Editing SSC/StabilityConjecture/Bipolytrope51 (section)

====Examples====
By trial-and-error, we have searched for accurate <math>(m_3, \ell_i)</math> pairs; this, of course gives us the desired <math>(\mu_e/\mu_c, \xi_i)</math> pairs.  When an accurate pair has been discovered, we should find that the LHS and RHS of the following expression should be equal to one another, to a very high degree of precision.
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\ell_i F (1 + \ell_i^2)^2 
\biggl\{
2m_3(1-m_3)\ell_i^2
- m_3 [1 + (1 - m_3)\ell_i^2 ]
\biggr\}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
F \cdot J \biggl\{ (H - m_3\ell_i) 
(3 + \ell_i^2) 
+  
2m_3\ell_i (1+\ell_i^2)\biggr\}
~- ~\ell_i^2(1 + \ell_i^2) H\cdot J\biggl[ m_3^2   + 2(1-m_3) + 2(1 - m_3)^2\ell_i^2 \biggr] \, .
</math>
  </td>
</tr>
</table>
In the following table, the first row of numbers (associated with <math>\mu_e/\mu_c = 1/4</math>) shows results from the relatively crude "trial" Example #2 that we used, above as we debugged our derivation of this analytic expression.  The second row of numbers improves on this initial guess, while the other rows give high-precision results for other selected values of <math>\mu_e/\mu_c</math>.
<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="9">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/Bipolytrope/Stability/qAndNuMax.xlsx --- worksheet = K-BK74]]<br />'''Example High-Precision Determinations of <math>\nu_\mathrm{max}(\mu_e/\mu_c)</math>'''
  </td>
</tr>
<tr>
  <td align="center"><math>\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>\xi_i</math></td>
  <td align="center">LHS</td>
  <td align="center">RHS:TERM1</td>
  <td align="center">RHS:TERM2</td>
  <td align="center">error<br />(TERM1 - TERM2 - LHS)</td>
  <td align="center"><math>q</math></td>
  <td align="center"><math>\nu_\mathrm{max}</math></td>
  <td align="center">earlier<br />fractional error</td>
</tr>
<tr>
  <td align="center"><math>\frac{1}{4}</math></td>
  <td align="center"><math>4.93827</math></td>
  <td align="center">2532.246285</td>
  <td align="center">56062.281392</td>
  <td align="center">53533.669713</td>
  <td align="center"><math>-3.6346</math></td>
  <td align="center"><math>0.084820</math></td>
  <td align="center"><math>0.1393701568</math></td>
  <td align="center"><math>-1.1 \times 10^{-2}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen"><math>\frac{1}{4}</math></td>
  <td align="center"><math>4.9379256</math></td>
  <td align="center">2530.312408401</td>
  <td align="center">56030.44257523</td>
  <td align="center">53500.13020660</td>
  <td align="center"><math>-0.0000398</math></td>
  <td align="center"><math>0.0848241365</math></td>
  <td align="center" bgcolor="lightgreen"><math>0.1393701572</math></td>
  <td align="center"><math>-1.2 \times 10^{-7}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen"><math>0.295</math></td>
  <td align="center"><math>7.07531489</math></td>
  <td align="center">22437.37085296</td>
  <td align="center">424789.588653918</td>
  <td align="center">402352.217777713</td>
  <td align="center"><math>+0.0000232</math></td>
  <td align="center"><math>0.0832775611</math></td>
  <td align="center" bgcolor="lightgreen"><math>0.2646775149</math></td>
  <td align="center"><math>6.3 \times 10^{-9}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen"><math>0.3</math></td>
  <td align="center"><math>7.569605936</math></td>
  <td align="center">34614.27130158</td>
  <td align="center">652591.38554202</td>
  <td align="center">617977.11415666</td>
  <td align="center"><math>+0.0000838</math></td>
  <td align="center"><math>0.0814202240</math></td>
  <td align="center" bgcolor="lightgreen"><math>0.2860557405</math></td>
  <td align="center"><math>1.5 \times 10^{-8}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen"><math>0.305</math></td>
  <td align="center"><math>8.193828507</math></td>
  <td align="center">57980.93749506</td>
  <td align="center">1095371.3718054</td>
  <td align="center">1037390.4343464</td>
  <td align="center"><math>-0.0000361</math></td>
  <td align="center"><math>0.0788994904</math></td>
  <td align="center" bgcolor="lightgreen"><math>0.3100155910</math></td>
  <td align="center"><math>9.6 \times 10^{-9}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen"><math>0.310</math></td>
  <td align="center"><math>9.014959766</math></td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center"><math>+0.000169</math></td>
  <td align="center"><math>0.0755022550</math></td>
  <td align="center" bgcolor="lightgreen"><math>0.3372170065</math></td>
  <td align="center"><math>-3.8 \times 10^{-9}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen"><math>0.320</math></td>
  <td align="center"><math>11.914571350</math></td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center"><math>-0.0000119</math></td>
  <td align="center"><math>0.0644564059</math></td>
  <td align="center" bgcolor="lightgreen"><math>0.4061310924</math></td>
  <td align="center"><math>1.4 \times 10^{-9}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen"><math>0.325</math></td>
  <td align="center"><math>15.0964057345</math></td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center"><math>-0.000216</math></td>
  <td align="center"><math>0.0549312331</math></td>
  <td align="center" bgcolor="lightgreen"><math>0.4531316008</math></td>
  <td align="center"><math>-4.7 \times 10^{-10}</math></td>
</tr>
<tr>
  <td align="center" bgcolor="lightgreen"><math>\frac{1}{3}</math></td>
  <td align="center"><math>\infty</math></td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center">-</td>
  <td align="center">--</td>
  <td align="center"><math>0.0</math></td>
  <td align="center" bgcolor="lightgreen"><math>0.63661977</math></td>
  <td align="center">-</td>
</tr>
</table>

In a [[SSC/Structure/BiPolytropes/Analytic51#Derivation|separate earlier derivation]], we determined that the analytic expression from which the value of <math>\nu_\mathrm{max}</math> can be derived is,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
\biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3] \, .
</math> 
  </td>
</tr>
</table>

The last column of the table &#8212; labeled "earlier fractional error" &#8212; shows the result of subtracting the LHS of this earlier expression from its RHS, then dividing by the LHS.  Because our derived "earlier fractional error" values are tiny, we are convinced that these two separately derived expressions are indeed identical.