Editing ParabolicDensity/Axisymmetric/Structure/Try8thru10 (section)

====Complete the Square====

Again, let's rewrite the term inside square brackets on the RHS of the expression for the gravitational potential, 

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

<tr>
  <td align="right">
<math>\biggl[ ~~ \biggr]_\mathrm{RHS}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[(A_{s s} a_\ell^2) \zeta^4
+ 2(A_{\ell s}a_\ell^2 )\chi^2 \zeta^2
+ (A_{\ell \ell} a_\ell^2)  \chi^4 \biggr]\, ,
</math>
  </td>
</tr>
</table>
in such a way that we effectively "complete the square."  Assuming that the desired expression takes the form,

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

<tr>
  <td align="right">
<math>\biggl[ ~~ \biggr]_\mathrm{RHS}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[(A_{s s} a_\ell^2)^{1 / 2} \zeta^2 + B\chi^2 \biggr]
\biggl[(A_{s s} a_\ell^2)^{1 / 2} \zeta^2 + C\chi^2 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(A_{s s} a_\ell^2) \zeta^4
+ (A_{s s} a_\ell^2)^{1 / 2} (B+C) \zeta^2\chi^2 
+ BC\chi^4 \, ,
</math>
  </td>
</tr>
</table>
we see that we must have,

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

<tr>
  <td align="right">
<math>(A_{s s} a_\ell^2)^{1 / 2} (B+C) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2(A_{\ell s}a_\ell^2 )
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ B </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} } - C \, ;
</math>
  </td>
</tr>
</table>
and we must also have,

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

<tr>
  <td align="right">
<math>BC </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(A_{\ell \ell} a_\ell^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ B </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(A_{\ell \ell} a_\ell^2)}{C} \, . 
</math>
  </td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>\frac{(A_{\ell \ell} a_\ell^2)}{C} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} } - C 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~  0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
C^2 - 2\biggl[ \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggr]C + (A_{\ell \ell} a_\ell^2) \, . 
</math>
  </td>
</tr>
</table>
The pair of roots of this quadratic expression are,

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

<tr>
  <td align="right">
<math>C_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggr]
\pm \frac{1}{2}\biggl\{
4\biggl[ \frac{(A_{\ell s}a_\ell^2 )^2}{(A_{s s} a_\ell^2) }\biggr]
- 4(A_{\ell \ell} a_\ell^2)
\biggr\}^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggl\{1
\pm \biggl[
1
- \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2}
\biggr]^{1 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{C_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl\{1
\pm \biggl[
1
- \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2}
\biggr]^{1 / 2} \biggr\} \, .
</math>
  </td>
</tr>
</table>
Also, then,

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

<tr>
  <td align="right">
<math>\frac{B_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }
-
\frac{C_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }
-
\frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl\{1
\pm \biggl[
1
- \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2}
\biggr]^{1 / 2} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl\{1
\mp \biggl[
1
- \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2}
\biggr]^{1 / 2} \biggr\} \, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" cellpadding="8" align="center"><tr><td align="left">
NOTE:  [[#Index_Symbol_Expressions|Given that]],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(A_{s s} a_\ell^2)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{3(1-e^2)} - \frac{2}{3}(A_{\ell s} a_\ell^2)
</math>
  </td>

  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>(A_{\ell \ell} a_\ell^2)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} - \frac{1}{4}(A_{\ell s} a_\ell^2)
\, ,
</math>
  </td>
</tr>
</table>
we can write,
[[File:LambdaVsEccentricity.png|250px|right|Lambda vs Eccentricity]]<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\Lambda \equiv \frac{(A_{s s} a_\ell^2)(A_{\ell \ell} a_\ell^2) }{(A_{\ell s}a_\ell^2 )^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1 }{(A_{\ell s}a_\ell^2 )^2} \biggl\{
\biggl[\frac{2}{3(1-e^2)} - \frac{2}{3}(A_{\ell s} a_\ell^2)\biggr]
\biggl[ \frac{1}{2} - \frac{1}{4}(A_{\ell s} a_\ell^2) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1 }{6(A_{\ell s}a_\ell^2 )^2} \biggl\{
\frac{1}{(1-e^2)} 
\biggl[ 2 - (A_{\ell s} a_\ell^2) \biggr]
-
(A_{\ell s} a_\ell^2)
\biggl[ 2 - (A_{\ell s} a_\ell^2) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1 }{6(A_{\ell s}a_\ell^2 )^2} \biggl\{
\biggl[\frac{1}{(1-e^2)} - (A_{\ell s} a_\ell^2)\biggr] 
\biggl[ 2 - (A_{\ell s} a_\ell^2) \biggr]
\biggr\}
</math>
  </td>
</tr>
</table>

</td></tr></table>

In summary, then, we can write,

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

<tr>
  <td align="right">
<math>\frac{B_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl[
1 \mp ( 1 - \Lambda )^{1 / 2} 
\biggr] 
</math>
  </td>

  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>\frac{C_\pm}{(A_{s s} a_\ell^2)^{1 / 2}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2) }\biggl[
1 \pm (1 - \Lambda )^{1 / 2} 
\biggr] 
\, ,
</math>
  </td>
</tr>
</table>
where, as illustrated by the inset "Lambda vs Eccentricity" plot, for all values of the eccentricity <math>(0 < e \leq 1)</math>, the quantity, <math>\Lambda</math>, is greater than unity.  It is clear, then, that both roots of the relevant quadratic equation are complex &#8212; i.e., they have imaginary components.  But that's okay because the coefficients that appear in the right-hand-side, bracketed quartic expression appear in the combinations,

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

<tr>
  <td align="right">
<math>(BC)_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(A_{\ell s}a_\ell^2 )^2}{(A_{s s} a_\ell^2) }
\biggl[ 1 - ( 1 - \Lambda )^{1 / 2} \biggr] 
\biggl[ 1 + ( 1 - \Lambda )^{1 / 2} \biggr] 
=
\frac{(A_{\ell s}a_\ell^2 )^2}{(A_{s s} a_\ell^2) }
\biggl[ \Lambda\biggr] 
=
(A_{\ell \ell}a_\ell^2 ) 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(B + C)_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggl[
1 \mp ( 1 - \Lambda )^{1 / 2} 
\biggr] 
+
\frac{(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }\biggl[
1 \pm (1 - \Lambda )^{1 / 2} 
\biggr] 
=
\frac{2(A_{\ell s}a_\ell^2 )}{(A_{s s} a_\ell^2)^{1 / 2} }
\, ,
</math>
  </td>
</tr>
</table>
both of which are real.