Editing Appendix/Ramblings/51BiPolytropeStability/NoAnalytic

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=Analog of Bonnor-Ebert Limiting Pressure=
As has been [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_(n=5)|demonstrated in an accompanying discussion]], the mass of a pressure-truncated, n = 5 polytrope is,

<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{M}{M_\mathrm{SWS}} </math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[  \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\xi_e^2/3)^3}{(1+\xi_e^2/3)^{4}} \biggr]^{1/2}
</math>
  </td>
</tr>
</table>
[[SSC/Structure/PolytropesEmbedded#Stahler's_Presentation|where]],

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

<tr>
  <td align="right">
<math>
M_\mathrm{SWS}
</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]_{n=5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2\cdot 3}{5G} \biggr)^{3/2} K^{5/3} P_\mathrm{e}^{-1/6} 
</math>
  </td>
</tr>
</table>

Now, as we have recorded in an [[SSC/Structure/PolytropesEmbedded#Some_Tabulated_Values|accompanying summary table]], the maximum (critical) mass arises precisely at <math>\xi_e = 3</math>.  That is,

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

<tr>
  <td align="right">
<math>M_\mathrm{crit}^6</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>M_\mathrm{SWS}^6
\biggl[  \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\xi_e^2/3)^3}{(1+\xi_e^2/3)^{4}} \biggr]^{3}_{\xi_e=3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl[ \biggl( \frac{2\cdot 3}{5G} \biggr)^{3/2} K^{5/3} P_\mathrm{e}^{-1/6} \biggr]^6
\biggl[  \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{3^3}{2^8} \biggr]^{3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl( \frac{2\cdot 3}{5} \biggr)^{9} \frac{K^{10}}{G^9 P_\mathrm{e} }
\biggl( \frac{3^4 \cdot 5^3}{2^{10}\pi} \biggr)^{3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl[\biggl( \frac{3}{2} \biggr)^7 \frac{1}{\pi}\biggr]^3 \frac{K^{10}}{G^9 P_\mathrm{e} } \, .
</math>
  </td>
</tr>
</table>

According to our [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Step_4:_Throughout_the_core|accompanying renormalization of the equilibrium configuration]] (also see below),

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

<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\mathcal{m}_\mathrm{surf}}{M_\mathrm{tot}}\biggr)^6 K_c^{10} G^{-9}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} 
\, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>M_r</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl( \frac{M_\mathrm{tot}}{\mathcal{m}_\mathrm{surf}}\biggr) \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} 
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr] \, .</math>
  </td>
</tr>
</table>

That is,

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

<tr>
  <td align="right"><math>M^6_\mathrm{core} \cdot P_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl( \frac{M_\mathrm{tot}}{\mathcal{m}_\mathrm{surf}}\biggr)^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{12} 
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{3} \biggl[ \xi_i^3 \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-3/2} \biggr]^6 
\cdot
\biggl(\frac{\mathcal{m}_\mathrm{surf}}{M_\mathrm{tot}}\biggr)^6 K_c^{10} G^{-9}\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} 
\biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math> 
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{3} \biggl[ \xi_i^{3} \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-2} \biggr]^6 
\frac{K_c^{10}}{ G^9} \, .
</math>
  </td>
</tr>
</table>
Setting these expressions equal to one another means,

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

<tr>
  <td align="right">
<math> 
\frac{3^2}{2^4}   
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\xi_i^{3} \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-2} \, .</math>
  </td>
</tr>
</table>
What do I make of this?

=Do Not Confine Search to Analytic Eigenvector=

==Maximum Mass Fraction (&nu;)==

<table border="1" align="center" cellpadding="5" width="80%">
<tr>
  <td align="center" colspan="5">
'''Maximum Mass Fraction''' <math>(\nu)</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>\Lambda_i</math></td>
  <td align="center"><math>\nu</math></td>
  <td align="center"><math>q</math></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"><math>\frac{2}{\pi}</math></td>
  <td align="center">0.0</td>
</tr>

<tr>
  <td align="center">0.3300</td>
  <td align="center">24.00496</td>
  <td align="center">0.2128753</td>
  <td align="center">0.52024552</td>
  <td align="center">0.038378833</td>
</tr>

<tr>
  <td align="center">0.316943</td>
  <td align="center">10.744571</td>
  <td align="center">0.4903393</td>
  <td align="center">0.382383875</td>
  <td align="center">0.068652715</td>
</tr>

<tr>
  <td align="center">0.3100<sup>&dagger;</sup></td>
  <td align="center">9.0149598</td>
  <td align="center">0.5983505</td>
  <td align="center">0.213039696</td>
  <td align="center">0.153835</td>
</tr>

<tr>
  <td align="center">0.3090</td>
  <td align="center">8.83017723</td>
  <td align="center">0.6130669</td>
  <td align="center">0.331475715</td>
  <td align="center">0.076265588</td>
</tr>

<tr>
  <td align="center"><math>\frac{1}{4}</math></td>
  <td align="center">4.9379256</td>
  <td align="center">1.4179907</td>
  <td align="center">0.139370157</td>
  <td align="center">0.084824137</td>
</tr>

<tr>
  <td align="left" colspan="5">
<sup>&dagger;</sup>This model also used in a [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings|related discussion]] where we investigate the relevance of the {{ B-KB74 }} conjecture.
  </td>
</tr>
</table>

==Overview==

<font color="red"><b>STEP01:</b></font><br />
Develop an algorithm (for Excel) that numerically integrates the LAWEs from the center to the surface of a <math>(n_c, n_e) = (5, 1)</math> bipolytrope, for an arbitrary specification of the three parameters: &nbsp; <math>\mu_e/\mu_c, \xi_i</math>, and <math>\sigma_c^2</math>.  
<ul>
  <li>Enforce the proper interface matching condition(s) at the interface location, <math>\xi_i</math>.</li>
  <li>Note that in general, for an arbitrarily chosen set of the three parameter values, the resulting ''surface'' displacement function will not match the desired boundary condition.</li>
</ul>

<font color="red"><b>STEP02:</b></font><br />
Fix your chosen value of the parameter pair, <math>(\mu_e/\mu_c, \xi_i)</math>, and vary <math>\sigma_c^2</math> until the proper surface boundary condition is realized.
<ul>
  <li>
In an [[SSC/Stability/BiPolytropes#Fundamental_Modes|accompanying discussion]], we claim to have identified at what point along various <math>\mu_e/\mu_c</math> sequences the fundamental mode of radial oscillation becomes unstable &#8212; that is, when <math>\sigma_c^2 = 0</math>.  For a given choice of <math>\mu_e/\mu_c</math>, it would be wise to begin our eigenvector search at a value of <math>\xi_i < [\xi_i]_\mathrm{FM}</math>, as specified in the following table:
<table border="1" cellpadding="10" align="center" width="40%">
<tr>
  <td align="center" colspan="2">Marginally Unstable Fundamental Modes</td>
</tr>
<tr>
  <td align="center"><math>\frac{\mu_e}{\mu_c}</math></td>
  <td align="center"><math>\biggl[ \xi_i \biggr]_{\mathrm{FM}}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="right">1.6686460157</td>
</tr>
<tr>
  <td align="center"><math>\tfrac{1}{2}</math></td>
  <td align="right">2.27925811317</td>
</tr>
<tr>
  <td align="center">0.345</td>
  <td align="right">2.560146865247</td>
</tr>
<tr>
  <td align="center"><math>\tfrac{1}{3}</math></td>
  <td align="right">2.582007485476</td>
</tr>
<tr>
  <td align="center">0.309</td>
  <td align="right">2.6274239687695</td>
</tr>
<tr>
  <td align="center"><math>\tfrac{1}{4}</math></td>
  <td align="right">2.7357711469398</td>
</tr>

<tr>
  <td align="left" colspan="2">See orange-colored triangular markers in the associated [[SSC/Stability/BiPolytropes#Figure4|Figure 4]]</td>
</tr>
</table> 
  </li>
  <li>
Keep steadily raising the value of the interface location until you find the 1<sup>st</sup> overtone mode; a related discussion (with animation) shows the results of this type of search in the context of [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Isolated_n_=_1_Polytrope|isolated n = 1 polytropes]].  Our expectation is that, if this mode is unstable, the model will coincide with the turning point along the equilibrium sequence and its eigenvector will essentially overlap with the eigenvector found using the [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Model_Pairings|B-KB74 conjecture]].  At the same time, the square-of-the-eigenfrequency for the fundamental mode will be ''very'' negative.
  </li>
</ul>

<font color="red"><b>STEP03:</b></font><br />
Regarding analytically specified eigenvectors that satisfy the governing LAWES &hellip;
<ul>
  <li>If we force <math>\sigma_c^2 = 0</math> in the core, we have shown that a parabolic-shaped eigenfunction satisfies the LAWE of the core.  We expect this eigenfunction to precisely overlay the numerically determined, marginally unstable displacement function in both the case of the unstable fundamental mode and the case of the unstable 1<sup>st</sup> overtone.
  </li>
  <li>If we force <math>\sigma_c^2 = 0</math> in the envelope, we have derived a different &#8212; <math>\cos(\eta - B)</math> dependent &#8212; eigenfunction that satisfies the LAWE of the envelope.  However, this proves to be irrelevant in the context of our bipolytrope because the derived eigenfunction does not match the physically relevant surface boundary condition.
  </li>
</ul>

==Renormalized LAWE==

As presented, for example, in a [[SSC/Stability/BiPolytropes/HeadScratching#LAWE|parallel discussion]], in terms of our original ( * ) parameter normalizations, the polytropic LAWE takes the form,
<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} + \biggl\{ \frac{4}{r^*} -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)^2}\biggr\} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{\omega^2}{\gamma_\mathrm{g} G\rho_c} + \biggl(\frac{4}{\gamma_\mathrm{g}} - 3\biggr)\frac{ M_r^*}{(r^*)^3}\biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x \, ,
</math>
  </td>
</tr>
</table>
where, <math>\alpha_g \equiv (3 - 4/\gamma_g)</math>, and
<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>

</table>

<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">

<div align="center"><b>New Normalization</b></div>

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

<tr>
  <td align="right"><math>\tilde\rho</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\rho \biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{-5} \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{P}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>P \biggl[K_c^{-10} G^{9} M_\mathrm{tot}^{6}  \biggr] \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{r}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>r \biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2}  \biggr]\, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{M}_r</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\frac{M_r}{M_\mathrm{tot}} \, ;</math></td>
</tr>

<tr>
  <td align="right"><math>\tilde{H}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>H \biggl[K_c^{-5 / 2} G^{3 / 2} M_\mathrm{tot} \biggr] \, .</math></td>
</tr>
</table>
</td></tr></table>

Switching to the new normalization, where [[SSC/Structure/BiPolytropes/Analytic51Renormalize|it is understood]] that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\biggl(\frac{G}{K_c}\biggr)^{3 / 2}M_\mathrm{tot} \rho_0^{1 / 5}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \mathcal{m}_\mathrm{surf} \, ,
  </math></td>
</tr>
</table>
where,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\mathcal{m}_\mathrm{surf}</math></td>
  <td align="center"><math>\equiv</math></td>
  <td align="left"><math>\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \theta_i^{-1}\biggl(-\eta^2 \frac{d\phi}{d\eta}\biggr)_s 
= \biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i} \, ,</math></td>
</tr>
</table>

we find the following relevant relations:

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

<tr>
  <td align="right"><math>(r^*)^2</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
r^2 \biggl[ \frac{G \rho_0^{4 / 5}}{K_c}\biggr]
=
\tilde{r}^2 \biggl[ \biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2}  \biggr]^{-2} \biggl[ \frac{G \rho_0^{4 / 5}}{K_c}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\tilde{r}^2 \biggl[ \biggl( \frac{G}{K_c} \biggr)^{3 / 2}  M_\mathrm{tot}\rho_0^{1 / 5} \biggr]^4 \, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\frac{\rho^*}{P^*}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{\rho}{\rho_0} \biggr) \biggl[\frac{K_c\rho_0^{6/5}}{P}\biggr]
=
\biggl(\frac{\rho}{P} \biggr) \biggl[K_c\rho_0^{1/5}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\frac{\tilde\rho}{\tilde{P}}
\biggl[\biggl( \frac{K_c}{G} \biggr)^{3 / 2} \frac{1}{M_\mathrm{tot}} \biggr]^{5}
\biggl[K_c^{-10} G^{9} M_\mathrm{tot}^{6}  \biggr] 
\biggl[K_c\rho_0^{1/5}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{\tilde\rho}{\tilde{P}}\biggl[ \biggl(\frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot} \rho_0^{1/5} \biggr] 
\, ;
</math>
  </td>
</tr>

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

</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{{\tilde{M}}_r}{\tilde{r}} 
\biggl[\biggl( \frac{K_c}{G} \biggr)^{5 / 2} M_\mathrm{tot}^{-2}  \biggr]
\biggl[
\biggl( \frac{G}{K_c} \biggr)\rho_0^{-1/5}
\biggr]M_\mathrm{tot}
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{{\tilde{M}}_r}{\tilde{r}} 
\biggl[ \biggl(\frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot} \rho_0^{1/5} \biggr]^{-1} \, .
</math>
  </td>
</tr>

</table>

Therefore, in terms of the renormalized variables the LAWE becomes,
<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} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} 
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}}  ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\}  x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5}  \biggr]^{-4}\frac{d^2x}{d\tilde{r}^2} 
+ \biggl\{ 4 - \biggl(\frac{\tilde\rho}{\tilde{P}}\biggr)\frac{ \tilde{M}_r}{\tilde{r}}\biggr\}
\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5}  \biggr]^{-4}\frac{1}{\tilde{r}} \frac{dx}{d\tilde{r}} 
+ \frac{\tilde\rho}{\tilde{P}}\biggl[ \biggl(\frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot} \rho_0^{1/5} \biggr]
\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} 
-~\frac{\alpha_\mathrm{g} \tilde{M}_r}
{(\tilde{r})^3}
\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5}  \biggr]^{-5}
\biggr\}  x \, .
</math>
  </td>
</tr>
</table>

----

Now let's change how "time" &#8212; and, hence, how frequency &#8212; is normalized.  Specifically, we employ the mapping,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2\pi}{3}\biggr) \frac{\sigma_c^2}{\gamma_g}</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>
\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5}  \biggr]^{-5} \biggl( \frac{2\pi}{3}\biggr) \frac{\tilde\sigma^2}{\gamma_g} \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tilde\sigma^2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3\omega^2}{2\pi}
\biggl[K_c^{-15/2} G^{+13/2} M_\mathrm{tot}^{5}  \biggr]  \, .
</math>
  </td>
</tr>
</table>
That is, the renormalized LAWE becomes,
<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[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5}  \biggr]^{-4}\frac{d^2x}{d\tilde{r}^2} 
+ \biggl\{ 4 - \biggl(\frac{\tilde\rho}{\tilde{P}}\biggr)\frac{ \tilde{M}_r}{\tilde{r}}\biggr\}
\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5}  \biggr]^{-4}\frac{1}{\tilde{r}} \frac{dx}{d\tilde{r}} 
+ \frac{\tilde\rho}{\tilde{P}}\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5}  \biggr]^{-4}
\biggl\{ \frac{2\pi \tilde\sigma^2}{3\gamma_\mathrm{g}} 
-~\frac{\alpha_\mathrm{g} \tilde{M}_r}
{(\tilde{r})^3}
\biggr\}  x \, .
</math>
  </td>
</tr>
</table>


----


After multiplying through by 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\biggl( \frac{G}{K_c} \biggr)^{3 / 2} M_\mathrm{tot}\rho_0^{1 / 5}  \biggr]^{4}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \mathcal{m}_\mathrm{surf} \biggr]^4
</math>
  </td>
</tr>
</table>
we have,
<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}{d\tilde{r}^2} 
+ \overbrace{\biggl\{ 4 - \biggl(\frac{\tilde\rho}{\tilde{P}}\biggr)\frac{ \tilde{M}_r}{\tilde{r}}\biggr\}}^{\tilde\mathcal{H}} 
\frac{1}{\tilde{r}}\frac{dx}{d\tilde{r}} 
+ \underbrace{\frac{\tilde\rho}{\tilde{P}}
\biggl\{ \frac{2\pi \tilde\sigma^2}{3\gamma_\mathrm{g}} 
-~\frac{\alpha_\mathrm{g} \tilde{M}_r}
{(\tilde{r})^3}
\biggr\}}_{\tilde\mathcal{K}}  x \, .
</math>
  </td>
</tr>
</table>

<table border="1" align="center" cellpadding="8" width="80%">
<tr>
  <td align="left">
From above, note that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \mathcal{m}_\mathrm{surf} \biggr]^5
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \biggl(\frac{2}{\pi}\biggr)^{5 / 2} \biggl[ \frac{A\eta_s}{\theta_i}\biggr]^5 \, ;
</math>
  </td>
</tr>
</table>

and, according to the [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|parameter values associated with the bipolytropic equilibrium configuration]],
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr) \frac{\rho_c}{\bar\rho}
</math>
  </td>
  <td align="center">
<math>
=
</math>
  </td>
  <td align="left">
<math>
\frac{\eta_s^2}{3A\theta_i^5} \, .
</math>
  </td>
</tr>
</table>

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

----


<font color="red">CAUTION:</font>

==Numerical Integration==

Here, the finite-difference representation of the LAWE parallel the approach used in a [[SSC/Stability/BiPolytropes#Numerical_Integration|closely related discussion]].

===General Approach===
The 2<sup>nd</sup>-order ODE that must be integrated to obtain the desired eigenvectors has the generic form,
<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>
x'' + \frac{\tilde\mathcal{H}}{\tilde{r}} x' + \tilde\mathcal{K}x \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x'</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{dx}{d\tilde{r}}</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>x''</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d^2x}{d\tilde{r}^2} \, .</math>
  </td>
</tr>
</table>
Adopting the same approach [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward|as before when we integrated the LAWE for pressure-truncated polytropes]], we will enlist the finite-difference approximations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x'</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2\delta \tilde{r}}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>x''</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ -2x_j + x_-}{(\delta \tilde{r})^2} \, .
</math>
  </td>
</tr>
</table>

The finite-difference representation of the LAWE is, therefore,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{x_+ -2x_j + x_-}{(\delta\tilde{r})^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\tilde\mathcal{H}}{\tilde{r}} \biggl[ \frac{x_+ - x_-}{2\delta \tilde{r}} \biggr] ~-~ \tilde\mathcal{K}x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ -2x_j + x_-</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\delta \tilde{r}}{2\tilde{r}} \biggl[ x_+ - x_- \biggr]\tilde\mathcal{H} ~-~ (\delta \tilde{r})^2 \tilde\mathcal{K}x_j 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_{j+1} \biggl[1 + \biggl( \frac{\delta \tilde{r}}{2\tilde{r}}\biggr) \tilde\mathcal{H} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 2 - (\delta \tilde{r})^2 \tilde\mathcal{K}\biggr] x_j ~
-~\biggl[ 1 - \biggl( \frac{\delta \tilde{r}}{2\tilde{r}} \biggr) \tilde\mathcal{H} \biggr]x_{j-1} \, .
</math>
  </td>
</tr>
</table>

In what follows we will also find it useful to rewrite <math>~\mathcal{K}</math> in the form,
<div align="center">
<math>\tilde\mathcal{K} ~\rightarrow ~\biggl(\frac{\tilde\sigma^2}{\gamma_\mathrm{g}}\biggr) \tilde\mathcal{K}_1 
- \alpha_\mathrm{g} \tilde\mathcal{K}_2 \, .</math>
</div>

The relevant coefficient expressions for ''all'' regions of the configuration are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\tilde\mathcal{H}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl\{ 4 -\biggl(\frac{\tilde\rho}{\tilde{P}}\biggr)\frac{ \tilde{M}_r}{\tilde{r}}\biggr\}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; , &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\tilde\mathcal{K}_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{2\pi }{3}\biggl(\frac{\tilde\rho}{ \tilde{P} } \biggr) 
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\tilde\mathcal{K}_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\tilde\rho}{ \tilde{P} } \biggr)\frac{\tilde{M}_r}{\tilde{r}^3} \, .
</math>
  </td>
</tr>
</table>

=See Also=


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