Editing SSC/StabilityConjecture/Bipolytrope51 (section)

===Model Pairings===

Guided by our [[Appendix/Ramblings/NonlinarOscillation#Summary|separate examination of the K-BK74 conjecture]] in the context of pressure-truncated n = 5 polytropes, we will adopt the small (and always positive) parameter,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\delta
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 - \biggl( \frac{\nu}{\nu_\mathrm{max}} \biggr)^2 \biggr]^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
(In our separate discussion, the small parameter was labelled, <math>\mu</math>, rather than <math>\delta</math>.)  For a given "m<sub>3</sub>" equilibrium sequence, we seek two different equilibrium models that have the same value of <math>\delta \ll 1</math>, but different values of the interface parameter, <math>\xi_+ > \xi_i</math> and <math>\xi_- < \xi_i</math>.  Guided also by the [[Appendix/Ramblings/NonlinarOscillation#Table1|''Selected Pairings'' table from this separate examination]], Table 2 (below) provides some model pairs (i.e., models with the same fractional core-mass) that lie close to the maximum value.

<table border="1" align="center" cellpadding="5">

<tr>
  <th align="center" colspan="9">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/Bipolytrope/Stability/qAndNuMax.xlsx --- worksheet = K-BK74 thru MinuPreparation]]Table 2<br />Bipolytrope with <math>(n_c, n_e) = (5, 1)</math><br />Selected Pairings along the <math>\mu_e/\mu_c = 0.31</math> Sequence</th>
</tr>

<tr>
  <td align="center">Pairing</td>
  <td align="center"><math>(\xi_i)_+</math></td>
  <td align="center"><math>\Lambda_i</math></td>
  <td align="center"><math>\nu</math></td>
  <td align="center"><math>q_+</math></td>
  <td align="center"><math>\delta_+</math></td>
  <td align="center"><math>(\xi_i)_-</math></td>
  <td align="center"><math>\delta_-</math></td>
  <td align="center"><math>q_-</math></td>
</tr>

<tr>
  <td align="center">'''A'''</td>
  <td align="center"><math>9.014959766</math></td>
  <td align="center"><math>0.59835053</math></td>
  <td align="center"><math>0.33721701</math></td>
  <td align="center"><math>0.075502255</math></td>
  <td align="center"><math>0.0</math></td>
  <td align="center">degenerate</td>
  <td align="center">degenerate</td>
  <td align="center">degenerate</td>
</tr>

<tr>
  <td align="center">'''B'''</td>
  <td align="center"><math>9.12744</math></td>
  <td align="center"><math>0.60069262</math></td>
  <td align="center"><math>0.337200144</math></td>
  <td align="center"><math>0.0746451491</math></td>
  <td align="center"><math>0.0100000169</math></td>
  <td align="center"><math>8.90394</math></td>
  <td align="center"><math>0.0100000123</math></td>
  <td align="center"><math>0.7636421328</math></td>
</tr>

<tr>
  <td align="center" colspan="9">
[[File:RadiusVsMassFraction310.png|450px|Radius vs. Mass for Maximum-Mass Bipoltrope having &mu;-ratio = 0.310]]
  </td>
</tr>
</table>

In a [[SSC/Structure/BiPolytropes/Analytic51|separate chapter]], we have described in detail how to construct equilibrium models of bipolytropes that have <math>(n_c, n_e) = (5, 1)</math>; each model is uniquely defined by the parameter-pair, <math>(\mu_e/\mu_c, \xi_i)</math>.  In [[SSC/Structure/BiPolytropes/Analytic51#Profile|Table 2 of that chapter]], we have provided plots that show how the density, pressure, and "interior mass" <math>M_r</math> vary with <math>r</math> throughout the interior of nine example equilibrium models.  We chose models having <math>\xi_i = 0.5, 1.0,</math> and <math>3.0</math>; and for each value of these interface locations, we illustrated <math>\mu_e/\mu_c = 1, 1/2,</math> and <math>1/4</math>.  We have followed the same sequence of steps to construct the two equilibrium models specified by the '''Pairing B'''  parameters that have been listed in our current Table 2 (immediately above); that is, we used <math>(\mu_e/\mu_c, \xi_i)_+ = (0.31, 9.12744)</math> and <math>(\mu_e/\mu_c, \xi_i)_- = (0.31, 8.90394)</math>. The plot at the bottom of our current Table 2 shows how the "plus" model's radial location varies with the enclosed mass-fraction, <math>M_r/M_\mathrm{tot}</math>.  (There is no need to show a plot of the "minus" model's mass profile because its structure was purposely chosen to be very similar to the "plus" model's profile.)