Editing Appendix/Ramblings/NonlinarOscillation (section)

===Identifying Equal-Mass Pairs===
====Determining the Relevant Values of <math>~\tilde\xi</math>====

<span id="n5MassRadiusSequence">
<table border="0" cellpadding="5" align="right">
<tr>
  <th align="center" colspan="1">Figure 1: &nbsp; n = 5 Mass-Radius Sequence</th>
</tr>
<tr>
<td align="right">[[File:N5MassRadius01.png|300px|n = 5 mass-radius equilibrium sequence]]</td>
</tr>
<tr>
  <th align="center" colspan="1">[[File:DataFileButton02.png|center|75px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/Workbook_n5.xlsx --- worksheet = Quartic]]</th>
</tr>
</table>
</span>
The mass-radius relationship for pressure-truncated, n = 5 polytropic configurations is displayed as a green solid curve, here on the right, in Figure 1. This sequence can be constructed either: (a) by choosing various values of the radius (between zero and the [[#extrema|above-specified maximum radius]]) and, for each choice, determining the two corresponding values of the equilibrium mass from the pair of [[#Roots_of_Quadratic_Equation|roots of the quadratic equation]]; or (b) by choosing various values of the mass (between zero and the [[#extrema|above-specified maximum mass]]) and, for each choice, using the [[#Roots_of_Quartic_Equation|two real roots of the quartic equation]] to determine the two corresponding values of the equilibrium radius.  The green curve shown in Figure 1 is identical to the orange-dashed curve, labeled n = 5, that is nested among six other polytropic equilibrium sequences in [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations|the righthand panel of Figure 3 in an accompanying discussion]].

Here we are interested in comparing the relative distribution of mass inside various pairs of models that have identical total masses.  We therefore will focus on method "b".  Specifically, given any value of the mass, <math>~M/M_\mathrm{SWS} < m_\mathrm{max}</math>, the roots of the quartic equation will give us the equilibrium radii of the two configurations that have the same, specified mass.  As examples, the second column of Table 1 lists ten separate values of the normalized mass that lie within the region of parameter space that is identified by the black-dashed rectangle drawn in Figure 1; Column three lists the corresponding value of <math>~\mu</math>; and columns four and five of Table 1 give values of the corresponding pair of equilibrium radii, <math>~\chi_\pm</math>. 

In each case, from these two values of the dimensionless radius, we can, in turn, determine the corresponding pair of values of <math>\ell_\pm</math> via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\chi_\pm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3 \cdot 5}{4\pi} \biggr)^{1/2} \ell_\pm (1+\ell_\pm^2)^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\ell_\pm^2-\biggl( \frac{3 \cdot 5}{4\pi \chi^2_\pm} \biggr)^{1/2} \ell_\pm +1 \, .
</math>
  </td>
</tr>
</table>
</div>
But this is a quadratic equation, meaning that for <math>\chi_+</math> there are two viable roots for <math>\ell_+</math>, and for <math>\chi_-</math> there are two viable roots for <math>\ell_-</math>.  We will deal with this by referring to the "plus" root as the "high" value, and by referring to the "minus" root as the "low" value.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\xi_\pm\biggr|_\mathrm{high} = \sqrt{3} \ell_\pm\biggr|_\mathrm{high}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sqrt{3} \biggl( \frac{3 \cdot 5}{2^4\pi \chi^2_\pm} \biggr)^{1/2} \biggl[ 1 + \biggl( 1 - \frac{2^4\pi \chi^2_\pm}{3 \cdot 5} \biggr)^{1 / 2} \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\xi_\pm\biggr|_\mathrm{low} = \sqrt{3} \ell_\pm\biggr|_\mathrm{low}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\sqrt{3} \biggl( \frac{3 \cdot 5}{2^4\pi \chi^2_\pm} \biggr)^{1/2} \biggl[ 1 - \biggl( 1 - \frac{2^4\pi \chi^2_\pm}{3 \cdot 5} \biggr)^{1 / 2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
This seems to work because, if we plug in a single value for <math>\chi_\pm</math> &#8212; for example, the degenerate case of <math>\chi_\pm = r_\mathrm{crit}</math> &#8212; we get the pair of values of <math>\xi</math> along the equilibrium sequence where the equilibrium radius has this selected value.  Specifically, when <math>\chi_\pm = r_\mathrm{crit}</math>, we find that, <math>\xi_\mathrm{high} = 3</math> and <math>\xi_\mathrm{low} = 1</math>.  Given that we are particularly interested in examining the region of parameter space that lies near the marginally unstable case &#8212; as identified by the black-dashed rectangle drawn in Figure 1 &#8212; columns six and seven of Table 1 list only values of <math>\xi_\pm</math> that correspond to the "high" roots.

<span id="Table1">&nbsp;</span>
<table border="1" align="center" cellpadding="5">
<tr>
  <th align="center" colspan="10">[[File:DataFileButton02.png|right|75px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/Workbook_n5.xlsx --- worksheet = Quartic]]<br />Table 1: &nbsp; Selected Pairings</th>
</tr>
<tr>
  <td align="center" rowspan="2">
Pairing (N)
  </td>
  <td align="center" rowspan="2">
<math>m \equiv \frac{M}{M_\mathrm{SWS}}</math>
  </td>
  <td align="center" rowspan="2">
<math>\mu \equiv \biggl[ 1 - \biggl( \frac{m}{m_\mathrm{max}} \biggr)^2 \biggr]^{1 / 2}</math>
  </td>
  <td align="center" rowspan="2">
<math>\chi_+  = \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)_+</math>
  </td>
  <td align="center" rowspan="2">
<math>\chi_- = \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)_-</math>
  </td>
  <td align="center" colspan="2">"high" roots</td>
  <td align="center" rowspan="2">
<math>\Delta{\tilde{C}}_2</math>
  </td>
  <td align="center" rowspan="2">
<math>\Delta{\tilde{C}}_1</math>
  </td>
  <td align="center" rowspan="2">
% Profile Difference<br /><math>\frac{1}{2}\biggl(\chi_+ - \chi_-\biggr) \biggl[ \frac{2^6 \pi}{3^2\cdot 5} \biggr]^{1 / 2}</math>
  </td>
</tr>
<tr>
   <td align="center" rowspan="1">
<math>\xi_+</math>
  </td>
  <td align="center" rowspan="1">
<math>\xi_-</math>
  </td> 
</tr>
<tr>
  <td align="center">
<b>A (11)</b>
  </td>
  <td align="center">
<math>\biggl( \frac{3^4\cdot 5^3}{2^{10} \pi} \biggr)^{1 / 2}</math> <p></p><font size="-1">&nbsp;</font>
  </td>
  <td align="center">
<math>0</math> <p></p><font size="-1">&nbsp;</font>
  </td>
  <td align="center" colspan="2">
<math>\biggl( \frac{3^2\cdot 5}{2^6 \pi} \biggr)^{1 / 2}</math><p></p><font size="-1">(degenerate)</font> 
  </td>
  <td align="center" colspan="2">
<math>3</math><p></p><p></p><font size="-1">(degenerate)</font>
  </td>
  <td align="center" colspan="2">
<math>0</math><p></p><p></p><font size="-1">(degenerate)</font>
  </td>
  <td align="center" colspan="1">
<math>0</math>
  </td>
</tr>
<tr>
  <td align="center">
<b>B (12)</b>
  </td>
  <td align="center">
1.7696424
  </td>
  <td align="center">
0.070666
  </td>
  <td align="center" colspan="1">
0.486212
  </td>
  <td align="center" colspan="1">
0.458911
  </td>
  <td align="center" colspan="1">
2.833124
  </td>
  <td align="center" colspan="1">
3.180242
  </td>
  <td align="center" colspan="1">
0.121273
  </td>
  <td align="center" colspan="1">
-0.110139
  </td>
  <td align="center" colspan="1">
2.9%
  </td>
</tr>
<tr>
  <td align="center">
<b>C (14)</b>
  </td>
  <td align="center">
1.7607720
  </td>
  <td align="center">
0.122245
  </td>
  <td align="center" colspan="1">
0.495129
  </td>
  <td align="center" colspan="1">
0.447886
  </td>
  <td align="center" colspan="1">
2.718303
  </td>
  <td align="center" colspan="1">
3.321996
  </td>
  <td align="center" colspan="1">
0.217999
  </td>
  <td align="center" colspan="1">
-0.184462
  </td>
  <td align="center" colspan="1">
5.0%
  </td>
</tr>
<tr>
  <td align="center">
<b>D (16)</b>
  </td>
  <td align="center">
1.7519016
  </td>
  <td align="center">
0.157619
  </td>
  <td align="center" colspan="1">
0.500918
  </td>
  <td align="center" colspan="1">
0.439985
  </td>
  <td align="center" colspan="1">
2.642460
  </td>
  <td align="center" colspan="1">
3.425043
  </td>
  <td align="center" colspan="1">
0.288919
  </td>
  <td align="center" colspan="1">
-0.232797
  </td>
  <td align="center" colspan="1">
6.4%
  </td>
</tr>
<tr>
  <td align="center">
<b>E (18)</b>
  </td>
  <td align="center">
1.7430312
  </td>
  <td align="center">
0.186263
  </td>
  <td align="center" colspan="1">
0.505407
  </td>
  <td align="center" colspan="1">
0.433378
  </td>
  <td align="center" colspan="1">
2.582586
  </td>
  <td align="center" colspan="1">
3.512395
  </td>
  <td align="center" colspan="1">
0.349376
  </td>
  <td align="center" colspan="1">
-0.270482
  </td>
  <td align="center" colspan="1">
7.6%
  </td>
</tr>
<tr>
  <td align="center">
<b>F (20)</b>
  </td>
  <td align="center">
1.7341608
  </td>
  <td align="center">
0.210935
  </td>
  <td align="center" colspan="1">
0.509128
  </td>
  <td align="center" colspan="1">
0.427533
  </td>
  <td align="center" colspan="1">
2.532015
  </td>
  <td align="center" colspan="1">
3.590722
  </td>
  <td align="center" colspan="1">
0.403816
  </td>
  <td align="center" colspan="1">
-0.301962
  </td>
  <td align="center" colspan="1">
8.6%
  </td>
</tr>
<tr>
  <td align="center">
<b>G (22)</b>
  </td>
  <td align="center">
1.7252904
  </td>
  <td align="center">
0.232903
  </td>
  <td align="center" colspan="1">
0.512327
  </td>
  <td align="center" colspan="1">
0.422206
  </td>
  <td align="center" colspan="1">
2.487708
  </td>
  <td align="center" colspan="1">
3.663068
  </td>
  <td align="center" colspan="1">
0.454266
  </td>
  <td align="center" colspan="1">
-0.329263
  </td>
  <td align="center" colspan="1">
9.5%
  </td>
</tr>
<tr>
  <td align="center">
<b>H (24)</b>
  </td>
  <td align="center">
1.7164200
  </td>
  <td align="center">
0.252871
  </td>
  <td align="center" colspan="1">
0.515138
  </td>
  <td align="center" colspan="1">
0.417261
  </td>
  <td align="center" colspan="1">
2.447976
  </td>
  <td align="center" colspan="1">
3.731126
  </td>
  <td align="center" colspan="1">
0.501855
  </td>
  <td align="center" colspan="1">
-0.353509
  </td>
  <td align="center" colspan="1">
10.3%
  </td>
</tr>
<tr>
  <td align="center">
<b>I (26)</b>
  </td>
  <td align="center">
1.7075496
  </td>
  <td align="center">
0.271282
  </td>
  <td align="center" colspan="1">
0.517648
  </td>
  <td align="center" colspan="1">
0.412612
  </td>
  <td align="center" colspan="1">
2.411770
  </td>
  <td align="center" colspan="1">
3.795950
  </td>
  <td align="center" colspan="1">
0.547286
  </td>
  <td align="center" colspan="1">
-0.375401
  </td>
  <td align="center" colspan="1">
11.1%
  </td>
</tr>
<tr>
  <td align="center">
<b>J (28)</b>
  </td>
  <td align="center">
1.6986793
  </td>
  <td align="center">
0.288433
  </td>
  <td align="center" colspan="1">
0.519913
  </td>
  <td align="center" colspan="1">
0.408203
  </td>
  <td align="center" colspan="1">
2.378383
  </td>
  <td align="center" colspan="1">
3.858252
  </td>
  <td align="center" colspan="1">
0.591032
  </td>
  <td align="center" colspan="1">
-0.395409
  </td>
  <td align="center" colspan="1">
11.8%
  </td>
</tr>
<tr>
  <td align="left" colspan="10">
NOTE:&nbsp;  The mass of a given configuration pair has been specified according to the expression,
<div align="center">
<math>\frac{M}{M_\mathrm{SWS}} = \biggl( \frac{3^4\cdot 5^3}{2^{10} \pi} \biggr)^{1 / 2}\biggl[1 - \frac{(N-11)}{400}\biggr] \, ,</math>
</div>
where, N is the integer that appears inside the parentheses in the first column of this table.
  </td>
</tr>
</table>

====Approximation Near the Maximum Mass====
We can rewrite the expression for the "high" roots of <math>~\xi_\pm</math> as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\epsilon_\pm \equiv 3 - \xi_\pm\biggr|_\mathrm{high} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 - \sqrt{3} \biggl( \frac{3 \cdot 5}{2^4\pi \chi^2_\pm} \biggr)^{1/2} \biggl[ 1 + \biggl( 1 - \frac{2^4\pi \chi^2_\pm}{3 \cdot 5} \biggr)^{1 / 2} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~3 - 2 \biggl( \frac{\chi_\pm}{r_\mathrm{crit}} \biggr)^{-1} \biggl\{ 
1 + \biggl[ 1 - \frac{3}{4} \biggl( \frac{\chi_\pm}{r_\mathrm{crit}} \biggr)^2 \biggr]^{1 / 2} 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
As desired, <math>~\epsilon_\pm \rightarrow 0</math> when <math>~\chi_\pm/r_\mathrm{crit} \rightarrow 1</math>.  Ultimately, we expect to find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta C_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde{C}_2 - 4</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{9 - (3-\epsilon_+)^2}{ (3-\epsilon_+)^2 } 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6\epsilon_+ - \epsilon_+^2}{ 9 -6\epsilon_+ + \epsilon_+^2 } \, .
</math>
  </td>
</tr>
</table>
</div>
And, similarly,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta C_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde{C}_1 - 4</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{6\epsilon_- - \epsilon_-^2}{ 9 -6\epsilon_- + \epsilon_-^2 } \, .
</math>
  </td>
</tr>
</table>
</div>

Now, let's work through power-series expansions for each.  For <math>~\epsilon_+</math> we need,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{\chi_+}{r_\mathrm{crit}} \biggr)^{-1}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\biggl\{
1 ~+ \biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2}{3^2}\biggr) \mu^2
~+~ \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) \biggr]
\biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~- 
\biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ 
\biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
-
\biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) \biggr]^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+
\biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) \biggr]^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl(\frac{2}{3^2}\biggr) \mu^2
- \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
+ \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ 
\biggl[ \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  \biggr]^2
\biggl\{ 1 - \biggl[
\biggl(\frac{2^3}{3^3}\biggr)^{1 / 2} \mu 
- \biggl( \frac{7^2}{2^6 \cdot 3^6} \biggr)^{1 / 2}\mu^2 
+ \biggl( \frac{2^5 \cdot 5^2}{3^{9} } \biggr)^{1 / 2} \mu^3  
+ \mathcal{O}(\mu^4)\biggr] 
\biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
-
\biggl[ \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  \biggr]^3
\biggl\{1 - \biggl[
\biggl(\frac{2^3}{3^3}\biggr)^{1 / 2} \mu 
- \biggl( \frac{7^2}{2^6 \cdot 3^6} \biggr)^{1 / 2}\mu^2 
+ \biggl( \frac{2^5 \cdot 5^2}{3^{9} } \biggr)^{1 / 2} \mu^3  
+ \mathcal{O}(\mu^4)\biggr]
\biggr\}^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+
\biggl[ \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  \biggr]^4
\biggl\{1 - \biggl[
\biggl(\frac{2^3}{3^3}\biggr)^{1 / 2} \mu 
- \biggl( \frac{7^2}{2^6 \cdot 3^6} \biggr)^{1 / 2}\mu^2 
+ \biggl( \frac{2^5 \cdot 5^2}{3^{9} } \biggr)^{1 / 2} \mu^3  
+ \mathcal{O}(\mu^4)\biggr] 
\biggr\}^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl(\frac{2}{3^2}\biggr) \mu^2
- \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
+ \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ 
\frac{\mu^2}{2 \cdot 3}  
\biggl\{ 1 - \biggl[
\biggl(\frac{2^3}{3^3}\biggr)^{1 / 2} \mu 
- \biggl( \frac{7^2}{2^6 \cdot 3^6} \biggr)^{1 / 2}\mu^2 
+ \mathcal{O}(\mu^3)\biggr] 
\biggr\}^2
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
-\biggl( \frac{1}{2 \cdot 3}\biggr)^{3 / 2}\mu^3
\biggl\{1 - \biggl[
\biggl(\frac{2^3}{3^3}\biggr)^{1 / 2} \mu 
+ \mathcal{O}(\mu^2)\biggr]
\biggr\}^3
+\biggl( \frac{1}{2 \cdot 3}\biggr)^{2}\mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl(\frac{2}{3^2}\biggr) \mu^2
- \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
+ \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ 
\frac{\mu^2}{2 \cdot 3}  
\biggl\{ 1 - 
\biggl(\frac{2^5}{3^3}\biggr)^{1 / 2} \mu 
+ \biggl[ \frac{7 + 2^3 \cdot 3^2}{2^2\cdot 3^3} \biggr] \mu^2 
\biggr\}
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
-\biggl( \frac{1}{2 \cdot 3}\biggr)^{3 / 2}\mu^3
+ 3\biggl(\frac{2^3}{3}\biggr)^{1 / 2}\biggl( \frac{1}{2 \cdot 3}\biggr)^{3 / 2}\mu^4
+\biggl( \frac{1}{2 \cdot 3}\biggr)^{2}\mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl[ \biggl(\frac{2}{3^2}\biggr) +   \frac{1}{2 \cdot 3} \biggr] \mu^2
- \biggl[\biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2} + \biggl(\frac{2^3}{3^5}\biggr)^{1 / 2}  
+ \biggl( \frac{1}{2 \cdot 3}\biggr)^{3 / 2} \biggr] \mu^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ \biggl[ \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr) + \biggl( \frac{79}{2^3 \cdot 3^4} \biggr)  
+ \biggl( \frac{5}{2^2 \cdot 3^2}\biggr)\biggr] \mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl( \frac{7}{2 \cdot 3^2} \biggr) \mu^2
- \biggl(\frac{139^2}{2^7 \cdot 3^7}\biggr)^{1 / 2} \mu^3
+ \biggl( \frac{23\cdot 29}{2^3 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>
</table>
</div>

And we need,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{\chi_+}{r_\mathrm{crit}} \biggr)^{2}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\biggl\{
1 ~+ \biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2}{3^2}\biggr) \mu^2
~+~ \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) \biggr]
\biggr\}^{2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~+ 2 
\biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ 
\biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^7 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^2 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~+ 
\biggl( \frac{2}{3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2^2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^5 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^3 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ \biggl[\biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu\biggr]^2
\biggl\{1  + \biggl[
- \biggl(\frac{2^3}{3^3}\biggr)^{1 / 2} \mu
+ \biggl( \frac{7^2}{2^6 \cdot 3^6} \biggr)^{1 / 2}\mu^2 
- \biggl( \frac{2^5 \cdot 5^2}{3^{9} } \biggr)^{1 / 2} \mu^3  
+ \mathcal{O}(\mu^4) \biggr] \biggr\}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~+ 
\biggl( \frac{2}{3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2^2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^5 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^3 \cdot 5}{3^5 } \biggr)\mu^4  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ \biggl( \frac{\mu^2}{2 \cdot 3}\biggr)
\biggl[ 1  - \biggl(\frac{2^5}{3^3}\biggr)^{1 / 2} \mu
+ \biggl( \frac{7^2}{2^4 \cdot 3^6} \biggr)^{1 / 2}\mu^2 
+ \biggl(\frac{2^3}{3^3}\biggr) \mu^2 \biggr]
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~+ 
\biggl( \frac{2}{3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{2^2}{3^2}\biggr) \mu^2
+ \biggl( \frac{7^2}{2^5 \cdot 3^7} \biggr)^{1 / 2}\mu^3 
- \biggl( \frac{2^3 \cdot 5}{3^5 } \biggr)\mu^4  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ 
\biggl( \frac{\mu^2}{2 \cdot 3}\biggr)  - \biggl( \frac{1}{2 \cdot 3}\biggr)\biggl(\frac{2^5}{3^3}\biggr)^{1 / 2} \mu^3
+ \biggl( \frac{1}{2 \cdot 3}\biggr)\biggl[ \biggl( \frac{7^2}{2^4 \cdot 3^6} \biggr)^{1 / 2} 
+ \biggl(\frac{2^6}{3^6}\biggr)^{1 / 2} \biggr]\mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~+ 
\biggl( \frac{2}{3}\biggr)^{1 / 2}\mu  
+ \biggl[ \biggl( \frac{1}{2 \cdot 3}\biggr)  
- \biggl(\frac{2^2}{3^2}\biggr) \biggr] \mu^2
+ \biggl[ \biggl( \frac{7^2}{2^5 \cdot 3^7} \biggr)^{1 / 2} 
- \biggl( \frac{1}{2 \cdot 3}\biggr)\biggl(\frac{2^5}{3^3}\biggr)^{1 / 2} \biggr] \mu^3
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ \biggl( \frac{1}{2 \cdot 3}\biggr)\biggl[ \biggl( \frac{7^2}{2^4 \cdot 3^6} \biggr)^{1 / 2} 
+ \biggl(\frac{2^6}{3^6}\biggr)^{1 / 2} \biggr]\mu^4
- \biggl( \frac{2^3 \cdot 5}{3^5 } \biggr)\mu^4  
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~+ 
\biggl( \frac{2}{3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{5}{2\cdot 3^2}\biggr)\mu^2
- \biggl(\frac{41^2}{2^5 \cdot 3^7}\biggr)^{1 / 2}  \mu^3
- \biggl( \frac{203}{2^3 \cdot 3^5 } \biggr) \mu^4  
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\biggl[ 1 - \frac{3}{4} \biggl( \frac{\chi_+}{r_\mathrm{crit}} \biggr)^2 \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 1 - \frac{3}{4} 
\biggl[ 
1 ~+ 
\biggl( \frac{2}{3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{5}{2\cdot 3^2}\biggr)\mu^2
- \biggl(\frac{41^2}{2^5 \cdot 3^7}\biggr)^{1 / 2}  \mu^3
- \biggl( \frac{203}{2^3 \cdot 3^5 } \biggr) \mu^4  
+ \mathcal{O}(\mu^5) 
\biggr] 
\biggr\}^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl\{ 1-3 
\biggl[ 
\biggl( \frac{2}{3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{5}{2\cdot 3^2}\biggr)\mu^2
- \biggl(\frac{41^2}{2^5 \cdot 3^7}\biggr)^{1 / 2}  \mu^3
- \biggl( \frac{203}{2^3 \cdot 3^5 } \biggr) \mu^4  
+ \mathcal{O}(\mu^5) 
\biggr] 
\biggr\}^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl\{ 1- 
\biggl[ 
\biggl( 2\cdot 3\biggr)^{1 / 2}\mu  
- \biggl(\frac{5}{2\cdot 3}\biggr)\mu^2
- \biggl(\frac{41^2}{2^5 \cdot 3^5}\biggr)^{1 / 2}  \mu^3
- \biggl( \frac{203}{2^3 \cdot 3^4 } \biggr) \mu^4  
+ \mathcal{O}(\mu^5) 
\biggr] 
\biggr\}^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl\{ 1- \frac{1}{2}
\biggl[ 
\biggl( 2\cdot 3\biggr)^{1 / 2}\mu  
- \biggl(\frac{5}{2\cdot 3}\biggr)\mu^2
- \biggl(\frac{41^2}{2^5 \cdot 3^5}\biggr)^{1 / 2}  \mu^3
- \biggl( \frac{203}{2^3 \cdot 3^4 } \biggr) \mu^4  
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{2^3} \biggl[ 
\biggl( 2\cdot 3\biggr)^{1 / 2}\mu  
- \biggl(\frac{5}{2\cdot 3}\biggr)\mu^2
- \biggl(\frac{41^2}{2^5 \cdot 3^5}\biggr)^{1 / 2}  \mu^3
\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{2^4} \biggl[ 
\biggl( 2\cdot 3\biggr)^{1 / 2}\mu  
- \biggl(\frac{5}{2\cdot 3}\biggr)\mu^2
\biggr]^3
- \frac{5}{2^7} \biggl[ 
\biggl( 2\cdot 3\biggr)^{1 / 2}\mu  
\biggr]^4
+ \mathcal{O}(\mu^5) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl\{ 1-  
\biggl( \frac{3}{2}\biggr)^{1 / 2}\mu  
+ \biggl(\frac{5}{2^2 \cdot 3}\biggr)\mu^2
+ \biggl(\frac{41^2}{2^7 \cdot 3^5}\biggr)^{1 / 2}  \mu^3
+ \biggl( \frac{203}{2^4 \cdot 3^4 } \biggr) \mu^4  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl( \frac{3}{2^2}\biggr) \mu^2 \biggl[1  
- \biggl(\frac{5^2}{2^3\cdot 3^3}\biggr)^{1 / 2} \mu
- \biggl(\frac{41^2}{2^6 \cdot 3^6}\biggr)^{1 / 2}  \mu^2
\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl( \frac{3^3}{2^5}\biggr)^{1 / 2} \mu^3  \biggl[ 
1 - \biggl(\frac{5^2}{2^3\cdot 3^3}\biggr)^{1 / 2} \mu \biggr]^3
- \biggl( \frac{3^2\cdot 5}{2^5} \biggr) \mu^4  
+ \mathcal{O}(\mu^5) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl\{ 1
-  \biggl( \frac{3}{2}\biggr)^{1 / 2}\mu  
+ \biggl[   \biggl(\frac{5}{2^2 \cdot 3}\biggr) - \biggl( \frac{3}{2^2}\biggr)\biggr] \mu^2
+ \biggl[ \biggl(\frac{2^2 \cdot 3^4 \cdot 5^2}{2^7\cdot 3^5}\biggr)^{1 / 2} 
- \biggl( \frac{2^2 \cdot 3^8}{2^7 \cdot 3^5}\biggr)^{1 / 2} + \biggl(\frac{41^2}{2^7 \cdot 3^5}\biggr)^{1 / 2}  \biggr] \mu^3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ \biggl( \frac{3^2\cdot 57}{2^5 \cdot 3^4}\biggr) 
- \biggl( \frac{3^5\cdot 5}{2^5 \cdot 3^4} \biggr) 
+ \biggl( \frac{2\cdot 203}{2^5 \cdot 3^4 } \biggr) \biggr]\mu^4  
\biggr\}
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{ 1
-  \biggl( \frac{3}{2}\biggr)^{1 / 2}\mu  
- \biggl(\frac{1}{3}\biggr) \mu^2
- \biggl( \frac{31^2}{2^7 \cdot 3^5}\biggr)^{1 / 2}   \mu^3  
- \biggl( \frac{296}{2^5 \cdot 3^4} \biggr) \mu^4  
\biggr\}
+ \mathcal{O}(\mu^5) \, ,
</math>
  </td>
</tr>
</table>
</div>
and, finally,
 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\epsilon_+ </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~3 - 2 \biggl( \frac{\chi_+}{r_\mathrm{crit}} \biggr)^{-1} \biggl\{ 
1 + \biggl[ 1 - \frac{3}{4} \biggl( \frac{\chi_+}{r_\mathrm{crit}} \biggr)^2 \biggr]^{1 / 2} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~3 - 2 \biggl\{ 
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl( \frac{7}{2 \cdot 3^2} \biggr) \mu^2
- \biggl(\frac{139^2}{2^7 \cdot 3^7}\biggr)^{1 / 2} \mu^3
+ \biggl( \frac{23\cdot 29}{2^3 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) 
\biggr\} \biggl\{ 
1 + 
\frac{1}{2}\biggl[ 1
-  \biggl( \frac{3}{2}\biggr)^{1 / 2}\mu  
- \biggl(\frac{1}{3}\biggr) \mu^2
- \biggl( \frac{31^2}{2^7 \cdot 3^5}\biggr)^{1 / 2}   \mu^3  
- \biggl( \frac{296}{2^5 \cdot 3^4} \biggr) \mu^4  
+ \mathcal{O}(\mu^5) 
\biggr] 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~3 - 3 \biggl\{ 
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl( \frac{7}{2 \cdot 3^2} \biggr) \mu^2
- \biggl(\frac{139^2}{2^7 \cdot 3^7}\biggr)^{1 / 2} \mu^3
+ \biggl( \frac{23\cdot 29}{2^3 \cdot 3^5} \biggr) \mu^4
+ \mathcal{O}(\mu^5) 
\biggr\} \biggl\{ 1
-  \biggl( \frac{1}{2\cdot 3}\biggr)^{1 / 2}\mu  
- \biggl(\frac{1}{3^2}\biggr) \mu^2
- \biggl( \frac{31^2}{2^7 \cdot 3^7}\biggr)^{1 / 2}   \mu^3  
- \biggl( \frac{296}{2^5 \cdot 3^5} \biggr) \mu^4  
+ \mathcal{O}(\mu^5) 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~3 - 3 \biggl\{ 
\biggl[ 1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl( \frac{7}{2 \cdot 3^2} \biggr) \mu^2
- \biggl(\frac{139^2}{2^7 \cdot 3^7}\biggr)^{1 / 2} \mu^3
+ \biggl( \frac{23\cdot 29}{2^3 \cdot 3^5} \biggr) \mu^4
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~-  \biggl( \frac{1}{2\cdot 3}\biggr)^{1 / 2}\mu  
\biggl[ 
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl( \frac{7}{2 \cdot 3^2} \biggr) \mu^2
- \biggl(\frac{139^2}{2^7 \cdot 3^7}\biggr)^{1 / 2} \mu^3
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~- \biggl(\frac{1}{3^2}\biggr) \mu^2  
\biggl[ 
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl( \frac{7}{2 \cdot 3^2} \biggr) \mu^2
\biggr] 
- \biggl(\frac{31^2}{2^7 \cdot 3^7}\biggr)^{1 / 2} \mu^3
\biggl[ 
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
\biggr] 
- \biggl( \frac{296}{2^5 \cdot 3^5} \biggr) \mu^4  
\biggr\}
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\biggl( \frac{3}{2}\biggr)^{1 / 2}\mu  
- \biggl( \frac{7}{2 \cdot 3} \biggr) \mu^2
+ \biggl(\frac{139^2}{2^7 \cdot 3^5}\biggr)^{1 / 2} \mu^3
- \biggl( \frac{23\cdot 29}{2^3 \cdot 3^4} \biggr) \mu^4
+ \biggl( \frac{3}{2}\biggr)^{1 / 2}\mu  
\biggl[ 
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl( \frac{7}{2 \cdot 3^2} \biggr) \mu^2
- \biggl(\frac{139^2}{2^7 \cdot 3^7}\biggr)^{1 / 2} \mu^3
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~+ \biggl(\frac{1}{3}\biggr) \mu^2  
\biggl[ 
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl( \frac{7}{2 \cdot 3^2} \biggr) \mu^2
\biggr] 
+ \biggl(\frac{31^2}{2^7 \cdot 3^5}\biggr)^{1 / 2} \mu^3
\biggl[ 
1 ~- \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
\biggr] 
+ \biggl( \frac{296}{2^5 \cdot 3^4} \biggr) \mu^4  
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\biggl[\biggl( \frac{3}{2}\biggr)^{1 / 2}  
+  \biggl( \frac{3}{2}\biggr)^{1 / 2} \biggr] \mu
- \biggl[ \biggl( \frac{7}{2 \cdot 3} \biggr)
+ \biggl( \frac{1}{2}\biggr) \biggr] \mu^2  
+ \biggl[ \biggl(\frac{139^2}{2^7 \cdot 3^5}\biggr)^{1 / 2} 
+ \biggl( \frac{7^2}{2^3 \cdot 3^3} \biggr)^{1 / 2}\biggr] \mu^3
- \biggl[ \biggl( \frac{23\cdot 29}{2^3 \cdot 3^4} \biggr) 
+ \biggl(\frac{139^2}{2^8 \cdot 3^6}\biggr)^{1 / 2}\biggr] \mu^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+  \biggl(\frac{1}{3}\biggr) \mu^2 ~
+ \biggl[ \biggl(\frac{31^2}{2^7 \cdot 3^5}\biggr)^{1 / 2} 
- \biggl(\frac{1}{3}\biggr) \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2} \biggr] \mu^3  
+ \biggl[ \biggl(\frac{1}{3}\biggr) \biggl( \frac{7}{2 \cdot 3^2} \biggr) 
- \biggl(\frac{31^2}{2^7 \cdot 3^5}\biggr)^{1 / 2} \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}  
+ \biggl( \frac{296}{2^5 \cdot 3^4} \biggr) \biggr]\mu^4  
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
\biggl[\biggl( \frac{3}{2}\biggr)^{1 / 2}  
+  \biggl( \frac{3}{2}\biggr)^{1 / 2} \biggr] \mu
+  \biggl[ \biggl(\frac{1}{3}\biggr) -\biggl( \frac{7}{2 \cdot 3} \biggr)
- \biggl( \frac{1}{2}\biggr) \biggr] \mu^2  
+ \biggl[ \biggl(\frac{139^2}{2^7 \cdot 3^5}\biggr)^{1 / 2} 
+ \biggl( \frac{7^2}{2^3 \cdot 3^3} \biggr)^{1 / 2} + \biggl(\frac{31^2}{2^7 \cdot 3^5}\biggr)^{1 / 2} 
- \biggl(\frac{1}{3}\biggr) \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2} \biggr] \mu^3  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
 <td align="left">
<math>~
+ \biggl[ \biggl(\frac{1}{3}\biggr) \biggl( \frac{7}{2 \cdot 3^2} \biggr) 
- \biggl(\frac{31^2}{2^7 \cdot 3^5}\biggr)^{1 / 2} \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}  
+ \biggl( \frac{296}{2^5 \cdot 3^4} \biggr) -\biggl( \frac{23\cdot 29}{2^3 \cdot 3^4} \biggr) 
- \biggl(\frac{139^2}{2^8 \cdot 3^6}\biggr)^{1 / 2}\biggr] \mu^4
+ \mathcal{O}(\mu^5) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
+ \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{191}{2\cdot 3^4}\biggr) \mu^4
+ \mathcal{O}(\mu^5) \, .
</math>
  </td>
</tr>
</table>
</div>

<!-- FIRST ATTEMPT AT LOWEST ORDER APPROXIMATION
Expanding to leading order, here is a quick idea of what to expect.
<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>~3 - 2 \biggl( \frac{\chi_\pm}{r_\mathrm{crit}} \biggr)^{-1} \biggl\{ 
1 + \biggl[ 1 - \frac{3}{4} \biggl( \frac{\chi_\pm}{r_\mathrm{crit}} \biggr)^2 \biggr]^{1 / 2} 
\biggr\} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~3 - 2 \biggl[ 1 \mp \frac{\mu}{\sqrt{6}} \biggr] \biggl\{ 
1 + \biggl[ 1 - \frac{3}{4} \biggl( 1 \pm \frac{2\mu}{\sqrt{6}}\biggr) \biggr]^{1 / 2} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~3 - 2 \biggl[ 1 \mp \frac{\mu}{\sqrt{6}} \biggr] \biggl\{ 
1 + \biggl[ \frac{1}{4} \mp \frac{3}{2} \frac{\mu}{\sqrt{6}} \biggr]^{1 / 2} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~3 - 2 \biggl[ 1 \mp \frac{\mu}{\sqrt{6}} \biggr] \biggl\{ 
1 + \frac{1}{2}\biggl[ 1 \mp \frac{6\mu}{\sqrt{6}} \biggr]^{1 / 2} 
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~3 - 2 \biggl[ 1 \mp \frac{\mu}{\sqrt{6}} \biggr] \biggl[ 
\frac{3}{2} \mp \frac{3\mu}{2\sqrt{6}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~3 - 3 \biggl[ 1 \mp \frac{\mu}{\sqrt{6}} \biggr] \biggl[ 
1\mp \frac{\mu}{\sqrt{6}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~3 - 3 \biggl[ 1 \mp \frac{2\mu}{\sqrt{6}} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\pm \sqrt{6}\mu \, . 
</math>
  </td>
</tr>
</table>
</div>
END FIRST STEP AT POWER-SERIES EXPANSION -->

Similarly we have determined that, for <math>~\epsilon_-</math>,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{\chi_-}{r_\mathrm{crit}} \biggr)^{-1}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~+ \biggl( \frac{1}{2 \cdot 3}\biggr)^{1 / 2}\mu  
+ \biggl[ \frac{7}{2 \cdot 3^2} \biggr] \mu^2
+ \biggl(\frac{139^2}{2^7 \cdot 3^7}\biggr)^{1 / 2} \mu^3
+ \biggl[ \frac{23\cdot 29}{2^3 \cdot 3^5} \biggr] \mu^4
+ \mathcal{O}(\mu^5) \, ,
</math>
  </td>
</tr>
</table>
</div>
and,

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

<tr>
  <td align="right">
<math>~\biggl( \frac{\chi_-}{r_\mathrm{crit}} \biggr)^{2}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
1 ~- 
\biggl( \frac{2}{3}\biggr)^{1 / 2}\mu  
+ \biggl[ - \biggl(\frac{5}{2\cdot 3^2}\biggr) \biggr] \mu^2
+  \biggl(\frac{41^2}{2^5 \cdot 3^7}\biggr)^{1 / 2}  \mu^3
+ \biggl[- \biggl( \frac{203}{2^3 \cdot 3^5 } \biggr) \biggr] \mu^4  
+ \mathcal{O}(\mu^5) \, ,
</math>
  </td>
</tr>
</table>
</div>
and,

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

<tr>
  <td align="right">
<math>~\biggl[ 1 - \frac{3}{4} \biggl( \frac{\chi_-}{r_\mathrm{crit}} \biggr)^2 \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}
\biggl\{ 1
+  \biggl( \frac{3}{2}\biggr)^{1 / 2}\mu  
- \biggl(\frac{1}{3}\biggr) \mu^2
+ \biggl( \frac{31^2}{2^7 \cdot 3^5}\biggr)^{1 / 2}   \mu^3  
- \biggl( \frac{296}{2^5 \cdot 3^4} \biggr) \mu^4  
\biggr\}
+ \mathcal{O}(\mu^5) \, ,
</math>
  </td>
</tr>
</table>
</div>

and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\epsilon_-</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
- 6^{1 / 2}  \mu
- \biggl(\frac{4}{3}\biggr)\mu^2  
- \biggl[ \frac{5\cdot 23}{(2^5 \cdot 3^5)^{1 / 2}} \biggr] \mu^3  
- \biggl(\frac{191}{2\cdot 3^4}\biggr) \mu^4
+ \mathcal{O}(\mu^5) \, .
</math>
  </td>
</tr>
</table>
</div>