Editing Appendix/Ramblings/Bordeaux (section)

====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>