Joel2: Created page with "FORCETOC =BiPolytrope with n_c = 5 and n_e = 1= Here we construct and analyze the relative stability of a bipolytrope in which the core has an $n_c=5$ polytropic index and the envelope has an $n_e=1$ polytropic index. ==Structure==

Individual model profiles, taken from:
SSC/Structure/BiPolytropes/Analytic51#Examples|SSC/Structure/BiPolytropes/Analyti..."

2024-07-15T23:10:07Z

Created page with "__FORCETOC__ =BiPolytrope with n<sub>c</sub> = 5 and n<sub>e</sub> = 1= Here we construct and analyze the relative stability of a 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/Analyti..."

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__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 /> <br />

<table border="1" align="center" cellpadding="8">
<tr>
<td align="center" colspan="12">
<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="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>
</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>
</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>
</tr>

<tr>
<td align="center">
0.31
</td>
<td align="right">
9.014959766 </td>
<td align="center">
--- </td>
<td align="center">
--- </td>
<td align="right">
0.59835053 </td>
<td align="center">
--- </td>
<td align="center">
--- </td>
<td align="center">
--- </td>
<td align="center">
--- </td>
<td align="right">
0.0755022550 </td>
<td align="right">
0.3372170064 </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>
</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>
</tr>

<tr>
<td align="left" colspan="11">
Recall that,
<div align="center">
<math>
\ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ;
</math>
      and      
<math>
m_3 \equiv 3 \biggl( \frac{\mu_e}{\mu_c} \biggr) \, .
</math>
</div>
</td>
</tr>
</table>
</li>

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

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

==Stability==

<ol>
<li>
LAWE taken from:   [[SSC/Stability/BiPolytropes/HeadScratching#Throughout_the_Configuration|SSC/Stability/BiPolytropes/HeadScratching#Throughout_the_Configuration]]
<ul>
<li>

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

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{d^2x}{dr*^2} + \frac{\mathcal{H}}{r^*} \frac{dx}{dr*}
+
\biggl[\biggl(\frac{\sigma_c^2}{\gamma_g}\biggr) \mathcal{K}_1 - \alpha_g \mathcal{K}_2\biggr] x \, ,
</math>
</td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\mathcal{H}</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}
</math>
</td>
<td align="center">      ,       </td>
<td align="right">
<math>~\mathcal{K}_1</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\frac{2\pi }{3}\biggl(\frac{\rho^*}{ P^* } \biggr)
</math>
</td>
<td align="center">      and       </td>
<td align="right">
<math>~\mathcal{K}_2</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~
\biggl(\frac{\rho^*}{ P^* } \biggr)\frac{M_r^*}{(r^*)^3} \, ,
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\sigma_c^2</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\frac{3\omega^2}{2\pi G\rho_c}
</math>
</td>
<td align="center">      ,       </td>
<td align="right">
<math>\alpha_g</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl(3 - \frac{4}{\gamma_g}\biggr) \, .
</math>
</td>
<td align="center" colspan="4">  </td>
</tr>
</table>
Core:   <math>\gamma_g = 6/5 ~~\Rightarrow ~~~ \alpha_g = -1/3</math><br />
Envelope:   <math>\gamma_g = 2 ~~\Rightarrow ~~~ \alpha_g = +1</math>
</li>
</ul>
<li>
Central Boundary condition drawn from:   [[SSC/Stability/BiPolytropes/HeadScratching#Central_Boundary_Condition|SSC/Stability/BiPolytropes/HeadScratching#Central_Boundary_Condition]]
<ul>
<li>

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

<tr>
<td align="right">
<math>
x_2
</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
x_1 \biggl[ 1 - \frac{(n+1) \mathfrak{F} \Delta_\xi^2}{60} \biggr]
=
x_1 \biggl[ 1 - \frac{\mathfrak{F} \Delta_\xi^2}{10} \biggr]
\, ,</math>
</td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>
\mathfrak{F}
</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>
\biggl[ \frac{\sigma_c^2}{\gamma_g} - 2\alpha_g \biggr]_\mathrm{core}
=
\frac{(8 + \sigma_c^2)}{(\gamma_g)_\mathrm{core}} - 6
=
\frac{1}{6}\biggl[ 5 \sigma_c^2 + 4 \biggr]
\, .</math>
</td>
</tr>
</table>

</li>
</ul>
</li>
<li>
Interface conditions taken from:  
<ul><li>[[SSC/Stability/BiPolytropes#Core:|SSC/Stability/BiPolytropes#Core:]]
As viewed from the perspective of the core, the slope at the interface is

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

<tr>
<td align="right">
<math>~\biggl[ \frac{dx_\mathrm{core}}{d\xi}\biggr]_i</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~
\frac{1}{2\delta \xi} \biggl\{
\biggl[ 2 - (\delta \xi)^2 \biggl( \frac{3\mathcal{K}}{2\pi} \biggr)\biggr] x_i ~-~2x_{i-1}
\biggr\}\biggl[1 + \biggl( \frac{\delta \xi}{2\xi}\biggr) \mathcal{H} \biggr]^{-1} \, .
</math>
</td>
</tr>
</table>

</li>
<li>[[SSC/Stability/BiPolytropes/HeadScratching#Interface|SSC/Stability/BiPolytropes/HeadScratching#Interface]]
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>0</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\biggl[ \gamma_c x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln r^*} \biggr)
- \gamma_e x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d \ln r^*} \biggr)\biggr]_i </math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl[ x_\mathrm{env} \biggl( 3 + \frac{d\ln x_\mathrm{env}}{d \ln r^*} \biggr)\biggr]_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{5}\biggl[ x_\mathrm{core} \biggl( 3 + \frac{d\ln x_\mathrm{core}}{d\ln r^*} \biggr) \biggr]_i
</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl[\frac{d\ln x_\mathrm{env}}{d \ln r^*} \biggr]_i</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{3}{5}\biggl[ \frac{d\ln x_\mathrm{core}}{d\ln r^*} -2\biggr]_i
\, .</math>
</td>
</tr>
</table>

</li></ul>
</li>
<li>
Surface boundary condition taken from:   [[SSC/Stability/BiPolytropes/HeadScratching#Surface_Boundary_Condition|SSC/Stability/BiPolytropes/HeadScratching#Surface_Boundary_Condition]]
<ul>
<li>
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>r_0 \frac{d\ln x}{dr_0}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>
        at         <math>~r_0 = R \, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ r^* \frac{d\ln x}{dr^*}</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-\alpha_g + \frac{R^3}{\gamma_g GM_\mathrm{tot}} \biggl[\frac{2\pi G \rho_c \sigma_c^2}{3} \biggr]
</math>        at         <math>~r^* = R^*
\, ,</math>
</td>
</tr>

<tr>
<td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{d\ln x}{d\ln r^*} \biggr]_s</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
- (\alpha_g)_\mathrm{env} + \frac{1}{2 (\gamma_g)_\mathrm{env}}\biggl[\frac{\sigma_c^2 \rho_c}{\bar\rho } \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">
 
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<math>
-1 + \frac{1}{4}\biggl[\frac{\sigma_c^2}{(\bar\rho)^*} \biggr]
\, .</math>
</td>
</tr>
</table>

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

==Example Eigenvectors Along 0.31 Sequence==

===First Analysis===

===Sigma-Squared Scaled to Mean Density, and Better Error Analysis===

====From the K-BK74 Analysis====

Keep in mind that, from the [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings|accompanying K-BK74 analysis]], we obtained the following eigenvector for the <math>\nu_\mathrm{max}</math> model along the <math>\mu_e/\mu_c = 0.31</math> equilibrium model sequence:

<table border="1" align="center" cellpadding="5"><tr><td align="center">
[[File:K-BK74eigenfunctionVsRadius.png|left||450px|File:K-BK74eigenfunctionVsRadius.png]]
</td></tr></table>

====ξ<sub>i</sub> = 2.804179398====


<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="3">
<b>Excel file:Models031 & pages …</b>
</td>
<td align="center" colspan="2"><b>xi02804gMode0</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 0.00000000</math></td>
<td align="center" colspan="2"><b>xi02804gMode1</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 11.60356417</math></td>
<td align="center" colspan="2"><b>xi02804gMode2</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 39.16484857</math></td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:Xi02804gsequence.png|300px|xi02804gsequence]]
</td>
<td align="center"><math>\xi_i</math></td>
<td align="right">2.804179398</td>
<td align="center" rowspan="8" bgcolor="white">Linear<br />
[[File:Xi02804gMode0.png|300px|xi02804gMode0]]
</td>
<td align="center" rowspan="8" bgcolor="white">Log10<br />
[[File:Xi02804gMode0Log.png|300px|xi02804gMode0Log]]
</td>
<td align="center" rowspan="8">Linear<br />
[[File:Xi02804gMode1.png|300px|xi02804gMode1]]
</td>
<td align="center" rowspan="8">Log10<br />
[[File:Xi02804gMode1Log.png|300px|xi02804gMode1Log]]
</td>
<td align="center" rowspan="8">Linear<br />
[[File:Xi02804gMode2.png|300px|xi02804gMode2]]
</td>
<td align="center" rowspan="8">Log10<br />
[[File:Xi02804gMode2Log.png|300px|xi02804gMode2Log]]
</td>
</tr>
<tr>
<td align="center"><math>q</math></td>
<td align="right">0.156742251</td>
</tr>
<tr>
<td align="center"><math>\nu</math></td>
<td align="right">0.199510368</td>
</tr>
<tr>
<td align="center"><math>\rho_c/\bar\rho</math></td>
<td align="right">3.57005807 × 10<sup>2</sup></td>
</tr>
<tr>
<td align="center" colspan="1">Mode</td>
<td align="center" colspan="1"><math>\sigma_c^2 \cdot (\rho_c/\bar\rho)</math></td>
</tr>
<tr>
<td align="center" bgcolor="yellow">0</td>
<td align="right" bgcolor="yellow">0.0</td>
</tr>
<tr>
<td align="center">1</td>
<td align="right">11.60356417</td>
</tr>
<tr>
<td align="center">2</td>
<td align="right">39.16484857</td>
</tr>
</table>

====ξ<sub>i</sub> = 4====

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="3">
<b>Excel file:Models031 & pages …</b>
</td>
<td align="center" colspan="2"><b>xi04gMode0</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = -192.1449647</math><br /><font color="red">CAUTION:  Surface B.C. not satisfied!</font></td>
<td align="center" colspan="2"><b>xi04gMode1</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 10.58972472</math></td>
<td align="center" colspan="2"><b>xi04gMode2</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 38.57111996</math></td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:Xi04gsequence.png|300px|xi04gsequence]]
</td>
<td align="center"><math>\xi_i</math></td>
<td align="right">4</td>
<td align="center" rowspan="8" bgcolor="white">Linear<br />
[[File:Xi04gMode0.png|300px|xi04gMode0]]
</td>
<td align="center" rowspan="8" bgcolor="white">Log10<br />
[[File:Xi04gMode0Log.png|300px|xi04gMode0Log]]
</td>
<td align="center" rowspan="8">Linear<br />
[[File:Xi04gMode1.png|300px|xi04gMode1]]
</td>
<td align="center" rowspan="8">Log10<br />
[[File:Xi04gMode1Log.png|300px|xi04gMode1Log]]
</td>
<td align="center" rowspan="8">Linear<br />
[[File:Xi04gMode2.png|300px|xi04gMode2]]
</td>
<td align="center" rowspan="8">Log10<br />
[[File:Xi04gMode2Log.png|300px|xi04gMode2Log]]
</td>
</tr>
<tr>
<td align="center"><math>q</math></td>
<td align="right">0.136805779</td>
</tr>
<tr>
<td align="center"><math>\nu</math></td>
<td align="right">0.266175482</td>
</tr>
<tr>
<td align="center"><math>\rho_c/\bar\rho</math></td>
<td align="right">1.656926473 × 10<sup>3</sup></td>
</tr>
<tr>
<td align="center" colspan="1">Mode</td>
<td align="center" colspan="1"><math>\sigma_c^2 \cdot (\rho_c/\bar\rho)</math></td>
</tr>
<tr>
<td align="center" bgcolor="white">0</td>
<td align="center" bgcolor="white">-192.1449647</td>
</tr>
<tr>
<td align="center">1</td>
<td align="right">+10.58972472</td>
</tr>
<tr>
<td align="center">2</td>
<td align="right">+38.57111996</td>
</tr>
</table>

====ξ<sub>i</sub> = 9====

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="3">
<b>Excel file:Models031 & pages …</b>
</td>
<td align="center" colspan="2"><b>xi09gMode0</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = -15553.34096</math><br /><font color="red">CAUTION:  Surface B.C. not satisfied!</font></td>
<td align="center" colspan="2"><b>xi09gMode1</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 11.16595582</math></td>
<td align="center" colspan="2"><b>xi09gMode2</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 39.57399551</math></td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:Xi09gsequence.png|300px|xi09gsequence]]
</td>
<td align="center"><math>\xi_i</math></td>
<td align="right">9</td>
<td align="center" rowspan="8" bgcolor="white">Linear<br />
[[File:Xi09gMode0.png|300px|xi09gMode0]]
</td>
<td align="center" rowspan="8" bgcolor="white">Log10<br />
[[File:Xi09gMode0Log.png|300px|xi09gMode0Log]]
</td>
<td align="center" rowspan="8">Linear<br />
[[File:Xi09gMode1.png|300px|xi09gMode1]]
</td>
<td align="center" rowspan="8">Log10<br />
[[File:Xi09gMode1Log.png|300px|xi09gMode1Log]]
</td>
<td align="center" rowspan="8">Linear<br />
[[File:Xi09gMode2.png|300px|xi09gMode2]]
</td>
<td align="center" rowspan="8">Log10<br />
[[File:Xi09gMode2Log.png|300px|xi09gMode2Log]]
</td>
</tr>
<tr>
<td align="center"><math>q</math></td>
<td align="right">0.075617469</td>
</tr>
<tr>
<td align="center"><math>\nu</math></td>
<td align="right">0.337216704</td>
</tr>
<tr>
<td align="center"><math>\rho_c/\bar\rho</math></td>
<td align="right">1.155525994 × 10<sup>5</sup></td>
</tr>
<tr>
<td align="center" colspan="1">Mode</td>
<td align="center" colspan="1"><math>\sigma_c^2 \cdot (\rho_c/\bar\rho)</math></td>
</tr>
<tr>
<td align="center" bgcolor="white">0</td>
<td align="center" bgcolor="white">-15553.34096</td>
</tr>
<tr>
<td align="center">1</td>
<td align="right">+11.16595582</td>
</tr>
<tr>
<td align="center">2</td>
<td align="right">+39.57399551</td>
</tr>
</table>

====ξ<sub>i</sub> = 14====

<table border="1" align="center" cellpadding="5">
<tr>
<td align="center" colspan="3">
<b>Excel file:Models031 & pages …</b>
</td>
<td align="center" colspan="2"><b>xi09gMode0</b></td>
<td align="center" colspan="2"><b>xi09gMode1</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 11.01793184</math></td>
<td align="center" colspan="2"><b>xi09gMode2</b><br /><math>\sigma_c^2 \cdot (\rho_c/\bar\rho) = 39.30533842</math></td>
</tr>
<tr>
<td align="center" rowspan="8">
[[File:Xi14gsequence.png|300px|xi14gsequence]]
</td>
<td align="center"><math>\xi_i</math></td>
<td align="right">14</td>
<td align="center" rowspan="8" bgcolor="white">N/A<br />
[[File:Xi14gMode0.png|300px|xi09gMode0]]
</td>
<td align="center" rowspan="8" bgcolor="white">N/A<br />
[[File:Xi14gMode0Log.png|300px|xi09gMode0Log]]
</td>
<td align="center" rowspan="8">Linear<br />
[[File:Xi14gMode1.png|300px|xi14gMode1]]
</td>
<td align="center" rowspan="8">Log10<br />
[[File:Xi14gMode1Log.png|300px|xi14gMode1Log]]
</td>
<td align="center" rowspan="8">Linear<br />
[[File:Xi14gMode2.png|300px|xi14gMode2]]
</td>
<td align="center" rowspan="8">Log10<br />
[[File:Xi14gMode2Log.png|300px|xi14gMode2Log]]
</td>
</tr>
<tr>
<td align="center"><math>q</math></td>
<td align="right">0.04880185</td>
</tr>
<tr>
<td align="center"><math>\nu</math></td>
<td align="right">0.316868628</td>
</tr>
<tr>
<td align="center"><math>\rho_c/\bar\rho</math></td>
<td align="right">1.4728825 × 10<sup>6</sup></td>
</tr>
<tr>
<td align="center" colspan="1">Mode</td>
<td align="center" colspan="1"><math>\sigma_c^2 \cdot (\rho_c/\bar\rho)</math></td>
</tr>
<tr>
<td align="center" bgcolor="white">0</td>
<td align="center" bgcolor="white">---</td>
</tr>
<tr>
<td align="center">1</td>
<td align="right">+11.01793184</td>
</tr>
<tr>
<td align="center">2</td>
<td align="right">+39.30533842</td>
</tr>
</table>

=See Also=

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SSC/Stability/BiPolytropes/SuccinctDiscussion - Revision history