Editing SSC/Stability/BiPolytropes/51Models (section)

====Marginally Unstable Model====
Numbers presented in the following table should be compared against our [[SSC/Stability/BiPolytropes#Other_Modes|earlier determinations]].  Various things to note:
<ol>
  <li>As discussed elsewhere &#8212; for example, [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Background|here]] &#8212; when <math>\sigma_c^2 = 0</math>, the radial displacement function for the core &#8212; that is, for all <math>\xi \le \xi_i</math> &#8212; should be given precisely by the expression,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>x_P\biggr|_{n=5}</math>
  </td>
  <td align="right"><math>=</math></td>
  <td align="right">
<math>1 - \frac{\xi^2}{15} \, .
</math>
  </td>
</tr>
</table>
Hence, given that <font color="green">&xi;<sub>i</sub> = 1.6639103365</font> as viewed from the perspective of the core, the magnitude of, and the logarithmic derivative of the radial displacement function should have the values, respectively,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>x_i</math>
  </td>
  <td align="right"><math>=</math></td>
  <td align="right">
<math>0.8154268 \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\biggl\{ \frac{d\ln x}{d\ln \xi}\biggr|_i\biggl\}_\mathrm{core}</math>
  </td>
  <td align="right"><math>=</math></td>
  <td align="right">
<math>
- \frac{2\xi^2}{15-\xi^2} = 
-0.45270322 \, .
</math>
  </td>
</tr>
</table>

  </li>
  <li>As discussed elsewhere &#8212; for example, [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Envelope|here]] &#8212; we expect,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\biggl\{ \frac{d\ln x}{d\ln \tilde{r}} \biggr|_i\biggr\}_\mathrm{env}</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>3\biggl(\frac{\gamma_c}{\gamma_e}-1 \biggr)
+ \frac{\gamma_c}{\gamma_e}\biggl\{ \frac{d\ln x}{d\ln \xi} \biggr|_i\biggr\}_\mathrm{core}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>3\biggl(\frac{3}{5}-1 \biggr)
+ \frac{3}{5}\biggl\{ \frac{d\ln x}{d\ln \xi} \biggr|_i\biggr\}_\mathrm{core}
= - 1.471622 \, .
  </math>
  </td>
</tr>
</table>
  </li>
</ol>

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="15">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51.xlsx --- worksheet = MuRatio100Fund]]'''Our <font color="green">September 2023</font> Determinations for Marginally Unstable Model Having <math>~\mu_e/\mu_c = 1</math>'''<br />&nbsp;<br />
<math>\xi_i = 1.6686460157 </math> &nbsp; &nbsp; <font color="green">NEW:</font>&nbsp;<math>\xi_i = 1.6639103365</math>
  </td>
</tr>
<tr>
  <td align="center" rowspan="2">Mode</td>
  <td align="center" rowspan="2"><math>~\sigma_c^2</math></td>
  <td align="center" rowspan="2"><math>~\Omega^2 \equiv \frac{\sigma_c^2}{2} \biggl( \frac{\rho_c}{\bar\rho}\biggr)</math></td>
  <td align="center" rowspan="2"><math>~x_i</math></td>
  <td align="center" colspan="2"><math>~\frac{d\ln x}{d\ln r^*}\biggr|_i</math></td>
  <td align="center" rowspan="2"><math>~x_\mathrm{surf}</math></td>
  <td align="center" colspan="2"><math>~\frac{d\ln x}{d\ln r^*}\biggr|_\mathrm{surf}</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_1</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_2</math></td>
  <td align="center" rowspan="2"><math>~\frac{r}{R}\biggr|_3</math></td>
  <td align="center" rowspan="2"><math>~1 - \frac{M_r}{M_\mathrm{tot}}\biggr|_3</math></td>
</tr>
<tr>
  <td align="center">core</td>
  <td align="center">env</td>
  <td align="center">''expected''</td>
  <td align="center">measured</td>
</tr>
<tr>
  <td align="center">1<br /><font size="-1">(Fundamental)</font></td>
  <td align="right">0.00</td>
  <td align="right">0.00</td>
  <td align="right">+0.81437470<br /><font color="green">0.8154268</font></td>
  <td align="right">-0.455872<br /><font color="green">-0.452703</font></td>
  <td align="right">-1.473523<br /><font color="green">-1.471622</font></td>
  <td align="right">+0.3820<br /><font color="green">0.3849493</font></td>
  <td align="right">-1</td>
  <td align="right">-0.999999992<br /><font color="green">-1.00618</font></td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="right">2.51513333<br /><font color="green">2.528013</font></td>
  <td align="right">10.7107538<br /><font color="green">10.720258</font></td>
  <td align="right">0.20482050<br /><font color="green">0.2069746</font></td>
  <td align="right">-7.09124<br /><font color="green">-7.000803</font></td>
  <td align="right">-5.4547441<br /><font color="green">-5.400482</font></td>
  <td align="right">- 0.9962<br /><font color="green">-1.018215</font></td>
  <td align="right">4.355376917<br /><font color="green">4.360129</font></td>
  <td align="right">4.35537692<br /><font color="green">4.3999485</font></td>
  <td align="right">0.64133<br /><font color="green">0.6456</font></td>
  <td align="right">0.3502<br /><font color="green">0.3444</font></td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="right">5.72371888<br /><font color="green">5.66087</font></td>
  <td align="right">24.3745901<br /><font color="green">24.0054</font></td>
  <td align="right">-0.14269277<br /><font color="green">-0.13587</font></td>
  <td align="right">+8.046019<br /><font color="green">+8.62053</font></td>
  <td align="right">+3.627611<br /><font color="green">+3.9723</font></td>
  <td align="right">+0.9308<br /><font color="green">+0.98810</font></td>
  <td align="right">11.18729505<br /><font color="green">11.0027</font></td>
  <td align="right">11.18729506<br /><font color="green">11.8164</font></td>
  <td align="right">0.4837<br /><font color="green">0.48395</font></td>
  <td align="right">0.5864<br /><font color="green">0.58326</font></td>
  <td align="right">0.842<br /><font color="green">0.84145</font></td>
  <td align="right">0.0854<br /><font color="green">0.08576</font></td>
  <td align="center">n/a</td>
  <td align="center">n/a</td>
</tr>
<tr>
  <td align="center">4</td>
  <td align="right">10.3458476</td>
  <td align="right">44.0622916</td>
  <td align="right">-0.20845197</td>
  <td align="right">-0.6949966</td>
  <td align="right">-1.61699793</td>
  <td align="right">-1.1443</td>
  <td align="right">21.03114578</td>
  <td align="right">21.03114577</td>
  <td align="right">0.3939</td>
  <td align="right">0.7154</td>
  <td align="right">0.6902</td>
  <td align="right">0.2777</td>
  <td align="center">0.9115</td>
  <td align="center">0.0284</td>
</tr>
<tr>
  <td align="center" colspan="15">
[[File:Mod0MuRatio100.png|550px|Our determination of eigenvector for mu_ratio = 1]]&nbsp;&nbsp;[[File:FourModesMuRatio100.png|550px|Our determination of multiple eigenvectors for mu_ratio = 1]]
  </td>
</tr>
</table>