Editing Appendix/Ramblings/NonlinarOscillation (section)

==Exploration==
===n = 5 Mass-Radius Relation===

So, for any <math>~\tilde\xi</math> configuration, the parametric relationship between <math>~m_\xi</math> and <math>~r_\xi</math> in pressure-truncated, <math>~n=5</math>  polytropes is,

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

<tr>
  <td align="right">
<math>~m_\xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{\xi}{\tilde\xi}\biggr)^3 \biggl(1 + \frac{\xi^2}{3}\biggr)^{-3/2} 
\biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{3/2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r_\xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\xi \biggl\{ 
\biggl[ \frac{4\pi}{2^5\cdot 3}\biggr]^{1/2} \tilde\xi^{-6}
\biggl( 1+\frac{\tilde\xi^2}{3} \biggr)^{3}\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
And this can be inverted analytically in the case of <math>~\tilde\xi = 3</math>.  Specifically,

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

<tr>
  <td align="right">
<math>~m_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2\xi}{3}\biggr)^3
\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ m_0^{2/3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{2^2\xi^2}{3^2}\biggr)
\biggl(1 + \frac{\xi^2}{3}\biggr)^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
2^2\xi^2 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3^2\biggl(1 + \frac{\xi^2}{3}\biggr) m_0^{2/3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\xi^2 (2^2 - 3 m_0^{2/3}) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3^2m_0^{2/3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\xi^2  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3^2m_0^{2/3}}{(2^2 - 3 m_0^{2/3})} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the radius-mass relationship in the configuration at the <math>~P_\mathrm{max}</math> turning point is,

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

<tr>
  <td align="right">
<math>~
r_0 (m_0)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{2^9 \pi}{3^{13}}\biggr]^{1/2} 
\biggl[\frac{3^2m_0^{2/3}}{2^2 - 3 m_0^{2/3}}\biggr]^{1/2} \, .
</math>
  </td>
</tr>
</table>
</div>

Actually, the inversion can be performed analytically for any choice of <math>~\tilde\xi</math> to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
r_\xi (m_\xi)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde{r}_\mathrm{edge}
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C} - 3 m_\xi^{2/3}}\biggr]^{1/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
<span id="DefineTildeC">where,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tilde{C}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{3^2}{\tilde\xi^2}\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr) \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tilde{r}_\mathrm{edge}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} {\tilde\xi}^{-6} \biggl(1+\frac{\tilde\xi^2}{3}\biggr)^3
\, .
</math>
  </td>
</tr>
</table>
</div>

===Finite Difference Representation of Radial Eigenfunction===
====Preamble====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tilde{C}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{3^2}{\tilde\xi^2}\biggl( 1 + \frac{\tilde\xi^2}{3} \biggr) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tilde{r}_\mathrm{edge}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} \biggl[ \frac{1}{ {\tilde\xi}^{2} }\biggl(1+\frac{\tilde\xi^2}{3}\biggr) \biggr] ^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\pi}{2^3\cdot 3}\biggr]^{1/2} \biggl[ \frac{\tilde{C} }{ 3^2 }\biggr] ^3
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\pi}{2^3\cdot 3^{13}}\biggr]^{1/2} {\tilde{C}}^3 \, .
</math>
  </td>
</tr>
</table>
</div>
Note that when <math>~\tilde\xi = 3</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tilde{C}_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\tilde{r}_{e3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{2^9\pi }{3^{13} } \biggr]^{1 / 2}
\, .
</math>
  </td>
</tr>
</table>
</div>


====Conjectures====
A first-cut examination of the structure of the radial eigenfunction associated with  the <math>~P_\mathrm{max}</math> turning point is given by simply subtracting one <math>~r_\xi(m_\xi)</math> profile from another ''at the same applied external pressure.''  (The ''specific'' choices of the two appropriate values of <math>~\tilde\xi</math> are discussed in the subsection titled, [[#Configuration_Pairing|"Configuration Pairing"]], which follows.)  The answer ''appears'' to be,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[ \Delta r_\xi ]_{21} = 
[r_\xi (m_\xi)]_2 - [r_\xi (m_\xi)]_1
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[\tilde{r}_\mathrm{edge}]_2
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C}_2 - 3 m_\xi^{2/3}}\biggr]^{1/2}
-
[\tilde{r}_\mathrm{edge}]_1
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C}_1 - 3 m_\xi^{2/3}}\biggr]^{1/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\pi}{2^3\cdot 3^{13}}\biggr]^{1 / 2} \biggl\{ {\tilde{C}_2}^3
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C}_2 - 3 m_\xi^{2/3}}\biggr]^{1/2}
-
 {\tilde{C}_1}^3
\biggl[\frac{3^2m_\xi^{2/3}}{\tilde{C}_1 - 3 m_\xi^{2/3}}\biggr]^{1 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\pi}{2^3\cdot 3^{13}}\biggr]^{1 / 2} \biggl\{ {\tilde{C}_2}^3
\biggl[\frac{3^2m_\xi^{2/3}/\tilde{C}_1 }{\tilde{C}_2/\tilde{C}_1 - 3 m_\xi^{2/3}/\tilde{C}_1 }\biggr]^{1 / 2}
-
 {\tilde{C}_1}^3
\biggl[\frac{3^2m_\xi^{2/3}/\tilde{C}_1 }{1 - 3 m_\xi^{2/3}/\tilde{C}_1 }\biggr]^{1 / 2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\pi}{2^3\cdot 3^{13}}\biggr]^{1 / 2} \biggl\{ {\tilde{C}_2}^3
\biggl[\frac{3/u }{\tilde{C}_2/\tilde{C}_1 - 1/u }\biggr]^{1 / 2}
-
 {\tilde{C}_1}^3
\biggl[\frac{3/u }{1 - 1/u }\biggr]^{1 / 2} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 {\tilde{C}_1}^3 \biggl[ \frac{\pi}{2^3\cdot 3^{12}}\biggr]^{1 / 2} \biggl\{ k_{21}^3
\biggl[\frac{1 }{k_{21} u - 1 }\biggr]^{1 / 2}
-
\biggl[\frac{1 }{u - 1 }\biggr]^{1 / 2} \biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~u \equiv \tilde{C}_1/(3 m_\xi^{2/3}) \, ,</math><p></p>

<math>~k_{21} \equiv  \frac{ \tilde{C}_2 }{ \tilde{C}_1 } \, .</math><p></p>
</div>

After examining the form of this last expression, it is clear that we can also write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~[r_\xi (m_\xi)]_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 {\tilde{C}_1}^3 \biggl[ \frac{\pi}{2^3\cdot 3^{12}}\biggr]^{1 / 2} 
\biggl[\frac{1 }{u - 1 }\biggr]^{1 / 2}  \, ,
</math>
  </td>
</tr>
</table>
</div>
in which case, the lopsided ''fractional'' eigenfunction takes the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{[ \Delta r_\xi ]_{21} }{[r_\xi (m_\xi)]_1}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
k_{21}^3 \biggl[\frac{u - 1 }{k_{21} u - 1 }\biggr]^{1 / 2} - 1 \, .
</math>
  </td>
</tr>
</table>
</div>
And the ''centered'' fractional eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{[ \Delta r_\xi ]_{32} }{[r_\xi (m_\xi)]_1}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ k_{31}^3 \biggl[\frac{u - 1 }{k_{31} u - 1 }\biggr]^{1 / 2} - 1 \biggr\}
-
\biggl\{ k_{21}^3 \biggl[\frac{u - 1 }{k_{21} u - 1 }\biggr]^{1 / 2} - 1 \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
k_{31}^3 \biggl[\frac{u - 1 }{k_{31} u - 1 }\biggr]^{1 / 2}  -  k_{21}^3 \biggl[\frac{u - 1 }{k_{21} u - 1 }\biggr]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
</div>

===Configuration Pairing===
====Setup====

Now, let's identify two <math>n=5</math> equilibrium states that sit very near the <math>P_\mathrm{max}</math> turning point on the two separate branches of the equilibrium sequence and that have identical external pressures.  We know [[SSC/FreeEnergy/Powerpoint#Case_M_Equilibrium_Conditions|from separate discussions]] that, in both cases,

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

<tr>
  <td align="right">
<math>
\frac{P_\mathrm{e}}{P_\mathrm{norm}}
</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{(n+1)^3}{4\pi}\biggr]^{(n+1)/(n-3)}
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{2\cdot 3^3}{\pi}\biggr]^{3}
\tilde\xi^{12} \tilde\theta_n^{6}( - \tilde\theta' )^{6} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~ \biggl[ \frac{\pi}{2\cdot 3^3}\biggr]^{1/2}\biggl[\frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]^{1/6}</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
\tilde\xi^{2} \tilde\theta_n( - \tilde\theta' )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
3\ell^2 (1+\ell^2)^{-1/2} \frac{\ell}{\sqrt{3}} (1+\ell^2)^{-3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\sqrt{3}\ell^3  (1+\ell^2)^{-2}
</math>
  </td>
</tr>
</table>
</div>
We can therefore write,

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

<tr>
  <td align="right">
<math>(1+\ell^2)^{2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
p_0\ell^3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~\ell^4 - p_0\ell^3 + 2\ell^2 + 1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0 \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>p_0 \equiv \biggl[ \frac{2\cdot 3^4}{\pi}\biggr]^{1/2}\biggl[\frac{P_\mathrm{e}}{P_\mathrm{norm}}\biggr]^{-1/6}</math>
</div>

So, in essence, we seek two real roots of this quartic equation that are near <math>P_\mathrm{max}</math>, that is, that are near <math>\ell = \sqrt{3}</math> &#8212; where <math>p_0 = (2^8/3^3)^{1/2}</math>.


Because we are hunting for equilibrium configurations near <math>P_\mathrm{max}</math>, it makes sense to make the variable substitution,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\ell</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>\rightarrow</math> &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>\sqrt{3}(1+\epsilon) \, ,</math>
  </td>
</tr>
</table>
</div>
and look for pairs of values, <math>~\epsilon_\pm</math> (both real, but one positive and the other negative).

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>3^2(1+\epsilon)^4 - 3^{3/2}p_0(1+\epsilon)^3 + 6(1+\epsilon)^2 + 1</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
3^2\biggl[1 + 4\epsilon + 6\epsilon^2 + 4\epsilon^3 + \epsilon^4\biggr] 
- 3^{3/2}p_0\biggl[ 1+ 3\epsilon + 3\epsilon^2 + \epsilon^3  )\biggr] 
+ 6\biggl[1 + 2\epsilon + \epsilon^2\biggr] 
+ 1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
\epsilon^4 \biggl[ 9 \biggr]
+\epsilon^3 \biggl[ 36 - 3^{3/2}p_0 \biggr]
+\epsilon^2 \biggl[ 54 - 3^{5/2}p_0 + 6 \biggr]
+\epsilon \biggl[36 - 3^{5/2}p_0 + 12 \biggr]
+(16- 3^{3/2}p_0)\, .
</math>
  </td>
</tr>
</table>
</div>
And, because we will only be examining values of the external pressure that are ''less'' than <math>P_\mathrm{max}</math>, and we know that ''at'' the point of maximum pressure, <math>3^{3/2}p_0 = 16</math>, it makes sense to make the substitution,
<div align="center">
<math>3^{3/2}p_0  ~~~\rightarrow ~~~ (16+\delta) \, .</math>
</div>

Hence, for a fixed choice of <math>\delta</math> (reasonably small, and positive), we seek two real roots (one positive and the other negative) of the quartic relation,

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

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
9\epsilon^4 
+\epsilon^3 \biggl[ 36 - (16+\delta ) \biggr]
+\epsilon^2 \biggl[ 60 - 3(16+\delta ) \biggr]
+\epsilon \biggl[48 - 3(16+\delta ) \biggr]
-\delta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=~</math>
  </td>
  <td align="left">
<math>
9\epsilon^4 
+\epsilon^3 (20-\delta  )
+\epsilon^2 ( 12 - 3\delta  )
-\epsilon (3\delta )
-\delta \, .
</math>
  </td>
</tr>
</table>
</div>

What are the reasonable limits on <math>~\delta</math>?  Well, first note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>p_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{(1+\ell^2)^2}{\ell^3}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 16+\delta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^{3/2}\biggl[\frac{(1+\ell^2)^2}{\ell^3}\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \delta </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^{3/2}\biggl[\frac{(1+\ell^2)^2}{\ell^3}\biggr] - 2^4 \, .</math>
  </td>
</tr>
</table>
</div>
Now, according to our [[SSC/FreeEnergy/PowerPoint#Case_M_Equilibrium_Conditions|accompanying discussion]], the relevant limits on <math>\ell</math> are <math>\sqrt{3}</math> (set by the maximum pressure turning point) and 2.223175 (set by the transition to dynamical instability).  The corresponding values of <math>\delta</math> are: &nbsp; 0 (by design) and 0.69938.

====Quartic Solution====
Here, we will draw from the [https://en.wikipedia.org/wiki/Quartic_function Wikipedia discussion of the quartic function].  The generic form is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>ax^4 + bx^3 + cx^2 + dx + e \,.</math>
  </td>
</tr>
</table>
</div>
Relating this to our specific quartic function, we should ultimately make the following assignments:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>9</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>b</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>20 - \delta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>12 -  3\delta  = 3(4-\delta )</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>d</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-3 \delta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>e</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-\delta</math>
  </td>
</tr>
</table>
</div>

We need to evaluate the following expressions:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>p</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{8ac-3b^2}{8a^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2^3 \cdot 3^3(4-\delta )-3(20-\delta )^2}{2^3\cdot 3^4}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>q</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{b^3 - 4abc + 8a^2d}{8a^3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{(20 - \delta )^3 - 2^2\cdot 3^3(4-\delta ) (20 - \delta ) - 2^3\cdot 3^5\delta }{2^3 \cdot 3^6}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Delta_0</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>c^2 - 3bd + 12ae</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^2(4-\delta )^2 + 3^2\delta (20 - \delta ) - 2^2\cdot 3^3\delta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>144</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Delta_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>2c^3 - 9bcd + 27b^2e+27ad^2 - 72ace</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\cdot 3^3(4-\delta)^3 + 3^4(20-\delta)(4-\delta)\delta - 3^3(20-\delta)^2 \delta+3^7\delta^2 + 2^3\cdot 3^5(4-\delta)\delta</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^3(128 + 32\delta + \delta^2) \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
Note:  &nbsp; <math>\Delta_1^2 - 4\Delta_0^3</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>3^6(2^{13}\delta + 2^8\cdot 5 \delta^2 + 2^6\delta^3 + \delta^4) \, .</math>
  </td>
</tr>
</table>
</div>

For a given value of <math>\delta</math>, then, the pair of real roots is:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\epsilon_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{b}{4a} + S \pm \frac{1}{2}\biggl[ -4S^2 - 2p - \frac{q}{S} \biggr]^{1/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>S</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{1}{2}\biggl[- \frac{2p}{3} + \frac{1}{3a}\biggl(Q + \frac{\Delta_0}{Q}\biggr)  \biggr]^{1/2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>Q</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\Delta_1 + \sqrt{\Delta_1^2 - 4\Delta_0^3}}{2} \biggr]^{1/3} \, .
</math>
  </td>
</tr>
</table>
</div>

We have used an Excel spreadsheet to evaluate these expressions.  The following table identifies <math>\epsilon_\pm</math> pairs (the middle two columns of numbers) for twenty different values of the external pressure; more specifically, for twenty values of <math>0 \le \delta \le 0.69938</math>, equally spaced between the two limits.  The corresponding pairs of <math>\tilde\xi_\pm</math> are also listed (rightmost pair of columns).

<div align="center">
<table border="0" align="center"><tr><th align="center">
<font size="+1">Table 1</font></th></tr>
<tr><td align="center">

<table border="1" align="center">
<tr><th align="center">
<font size="+1">Sets of Paired Models from Quartic Solution</font>
</th></tr>
<tr><td align="left">
<pre>
P_e/P_norm   delta	  eps_+   eps_-	         xi_+ 	 xi_-
160.867	    0.00000	0.00000  0.00000	3.00000	3.00000
158.664	    0.03681	0.05747	-0.05338	3.17241	2.83986
156.497	    0.07362	0.08253	-0.07435	3.24759	2.77695
154.363	    0.11043	0.10227	-0.09000	3.30681	2.72999
152.264	    0.14724	0.11927	-0.10291	3.35781	2.69127
150.198	    0.18405	0.13452	-0.11407	3.40355	2.65780
148.164	    0.22086	0.14852	-0.12398	3.44556	2.62805
146.163	    0.25767	0.16159	-0.13296	3.48476	2.60111
144.193	    0.29448	0.17391	-0.14120	3.52174	2.57640
142.254	    0.33129	0.18563	-0.14883	3.55690	2.55351
140.345	    0.36809	0.19685	-0.15596	3.59055	2.53212
138.466	    0.40490	0.20764	-0.16266	3.62291	2.51203
136.617	    0.44171	0.21805	-0.16898	3.65415	2.49305
134.796	    0.47852	0.22814	-0.17498	3.68442	2.47505
133.003	    0.51533	0.23794	-0.18070	3.71382	2.45791
131.239	    0.55214	0.24748	-0.18615	3.74245	2.44154
129.501	    0.58895	0.25680	-0.19138	3.77039	2.42586
127.790	    0.62576	0.26590	-0.19640	3.79770	2.41081
126.106	    0.66257	0.27481	-0.20122	3.82444	2.39634
124.447	    0.69938	0.28355	-0.20587	3.85065	2.38238
</pre>
</td></tr>
</table>

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

===Two Example Eigenfunctions===

The following figure is fundamentally a reproduction of [[SSC/FreeEnergy/Powerpoint#Figure3|Figure 3 from an accompanying discussion]].  It presents the "Case M" equilibrium sequence from both an order-of-magnitude analysis (marked by light-blue squares) and a detailed force-balance analysis (light-green triangles).  The dark green circular dot identifies the configuration at the pressure maximum of the sequence &#8212; <math>P_\mathrm{max}/P_\mathrm{norm} = 160.867</math> &#8212; and the red circular dot identifies the location along the sequence where the transition from stable to dynamically unstable configurations occurs  &#8212; <math>P_e/P_\mathrm{norm} = 124.447</math>.  (All pressures have been normalized to <math>P_\mathrm{max}</math> in the figure.)

<div align="center">
<table border="0" align="center">
<tr><th align="center"><font size="+1">Figure 1</font></th></tr>
<tr><td align="center">
[[File:CaseMpairings0.png|400px|center|Case M equilibrium sequences]]
</td></tr>
</table>
</div>


At any <math>P_e</math> between these two limiting values, a pair of ''stable'' equilibrium configurations exist; approximately twenty example pairings are listed in Table 1.  The horizontal, black dashed line in the figure has been drawn at <math>P_e/P_\mathrm{norm} = 158.664</math>.  The pair of equilibrium configurations associated with this pressure is identified graphically by the two points at which this dashed line intersects the detailed force-balance equilibrium sequence; as is detailed in the second row of Table 1, the configurations correspond to models having <math>\tilde\xi = 2.83986</math> (right intersection) and <math>\tilde\xi = 3.17241</math> (left intersection).  The left-hand panel of Figure 2 shows how the Lagrangian radial coordinate varies with mass, <math>r_\xi(m_\xi)</math>, throughout the interior of these two equilibrium configurations (the locus of green and orange dots, respectively); for reference, the profile of the configuration at <math>P_\mathrm{max}</math> is presented as well (locus of black dots). The right-hand panel of Figure 2 shows the same paired configuration profiles, but ''relative'' to the profile of the configuration at <math>P_\mathrm{max}</math>.

<div align="center">
<table border="0" align="center">
<tr><th align="center"><font size="+1">Figure 2</font></th></tr>
<tr><td align="center">
[[File:CaseMeigen2.png|600px|center|Case M eigenfunction#1]]
</td></tr>
</table>
</div>


The horizontal, black dot-dash line in Figure 1 has been drawn at <math>P_e/P_\mathrm{norm} = 124.447</math>.  The pair of equilibrium configurations associated with this pressure is identified graphically by the two points at which this dot-dash line intersects the detailed force-balance equilibrium sequence; as is detailed in the last row of Table 1, the configurations correspond to models having <math>\tilde\xi = 2.83986</math> (right intersection) and  <math>\tilde\xi = 3.17241</math> (left intersection), which is the configuration that marks the onset of a dynamical instability.  The left-hand panel of Figure 3 shows how the Lagrangian radial coordinate varies with mass, <math>r_\xi(m_\xi)</math>, throughout the interior of these two ''paired'' equilibrium configurations (the locus of green and orange dots, respectively); again, for reference, the profile of the configuration at <math>P_\mathrm{max}</math> is presented as well (locus of black dots). The right-hand panel of Figure 3 shows the same paired configuration profiles, but ''relative'' to the profile of the configuration at <math>P_\mathrm{max}</math>.


<div align="center">
<table border="0" align="center">
<tr><th align="center"><font size="+1">Figure 3</font></th></tr>
<tr><td align="center">
[[File:CaseMeigen1.png|600px|center|Case M eigenfunction#2]]
</td></tr>
</table>
</div>

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