Editing SSC/Stability/BiPolytropes/51Models

__FORCETOC__
=BiPolytrope with n<sub>c</sub> = 5 and n<sub>e</sub> = 1=

Here we construct and analyze the relative stability of a [[SSC/Structure/BiPolytropes#BiPolytropes|bipolytrope]] in which the core has an <math>n_c=5</math> polytropic index and the envelope has an <math>n_e=1</math> polytropic index.  

==Structure==

<ol>
  <li>Individual model profiles, taken from: 
      <ul><li>[[SSC/Structure/BiPolytropes/Analytic51#Examples|SSC/Structure/BiPolytropes/Analytic51#Examples]]</li></ul>
  </li>

  <li><math>(q, \nu)</math> sequences of fixed <math>\mu_e/\mu_c</math>, taken from: 
      <ul><li>[[SSC/Structure/BiPolytropes/Analytic51#Model_Sequences|SSC/Structure/BiPolytropes/Analytic51#Model_Sequences]]</li></ul>
  </li>

  <li><math>\nu_\mathrm{max}</math> model, taken from:
      <ul><li>[[SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass|SSC/Structure/BiPolytropes/Analytic51#Limiting_Mass]]
<br />&nbsp;<br />

<table border="1" align="center" cellpadding="8">
<tr>
  <td align="center" colspan="13">
<b>Maximum Fractional Core Mass, <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math> (solid green circular markers)<br />for Equilibrium Sequences having Various Values of <math>\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">
<math>\theta_i</math>
  </td>
  <td align="center">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>\Lambda_i</math>
  </td>
  <td align="center">
<math>A</math>
  </td>
  <td align="center">
<math>\eta_s</math>
  </td>
  <td align="center">
LHS
  </td>
  <td align="center">
RHS
  </td>
  <td align="center">
<math>q \equiv \frac{r_\mathrm{core}}{R}</math>
  </td>
  <td align="center">
<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math>
  </td>
  <td align="right">
<math>[\xi_i]_\mathrm{smooth}</math>  </td>
  <td align="center" rowspan="7">[[File:TurningPoints51BipolytropesLabels.png|450px|Extrema along Various Equilibrium Sequences]]</td>
</tr>

<tr>
  <td align="center">
<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">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">0.0 </td>
  <td align="center">
<math>\frac{2}{\pi}</math> </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
0.33
  </td>
  <td align="right">
24.00496  </td>
  <td align="right">
0.0719668  </td>
  <td align="right">
0.0710624  </td>
  <td align="right">
0.2128753  </td>
  <td align="right">
0.0726547  </td>
  <td align="right">
1.8516032  </td>
  <td align="right">
-223.8157  </td>
  <td align="right">
-223.8159  </td>
  <td align="right">
0.038378833  </td>
  <td align="right">
0.52024552  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
0.316943
  </td>
  <td align="right">
10.744571  </td>
  <td align="right">
0.1591479  </td>
  <td align="right">
0.1493938  </td>
  <td align="right">
0.4903393  </td>
  <td align="right">
0.1663869  </td>
  <td align="right">
2.1760793  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
-31.55254  </td>
  <td align="right">
0.068652714  </td>
  <td align="right">
0.382383875  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
0.31
  </td>
  <td align="right">
9.014959766  </td>
  <td align="center">
0.1886798  </td>
  <td align="center">
0.172320503  </td>
  <td align="right">
0.59835053  </td>
  <td align="center">
0.20081242  </td>
  <td align="center">
2.2823226  </td>
  <td align="center">
---  </td>
  <td align="center">
---  </td>
  <td align="right">
0.0755022550  </td>
  <td align="right">
0.3372170064  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
0.3090
  </td>
  <td align="right">
8.8301772  </td>
  <td align="right">
0.1924833  </td>
  <td align="right">
0.1750954  </td>
  <td align="right">
0.6130669  </td>
  <td align="right">
0.2053811  </td>
  <td align="right">
2.2958639  </td>
  <td align="right">
-18.47809  </td>
  <td align="right">
-18.47808  </td>
  <td align="right">
0.076265588  </td>
  <td align="right">
0.331475715  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="center">
<math>\frac{1}{4}</math>
  </td>
  <td align="right">
4.9379256  </td>
  <td align="right">
0.3309933  </td>
  <td align="right">
0.2342522  </td>
  <td align="right">
1.4179907  </td>
  <td align="right">
0.4064595  </td>
  <td align="right">
2.761622  </td>
  <td align="right">
-2.601255  </td>
  <td align="right">
-2.601257  </td>
  <td align="right">
0.084824137  </td>
  <td align="right">
0.139370157  </td>
  <td align="right">
0.0  </td>
</tr>

<tr>
  <td align="left" colspan="13">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}}  \, ;
</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
Also, go [[Appendix/Ramblings/PatrickMotl#Pick_a_Different_Molecular-Weight_Ratio|here]] for definition of <math>[\xi_i]_\mathrm{smooth}</math>, which identifies the location of the specific-entropy step function; stability against convection is ensured whenever <math>\xi_i > [\xi_i]_\mathrm{smooth}</math>.
  </td>
</tr>
</table>
</li>

<br />&nbsp;</br />
      </li>
      <li>
[[SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings|SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings]]
<br />&nbsp;<br />

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

<tr>
  <th align="center" colspan="5">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/Bipolytrope/Stability/qAndNuMax.xlsx --- worksheet = B-KB74 thru MinuPreparation]]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>
</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>0.3372170064</math></td>
  <td align="center"><math>0.0755022550</math></td>
</tr>

<tr>
  <td align="center">'''B1'''</td>
  <td align="center"><math>9.12744</math></td>
  <td align="center"><math>0.60069262</math></td>
  <td align="center"><math>0.3372001445</math></td>
  <td align="center"><math>0.0746451491</math></td>
</tr>

<tr>
  <td align="center">'''B2'''</td>
  <td align="center"><math>8.90394</math></td>
  <td align="center"><math>0.59610192</math></td>
  <td align="center"><math>0.33720014467</math></td>
  <td align="center"><math>0.0763642133</math></td>
</tr>
</table>


<table border="1" align="center" cellpadding="10">
<tr>
  <td align="center">[[File:TurningPoints51BipolytropesLabels.png|right|350px|Bipolytropic (5, 1) Equilibrium Sequences]]</td>
  <td align="center">[[File:TurningPoints51Bpairing.png|right|350px|Bipolytropic (5, 1) Equilibrium Sequences]]</td>
</tr>
</table>

<table border="1" align="center" cellpadding="10">
<tr>
  <td align="center">[[File:FundModeLocations02Labels.png|right|350px|Bipolytropic (5, 1) Neutral Fundamental Mode Locations]]</td>
  <td align="center">[[File:ConvectiveBoundary2Labeled.png|right|350px|Bipolytropic (5, 1) Equilibrium Sequences]]</td>
</tr>
</table>

      </li>
      </ul>
</li>
</ol>

==Yet Another Normalization==

===Fixed Core Mass===
Initially, our normalization was based on [[SSC/Structure/BiPolytropes/Analytic51#Normalization|holding <math>K_c</math> and the central density <math>(\rho_0)</math> constant]].  Specifically,

<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{\rho}{\rho_0}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>M_r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>H^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{H}{K_c\rho_0^{1/5}}</math>
  </td>

  <td align="center">. &nbsp;&nbsp;&nbsp;</td>

  <td align="right" colspan="3">
&nbsp;
  </td>
</tr>

</table>
We also have explored a [[SSC/Structure/BiPolytropes/51RenormaizePart2#Basic_Equilibrium_Structure|"new normalization"]] based on holding <math>K_c</math> and <math>M_\mathrm{tot}</math> constant.  Here we want to perform a Bonnor-Ebert-type analysis, examining how <math>P_i</math> varies with radius if we hold <math>K_c</math> and the ''core mass'' constant along an equilibrium sequence.  According to our initial normalization &#8212; see, for example, [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Step_4:_Throughout_the_core|here]] &#8212; we can write,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>M_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \rho_0^{1 / 5}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3</math>
  </td>
</tr>
</table>

Therefore, from the analytic profiles that describe the core, we have,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\rho_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \theta_i^5</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \theta_i^5</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \theta_i^6</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \theta_i^6</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i)^3 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i)^3 </math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right"><math>\rho_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\rho_0 \theta_i^5
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3
\biggr\}^5 \theta_i^5
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{5/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{5/2} \xi_i^{15} \theta_i^{20} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>P_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
K_c \rho_0^{6/5} \theta_i^6
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
K_c\biggl\{
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3
\biggr\}^6 \theta_i^6
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^{10}}{G^9 M_\mathrm{core}^6 } \biggr] \biggl( \frac{2\cdot 3}{\pi } \biggr)^{3} \xi_i^{18} \theta_i^{24} \, , 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>r_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c}{G}\biggr]^{1/2} \biggl\{ 
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3
\biggr\}^{-2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{G}{K_c}\biggr]^{5/2} M_\mathrm{core}^{-1} 
\biggl(\frac{\pi}{2^3 3}\biggr)^{1/2} \xi_i^{-5} \theta_i^{-6}
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \mathrm{volume}~=\biggl(\frac{2^2\pi}{3}\biggr)r_i^3</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{2^2\pi}{3}\biggr)\biggl\{
\biggl[ \frac{G}{K_c}\biggr]^{5/2} M_\mathrm{core}^{-1} 
\biggl(\frac{\pi}{2^3 3}\biggr)^{1/2} \xi_i^{-5} \theta_i^{-6}
\biggr\}^3
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{G}{K_c}\biggr]^{15/2} M_\mathrm{core}^{-3} 
\biggl(\frac{\pi}{2 \cdot 3}\biggr)^{5/2}
\xi_i^{-15} \theta_i^{-18} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>M_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i)^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^3}{G^3 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1 / 2}
\biggl\{
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3
\biggr\}^{-1} ( \xi_i \theta_i)^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
M_\mathrm{core} \, .
</math>
  </td>
</tr>
</table>

Immediately below we reproduce [[SSC/Structure/PolytropesEmbedded#Fig3|Figure 3 from our accompanying discussion of ''embedded (pressure-truncated) polytropes'' having <math>n=5</math>]].  Notice that frame (a) contains a plot that displays our "yet another normalization" expressions for <math>P_i</math> vs. volume.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
  <td align="center" colspan="6">
Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
  </td>
</tr>
<tr>
  <td align="center"><font color="black" size="+2">&#x25CF;</font></td><td align="center"><math>~\xi_e</math></td>
  <td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
  <td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="yellow" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">&radic;3</td>
  <td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="darkgreen" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">3</td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="purple" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">&radic;15</td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="red" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">9.01</td>
</tr>
<tr>
  <td align="center" colspan="2">&nbsp;</td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. 
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
  </td>
  <td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[  \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
  </td>
</tr>
</table>
</div>

===Fixed Radius===
Given that &hellip;
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{\rho}{\rho_0}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>M_r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>H^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{H}{K_c\rho_0^{1/5}}</math>
  </td>

  <td align="center">. &nbsp;&nbsp;&nbsp;</td>

  <td align="right" colspan="3">
&nbsp;
  </td>
</tr>

</table>
we can flip from holding <math>\rho_0</math> fixed to holding <math>R</math> fixed via the relation,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>R = [K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]R^*</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl[ \frac{K_c}{(G\rho_0^{4/5})}\biggr]^{1 / 2}
\biggl(\frac{1}{2\pi}\biggr)^{1 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1} \frac{\eta_s}{\theta_i^2}
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ R^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G\rho_0^{4/5}}\biggr]
\biggl(\frac{1}{2\pi}\biggr) \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \eta_s^2 \theta_i^{-4}
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \rho_0^{4 / 5}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c}{GR^2}\biggr]
\biggl(\frac{1}{2\pi}\biggr) \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \eta_s^2 \theta_i^{-4}
</math>
  </td>
</tr>
</table>

As a result,

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

<tr>
  <td align="right"><math>M = [K_c^{3 /2}/(G^{3 /2}\rho_0^{1/5})]M^*</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^{3}}{G^{3}\rho_0^{2/5}}\biggr]^{1 / 2}M^*</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow~~~ M^4</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^{6}}{G^{6}}\biggr]\rho_0^{-4 / 5}(M^*)^4</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^{6}}{G^{6}}\biggr]\biggl\{
\biggl[ \frac{K_c}{GR^2}\biggr]
\biggl(\frac{1}{2\pi}\biggr) \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \eta_s^2 \theta_i^{-4}
\biggr\}^{-1}(M^*)^4</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\pi \biggl[ \frac{K_c^{5}R^2}{G^{5}}\biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \eta_s^{-2} \theta_i^{4}
(M^*)^4 \, .</math>
  </td>
</tr>
</table>
If we want to see the behavior along a sequence of the core mass, the expression is,

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

<tr>
  <td align="right"><math>M_\mathrm{core}^4</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\pi \biggl[ \frac{K_c^{5}R^2}{G^{5}}\biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \eta_s^{-2} \theta_i^{4}
\biggl[
\biggl(\frac{6}{\pi}\biggr)^{1 / 2} \xi_i^3 \theta_i^3
\biggr]^4 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{2^3\cdot 3^2}{\pi}\biggr) \biggl[ \frac{K_c^{5}R^2}{G^{5}}\biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \eta_s^{-2} 
\biggl[
\xi_i^{12} \theta_i^{16}
\biggr] \, ;</math>
  </td>
</tr>
</table>
while the expression for the total mass is,

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

<tr>
  <td align="right"><math>M_\mathrm{tot}^4</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2\pi \biggl[ \frac{K_c^{5}R^2}{G^{5}}\biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} \eta_s^{-2} \theta_i^{4}
\biggl[
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2}\biggl(\frac{2}{\pi}\biggr)^{1 / 2} A\eta_s \theta_i^{-1}
\biggr]^4 </math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{2^3}{\pi}\biggr) \biggl[ \frac{K_c^{5}R^2}{G^{5}}\biggr]
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-6} A^4 \eta_s^2 
\, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" width="60%" cellpadding="8"><tr><td align="left">
<div align="center"><b>Summary:</b>&nbsp; For fixed <math>K_c</math> and <math>R</math></div>
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\rho_0</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c}{GR^2}\biggr]^{5 / 4}
\biggl(\frac{1}{2\pi}\biggr)^{5 / 4} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5 / 2} \eta_s^{5 / 2} \theta_i^{-5}
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>M_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{2^3\cdot 3^2}{\pi}\biggr)^{1 / 4} \biggl[ \frac{K_c^{5}R^2}{G^{5}}\biggr]^{1 / 4}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{1 / 2} \eta_s^{-1 / 2} 
\biggl[
\xi_i^{3} \theta_i^{4}
\biggr] \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>M_\mathrm{tot}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{2^3}{\pi}\biggr)^{1 / 4} \biggl[ \frac{K_c^{5}R^2}{G^{5}}\biggr]^{1 / 4}
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-3 / 2} A \eta_s^{1 / 2} 
\, .
</math>
  </td>
</tr>
</table>

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

==Stability==

===Introduction &amp; Summary===

Here we solve the LAWE numerically (on a uniformly zoned mesh &#8212; different <math>\Delta\tilde{r}</math> for the separate core/envelope regions) using a 2<sup>nd</sup>-order accurate,  [[Appendix/Ramblings/51BiPolytropeStability/BetterInterfacePt2#Convert_to_Implicit_Approach|implicit integration scheme]] in which the LAWE is broken into a pair of 1<sup>st</sup>-order ODEs.  These results should be compared against a separate [[SSC/Stability/BiPolytropes/SuccinctDiscussion#Stability|succinct discussion]] of our analysis obtained from integrating the LAWE in its standard 2<sup>nd</sup>-order ODE form.

<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center" colspan="7"><b>Properties of ''Neutral'' Fundamental Mode for Various Sequences</b></td>
  <td align="center" colspan="2"><b>&sigma;<sub>c</sub><sup>2</sup> for Overtones</b></td>
  <td align="center" colspan="2"><b>&Omega;<sup>2</sup> for Overtones</b></td>
</tr>

<tr>
  <td align="center" rowspan="7">[[File:FundModeLocations01Labels.png|300px|Fundamental Model Locations]]</td>
  <td align="center"><math>\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>\xi_i</math></td>
  <td align="center"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center"><math>\nu \equiv \frac{M_c}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>q \equiv \frac{r_c}{R}</math></td>
  <td align="center"><math>\sigma_c^2</math></td>
  <td align="center">1<sup>st</sup></td>
  <td align="center">2<sup>nd</sup></td>
  <td align="center">1<sup>st</sup></td>
  <td align="center">2<sup>nd</sup></td>
</tr>

<tr>
  <td align="right">1.000</td>
  <td align="right">1.6639103365</td>
  <td align="right">8.4811731</td>
  <td align="right">0.49622717</td>
  <td align="right">0.53833097</td>
  <td align="right">0.000000</td>
  <td align="right">2.528013</td>
  <td align="right">5.66087</td>
  <td align="right">10.72026</td>
  <td align="right">24.0054</td>
</tr>

<tr>
  <td align="right">0.500</td>
  <td align="right">2.2703111897</td>
  <td align="right">62.666493</td>
  <td align="right">0.399760079</td>
  <td align="right">0.305764976</td>
  <td align="right">0.000000</td>
  <td align="right"> 0.2659116 </td>
  <td align="center">0.73022</td>
  <td align="right">8.33187</td>
  <td align="right">22.8802</td>
</tr>

<tr>
  <td align="right">0.345</td>
  <td align="right">2.546385206</td>
  <td align="right">205.77394</td>
  <td align="right">0.232779379</td>
  <td align="right">0.185262833</td>
  <td align="right">0.000000</td>
  <td align="right">0.06741185</td>
  <td align="right">0.198075</td>
  <td align="right">6.93580</td>
  <td align="right">20.3793</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{1}{3}</math></td>
  <td align="right">2.5675774773</td>
  <td align="right">225.75664</td>
  <td align="right">0.216806201</td>
  <td align="right">0.176420918</td>
  <td align="right">0.000000</td>
  <td align="right">0.0602615</td>
  <td align="right">0.178432</td>
  <td align="right">6.80222</td>
  <td align="right">20.1411</td>
</tr>

<tr>
  <td align="center"><math>0.310</math></td>
  <td align="right">2.6095097538</td>
  <td align="right">270.59221</td>
  <td align="right">0.184909369</td>
  <td align="right">0.159274</td>
  <td align="right">0.000000</td>
  <td align="right">0.04821396</td>
  <td align="right">0.145248</td>
  <td align="right">6.52316</td>
  <td align="right">19.6515</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{1}{4}</math></td>
  <td align="right">2.712384289</td>
  <td align="right">415.67338</td>
  <td align="right">0.109935743</td>
  <td align="right">0.1192667</td>
  <td align="right">0.000000</td>
  <td align="right">0.02772424</td>
  <td align="right">0.088472</td>
  <td align="right">5.76211</td>
  <td align="right">18.3877</td>
</tr>
</table>

===Model Sequence: &nbsp;&mu;<sub>e</sub>/&mu;<sub>c</sub> = 1.00===

====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>

===Model Sequence: &nbsp;&mu;<sub>e</sub>/&mu;<sub>c</sub> = 0.31===

Here we examine how the frequency of the 1<sup>st</sup> overtone varies as <math>\xi_i</math> is increased.

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

<tr>
  <td align="center" colspan="9">Frequency Variation Along the Sequence having <math>\mu_e/\mu_c = 0.31</math></td>
</tr>

<tr>
  <td align="center" rowspan="13">[[File:Evolve031B.png|500px|Overtone Frequencies]]</td>
  <td align="center" rowspan="2">Note</td>
  <td align="center" rowspan="2"><math>\xi_i</math></td>
  <td align="center" rowspan="2"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center" rowspan="1" colspan="2">1<sup>st</sup> Overtone</td>
  <td align="center" rowspan="13">[[File:Omega2for1stOvertone4.png|500px|Overtone Frequencies]]</td>
  <td align="center" rowspan="1" colspan="2">Fundamental</td>
</tr>

<tr>
  <td align="center" rowspan="1"><math>\sigma_c^2</math></td>
  <td align="center" rowspan="1" bgcolor="lightblue"><math>\Omega^2 = \frac{\sigma_c^2}{2}\biggl(\frac{\rho_c}{\bar\rho}\biggr)</math></td>
  <td align="center" rowspan="1"><math>\sigma_c^2</math></td>
  <td align="center" rowspan="1" bgcolor="#FF5733"><math>\Omega^2 = \frac{\sigma_c^2}{2}\biggl(\frac{\rho_c}{\bar\rho}\biggr)</math></td>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">1.6</td>
  <td align="center">58.39858647</td>
  <td align="center">0.498473</td>
  <td align="center">14.5550593</td>
  <td align="center">0.1333725</td>
  <td align="center">3.8943827</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">2.0000</td>
  <td align="center">108.69129</td>
  <td align="center">0.236047</td>
  <td align="center">12.82812694</td>
  <td align="center">0.07011655</td>
  <td align="center">3.8105293</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">2.4000</td>
  <td align="center">199.16363</td>
  <td align="center">0.0870005</td>
  <td align="center">8.6636677</td>
  <td align="center">0.028066485</td>
  <td align="center">2.794911541</td>
</tr>

<tr>
  <td align="right" bgcolor="orange">Neutral Fundamental ==></td>
  <td align="center">2.6095097538</td>
  <td align="center">270.5922</td>
  <td align="center">0.04821396</td>
  <td align="center">6.523161</td>
  <td align="center">0.0</td>
  <td align="center">0.0</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">3.0000</td>
  <td align="center">468.1500</td>
  <td align="center">0.02329066</td>
  <td align="center">5.451761</td>
  <td align="center">-0.056763527</td>
  <td align="center">-13.2869232</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">3.5</td>
  <td align="center">902.640279</td>
  <td align="center">0.011747773</td>
  <td align="center">5.302006549</td>
  <td align="center">- 0.098905428</td>
  <td align="center">-44.63801154</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">4.0000</td>
  <td align="center">1656.926</td>
  <td align="center">0.006427613</td>
  <td align="center">5.325041</td>
  <td align="center">-0.118551256677297</td>
  <td align="center">-98.21535777</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">5.0000</td>
  <td align="center">4900.105</td>
  <td align="center">0.002215415</td>
  <td align="center">5.4279</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">6.0000</td>
  <td align="center">12544.67</td>
  <td align="center">0.000878472</td>
  <td align="center">5.510074</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="right" bgcolor="lightgreen"><math>\nu_\mathrm{max}</math> ==></td>
  <td align="center">9.014959766</td>
  <td align="center"><math>1.1664159 \times 10^{5}</math></td>
  <td align="center"><math>9.60837 \times 10^{-5}</math></td>
  <td align="center">5.60367789</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>

<tr>
  <td align="center">&nbsp;</td>
  <td align="center">12.0000</td>
  <td align="center"><math>6.0066416 \times 10^{5}</math></td>
  <td align="center"><math>1.857813 \times 10^{-5}</math></td>
  <td align="center">5.579608</td>
  <td align="center">---</td>
  <td align="center">---</td>
</tr>
</table>

===SearchMuRatio===

Adding models to the [[#Introduction_&_Summary|above table]], here we choose <math>\xi_i</math> and iterate until we have found the value of <math>\mu_e/\mu_c</math> that corresponds to the fundamental-mode.  At the interface, we expect,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\gamma_e \biggl[3 + \biggl(\frac{d\ln x}{d \ln \xi}\biggr)_\mathrm{env}  \biggr]_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\gamma_c \biggl[3 + \biggl(\frac{d\ln x}{d \ln \xi}\biggr)_\mathrm{core}  \biggr]_i \, .
</math>
  </td>
</tr>
</table>
Throughout the core, for the ''neutral'' (i.e., <math>\sigma_c^2 = 0</math>) fundamental mode of oscillation, we expect that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>x_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
1 - \frac{\xi^2}{15}</math> &nbsp; &nbsp; &nbsp; <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp; <math> \frac{dx_\mathrm{core}}{d\xi} = -\frac{2\xi}{15}\, .
</math>
  </td>
</tr>
</table>
Given that <math>(\gamma_c, \gamma_e) = (\tfrac{6}{5}, 2)</math> at the interface, we expect,
<table border="0" align="center" cellpadding="8">

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

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{3}{5} \biggl[3 - \frac{15\xi}{(15-\xi^2)}\biggl(\frac{2\xi}{15}\biggr)  \biggr]_i -3
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
-\frac{3}{5} \biggl[2+ \frac{2\xi^2}{(15-\xi^2)} \biggr]_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[\frac{18}{\xi_i^2-15} \biggr] \, . 
</math>
  </td>
</tr>
</table>
Similarly at the surface of the envelope for the ''neutral'' (i.e., <math>\sigma_c^2 = 0</math>) fundamental mode of oscillation, we expect that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right"><math>\biggl[\biggl(\frac{d\ln x}{d \ln \xi}\biggr)_\mathrm{env}  \biggr]_\mathrm{surf}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\cancelto{0}{\frac{\sigma_c^2}{4}} \biggl(\frac{\rho_c}{\bar\rho}\biggr) - 1 = -1 \, .
</math>
  </td>
</tr>
</table>


<table border="1" align="center" cellpadding="8">

<tr>
  <td align="center" colspan="13">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/TwoFirstOrderODEs/Bipolytrope51New.xlsx --- worksheet = ConvectiveBoundary]]<b>Properties of ''Neutral'' Fundamental Mode for Various Sequences</b></td>
</tr>

<tr>
  <td align="center" rowspan="14">[[File:FundModeLocations05Labels.png|500px|Fundamental Model Locations]]</td>
  <td align="center" rowspan="3"><math>\frac{\mu_e}{\mu_c}</math></td>
  <td align="center" rowspan="3"><math>\xi_i</math></td>
  <td align="center" rowspan="3"><math>\frac{\rho_c}{\bar\rho}</math></td>
  <td align="center" rowspan="3"><math>\nu \equiv \frac{M_c}{M_\mathrm{tot}}</math></td>
  <td align="center" rowspan="3"><math>q \equiv \frac{r_c}{R}</math></td>
  <td align="center" rowspan="3"><math>\sigma_c^2</math></td>
  <td align="center" colspan="4"><math>[d\ln x/d\ln\xi]_\mathrm{env}</math></td>
</tr>

<tr>
  <td align="center" colspan="2">Interface</td>
  <td align="center" colspan="2">Surface</td>
</tr>

<tr>
  <td align="center" colspan="1">expected<br /><math>18/(\xi_i^2-15)</math></td>
  <td align="center" colspan="1">measured</td>
  <td align="center" colspan="1">expected<br /><math>-1</math></td>
  <td align="center" colspan="1">measured</td>
</tr>

<tr>
  <td align="right">1.000</td>
  <td align="right">1.6639103365</td>
  <td align="right">8.4811731</td>
  <td align="right">0.49622717</td>
  <td align="right">0.53833097</td>
  <td align="right">0.000000</td>
  <td align="right">-1.471622</td>
  <td align="right">-1.471622</td>
  <td align="right">-1</td>
  <td align="right">-1.0062</td>
</tr>

<tr>
  <td align="right">0.681590377</td>
  <td align="right">2.0</td>
  <td align="right">23.176456</td>
  <td align="right">0.476716895</td>
  <td align="right">0.418529653</td>
  <td align="right">0.000000</td>
  <td align="right">-1.636364</td>
  <td align="right">-1.636364</td>
  <td align="right">-1</td>
  <td align="right">-1.0078</td>
</tr>

<tr>
  <td align="right">0.500</td>
  <td align="right">2.2703111897</td>
  <td align="right">62.666493</td>
  <td align="right">0.399760079</td>
  <td align="right">0.305764976</td>
  <td align="right">0.000000</td>
  <td align="right">-1.828212</td>
  <td align="right">-1.828212</td>
  <td align="right">-1</td>
  <td align="right">-1.0093</td>
</tr>

<tr>
  <td align="right">0.425426009</td>
  <td align="right">2.4</td>
  <td align="right">108.10495</td>
  <td align="right">0.332967203</td>
  <td align="right">0.248624189</td>
  <td align="right">0.000000</td>
  <td align="right">-1.948052</td>
  <td align="right">-1.948052 </td>
  <td align="right">-1</td>
  <td align="right">-1.0100</td>
</tr>

<tr>
  <td align="right">0.345</td>
  <td align="right">2.546385206</td>
  <td align="right">205.77394</td>
  <td align="right">0.232779379</td>
  <td align="right">0.185262833</td>
  <td align="right">0.000000</td>
  <td align="right">-2.113688</td>
  <td align="right">-2.113688</td>
  <td align="right">-1</td>
  <td align="right">-1.0108</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{1}{3}</math></td>
  <td align="right">2.5675774773</td>
  <td align="right">225.75664</td>
  <td align="right">0.216806201</td>
  <td align="right">0.176420918</td>
  <td align="right">0.000000</td>
  <td align="right">-2.140934</td>
  <td align="right">-2.140934</td>
  <td align="right">-1</td>
  <td align="right">-1.0110</td>
</tr>

<tr>
  <td align="center"><math>0.310</math></td>
  <td align="right">2.6095097538</td>
  <td align="right">270.59221</td>
  <td align="right">0.184909369</td>
  <td align="right">0.159274</td>
  <td align="right">0.000000</td>
  <td align="right">-2.197679</td>
  <td align="right">-2.197679</td>
  <td align="right">-1</td>
  <td align="right">-1.0112</td>
</tr>

<tr>
  <td align="center"><math>\tfrac{1}{4}</math></td>
  <td align="right">2.712384289</td>
  <td align="right">415.67338</td>
  <td align="right">0.109935743</td>
  <td align="right">0.1192667</td>
  <td align="right">0.000000</td>
  <td align="right">-2.355105</td>
  <td align="right">-2.355105</td>
  <td align="right">-1</td>
  <td align="right">-1.0117</td>
</tr>

<tr>
  <td align="center"><math>0.156419569</math></td>
  <td align="right">2.85</td>
  <td align="right">757.45344</td>
  <td align="right">0.034014631</td>
  <td align="right">0.068440082</td>
  <td align="right">0.000000</td>
  <td align="right">-2.61723</td>
  <td align="right">-2.61723 </td>
  <td align="right">-1</td>
  <td align="right">-1.0123</td>
</tr>

<tr>
  <td align="center"><math>0.067984979</math></td>
  <td align="right">2.95</td>
  <td align="right">1688.1377</td>
  <td align="right">0.005065202</td>
  <td align="right">0.028486668</td>
  <td align="right">0.000000</td>
  <td align="right">-2.858277</td>
  <td align="right">-2.858277</td>
  <td align="right">-1</td>
  <td align="right">-1.0148</td>
</tr>

<tr>
  <td align="center"><math>0.012591194</math></td>
  <td align="right">2.995</td>
  <td align="right">8547.1981</td>
  <td align="right">0.000151797</td>
  <td align="right">0.005211544</td>
  <td align="right">0.000000</td>
  <td align="right">-2.985087</td>
  <td align="right">-2.985087 </td>
  <td align="right">-1</td>
  <td align="right">-1.0132</td>
</tr>
</table>

=See Also=

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