.
 +
 +
 +
<table align="center"  border="1" width="100%" cellpadding="8"><tr><td align="left">
 +
 +
<div align="center">
 +
<math>~\Upsilon_{W0}</math>, in the limit of <math>~e \rightarrow 0</math>.
 +
</div>
 +
First, drawing from our [[User:Tohline/Apps/Wong1973Potential#Phase_0C|separate examination of the behavior of complete elliptic integral functions]], we appreciate that,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  E(k_0) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 ~+~\frac{1}{2^5} ~k_0^4
 +
~+~\frac{1}{2^5} ~ k_0^6
 +
+ \mathcal{O}(k_0^{8}) \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  K(k_0) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 + \frac{1}{2} k_0^2
 +
+ \frac{11}{2^5} ~k_0^4
 +
+ \frac{17}{2^6} ~ k_0^6
 +
+ \mathcal{O}(k_0^{8}) \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot  E(k_0) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~-~\frac{1}{2^6} ~ k_0^6
 +
+ \mathcal{O}(k_0^{8}) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
Next, employing the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2e(1+e)^{-1}</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2e(  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
and,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0^4</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~4e^2(1+e)^{-2}</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~4e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
Hence, we have,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1-e) \biggl[ 1 + \frac{1}{2} k_0^2
 +
+ \frac{11}{2^5} ~k_0^4
 +
+ \mathcal{O}(k_0^{6})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4
 +
+ \mathcal{O}(k_0^{6})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~
 +
+ \mathcal{O}(k_0^{6})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1-e) \biggl[ 1 + e (  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots )
 +
+ \frac{11}{2^3} \cdot ~e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots )
 +
+ \mathcal{O}(e^{3})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots )
 +
+ \mathcal{O}(e^{3})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1+e) \biggl[ 1 - e ~(  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) ~
 +
-~ \frac{1}{2^3} \cdot~ e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) ~
 +
+ \mathcal{O}(e^{3})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1-e) \biggl[ 1 + e (  1 - e  )
 +
+ \frac{11}{2^3} \cdot ~e^2
 +
\biggr]
 +
+ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2
 +
\biggr]
 +
- (1+e) \biggl[ 1 - e ~(  1 - e  ) ~
 +
-~ \frac{1}{2^3} \cdot~ e^2 ~
 +
\biggr]
 +
+ \mathcal{O}(e^{3})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \biggl[ 1 + e (  1 - e  )
 +
+ \frac{11}{2^3} \cdot ~e^2
 +
\biggr]
 +
+e \biggl[ 1 + e
 +
\biggr]
 +
+ 2 \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2
 +
\biggr]
 +
+ 2 e^2
 +
- \biggl[ 1 - e ~(  1 - e  ) ~
 +
-~ \frac{1}{2^3} \cdot~ e^2 ~
 +
\biggr]
 +
- e \biggl[ 1 - e
 +
\biggr]
 +
+ \mathcal{O}(e^{3})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-1 - e + e^2
 +
- \frac{11}{2^3} \cdot ~e^2
 +
+ e + e^2
 +
+ 2 ~+~\frac{1}{2^2} \cdot~e^2
 +
+ 2 e^2
 +
-1 + e - e^2  ~
 +
+~ \frac{1}{2^3} \cdot~ e^2 ~
 +
- e +e^2
 +
+ \mathcal{O}(e^{3})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{e^2}{2^3}
 +
\biggl[ 2^5
 +
- 11~
 +
~+~3 \biggr]~
 +
+ \mathcal{O}(e^{3})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
3e^2
 +
+ \mathcal{O}(e^{3})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
[1 + \mathcal{O}(e^{1})]\cdot (1 - e^2)^{1 / 2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ \biggl[ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0) \biggr]_{e\rightarrow 0}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
</td></tr></table>
 +
 +
Given that,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~ \biggl[ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0) \biggr]_{e\rightarrow 0}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 \, ,</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
we conclude that,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{\Phi_\mathrm{W0}}{GM} \biggr|_{e\rightarrow 0}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{\Psi_0}{GM}  \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
that is, we conclude that <math>~\Psi_0</math> matches <math>~\Phi_{W0}</math> in the limit of, <math>~e\rightarrow 0</math>.
 +
 +
===Go to Higher Order===
 +
 +
Let's keep higher order terms in Wong's n = 0 component, and let's examine contributions to the same order that come from Wong's n = 1 and (if necessary) n = 2 components.
 +
 +
<span id="Step02">First, note that,</span>
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\Delta_0 \biggl( \frac{2^{2} }{3\pi^2} \biggr)
 +
\Upsilon_{W0}(\eta_0) \biggl\{
 +
\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
 +
====Keeping Higher Order in Wong's First Component====
 +
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  E(k_0) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 ~+~\frac{1}{2^5} ~k_0^4
 +
~+~\frac{1}{2^5} ~ k_0^6
 +
~+~\frac{231}{2^{13}} ~ k_0^8
 +
+ \mathcal{O}(k_0^{10}) \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  K(k_0) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 + \frac{1}{2} k_0^2
 +
+ \frac{11}{2^5} ~k_0^4
 +
+ \frac{17}{2^6} ~ k_0^6
 +
+ \frac{1787}{2^{13}} ~k^8
 +
+ \mathcal{O}(k_0^{10})
 +
\, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot  E(k_0) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4
 +
~-~\frac{1}{2^6} ~ k_0^6
 +
~-~\frac{77}{2^{13}} ~ k_0^8
 +
+ \mathcal{O}(k_0^{10}) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<table border="1" width="80%" cellpadding="5" align="center">
 +
<tr><td align="center">'''Add One Additional Term'''</td></tr>
 +
<tr><td align="center">
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[K(k) \cdot  E(k) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 + \biggl( \frac{1}{2} \biggr)^2k^2
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
 +
+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
 +
+ \cdots
 +
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
 +
+ \cdots
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\times~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
 +
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
 +
- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
 +
~-~ \cdots
 +
\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
 +
~-~ \cdots
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
 +
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
 +
- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\biggl( \frac{1}{2} \biggr)^2k^2
 +
\biggr\}
 +
\times~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
 +
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~+~
 +
\biggl\{
 +
\biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
 +
\biggr\}
 +
\times~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
 +
\biggr\}\times
 +
\biggl\{1 - \frac{1}{2^2} ~k^2
 +
\biggr\}
 +
+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{3}{2^6}~ k^4
 +
- \biggl(\frac{5}{2^8}\biggr)~k^6
 +
- \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr)~k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\biggl( \frac{1}{2^2} \biggr)k^2
 +
- \frac{1}{2^4} ~k^4
 +
- \frac{3}{2^8}~ k^6
 +
-\frac{5}{2^{10}} ~k^8
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
~+~
 +
\biggl( \frac{3^2}{2^6}\biggr) k^4
 +
~-~
 +
\biggl( \frac{3^2}{2^8}\biggr) k^6
 +
~-~ \biggl(\frac{3^3}{2^{12}}\biggr) ~k^8
 +
~+~
 +
\biggl( \frac{5^2}{2^8}\biggr) k^6
 +
~-~\biggl(\frac{5^2}{2^{10}}\biggr) ~k^8
 +
~+~\biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) ~k^8
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1
 +
~+~\biggl[ \frac{1}{2^2} 
 +
- \frac{1}{2^2} \biggr] ~k^2
 +
~+~\biggl[ \frac{3^2}{2^6}
 +
- \frac{3}{2^6}
 +
- \frac{1}{2^4} \biggr]~k^4
 +
~+~\biggl[ \frac{5^2}{2^8} 
 +
- \frac{5}{2^8}
 +
- \frac{3}{2^8}
 +
~-~\frac{3^2}{2^8} \biggr]~ k^6
 +
+ \biggl[
 +
\biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) - \biggl(\frac{5^2}{2^{10}}\biggr) - \biggl(\frac{3^3}{2^{12}}\biggr) - \frac{5}{2^{10}} - \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr)
 +
\biggr]~k^8
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 ~+~\frac{1}{2^5} ~k^4
 +
~+~\frac{1}{2^5} ~ k^6
 +
~+~\frac{231}{2^{13}} ~ k^8
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[K(k) \cdot  K(k) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 + \biggl( \frac{1}{2} \biggr)^2k^2
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
 +
+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
 +
+ \cdots
 +
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
 +
+ \cdots
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\times~
 +
\biggl\{
 +
1 + \biggl( \frac{1}{2} \biggr)^2k^2
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
 +
+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
 +
+ \cdots
 +
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
 +
+ \cdots
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1
 +
+ \frac{1}{2^2} k^2
 +
+ \frac{3^2}{2^6} k^4
 +
+ \frac{5^2}{2^8} k^6
 +
+ \frac{5^2 \cdot 7^2}{2^{14}} k^8
 +
\biggr\}
 +
\times~
 +
\biggl\{
 +
1
 +
+ \frac{1}{2^2} k^2
 +
+ \frac{3^2}{2^6} k^4
 +
+ \frac{5^2}{2^8} k^6
 +
+ \frac{5^2 \cdot 7^2}{2^{14}} k^8
 +
\biggr\}
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1
 +
+ \frac{1}{2^2} k^2
 +
+ \frac{3^2}{2^6} k^4
 +
+ \frac{5^2}{2^8} k^6
 +
+ \frac{5^2 \cdot 7^2}{2^{14}} k^8
 +
\biggr\}
 +
~+~\frac{1}{2^2} k^2
 +
\biggl\{
 +
1
 +
+ \frac{1}{2^2} k^2
 +
+ \frac{3^2}{2^6} k^4
 +
+ \frac{5^2}{2^8} k^6
 +
\biggr\}
 +
~+~\frac{3^2}{2^6} k^4
 +
\biggl\{
 +
1
 +
+ \frac{1}{2^2} k^2
 +
+ \frac{3^2}{2^6} k^4
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\frac{5^2}{2^8} k^6
 +
\biggr\}
 +
\biggl\{
 +
1
 +
+ \frac{1}{2^2} k^2
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\frac{5^2 \cdot 7^2}{2^{14}} k^8
 +
\biggr\}
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 + \frac{1}{2^2} k^2
 +
+ \frac{3^2}{2^6} k^4
 +
+ \frac{5^2}{2^8} k^6
 +
+ \frac{5^2 \cdot 7^2}{2^{14}} k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\frac{1}{2^2} k^2
 +
+ \frac{1}{2^4} k^4
 +
+ \frac{3^2}{2^8} k^6
 +
+ \frac{5^2}{2^{10}} k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\frac{3^2}{2^6} k^4
 +
~+~\frac{3^2}{2^8} k^6
 +
~+~\frac{3^4}{2^{12}} k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\frac{5^2}{2^8} k^6
 +
~+~\frac{5^2}{2^{10}} k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
\frac{5^2 \cdot 7^2}{2^{14}} k^8
 +
\biggr\}
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 + \biggl[ \frac{1}{2^2}
 +
~+~ \frac{1}{2^2} \biggr] ~k^2
 +
+ \biggl[ \frac{3^2}{2^6}
 +
+ \frac{1}{2^4}
 +
~+~\frac{3^2}{2^6} \biggr]~k^4
 +
+ \biggl[ \frac{5^2}{2^8}
 +
+ \frac{3^2}{2^8}
 +
~+~\frac{3^2}{2^8} 
 +
~+~\frac{5^2}{2^8} \biggr]~ k^6
 +
~+~\biggl[
 +
\frac{5^2 \cdot 7^2}{2^{14}}+ \frac{5^2}{2^{10}} ~+~\frac{3^4}{2^{12}} ~+~\frac{5^2}{2^{10}} ~+~\frac{5^2 \cdot 7^2}{2^{14}}
 +
\biggr]k^8
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 + \frac{1}{2} k^2
 +
+ \frac{11}{2^5} ~k^4
 +
+ \frac{17}{2^6} ~ k^6
 +
+ \frac{1787}{2^{13}} ~k^{8}
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[E(k) \cdot  E(k) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
 +
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
 +
- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
 +
~-~ \cdots
 +
\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
 +
~-~ \cdots
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\times~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
 +
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
 +
- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
 +
~-~ \cdots
 +
\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
 +
~-~ \cdots
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{3}{2^6}~ k^4
 +
- \frac{5}{2^8}~ k^6
 +
- \frac{5^2\cdot 7}{2^{14}} ~k^8
 +
\biggr\}
 +
\times
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{3}{2^6}~ k^4
 +
- \frac{5}{2^8}~ k^6
 +
- \frac{5^2\cdot 7}{2^{14}} ~k^8
 +
\biggr\}
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{3}{2^6}~ k^4
 +
- \frac{5}{2^8}~ k^6
 +
- \frac{5^2\cdot 7}{2^{14}} ~k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{- \frac{1}{2^2} ~k^2
 +
\biggr\}
 +
\times
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{3}{2^6}~ k^4
 +
- \frac{5}{2^8}~ k^6
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
~+~
 +
\biggl\{
 +
- \frac{3}{2^6}~ k^4
 +
\biggr\}
 +
\times
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{3}{2^6}~ k^4
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
- \frac{5}{2^8}~ k^6
 +
\biggr\}
 +
\times
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
- \frac{5^2\cdot 7}{2^{14}}k^8
 +
\biggr\}
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
1 - \frac{1}{2^2} ~k^2
 +
- \frac{3}{2^6}~ k^4
 +
- \frac{5}{2^8}~ k^6
 +
- \frac{5^2\cdot 7}{2^{14}} ~k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
- \frac{1}{2^2} ~k^2  + \frac{1}{2^4} ~k^4
 +
+ \frac{3}{2^8}~ k^6
 +
+ \frac{5}{2^{10}}~k^8
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
~+~
 +
\biggl\{
 +
~-~ \frac{3}{2^6}~ k^4
 +
~+~ \frac{3}{2^8}~ k^6
 +
+ \frac{3^2}{2^{12}}~k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
~-~\frac{5}{2^8}~ k^6
 +
+ \frac{5}{2^{10}}~k^8
 +
\biggr\}
 +
~+~
 +
\biggl\{
 +
~-~ \frac{5^2\cdot 7}{2^{14}}k^8
 +
\biggr\}
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 + \biggl[ - \frac{1}{2^2}- \frac{1}{2^2} \biggr]k^2
 +
+ \biggl[ - \frac{3}{2^6}+ \frac{1}{2^4}~-~ \frac{3}{2^6} \biggr] k^4
 +
+ \biggl[ - \frac{5}{2^8}+ \frac{3}{2^8}~+~ \frac{3}{2^8}~-~\frac{5}{2^8} \biggr] k^6
 +
+ \biggl[ - \frac{5^2\cdot 7}{2^{14}} + \frac{5}{2^{10}}+ \frac{3^2}{2^{12}}+ \frac{5}{2^{10}}~-~ \frac{5^2\cdot 7}{2^{14}} \biggr] k^8
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 + \biggl[ - \frac{2}{2^2} \biggr]k^2
 +
+ \biggl[ - \frac{3}{2^5}+ \frac{2}{2^5} \biggr] k^4
 +
+ \biggl[ - \frac{5}{2^7}+ \frac{3}{2^7} \biggr] k^6
 +
+ \biggl[ - \frac{5^2\cdot 7}{2^{13}} + \frac{5\cdot 2^4}{2^{13}}+ \frac{2 \cdot 3^2}{2^{13}}\biggr] k^8
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 - \frac{1}{2} ~k^2 ~-~ \frac{1}{2^5} ~ k^4 ~-~\frac{1}{2^6} ~ k^6 ~-~\frac{77}{2^{13}} ~ k^8
 +
+ \mathcal{O}(k^{10})
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
</td></tr>
 +
</table>
 +
 +
 +
Next, employing the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2e(1+e)^{-1}</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2e(  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0^4</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~4e^2(1+e)^{-2}</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~4e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, ;</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0^6</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2^3e^3(1+e)^{-3}</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2^3e^3(  1 - 3e + 6e^2 - 10e^3 + 15e^4 - 21e^5 + \cdots ) \, ;</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~k_0^8</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2^4e^4(1+e)^{-4}</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2^4e^4(1 - 4e + 10 e^2 - 20e^3 + 35e^4 - 56e^5 + \cdots) \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
<span id="Step03">Hence, we have,</span>
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1-e) \biggl[ 1 + \frac{1}{2} k_0^2
 +
+ \frac{11}{2^5} ~k_0^4
 +
+ \frac{17}{2^6} ~ k_0^6
 +
+ \frac{1787}{2^{13}} ~k_0^8
 +
+ \mathcal{O}(k_0^{10})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4
 +
~+~\frac{1}{2^5} ~ k_0^6
 +
~+~ \frac{231}{2^{13}}~k_0^8
 +
+ \mathcal{O}(k_0^{10})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~
 +
~-~\frac{1}{2^6} ~ k_0^6
 +
~-~\frac{77}{2^{13}}~k_0^8
 +
+ \mathcal{O}(k_0^{10})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1-e) \biggl[ 1 + e (  1 - e +e^2 -e^3 )
 +
+ \frac{11}{2^3} \cdot ~e^2(  1 - 2e  + 3e^2)
 +
+ \frac{17}{2^6} ~\cdot 2^3 e^3 ( 1 - 3e)
 +
+ \frac{1787}{2^{13}} \cdot 2^4e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} \cdot~4e^2(  1 - 2e  + 3e^2)
 +
~+~\frac{1}{2^5} ~ 2^3e^3 (1-3e)
 +
~+~\frac{231}{2^{13}} \cdot 2^4e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1+e) \biggl[ 1 - e ~(  1 - e +e^2 -e^3) ~
 +
-~ \frac{1}{2^5} \cdot~ 4e^2(  1 - 2e  +3e^2) ~
 +
~-~\frac{1}{2^6} ~ 2^3e^3(1 - 3e)
 +
~-~\frac{77}{2^{13}}~2^4e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^{-9}(1-e) \biggl[ 512 (1+  e - e^2 +e^3 -e^4 )
 +
+ 704  \cdot ~(  e^2 - 2e^3  + 3e^4)
 +
+ 1088 ~\cdot ( e^3 - 3e^4)
 +
+ 1787 \cdot e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ (1+e^2) 2^{-9}\biggl[ 1024
 +
~+~128 \cdot~(  e^2 - 2e^3  + 3e^4)
 +
~+~256 ~ (e^3-3e^4)
 +
~+~462 \cdot e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^{-9}(1+e) \biggl[ 512 ~(1  -e + e^2 -e^3 + e^4) ~
 +
-~ 64 \cdot~ (  e^2 - 2e^3  +3e^4) ~
 +
~-~64 ~ (e^3 - 3e^4)
 +
~-~77~e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^{-9}(1-e) \biggl[
 +
512 + 512e + 192e^2 + e^3(512 - 1408 + 1088) + e^4(704-512 - 3264 + 1787)
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2^{-9}(1+e^2) \biggl[
 +
1024 + 128e^2 + e^3(-256 + 256  ) + e^4(384 -768 + 462)
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^{-9}(1+e) \biggl[
 +
512 - 512e + e^2(512 - 64  ) + e^3(-512 +128 -64  ) + e^4(512 - 192 - 192 - 77)
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^{-9}(1-e) \biggl[
 +
512 + 512e + 192e^2 + 192e^3 - 1285 e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2^{-9}(1+e^2) \biggl[
 +
1024 + 128e^2 + 78e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^{-9}(1+e) \biggl[
 +
512 - 512e + 448e^2 - 448 e^3 + 51e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^{-9}\biggl[
 +
(- 192+ 128 - 448)e^2 
 +
+ (- 192 + 448) e^3
 +
+ (1285 + 78- 51)e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2^{-9} e \biggl[
 +
1024e + (192- 448)e^2 
 +
+ (192+ 448) e^3 
 +
+ \mathcal{O}(e^{4})
 +
\biggr]
 +
+ 2^{-9} e^2 \biggl[
 +
1024 + 128e^2
 +
+ \mathcal{O}(e^{3})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^{-9} \biggl[
 +
(- 192+ 128 - 448)e^2  + 2048 e^2
 +
+ (1285 + 78- 51)e^4
 +
+ (192+ 448) e^4 
 +
+ 128e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^{-9} \biggl[
 +
1536e^2 
 +
+ 2080e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
3e^2 
 +
+ \biggl[ \frac{5\cdot 13}{2^4}\biggr] e^4
 +
+ \mathcal{O}(e^{5})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
Hence,
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
[1 + \mathcal{O}(e^{2})]\cdot (1 - e^2)^{1 / 2}
 +
\biggl\{
 +
\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\Delta_0  \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
or, more precisely,
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2}
 +
\biggl\{
 +
\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\Delta_0  \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
====Next Factors====
 +
 +
 +
Now,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Delta_0^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
(\varpi_W + R_c)^2 + z_W^2 \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~r_1^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2  \, ,</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ r_1^2 - \Delta_0^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2  - \biggl[ (\varpi_W + R_c)^2 + z_W^2 \biggr]</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\varpi_W^2 + 2\varpi_W R_c (1 - e^2 )^{1 / 2} + R_c^2 (1 - e^2 ) + z_W^2  - [\varpi_W^2 +  2\varpi_W R_c + R_c^2 + z_W^2 ]</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~2\varpi_W R_c [(1 - e^2 )^{1 / 2}  - 1]  -e^2 R_c^2 \, .  </math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
----
 +
 +
Again, drawing from the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we have,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~(1 -e^2)^{1 / 2}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 - \frac{1}{2}e^2 + \biggl[\frac{ \tfrac{1}{2}(-\tfrac{1}{2}) }{ 2 } \biggr]e^4 - \biggl[ \frac{ \tfrac{1}{2}(-\tfrac{1}{2} )(-\tfrac{3}{2} ) }{ 3! } \biggr]e^6
 +
+ \biggl[ \frac{ \tfrac{1}{2} (-\tfrac{1}{2})(-\tfrac{3}{2})(-\tfrac{5}{2})  }{ 4! } \biggr]e^8 + \cdots
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 - \frac{1}{2}e^2 - \frac{1}{2^3} e^4 - \frac{1}{2^4}e^6 - \frac{5}{2^7} e^8 - \mathcal{O}(e^{10}) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
----
 +
 +
<div align="center" id="Step04">
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ r_1^2 - \Delta_0^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\varpi_W R_c \biggl[- e^2 - \frac{1}{2^2} e^4 - \frac{1}{2^3}e^6 - \frac{5}{2^6} e^8 - \mathcal{O}(e^{10})  \biggr]  -e^2 R_c^2  </math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~\frac{ r_1^2}{\Delta_0^2} </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~1
 +
-e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{\Delta_0^2}\biggr]
 +
- \frac{\varpi_W R_c}{\Delta_0^2} \biggl[\frac{1}{2^2} e^4 + \frac{1}{2^3}e^6 + \frac{5}{2^6} e^8 + \mathcal{O}(e^{10})  \biggr] </math>
 +
  </td>
 +
</tr>
 +
</table>
 +
</div>
 +
 +
 +
 +
====Now Work on Elliptic Integral Expressions====
 +
 +
 +
From a [[User:Tohline/2DStructure/ToroidalGreenFunction#Series_Expansions|separate discussion]] we can draw the series expansion of <math>~\boldsymbol{K}(k)</math>, specifically,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2K(k)}{\pi}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 + \biggl( \frac{1}{2} \biggr)^2k^2
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
 +
+ \cdots
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
where,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{k^2}{4}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{1}{4} \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]
 +
= \biggl[ \frac{a\varpi}{r_1^2} \biggr]
 +
= \biggl[ \frac{a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{\varpi_W R_c(1-e^2)^{1 / 2}}{[ \varpi_W + R_c(1-e^2)^{1 / 2} ]^2 + z_W^2}  \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
Also,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2K(k_H)}{\pi}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1 + \biggl( \frac{1}{2} \biggr)^2k_H^2
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6
 +
+ \cdots
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
where,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{k_H^2}{4}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{1}{\Delta^2}\biggl[ R (R_c + b\cos\theta_H) \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{\varpi_W (R_c + b\cos\theta_H)}{[\varpi_W + (R_c + b\cos\theta_H)]^2 + [z_W - b\sin\theta_H]^2}  \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
What we want to do is write <math>~K(k)</math> in terms of <math>~K(k_H)</math>.  Let's try &hellip;
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2K(k)}{\pi}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{2K(k_H)}{\pi} + \delta_K \, ,</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
 +
where,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\delta_K</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\frac{2K(k)}{\pi} - \frac{2K(k_H)}{\pi} </math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl\{1 + \frac{k^2}{4}
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
 +
+ \cdots
 +
\biggr\}
 +
- \biggl\{1 + \frac{k_H^2}{4}
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6
 +
+ \cdots
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl\{1 + \frac{k^2}{4}
 +
\biggr\}
 +
- \biggl\{1 + \frac{k_H^2}{4}
 +
\biggr\}
 +
= \frac{k^2}{4} - \frac{k_H^2}{4}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{\varpi_W R_c(1-e^2)^{1 / 2}}{[ \varpi_W + R_c(1-e^2)^{1 / 2} ]^2 + z_W^2}
 +
-
 +
\frac{\varpi_W (R_c + b\cos\theta_H)}{[\varpi_W + (R_c + b\cos\theta_H)]^2 + [z_W - b\sin\theta_H]^2} 
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{ \varpi_W R_c(1-e^2)^{1 / 2} \biggr\} \biggl\{\varpi_W^2 +R_c^2 + z_W^2  + 2\varpi_W R_c(1-e^2)^{1 / 2} - R_c^2 e^2 \biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-
 +
\biggl\{ \varpi_W R_c (1 + e\cos\theta_H) \biggr\} \biggl\{[\varpi_W + R_c(1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2 \biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{ \varpi_W R_c \biggl[ 1-\frac{1}{2} e^2 + \mathcal{O}(e^4)\biggr] \biggr\} \biggl\{\varpi_W^2 +R_c^2 + z_W^2  + 2\varpi_W R_c\biggl[ 1-\frac{1}{2} e^2 + \mathcal{O}(e^4)\biggr] - R_c^2 e^2 \biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-
 +
\biggl\{ \varpi_W R_c \biggl[ 1 + e\cos\theta_H \biggr] \biggr\}
 +
\biggl\{\varpi_W^2 + 2\varpi_W R_c(1+e\cos\theta_H)
 +
+ R_c^2(1 + 2e\cos\theta_H + e^2\cos^2\theta_H)  + z_W^2 - 2 z_W R_c e\sin\theta_H + R_c^2 e^2 \sin\theta^2_H  \biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{\varpi_W R_c  -  e^2  \biggl[ \frac{\varpi_W R_c}{2} \biggr]  + \mathcal{O}(e^4) \biggr\}
 +
\biggl\{ [ (\varpi_W +R_c)^2 + z_W^2 ] - e^2 \biggl[  \varpi_W R_c + R_c^2 \biggr]  + \mathcal{O}(e^4) \biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-
 +
\biggl\{ \varpi_W R_c  +  e \biggl[ \varpi_W R_c \cos\theta_H \biggr]\biggr\}
 +
\biggl\{ [ (\varpi_W^2 + R_c)^2 + z_W^2 ]
 +
+ 2R_c e(R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H) + R_c^2 e^2  \biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{ \frac{\varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } \biggl[1 - \frac{1}{2}e^2 \biggr]  \biggr\}
 +
\biggl\{ 1 - e^2 \biggl[ \frac{ \varpi_W R_c + R_c^2  }{ (\varpi_W +R_c)^2 + z_W^2 } \biggr] \biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~
 +
\biggl\{ \frac{ \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ]  } \biggl[ 1 + e\cos\theta_H \biggr] \biggr\}
 +
\biggl\{ 1
 +
+ e\biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr]
 +
+ e^2 \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr]  \biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{\varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } \biggl[1 - \frac{1}{2}e^2 \biggr] 
 +
\biggl\{ 1 + e^2 \biggl[ \frac{ \varpi_W R_c + R_c^2  }{ (\varpi_W +R_c)^2 + z_W^2 } \biggr] \biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~\frac{ \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ]  } \biggl[ 1 + e\cos\theta_H \biggr]
 +
\biggl\{ 1
 +
- e\biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr]
 +
- e^2 \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr]  \biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~\frac{ e\cdot \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ]  }
 +
\biggl[ \cos\theta_H - \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~\frac{e^2 \cdot \varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] }
 +
\biggl\{ \biggl[ \frac{ \varpi_W R_c + R_c^2  }{ (\varpi_W +R_c)^2 + z_W^2 } - \frac{1}{2} \biggr]
 +
- \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr] 
 +
- \cos\theta_H \biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr]
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~\frac{ e\cdot \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ]  [ (\varpi_W^2 + R_c)^2 + z_W^2 ] }
 +
\biggl[ \cos\theta_H [(\varpi_W^2 + R_c)^2 + z_W^2 ]  -2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H) \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~\frac{\tfrac{1}{2}e^2 \cdot \varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ][ (\varpi_W^2 + R_c)^2 + z_W^2 ] }
 +
\biggl\{ 2 (\varpi_W R_c + R_c^2 )  - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] 
 +
- 2R_c^2   
 +
- 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~\frac{ e\cdot \varpi_W R_c }{ \Delta_0^2 }
 +
\biggl[ \cos\theta_H  - \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{\Delta_0^2} \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~\frac{ e^2 \cdot \varpi_W R_c}{2 \Delta_0^4 }
 +
\biggl\{ 2 (\varpi_W R_c + R_c^2 )  - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] 
 +
- 2R_c^2   
 +
- 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
 +
\biggr\} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
 +
Let's subtract <math>~K([k_H]_0)</math> from the potential expression.  But first, let's adopt the shorthand notation &hellip;
 +
 +
<table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left">
 +
Given that,
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{
 +
\boldsymbol{K}(k)  \biggr\} \frac{\Delta_0}{r_1}~\biggl( \frac{2^2}{3\pi^2} \biggr) \Upsilon_{W0}(\eta_0)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl\{ \boldsymbol{K}(k) \biggr\} \biggl[\frac{r_1^2 }{ \Delta_0^2 } \biggr]^{-1 / 2}
 +
\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl\{ \boldsymbol{K}(k) \biggr\} \biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \mathcal{O}(e^4)\biggr\}
 +
\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot \biggl[ 1 - \frac{1}{2}e^2  - \mathcal{O}(e^{4}) \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
let's define the variable, <math>~\mathcal{A}</math>, such that,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{\boldsymbol{K}(k)\biggr\} \{1 + e^2\mathcal{A}\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ \{ 1 + e^2 \cdot \mathcal{A} \}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \mathcal{O}(e^4)\biggr\}
 +
\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot \biggl[ 1 - \frac{1}{2}e^2  - \mathcal{O}(e^{4}) \biggr]</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] \biggr\}
 +
\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2  \biggr] \biggl[ 1 - \frac{1}{2}e^2  \biggr]</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2  - \frac{1}{2}e^2 \biggr\}</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ 2\mathcal{A}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \biggl( \frac{41}{2^3\cdot 3}\biggr) \, . </math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
</td></tr></table>
 +
 +
 +
We can therefore write,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] - K([k_H]_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~- K([k_H]_0) +
 +
\biggl\{ K(k_H) + \frac{\pi}{2} \cdot \delta_K\biggr\}
 +
\{ 1 + e^2 \cdot \mathcal{A} \}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~- K([k_H]_0)
 +
+
 +
K(k_H)
 +
\{ 1 + e^2 \cdot \mathcal{A} \}
 +
+
 +
\frac{\pi}{2} \cdot \delta_K \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
where we should keep in mind that <math>~\delta_k</math> is <math>~\mathcal{O}(e^1)</math>.  So, let's examine the piece,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{1 + \frac{k_H^2}{4}
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6
 +
+ \cdots
 +
\biggr\} -
 +
\biggl\{1 + \frac{k_H^2}{4}
 +
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4
 +
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6
 +
+ \cdots
 +
\biggr\}_{e\rightarrow 0}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{k_H^2}{4} - \biggl[ \frac{k_H^2}{4} \biggr]_{e\rightarrow 0}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{\varpi_W R_c (1 + e\cos\theta_H)}{[\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2} \biggr]
 +
-
 +
\biggl[  \frac{\varpi_W R_c (1 + e\cos\theta_H)}{[\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2} \biggr]_{e\rightarrow 0}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\varpi_W R_c (1 + e\cos\theta_H)
 +
\biggl\{ [\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2 \biggr\}^{-1}
 +
-
 +
\biggl[  \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\varpi_W R_c (1 + e\cos\theta_H)
 +
\biggl\{
 +
\varpi_W^2 + 2\varpi_W R_c(1 + e\cos\theta_H) + R_c^2(1 + 2e\cos\theta_H + e^2\cos^2\theta_H) + z_W^2 -2z_W R_c e\sin\theta_H + R_c^2 e^2\sin^2\theta_H
 +
\biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-
 +
\biggl[  \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-\biggl[  \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr]
 +
+ \biggl[ \frac{ \varpi_W R_c (1 + e\cos\theta_H)}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr]
 +
\biggl\{1
 +
+ \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{(\varpi_W + R_c)^2 + z_W^2}
 +
+ \frac{e^2 [R_c^2\cos^2\theta_H  + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2}
 +
\biggr\}^{-1}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-\biggl[  \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr]
 +
+\biggl[ \frac{ \varpi_W R_c (1 + e\cos\theta_H)}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr]
 +
\biggl\{1
 +
- \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{(\varpi_W + R_c)^2 + z_W^2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{e^2 [R_c^2\cos^2\theta_H  + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2}
 +
+ \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{ [(\varpi_W + R_c)^2 + z_W^2]^2}
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{ \varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr]
 +
\biggl\{
 +
- \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{(\varpi_W + R_c)^2 + z_W^2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{e^2 [R_c^2\cos^2\theta_H  + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2}
 +
+ \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{ [(\varpi_W + R_c)^2 + z_W^2]^2}
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+\biggl[ \frac{ \varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr]
 +
\biggl\{ \frac{e\cos\theta_H [(\varpi_W + R_c)^2 + z_W^2] -e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{(\varpi_W + R_c)^2 + z_W^2}
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{ \varpi_W R_c}{ \Delta_0^2 } \biggr]
 +
\biggl\{
 +
e\cos\theta_H
 +
- \frac{ eR_c\cos\theta_H [2\varpi_W  + 2 R_c - 2z_W  \tan\theta_H]  }{ \Delta_0^2}
 +
- \frac{e^2 R_c^2 }{ \Delta_0^2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{ \Delta_0^2}
 +
+ \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^4}
 +
\biggr\} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
Now we have,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] - K([k_H]_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~- K([k_H]_0)
 +
+
 +
K(k_H)
 +
\{ 1 + e^2 \cdot \mathcal{A} \}
 +
+
 +
\frac{\pi}{2} \cdot \delta_K
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ - \Delta_0 \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\frac{2}{\pi} K([k_H]_0)
 +
+ \frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]
 +
+
 +
\delta_K
 +
+
 +
\frac{2}{\pi} K(k_H) e^2 \cdot \mathcal{A} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
But, as we have just demonstrated,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]+ \delta_K
 +
</math>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{ \varpi_W R_c}{ \Delta_0^2 } \biggr]
 +
\biggl\{
 +
e\cos\theta_H
 +
- \frac{ eR_c\cos\theta_H [2\varpi_W  + 2 R_c - 2z_W  \tan\theta_H]  }{ \Delta_0^2}
 +
- \frac{e^2 R_c^2 }{ \Delta_0^2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]  }{ \Delta_0^2}
 +
+ \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^4}
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
-~\frac{ e\cdot \varpi_W R_c }{ \Delta_0^2 }
 +
\biggl[ \cos\theta_H  - \frac{2R_c (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)}{\Delta_0^2} \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~\frac{ e^2 \cdot \varpi_W R_c}{2 \Delta_0^4 }
 +
\biggl\{ 2 (\varpi_W R_c + R_c^2 )  - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] 
 +
- 2R_c^2   
 +
- 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
 +
\biggr\} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[ \frac{e^2 \cdot  \varpi_W R_c}{ \Delta_0^4 } \biggr]
 +
\biggl\{
 +
- R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~(\varpi_W R_c + R_c^2 )  - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] 
 +
- R_c^2   
 +
- 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ \frac{  [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^2}
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr><td align="center" colspan="3"><font color="red">TEMPORARY BREAK HERE</font></td></tr>
 +
</table>
 +
 +
Hence,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~ - \frac{\pi  \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
K([k_H]_0)
 +
+
 +
K(k_H) e^2 \cdot \mathcal{A}
 +
+~\frac{\pi}{2} \biggl[ \frac{e^2 \cdot  \varpi_W R_c}{ \Delta_0^4 } \biggr]
 +
\biggl\{
 +
- R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~(\varpi_W R_c + R_c^2 )  - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] 
 +
- R_c^2   
 +
- 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ \frac{  [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^2}
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
===Include Second Wong Term===
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
 +
\Upsilon_{W1}(\eta_0) \times \cos\theta
 +
\biggl\{ \frac{a}{r_2} \cdot
 +
\boldsymbol{E}(k) \biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ - \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W1}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl( \frac{2^{2} }{3\pi^2} \biggr)
 +
\Upsilon_{W1}(\eta_0) \times \cos\theta
 +
\biggl\{ \frac{\Delta_0}{r_2} \cdot
 +
\boldsymbol{E}(k) \biggr\} \, ;
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
====Leading (Upsilon) Coefficient====
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Upsilon_{W1}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\equiv</math>
 +
  </td>
 +
  <td align="left" colspan="2">
 +
<math>~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] 
 +
\biggl\{
 +
K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)]
 +
+~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)]
 +
-~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)]
 +
\biggr\}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow~~~ \biggl[ \frac{e^2}{ (1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left" colspan="2">
 +
<math>~ 
 +
- (1-e)K(k_0)\cdot K(k_0)
 +
+~2(3-e^2)K(k_0)\cdot E(k_0) 
 +
-~5(1+e) E(k_0)\cdot E(k_0)  \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ \frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1-e) \biggl[ 1 + \frac{1}{2} k_0^2
 +
+ \frac{11}{2^5} ~k_0^4
 +
+ \frac{17}{2^6} ~ k_0^6
 +
+ \frac{1787}{2^{13}} ~k_0^8
 +
+ \mathcal{O}(k_0^{10})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2 (3-e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4
 +
~+~\frac{1}{2^5} ~ k_0^6
 +
~+~ \frac{231}{2^{13}}~k_0^8
 +
+ \mathcal{O}(k_0^{10})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 5(1+e) \biggl[ 1
 +
- ~\frac{1}{2} ~k_0^2 ~
 +
-~ \frac{1}{2^5} ~ k_0^4 ~
 +
~-~\frac{1}{2^6} ~ k_0^6
 +
~-~\frac{77}{2^{13}}~k_0^8
 +
+ \mathcal{O}(k_0^{10})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- (1-e) \biggl[ 1 + \frac{1}{2} \cdot 2e(  1 - e +e^2 - e^3 )
 +
+ \frac{11}{2^5} ~\cdot 4e^2(  1 - 2e + 3e^2 )
 +
+ \frac{17}{2^6} ~ \cdot 2^3e^3(  1 - 3e  )
 +
+ \frac{1787}{2^{13}} ~\cdot 2^4e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2 (3-e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~\cdot 4e^2(  1 - 2e + 3e^2 )
 +
~+~\frac{1}{2^5} ~ \cdot 2^3e^3(  1 - 3e  )
 +
~+~ \frac{231}{2^{13}}~\cdot 2^4e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 5(1+e) \biggl[ 1
 +
- ~\frac{1}{2} ~\cdot 2e(  1 - e +e^2 - e^3 ) ~
 +
-~ \frac{1}{2^5} ~ \cdot 4e^2(  1 - 2e + 3e^2 )
 +
~-~\frac{1}{2^6} ~ \cdot 2^3e^3(  1 - 3e  )
 +
~-~\frac{77}{2^{13}}~\cdot 2^4e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^{-9}(1-e) \biggl[ 2^9 + 2^9e(  1 - e +e^2 - e^3 )
 +
+ 2^6 \cdot 11 ~\cdot e^2(  1 - 2e + 3e^2 )
 +
+ 2^6\cdot 17 ~ \cdot e^3(  1 - 3e  )
 +
+ 1787 ~\cdot e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2^{-9} (6- 2e^2) \biggl[ 2^9 ~+~2^6~\cdot e^2(  1 - 2e + 3e^2 )
 +
~+~2^7 \cdot e^3(  1 - 3e  )
 +
~+~ 231~\cdot e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^{-9}(5+ 5e) \biggl[ 2^9
 +
- ~2^9 \cdot e(  1 - e +e^2 - e^3 ) ~
 +
-~ 2^6 \cdot e^2(  1 - 2e + 3e^2 )
 +
~-~2^6 \cdot e^3(  1 - 3e  )
 +
~-~77~\cdot e^4
 +
+ \mathcal{O}(e^{5})
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~\frac{2^{11}}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^9 - 2^9e(  1 - e +e^2 - e^3 )
 +
- 2^6 \cdot 11 ~\cdot e^2(  1 - 2e + 3e^2 )
 +
- 2^6\cdot 17 ~ \cdot e^3(  1 - 3e  )
 +
- 1787 ~\cdot e^4
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 2^9e + 2^9e^2(  1 - e +e^2 )
 +
+ 2^6 \cdot 11 ~\cdot e^3(  1 - 2e )
 +
+ 2^6\cdot 17 ~ \cdot e^4
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 6\biggl[ 2^9 ~+~2^6~\cdot e^2(  1 - 2e + 3e^2 )
 +
~+~2^7 \cdot e^3(  1 - 3e  )
 +
~+~ 231~\cdot e^4
 +
\biggr]
 +
-2^{10}e^2 ~-~2^7~\cdot e^4
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 5 \biggl[ -2^9
 +
+ ~2^9 \cdot e(  1 - e +e^2 - e^3 ) ~
 +
+~ 2^6 \cdot e^2(  1 - 2e + 3e^2 )
 +
~+~2^6 \cdot e^3(  1 - 3e  )
 +
~+~77~\cdot e^4
 +
\biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~5 \biggl[ -2^9e
 +
+ ~2^9 \cdot e^2(  1 - e +e^2 ) ~
 +
+~ 2^6 \cdot e^3(  1 - 2e )
 +
~+~2^6 \cdot e^4
 +
\biggr]
 +
+ \mathcal{O}(e^{5})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^9( e^2 - e^3 + e^4 )
 +
- 2^6 \cdot 11 ~(  e^2 - 2e^3 + 3e^4 )
 +
- 2^6\cdot 17 ~(  e^3 - 3e^4  )
 +
- 1787 ~e^4
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+2^9 (  e^2 - e^3 +e^4 )
 +
+ 2^6 \cdot 11 (  e^3 - 2e^4 )
 +
+ 2^6\cdot 17 ~ e^4
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~ 3\cdot 2^7~(  e^2 - 2e^3 + 3e^4 )
 +
~+~3\cdot 2^8 (  e^3 - 3e^4  )
 +
~+~ 2\cdot 3^2 \cdot 7\cdot 11~\cdot e^4
 +
-2^{10}e^2 ~-~2^7~\cdot e^4
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ 5\cdot 2^9 (  - e^2 +e^3 - e^4 ) ~
 +
+~ 5\cdot 2^6 (  e^2 - 2e^3 + 3e^4 )
 +
~+~5\cdot 2^6 (  e^3 - 3e^4  )
 +
~+~5\cdot 77~e^4
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+5\cdot 2^9 (  e^2 - e^3 +e^4 ) ~
 +
+~ 5\cdot 2^6 (  e^3 - 2e^4 )
 +
~+~5\cdot 2^6 e^4
 +
+ \mathcal{O}(e^{5})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
e^2 [ 2^9 - 2^6\cdot 11 + 2^9 + 3\cdot 2^7 -2^{10} -5\cdot 2^9 + 5\cdot 2^6 + 5\cdot 2^9 ]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ e^3 [ -2^9 - 2^7\cdot 11 - 2^6\cdot 17 - 2^9 +2^6\cdot 11 - 3\cdot 2^8+3\cdot 2^8  + 5\cdot 2^9-5\cdot 2^7 +5\cdot 2^6 -5\cdot 2^9 + 5\cdot 2^6]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ e^4 [ 2^9 - 3\cdot 11\cdot 2^6+3\cdot 17\cdot 2^6 - 1787 + 2^9 - 11\cdot 2^7 + 17\cdot 2^6 + 3^2\cdot 2^7 - 3^2\cdot 2^8+ 2\cdot 3^2\cdot 7\cdot 11 - 2^7
 +
- 5\cdot 2^9 + 3\cdot 5\cdot 2^6 - 3\cdot 5\cdot 2^6 + 5\cdot 7\cdot 11  + 5\cdot 2^9 - 5\cdot 2^7 +5\cdot 2^6]
 +
+ \mathcal{O}(e^{5})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
e^2 [ - 2^6\cdot 11  + 3\cdot 2^7  + 5\cdot 2^6  ]
 +
+ e^3 [ -2^{10} - 2^7\cdot 11 - 2^6\cdot 17 +2^6\cdot 11  ]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ e^4 [ 2^{10}~-~ 1787~+~ 2\cdot 3^2\cdot 7\cdot 11~+~ 5\cdot 7\cdot 11 ~+~ 2^6\cdot (3\cdot 17 - 3\cdot 11 - 22 + 17 + 18 - 36 - 2 - 10 + 5)  ]
 +
+ \mathcal{O}(e^{5})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
2^6 e^2 [ 0  ] ~- 2^8 \cdot 11 e^3 + e^4 [ 2^{10}~-~ 1787~+~ 7\cdot 11\cdot 23 ~-~ 2^8\cdot 3  ]
 +
+ \mathcal{O}(e^{5})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- 2^8 \cdot 11 e^3 + 2^4\cdot 3\cdot 5e^4
 +
+ \mathcal{O}(e^{5})
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~\frac{2^{2}}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
- \frac{11}{2} e^3 + \frac{3\cdot 5}{2^5} e^4
 +
+ \mathcal{O}(e^{5}) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
</table>
 +
 +
<table border="1" cellpadding="8" align="center" width="90%">
 +
<tr>
 +
  <td align="center" bgcolor="black"><font color="white">'''Floating Comparison Summary'''</font></td>
 +
</tr>
 +
<tr><td align="left">
 +
As [[#Step01|shown above]], the first three terms of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Hur&eacute;, et al. (2020)] series expression may be written as,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\boldsymbol{K}([k_H]_0)
 +
-
 +
\frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H)
 +
+  \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr]  \boldsymbol{E}(k_H) \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
Let's see how it compares to the first term of Wong's (1973) expression which, as [[#Step02|shown separately above]], can be written in the form,
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\Delta_0 \biggl( \frac{2^{2} }{3\pi^2} \biggr)
 +
\Upsilon_{W0}(\eta_0) \biggl\{
 +
\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
----
 +
First, as [[#Step03|shown above]],
 +
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl[1 + \biggl(\frac{5\cdot 13}{2^4\cdot 3} \biggr)e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
Note that, in order to determine the functional form of the <math>~\mathcal{O}(e^{2})</math> term in this expression, we will have to include <math>~k_0^8</math> terms in the various expressions for products of elliptic integrals.  Second, [[#Step04|we have shown that]],
 +
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~\frac{ r_1^2}{\Delta_0^2} </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~1
 +
-e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{\Delta_0^2}\biggr]
 +
- \frac{\varpi_W R_c}{\Delta_0^2} \biggl[\frac{1}{2^2} e^4 + \frac{1}{2^3}e^6 + \frac{5}{2^6} e^8 + \mathcal{O}(e^{10})  \biggr] </math>
 +
  </td>
 +
</tr>
 +
<tr>
 +
  <td align="right">
 +
<math>~\Rightarrow ~~~ \frac{\Delta_0}{r_1} </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
1  + e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{2\Delta_0^2}\biggr]
 +
\, ,</math> &nbsp; &nbsp; &nbsp; and we are defining <math>~\delta_K</math> such that,
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~K(k)</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~K(k_H) + \frac{\pi}{2} \cdot \delta_K \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
----
 +
 +
Hence,
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{ K(k_H) + \frac{\pi}{2} \cdot \delta_K\biggr\}
 +
\biggl\{ 1  + e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{2\Delta_0^2}\biggr] \biggr\}\biggl[1 + \biggl(\frac{5\cdot 13}{2^4\cdot 3} \biggr)e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2}
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\boldsymbol{K}([k_H]_0)
 +
+
 +
\boldsymbol{K}(k_H) e^2 \cdot \mathcal{A}
 +
+~\frac{\pi}{2} \biggl[ \frac{e^2 \cdot  \varpi_W R_c}{ \Delta_0^4 } \biggr]
 +
\biggl\{
 +
- R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+~(\varpi_W R_c + R_c^2 )  - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] 
 +
- R_c^2   
 +
- 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W  \cos\theta_H - z_W \sin\theta_H)
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
&nbsp;
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
+ \frac{  [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2  }{  \Delta_0^2}
 +
\biggr\} \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
and,
 +
<table border="0" cellpadding="5" align="center">
 +
<tr>
 +
  <td align="right">
 +
<math>~\mathcal{A}</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~\approx</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[ \frac{R_c(R_c + \varpi_W)}{2\Delta_0^2}\biggr] + \biggl( \frac{41}{2^4\cdot 3}\biggr) \, . </math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
----
 +
 +
Second,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~- \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W1}}{GM} \biggr] </math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\boldsymbol{E}(k)
 +
\biggl\{ \frac{\Delta_0 \cdot \cos\theta}{r_2} \biggr\}
 +
\biggl[ \biggl( \frac{2^{2} }{3\pi^2} \biggr) \Upsilon_{W1}(\eta_0) \biggr]
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
&nbsp;
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
\biggl\{ \boldsymbol{E}(k_H) + \frac{\pi}{2}\cdot \delta_E \biggr\}
 +
\biggl\{ \frac{\Delta_0 \cdot \cos\theta}{r_2} \biggr\}
 +
\biggl[- \biggl(\frac{11}{2\cdot 3} \biggr) e + \biggl(\frac{5}{2^5}\biggr) e^2 + \mathcal{O}(e^{5}) \biggr]\cdot (1 - e^2)^{1 / 2} \, .
 +
</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
 +
</td></tr>
 +
</table>
 +
 +
====Geometric Factor====
 +
 +
By definition,
 +
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\Delta_0^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~
 +
(\varpi_W + R_c)^2 + z_W^2 \, ,
 +
</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~r_1^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2  \, ,</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~r_2^2</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[ \varpi_W - R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2  \, ,</math>
 +
  </td>
 +
</tr>
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~\cos^2\theta</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~\biggl[ \frac{r_1^2 + r_2^2 - 4R_c^2(1-e^2)}{2r_1 r_2} \biggr]^2  \, .</math>
 +
  </td>
 +
</tr>
 +
</table>
 +
Hence,
 +
<table border="0" cellpadding="5" align="center">
 +
 +
<tr>
 +
  <td align="right">
 +
<math>~r_2^2 - \Delta_0^2 \cdot \cos^2\theta</math>
 +
  </td>
 +
  <td align="center">
 +
<math>~=</math>
 +
  </td>
 +
  <td align="left">
 +
<math>~</math>
   </td>
   </td>
</tr>
</tr>

Current revision as of 21:56, 23 July 2020

Contents

Université de Bordeaux (Part 1)

Whitworth's (1981) Isothermal Free-Energy Surface
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Spheroid-Ring Systems

Through a research collaboration at the Université de Bordeaux, B. Basillais & J. -M. Huré (2019), MNRAS, 487, 4504-4509 have published a paper titled, Rigidly Rotating, Incompressible Spheroid-Ring Systems: New Bifurcations, Critical Rotations, and Degenerate States.

We discuss this topic in a separate, accompanying chapter.

Exterior Gravitational Potential of Toroids

J. -M. Huré, B. Basillais, V. Karas, A. Trova, & O. Semerák (2020), MNRAS, 494, 5825-5838 have published a paper titled, The Exterior Gravitational Potential of Toroids. Here we examine how their work relates to the published work by C.-Y. Wong (1973, Annals of Physics, 77, 279), which we have separately discussed in detail.

Our Presentation of Wong's (1973) Result

Summary:  First three terms in Wong's (1973) expression for the gravitational potential at any point, P(ϖ, z), outside of a uniform-density torus.
Wong diagram

~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}

~=

~
- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
\Upsilon_{W0}(\eta_0) \biggl\{
\frac{a}{ r_1 } \cdot  \boldsymbol{K}(k) \biggr\}\, ,

~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}

~=

~
- \biggl( \frac{2^{3} }{3\pi^3} \biggr)
\Upsilon_{W1}(\eta_0) \times \cos\theta
\biggl\{ \frac{a}{r_2} \cdot
\boldsymbol{E}(k) \biggr\} \, ,

~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\varpi, z)\biggr|_\mathrm{exterior}

~=

~
- \biggl( \frac{2^{3} }{3\pi^3} \biggr)\Upsilon_{W2}(\eta_0)
\times \cos(2\theta)
\biggl\{ 
\biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot  \boldsymbol{E}(k)  
- 
\frac{a}{r_1}  \cdot \boldsymbol{K}(k) 
\biggr\} \, ,

where, once the major ( R ) and minor ( d ) radii of the torus — as well as the vertical location of its equatorial plane (Z0) — have been specified, we have,

~a^2

~\equiv

~
R^2 - d^2       and,       ~\cosh\eta_0 \equiv \frac{R}{d} ~~~~\Rightarrow ~~~\sinh\eta_0 = \frac{a}{d}
\, ,

~r_1^2

~\equiv

~(\varpi + a)^2 + (z - Z_0)^2 \, ,

~r_2^2

~\equiv

~(\varpi - a)^2 + (z - Z_0)^2 \, ,

~\cos\theta

~\equiv

~\biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \, ,

~k

~\equiv

~
\biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2}
= \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2}
= \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} 
\, .


  Leading Coefficient Expressions … … evaluated for:    ~\frac{R}{d} = \cosh\eta_0 = 3

~\Upsilon_{W0}(\eta_0)

~\equiv

~
\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]  \biggl\{
K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] 
+ 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0  + 1 ] 
- E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0)  ]
\biggr\}  \, ,

7.134677

~\Upsilon_{W1}(\eta_0)

~\equiv

~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr]  
\biggl\{
K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] 
+~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] 
-~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] 
\biggr\}
\, ,

0.130324

~\Upsilon_{W2}(\eta_0)

~\equiv

~
\frac{2^{3 / 2}}{3^2}  \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{
K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13]
+  2  K ( k_0 ) \cdot E(k_0)  [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ]

 

 

 

~
-~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ] 
\biggr\} \, ,

0.003153
where,

~k_0

~\equiv

~
\biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, .

0.707106781

NOTE: In evaluating these "leading coefficient expressions" for the case, ~R/d = 3, we have used the complete elliptic integral evaluations, K(k0) = 1.854074677 and E(k0) = 1.350643881.

Setup

From our accompanying discussion of Wong's (1973) derivation, the exterior potential is given by the expression,

~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W}(\eta,\theta)

~=

~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~\sum_{n=0}^{\mathrm{nmax}} \epsilon_n \cos(n\theta) C_n(\cosh\eta_0)P_{n-\frac{1}{2}}(\cosh\eta) \, ,

Wong (1973), §II.D, p. 294, Eqs. (2.59) & (2.61)

where,

~D_0

~\equiv

~
\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{(R^2 - d^2)^{3 / 2}}{d^2 R}  \biggr] 
\, ,

~C_n(\cosh\eta_0)

~\equiv

~(n+\tfrac{1}{2})Q_{n+\frac{1}{2}}(\cosh \eta_0) Q_{n - \frac{1}{2}}^2(\cosh \eta_0) 
- (n - \tfrac{3}{2}) Q_{n - \frac{1}{2}}(\cosh \eta_0)~Q^2_{n + \frac{1}{2}}(\cosh \eta_0) \,

Wong (1973), §II.D, p. 294, Eq. (2.63)

and where, in terms of the major ( R ) and minor ( d ) radii of the torus — or their ratio, ε ≡ d/R,

~\cosh\eta_0

~=

~\frac{R}{d} = \frac{1}{\epsilon} \, ,

~\sinh\eta_0

~=

~\frac{a}{d} = \frac{1}{d}\biggl[ R^2 - d^2 \biggr]^{1 / 2} = \frac{1}{\epsilon} \biggl[1 - \epsilon^2 \biggr]^{1 / 2} \, .

These expressions incorporate a number of basic elements of a toroidal coordinate system. In what follows, we will also make use of the following relations:

Once the primary scale factor, ~a, has been specified, the illustration shown at the bottom of this inset box — see also our accompanying set of similar figures used by other researchers — helps in explaining how transformations can be made between any two of the referenced coordinate pairs: ~(\varpi, z), ~(\eta, \theta), ~(r_1, r_2).

~\varpi

~=

~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)}

     ~\Rightarrow ~     

~\cos\theta

~=

~\cosh\eta - \frac{a\sinh\eta}{\varpi}

~z - Z_0

~=

~\frac{a\sin\theta}{(\cosh\eta - \cos\theta)}

     ~\Rightarrow ~     

~\sin\theta

~=

~\frac{(z - Z_0)}{\varpi} \cdot \sinh\eta

Given that (sin2θ + cos2θ) = 1, we have,

~1

~=

~
\biggl[ \frac{(z - Z_0)}{\varpi} \cdot \sinh\eta \biggr]^2 + \biggl[\cosh\eta - \frac{a\sinh\eta}{\varpi}\biggr]^2

~\Rightarrow ~~~ \coth\eta

~=

~
\frac{1}{2a\varpi}\biggl[\varpi^2 + a^2 + (z - Z_0)^2  \biggr] \, .

We deduce as well that,

~\frac{2}{\coth\eta + 1}

~=

~
\frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \, ,
        and,

~\sinh\eta + \cosh\eta

~=

~
\frac{\varpi^2 + a^2 + (z - Z_0)^2}{(\varpi + a)^2 + (z - Z_0)^2} \, .


Given the definitions,

~r_1^2

~=

~(\varpi + a)^2 + (z - Z_0)^2 \, ,

~r_2^2

~=

~(\varpi - a)^2 + (z - Z_0)^2 \, ,

we can use the transformations,

~\varpi

~=

~\frac{(r_1^2 - r_2^2)}{4a}     and,

~(z - Z_0)^2

~=

~r_2^2 - \frac{1}{16a^2}\biggl[ r_1^2 - r_2^2 - 4a^2 \biggr]^2 \, ,     or,

~(z - Z_0)^2

~=

~r_1^2 - \frac{1}{16a^2}\biggl[ r_1^2 - r_2^2 + 4a^2 \biggr]^2 \, .

Or we can use the transformations,

~\eta

~=

~\ln \biggl(\frac{r_1}{r_2}\biggr) \, ,

~\cos\theta

~=

~\frac{r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \, .


Additional potentially useful relations can be found in an accompanying chapter wherein we present a variety of basic elements of a toroidal coordinate system.

Wong diagram

Leading (n = 0) Term

Wong's Expression

Now, from our separate derivation we have,

~P_{-1 / 2}(\cosh\eta)

~=

~
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} Q_{-1 / 2}(\coth\eta) \, .

And if we make the function-argument substitution, ~z \rightarrow \coth\eta, in the "Key Equation,"

~Q_{-\frac{1}{2}}(z)

~=

~
\sqrt{ \frac{2}{z+1} } ~K\biggl( \sqrt{ \frac{2}{z+1}} \biggr)

Abramowitz & Stegun (1995), p. 337, eq. (8.13.3)

we can write,

~P_{-1 / 2}(\cosh\eta)

~=

~
\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) \, ,

where, from above, we recognize that,

~
k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2}  
=
\biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} \, .

So, the leading (n = 0) term gives,

~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)

~=

~
-D_0
(\cosh\eta - \cos\theta)^{1 / 2} ~C_0(\cosh\eta_0)P_{-\frac{1}{2}}(\cosh\eta)

 

~=

~
-D_0~C_0(\cosh\eta_0)
\biggl[ \frac{a \sinh\eta}{\varpi} \biggr]^{1 / 2} ~\frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)

 

~=

~
-\frac{D_0~C_0(\cosh\eta_0)}{\pi}
\biggl[ \frac{2a }{\varpi} \biggr]^{1 / 2} ~ k \boldsymbol{K}(k)

 

~=

~
- C_0(\cosh\eta_0) \cdot \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] 
\frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot  \boldsymbol{K}(k) \, .

Thin-Ring Evaluation of C0

In an accompanying discussion of the thin-ring approximation, we showed that as ~\cosh\eta_0 \rightarrow \infty

~C_0(x)\biggr|_{x\rightarrow \infty}

~=

~\biggl( \frac{3 \pi^2}{2^2} \biggr) \frac{1}{\cosh^2\eta_0} \, .

Hence, in this limit we can write,

~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\eta,\theta)\biggr|_\mathrm{thin-ring}

~=

~
- \frac{2 }{\pi}  \cancelto{1}{\biggl[\frac{\sinh\eta_0}{\cosh\eta_0}\biggr]^3 } 
\frac{a}{ [ (\varpi + a)^2 + (z - Z_0)^2 ]^{1 / 2} } \cdot  \boldsymbol{K}(k) \, .

More General Evaluation of C0

NOTE of CAUTION: In our above evaluation of the toroidal function, ~Q_{-\frac{1}{2}}(z), we appropriately associated the function argument, ~z, with the hyperbolic-cotangent of ~\eta; that is, we made the substitution, ~z \rightarrow \coth\eta. Here, as we assess the behavior of, and evaluate, the leading coefficient, ~C_0, an alternate substitution is appropriate, namely, ~z_0 \rightarrow \cosh\eta_0; we affix the subscript zero to this function argument in an effort to minimize possible confusion with the argument, ~z.

Drawing from our accompanying tabulation of Toroidal Function Evaluations, we have more generally,

~2C_0(\cosh\eta_0)

~=

~
\biggl[ Q_{+\frac{1}{2}}(\cosh \eta_0) \biggr]
\biggl[ Q_{ - \frac{1}{2}}^2(\cosh \eta_0) \biggr]
+ 
3 \biggl[ Q_{ - \frac{1}{2}}(\cosh \eta_0) \biggr]
\biggl[ Q^2_{ + \frac{1}{2}}(\cosh \eta_0) \biggr]

 

~=

~
\biggl[ \cosh\eta_0 ~k_0~K(k_0) ~-~ [2(\cosh\eta_0+1)]^{1 / 2} E(k_0) \biggr]
\times \biggl\{ \frac{ 4 \cosh\eta_0 ~E(k_0) - (\cosh\eta_0-1) K(k_0) }{ [2^{3} (\cosh\eta_0+1) (\cosh\eta_0-1)^{2} ]^{1 / 2}} \biggr\}