Editing SSC/Structure/BiPolytropes/Analytic51Renormalize

__FORCETOC__
=BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Renormalized=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize|Part I: &nbsp; (5, 1) Analytic Renormalize]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2|Part II:&nbsp; Envelope]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt3|III:&nbsp; Interface Pressure Gradient]]<br />&nbsp;</td>
</tr>
</table>

This chapter very closely parallels our [[SSC/Structure/BiPolytropes/Analytic51|original analytic derivation]] &#8212; see also, {{ EFC98full }} &#8212; of the structure of bipolytropes 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.  Our primary objective, here, is to renormalize the principal set of variables, replacing the central density with the configuration's total mass, so that the mass is held fixed along each model ''sequence''.   

From [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|Table 1 of our original analytic derivation]], we see that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 M_\mathrm{tot}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \rho_0^{-1 / 5}
  </math></td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \rho_0 </math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\biggl\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^5
\, ,</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>

==Steps 2 &amp; 3==
Based on the discussion [[SSC/Structure/Polytropes#n_.3D_5_Polytrope|presented elsewhere of the structure of an isolated <math>n=5</math> polytrope]], the core of this bipolytrope will have the following properties:
<div align="center">
<math>
\theta(\xi) = \biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-1/2} ~~~~\Rightarrow ~~~~  \theta_i = \biggl[ 1 + \frac{1}{3}\xi_i^2 \biggr]^{-1/2} ;
</math>

<math>
\frac{d\theta}{d\xi} = - \frac{\xi}{3}\biggl[ 1 + \frac{1}{3}\xi^2 \biggr]^{-3/2} ~~~~\Rightarrow ~~~~  \biggl(\frac{d\theta}{d\xi}\biggr)_i = - \frac{\xi_i}{3}\biggl[ 1 + \frac{1}{3}\xi_i^2 \biggr]^{-3/2} \, .
</math>
</div>
The first zero of the function <math>~\theta(\xi)</math> and, hence, the surface of the corresponding isolated <math>~n=5</math> polytrope is located at <math>~\xi_s = \infty</math>.  Hence, the interface between the core and the envelope can be positioned anywhere within the range, <math>~0 < \xi_i < \infty</math>.

==Step 4:  Throughout the core==
<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="center" colspan="3">
Specify:  <math>K_c</math> and <math>\rho_0 ~\Rightarrow</math>
  </td>
  <td colspan="2">
&nbsp;
  </td>
</tr>
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \theta^{n_c}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2}</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^{1+1/n_c} \theta^{n_c + 1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3}</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</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} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]</math>
  </td>
</tr>
</table>

</div>

----

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="center">
Specify:  <math>K_c</math> and <math>M_\mathrm{tot} ~\Rightarrow</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right"><math>\rho</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^5
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2}</math>
  </td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl(\frac{\mathcal{m}_\mathrm{surf}}{M_\mathrm{tot}}\biggr)^5 \biggl(\frac{K_c}{G}\biggr)^{15 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>P</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>K_c \biggl\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^6
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} </math>
  </td>
</tr>
<tr>
  <td align="right">&nbsp;</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>r</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^{-2}
\biggl[ \frac{K_c}{G} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi</math>
  </td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl( \frac{\mathcal{m}_\mathrm{surf}}{M_\mathrm{tot}}\biggr)^{-2} \biggl(\frac{K_c}{G}\biggr)^{-5 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} 
\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>M_r</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\biggl\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^{-1}
\biggl[ \frac{K_c^3}{G^3 } \biggr]^{1/2} \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>
<tr>
  <td align="right">&nbsp;</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>

<span id="NewNormalization">&nbsp;</span>
<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>

<span id="CoreParameters">After applying this new normalization, we have throughout the core,</span>

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

<tr>
  <td align="right"><math>\tilde{\rho}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} 
\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\tilde{P}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^6 \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>\tilde{r}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} 
\biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi \, ;</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\tilde{M}_r</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>\mathcal{m}_\mathrm{surf}^{-1} \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>

==Step 8:  Throughout the envelope==
Given (from above) that,
<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\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^5
\, ,</math></td>
</tr>
</table>
we have throughout the envelope,

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i \phi</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^5
 \biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{5}_i \phi</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{\biggl(\frac{K_c}{G}\biggr)^{15 / 2} M_\mathrm{tot}^{-5}\biggr\}
\mathcal{m}_\mathrm{surf}^5 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-9} \theta^{5}_i \phi \, ;</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^{6}_i \phi^{2}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \biggl\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^6
 \theta^{6}_i \phi^{2}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl\{ K_c^{10} G^{-9}
M_\mathrm{tot}^{-6} \biggr\}
\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \theta^{6}_i \phi^{2} \, ;</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{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta</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\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^{-2}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl\{\biggl(\frac{K_c}{G}\biggr)^{-5 / 2}  M_\mathrm{tot}^{2}\biggr\}
\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r</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{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</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\{\mathcal{m}_\mathrm{surf} \biggl(\frac{K_c}{G}\biggr)^{3 / 2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} M_\mathrm{tot}^{-1}\biggr\}^{-1}
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
M_\mathrm{tot} ~
\mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>
<span id="EnvelopeParameters">Adopting the new normalization then gives,</span>

<div align="center">
<table border="0" cellpadding="3">
<tr>
  <td align="right">
<math>\tilde\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^5 \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-9} \theta^{5}_i \phi \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde{P}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^6 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-12} \theta^{6}_i \phi^{2} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde{r}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde{M}_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>

==Behavior of Central Density Along Equilibrium Sequence==
Each equilibrium sequence will be defined as a sequence of models having the same jump in the mean-molecular weight, <math>\mu_e/\mu_c</math>.  Along a given sequence, we vary the location of the core/envelope interface, <math>\xi_i</math>.  Our desire is to analyze the behavior of the central density, while holding the total mass fixed, as the location of the interface is varied.

The central density is given by the expression,

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

<tr>
  <td align="right"><math>{\tilde\rho}_c</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} 
\biggl[ \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-5/2} \biggr]_{\xi = 0}
=
\mathcal{m}_\mathrm{surf}^5 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} \, ,
</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>=</math></td>
  <td align="left"><math>\biggl(\frac{2}{\pi}\biggr)^{1 / 2} \frac{A\eta_s}{\theta_i} \, .</math></td>
</tr>
</table>
In order to evaluate <math>\mathcal{m}_\mathrm{surf}</math> for a given specification of the interface location, <math>\xi_i</math>, we need to know that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\theta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(1+\frac{1}{3}\xi_i^2\biggr)^{-1 / 2} \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\eta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3} \theta_i^2 \xi_i \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\Lambda_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\frac{\xi_i}{\sqrt{3}} 
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr] \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>A</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\eta_i\biggl(1 + \Lambda_i^2\biggr)^{1 / 2} \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\eta_s</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math> 
\frac{\pi}{2} + \eta_i + \tan^{-1}(\Lambda_i) \, .</math></td>
</tr>
</table>
Keep in mind, as well, that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)^2 \sqrt{3} ~ \biggl[ \frac{\xi_i^3 \theta_i^4}{A\eta_s}\biggr] \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>q \equiv \frac{r_\mathrm{core}}{R}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr) \sqrt{3} ~ \biggl[ \frac{\xi_i \theta_i^2}{\eta_s}\biggr] \, .</math></td>
</tr>
</table>

<span id="Sequences">&nbsp;</span>
<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:RhoC.png|350px|Central Density versus xi_i (mu_ratio = 0.3100)]]</td>
</tr>
</table>

==Model Pairings==
Here we work in the context of the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_=_5_Polytropes|B-KB74 conjecture]]. We will stick with the sequence corresponding to <math>\mu_e/\mu_c = 0.31</math>, and [[SSC/StabilityConjecture/Bipolytrope51#Model_Pairings|continue to examine the model pairings]] (<b>B1</b> and <b>B2</b>) associated with the degenerate model (<b>A</b>) at <math>\nu_\mathrm{max}</math>.  Specifically &hellip;

<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:TurningPoints51Magnify.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="5">
<tr>
  <td align="center" colspan="6"><b>Core</b></td>
  <td align="center" rowspan="13">
[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/BiPolytrope/qAndNuMaxAug21.xlsx --- worksheet = B1renorm]]<br />
[[File:K-BK74eigenfunction.png|350px|B-KB74 Eigenfunction]]
</td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>m_r</math></td>
  <td align="center" colspan="2"><b>B1</b></td>
  <td align="center" colspan="2"><b>B2</b></td>
  <td align="center" rowspan="2"><math>\frac{\delta r}{r} = \frac{\tilde{r}_\mathrm{B1} - \tilde{r}_\mathrm{B2}}{2(\tilde{r}_\mathrm{B1} + \tilde{r}_\mathrm{B2})}</math></td>
</tr>
<tr>
  <td align="center"><math>\xi</math></td>
  <td align="center"><math>\tilde{r}(m_r)</math></td>
  <td align="center"><math>\xi</math></td>
  <td align="center"><math>\tilde{r}(m_r)</math></td>
</tr>
<tr>
  <td align="right">0.0</td>
  <td align="right">0.0</td>
  <td align="right">0.0</td>
  <td align="right">0.0</td>
  <td align="right">0.0</td>
  <td align="right">--</td>
</tr>
<tr>
  <td align="right">0.005</td>
  <td align="right">0.430797</td>
  <td align="right">0.0007299</td>
  <td align="right">0.430395</td>
  <td align="right">0.0007331</td>
  <td align="right">-0.00109</td>
</tr>
<tr>
  <td align="right">0.05</td>
  <td align="right">1.054468</td>
  <td align="right">0.0017865</td>
  <td align="right">1.053194</td>
  <td align="right">0.0017938</td>
  <td align="right">-0.00102</td>
</tr>
<tr>
  <td align="right">0.1</td>
  <td align="right">1.502081</td>
  <td align="right">0.0025449</td>
  <td align="right">1.499761</td>
  <td align="right">0.0025544</td>
  <td align="right">-0.00093</td>
</tr>
<tr>
  <td align="right">0.15</td>
  <td align="right">1.963871</td>
  <td align="right">0.0033273</td>
  <td align="right">1.959917</td>
  <td align="right">0.0033382</td>
  <td align="right">-0.00082</td>
</tr>
<tr>
  <td align="right">0.20</td>
  <td align="right">2.5329785</td>
  <td align="right">0.0042915</td>
  <td align="right">2.525985</td>
  <td align="right">0.0043023</td>
  <td align="right">-0.00063</td>
</tr>
<tr>
  <td align="right">0.25</td>
  <td align="right">3.366385</td>
  <td align="right">0.0057034</td>
  <td align="right">3.352268</td>
  <td align="right">0.0057097</td>
  <td align="right">-0.00028</td>
</tr>
<tr>
  <td align="right">0.30</td>
  <td align="right">5.000525</td>
  <td align="right">0.0084721</td>
  <td align="right">4.959790</td>
  <td align="right">0.0084477</td>
  <td align="right">+0.00072</td>
</tr>
<tr>
  <td align="right">0.33715</td>
  <td align="right">9.11445</td>
  <td align="right">0.015442</td>
  <td align="right">8.89185</td>
  <td align="right">0.0151449</td>
  <td align="right">+0.00486</td>
</tr>
<tr>
  <td align="right" bgcolor="lightgrey">0.3372001</td>
  <td align="right" bgcolor="lightgrey">9.12744</td>
  <td align="right" bgcolor="lightgrey">0.015464</td>
  <td align="right" bgcolor="lightgrey">8.90394</td>
  <td align="right" bgcolor="lightgrey">0.0151654</td>
  <td align="right" bgcolor="lightgrey">+0.00487</td>
</tr>
</table>



<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="6"><b>Envelope</b></td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>m_r</math></td>
  <td align="center" colspan="2"><b>B1</b></td>
  <td align="center" colspan="2"><b>B2</b></td>
  <td align="center" rowspan="2"><math>\frac{\delta r}{r} = \frac{\tilde{r}_\mathrm{B1} - \tilde{r}_\mathrm{B2}}{2(\tilde{r}_\mathrm{B1} + \tilde{r}_\mathrm{B2})}</math></td>
</tr>
<tr>
  <td align="center"><math>\eta</math></td>
  <td align="center"><math>\tilde{r}(m_r)</math></td>
  <td align="center"><math>\eta</math></td>
  <td align="center"><math>\tilde{r}(m_r)</math></td>
</tr>
<tr>
  <td align="right" bgcolor="lightgrey">0.3372001</td>
  <td align="right" bgcolor="lightgrey">0.1703455</td>
  <td align="right" bgcolor="lightgrey">0.015464</td>
  <td align="right" bgcolor="lightgrey">0.1743134</td>
  <td align="right" bgcolor="lightgrey">0.0151654</td>
  <td align="right" bgcolor="lightgrey">+0.00487</td>
</tr>
<tr>
  <td align="right">0.35</td>
  <td align="right">0.3073375</td>
  <td align="right">0.0279002</td>
  <td align="right">0.309463</td>
  <td align="right">0.0269236</td>
  <td align="right">+0.00891</td>
</tr>
<tr>
  <td align="right">0.40</td>
  <td align="right">0.5753765</td>
  <td align="right">0.0522328</td>
  <td align="right">0.576515</td>
  <td align="right">0.0501574</td>
  <td align="right">+0.01013</td>
</tr>
<tr>
  <td align="right">0.45</td>
  <td align="right">0.748189</td>
  <td align="right">0.0679208</td>
  <td align="right">0.749101</td>
  <td align="right">0.0651726</td>
  <td align="right">+0.01032</td>
</tr>
<tr>
  <td align="right">0.50</td>
  <td align="right">0.8885645</td>
  <td align="right">0.0806641</td>
  <td align="right">0.8893695</td>
  <td align="right">0.0773761</td>
  <td align="right">+0.01040</td>
</tr>
<tr>
  <td align="right">0.55</td>
  <td align="right">1.0122575</td>
  <td align="right">0.091893</td>
  <td align="right">1.012999</td>
  <td align="right">0.088132</td>
  <td align="right">+0.01045</td>
</tr>
<tr>
  <td align="right">0.60</td>
  <td align="right">1.126297</td>
  <td align="right">0.1022455</td>
  <td align="right">1.1269968</td>
  <td align="right">0.0980499</td>
  <td align="right">+0.01047</td>
</tr>
<tr>
  <td align="right">0.65</td>
  <td align="right">1.2347644</td>
  <td align="right">0.1120922</td>
  <td align="right">1.2354345</td>
  <td align="right">0.1074841</td>
  <td align="right">+0.01049</td>
</tr>
<tr>
  <td align="right">0.70</td>
  <td align="right">1.3405518</td>
  <td align="right">0.1216956</td>
  <td align="right">1.3411998</td>
  <td align="right">0.1166858</td>
  <td align="right">+0.01051</td>
</tr>
<tr>
  <td align="right">0.75</td>
  <td align="right">1.4461523</td>
  <td align="right">0.131282</td>
  <td align="right">1.4467833</td>
  <td align="right">0.1258716</td>
  <td align="right">+0.01052</td>
</tr>
<tr>
  <td align="right">0.80</td>
  <td align="right">1.5542198</td>
  <td align="right">0.1410924</td>
  <td align="right">1.5548378</td>
  <td align="right">0.1352725</td>
  <td align="right">+0.01053</td>
</tr>
<tr>
  <td align="right">0.85</td>
  <td align="right">1.6683004</td>
  <td align="right">0.1514487</td>
  <td align="right">1.668908</td>
  <td align="right">0.1451967</td>
  <td align="right">+0.01054</td>
</tr>
<tr>
  <td align="right">0.90</td>
  <td align="right">1.794487</td>
  <td align="right">0.1629039</td>
  <td align="right">1.7950862</td>
  <td align="right">0.1561743</td>
  <td align="right">+0.01055</td>
</tr>
<tr>
  <td align="right">0.95</td>
  <td align="right">1.94764</td>
  <td align="right">0.1768072</td>
  <td align="right">1.9482325</td>
  <td align="right">0.1694982</td>
  <td align="right">+0.01055</td>
</tr>
<tr>
  <td align="right">1.00</td>
  <td align="right">2.2820704</td>
  <td align="right">0.2071669</td>
  <td align="right">2.282658</td>
  <td align="right">0.1985936</td>
  <td align="right">+0.01056</td>
</tr>
</table>

<div align="center">
[[File:K-BK74eigenfunctionVsRadius.png|700px|B-KB74 Eigenfunction]]
</div>

==Attempt at Constructing Analytic Eigenfunction Expression==
===Background===

In our [[SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|accompanying discussion of eigenvectors associated with the radial oscillation of pressure-truncated polytropes]], we derived the following,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3"><font color="maroon"><b>Exact Solution to the Polytropic LAWE</b></font></td>
</tr>
<tr>
  <td align="right">
<math>\sigma_c^2 = 0</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>x_P \equiv \frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, .</math>
  </td>
</tr>
</table>
Drawing on the definition of <math>\theta(\xi)</math> for n = 5 polytropes, as given [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|in an accompanying chapter]], we deduce that,

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

<tr>
  <td align="right">
<math>x_P\biggr|_{n=5}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{5}\biggl[1 + \frac{1}{2}\biggl( \frac{1}{\xi \theta^{5}}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{5} - \frac{3}{5\xi} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{5/2} \frac{\xi}{3} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{5} - \frac{1}{5} \biggl( 1 + \frac{\xi^2}{3} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \frac{\xi^2}{15} \, .
</math>
  </td>
</tr>
</table>

And, given that [[SSC/Structure/Polytropes#Primary_E-Type_Solution_2|for n = 1 polytropes]], 
<div align="center">
<math>\theta(\xi) = \frac{\sin\xi}{\xi} \, ,</math>
</div>

we also find,

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

<tr>
  <td align="right">
<math>x_P\biggr|_{n=1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-3 \biggl[ \biggl( \frac{1}{\xi \theta}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=1} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi  \biggr] \, .
</math>
  </td>
</tr>
</table>

===Core===

Allowing for an overall leading scale factor, <math>\alpha</math>, a viable displacement function for the <math>(n = 5)</math> core of our bipolytropic configuration is,

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

<tr>
  <td align="right">
<math>\frac{x_\mathrm{core}}{\alpha}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ 1 - \frac{\xi^2}{15} \biggr] \, .
</math>
  </td>
</tr>
</table>

Throughout the core, the corresponding Lagrangian radial coordinate, <math>\tilde{r}</math>, is given by the expression,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\tilde{r}_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi 
\, .</math>
  </td>
</tr>
</table>

For "model <b>A</b>" the range is,
<div align="center">
<math>0 \le \xi \le \xi_i = 9.0149598 \, .</math>
</div>

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

<font color="red">SLOPE:</font>&nbsp;  What is the slope of the function, <math>x_\mathrm{core}(\tilde{r})</math>, at the interface?
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{dx_\mathrm{core}}{d\xi} \biggr|_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>- \frac{2\alpha \xi_i}{15}
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{dx_\mathrm{core}}{d\tilde{r}} \biggr|_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>- \frac{2\alpha \xi_i}{15} \biggl[ 
\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2} 
\biggr]^{-1}
= -~707.53765~\alpha
\, ,</math>
  </td>
</tr>
</table>
where, for "model <b>A</b>," we have set <math>(\mu_e/\mu_c) = 0.31</math> and <math>\mathcal{m}_\mathrm{surf} = 1.938127063</math>.  Note as well that,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>\frac{d\ln (x_\mathrm{core})}{d\ln\tilde{r}} \biggr|_i = \frac{d\ln (x_\mathrm{core})}{d\ln\xi} \biggr|_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>- \frac{2 \xi_i^2}{15}\biggl[1 - \frac{\xi_i^2}{15}\biggr]^{-1}
= +2.452697
\, .</math>
  </td>
</tr>
</table>

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

===Envelope===
As we have demonstrated [[SSC/Structure/BiPolytropes/Analytic51#Step_6:_Envelope_Solution|in a separate ''structure'' discussion]], the radial profile of the <math>(n = 1)</math> envelope of our bipolytropic configuration is governed by the ''modified'' sinc-function,

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

<tr>
  <td align="right">
<math>\phi(\eta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
A \biggl[ \frac{\sin(\eta-B)}{\eta}\biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ -\frac{d\phi}{d\eta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{A}{\eta^2} \biggl[\sin(\eta-B) - \eta\cos(\eta - B) \biggr] \, .
</math>
  </td>
</tr>
</table>
where, for "model <b>A</b>," <math>A = 0.200812422</math> and <math>B = -0.859270052</math>.

Again allowing for an overall leading scale factor, <math>\beta</math>, a viable displacement function for the <math>(n = 1)</math> envelope of our bipolytropic configuration is,

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

<tr>
  <td align="right">
<math>\frac{x_\mathrm{env}}{3\beta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta \phi} \biggl( - \frac{d\phi}{d\eta} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{A}{\eta^3 } \biggl[\sin(\eta-B) - \eta\cos(\eta - B) \biggr] \cdot  \biggl[\frac{\eta}{A\sin(\eta - B)}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta^2 } \biggl[1 - \eta\cot(\eta - B) \biggr]\, . 
</math>
  </td>
</tr>
</table>

Throughout the envelope, the corresponding Lagrangian radial coordinate is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\tilde{r}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta \, .</math>
  </td>
</tr>
</table>

For "model <b>A</b>" the range is,
<div align="center">
<math>(\eta_i = 0.1723205) \le \eta \le (\eta_s = 2.282322601) \, .</math>
</div>

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

<font color="red">SLOPE:</font>&nbsp;  As we have [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Eigenfunction_Details|detailed elsewhere]], the slope of the function, <math>x_\mathrm{env}(\tilde{r})</math>, is related to the slope of <math>x_\mathrm{core}(\tilde{r})</math> at the interface via the expression,

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

<tr>
  <td align="right">
<math>~
\biggl\{ \frac{d\ln x}{d\ln r}\biggr|_i \biggr\}_\mathrm{env}
=
\biggl\{ \frac{d\ln x}{d\ln \eta}\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>
</table>
In our case, <math>\gamma_c = 6/5</math> and <math>\gamma_e = 2 ~~\Rightarrow \gamma_c/\gamma_e = 3/5</math>.  Hence, from the point of view of the envelope displacement function, at the interface,

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

<tr>
  <td align="right">
<math>
\biggl[ \frac{\tilde{r}}{x_\mathrm{env}} \cdot \frac{d x_\mathrm{env}}{d \tilde{r} } \biggr]_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{5} \biggl\{ \biggl[ \frac{d\ln (x_\mathrm{core})}{d\ln \xi}\biggr]_i  - 2\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3}{5} \biggl\{ \biggl[ 2.452697  - 2\biggr\} = 0.271618 \, .
</math>
  </td>
</tr>
</table>
Now, at the interface of any bipolytrope, the ratio <math>\tilde{r}/x</math> should have the same numerical value whether it is viewed from the point of view of the core or the envelope.  Given that, for our particular "model <b>A</b>",
<div align="center">
<math>\biggl[ \frac{\tilde{r}}{x} \biggr]_i = \frac{0.015315}{0.00485976} = 3.15139 \, ,</math>
</div>
we should expect the slope of the envelope's displacement function at the interface to be,

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

<tr>
  <td align="right">
<math>
\frac{d x_\mathrm{env}}{d \tilde{r} } \biggr|_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0.08619 \, .
</math>
  </td>
</tr>
</table>


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

===Trial Displacement Function===

The blue curve in the following figure results from plotting <math>x_\mathrm{core}</math> versus <math>\tilde{r}_\mathrm{core}</math> after setting the leading coefficient, <math>\alpha = - 0.0011</math>.  The red-dotted curve results from plotting <math>(x_\mathrm{env} + x_\mathrm{shift})</math> versus <math>\tilde{r}_\mathrm{env}</math> after setting the leading coefficient, <math>\beta = - 0.000062</math>, and <math>x_\mathrm{shift} = + 0.0105</math>.

<table border="1" align="center" cellpadding="2">
<tr>
  <td align="center">[[File:TrialAnalyticEigenfunction01.png|700px|Trial Analytic Eigenfunction]]</td>
</tr>
</table>

ASSESSMENT:
<ul>
  <li>Our analytically specified displacement function, <math>x_\mathrm{core}</math>, appears to be an excellent match to the displacement function obtained throughout the core by implementing the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_=_5_Polytropes|B-KB74 conjecture]].</li>
  <li>At first glance, the plot of <math>(x_\mathrm{env} + x_\mathrm{shift})</math> appears to provide a reasonably good fit to the ''approximate'' displacement function that we have obtained throughout the envelope by implementing the B-KB74 conjecture.   But, in reality, there are two fatal flaws:
<ol type="1">
<li>We have presented the behavior of our analytically specified envelope displacement function only up to the radial coordinate, <math>\eta = 2.19707 ~~ (\tilde{r}_\mathrm{env} = 0.19526)</math>. Between this point and the surface, <math>\eta_s = 2.2823226 ~~ (\tilde{r}_\mathrm{env} = 0.2028415)</math> &#8212; where the argument of the cotangent, <math>(\eta_s - B) \rightarrow \pi</math> &#8212; the analytic function dives steeply to negative infinity.  This violently departs from the behavior derived via the [[Appendix/Ramblings/NonlinarOscillation#Radial_Oscillations_in_Pressure-Truncated_n_=_5_Polytropes|B-KB74 conjecture]].</li>
<li>While our analytically specified displacement function, <math>x_\mathrm{env}</math>, satisfies the "n = 1" polytropic LAWE, this satisfaction is destroyed by adding <math>x_\mathrm{shift}</math> to the displacement function.</li>
</ol>
  </li>
</ul>

Let's examine the slope of the displacement function at the interface.  From the perspective of the core, our analytic prescription for the displacement function matches the K-BK74-derived displacement function very well.  An analytic evaluation of the slope at the inferface &#8212; as derived [[#Core|above]] &#8212; gives,
<div align="center">
<math>\frac{dx_\mathrm{core}}{d\tilde{r}}\biggr|_i = -707.53765\alpha = +0.77829</math>.
</div>
The black-dashed line segment that appears in the following figure has this slope and goes through the point of intersection; it appears to be tangent to the analytic displacement function, as expected.  Alternatively, the orange-dashed line segment that appears in this same figure, also goes through the point of intersection, but it has a slope that matches our ''expectation'' for the envelope's displacement function; that is, it has a slope [[#Envelope|as derived]] of,
<div align="center">
<math>\frac{dx_\mathrm{env}}{d\tilde{r}}\biggr|_i = +0.08619</math>.
</div>
This orange-dashed line segment does ''not'' appear to lie tangent to the K-BK74-derived displacement function for the envelope.

<table border="1" align="center" cellpadding="2">
<tr>
  <td align="center">[[File:TrialEigenfunctionSlopes01.png|700px|Trial Analytic Eigenfunction with Intersection Slopes]]</td>
</tr>
</table>

===2<sup>nd</sup> Trial===

The relevant LAWE for the envelope is,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right"><math>\biggl[ \frac{A\sin(\eta-B)}{\eta}\biggr]\frac{d^2x}{d\eta^2} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{2A}{\eta}\biggl\{ \sin(\eta - B) + \eta\cos(\eta - B) \biggr\} \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ \frac{2A}{\eta} \biggl\{ \sin(\eta - B) - \eta\cos(\eta - B) \biggr\}  \frac{x}{\eta^2}
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) x 
</math>
</td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \frac{1}{3\beta} \cdot \frac{d^2x}{d\eta^2} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{2}{\eta}\biggl\{ 1 + \eta\cot(\eta - B) \biggr\} \biggl[ \frac{1}{3\beta } \cdot \frac{dx}{d\eta} \biggr] 
+ 2 \biggl\{ 1 - \eta\cot(\eta - B) \biggr\}  \frac{x}{3\beta\eta^2}
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) \biggl[ \frac{\eta}{3 \beta A\sin(\eta-B)}\biggr] x
\, . 
</math>
</td>
</tr>
</table>
Here, we will ''guess'' a displacement function, <math>x_\mathrm{env}</math>, of the form,

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

<tr>
  <td align="right">
<math>\frac{x_\mathrm{env}}{3\beta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\eta^2 } \biggl[1 - \eta\cot(\eta - B_x) \biggr]
=
\frac{1}{\eta^2 }  - \frac{\cos(\eta - B_x)}{\eta\sin(\eta - B_x)} 
\, , 
</math>
  </td>
</tr>
</table>
where we will assume, quite generally, that <math>B_x \ne B</math>.  The first and second derivatives of <math>x_\mathrm{env}</math> are,

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

<tr>
  <td align="right">
<math>\frac{1}{3\beta} \biggl[ \frac{dx_\mathrm{env}}{d\eta} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{2}{\eta^3 }  
+ 
\frac{\cos(\eta - B_x)}{\eta^2\sin(\eta - B_x)} 
+ 
\frac{\sin(\eta - B_x)}{\eta\sin(\eta - B_x)} 
+ 
\frac{\cos^2(\eta - B_x)}{\eta\sin^2(\eta - B_x)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{2}{\eta^3 }  
+ 
\frac{1}{\eta} 
+ 
\frac{\cos(\eta - B_x)}{\eta^2\sin(\eta - B_x)} 
+ 
\frac{\cos^2(\eta - B_x)}{\eta\sin^2(\eta - B_x)}
\, ; 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{1}{3\beta} \biggl[ \frac{d^2x_\mathrm{env}}{d\eta^2} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{\eta^4} - \frac{1}{\eta^2}
- \frac{2\cos(\eta-B_x)}{\eta^3\sin(\eta-B_x)}
+ \frac{1}{\eta^2}\biggl[
- \frac{\sin(\eta - B_x)}{\sin(\eta - B_x)}
-
\frac{\cos^2(\eta - B_x)}{\sin^2(\eta - B_x)}
\biggr]
- 
\frac{\cos^2(\eta - B_x)}{\eta^2\sin^2(\eta - B_x)}
+ \frac{1}{\eta} \biggl[
-
\frac{2\cos(\eta - B_x)}{\sin(\eta - B_x)}
-
\frac{2\cos^3(\eta - B_x)}{\sin^3(\eta - B_x)}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{\eta^4} - \frac{1}{\eta^2}
- \frac{2\cos(\eta-B_x)}{\eta^3\sin(\eta-B_x)}
- \frac{1}{\eta^2}\biggl[
1
+
\frac{\cos^2(\eta - B_x)}{\sin^2(\eta - B_x)}
\biggr]
- 
\frac{\cos^2(\eta - B_x)}{\eta^2\sin^2(\eta - B_x)}
- \frac{1}{\eta}\biggl[
\frac{2\cos(\eta - B_x)}{\sin(\eta - B_x)}
+
\frac{2\cos^3(\eta - B_x)}{\sin^3(\eta - B_x)}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{\eta^4} - \frac{2}{\eta^2}
- \frac{2\cos(\eta-B_x)}{\eta^3\sin(\eta-B_x)}
- \frac{2}{\eta^2}\biggl[
\frac{\cos^2(\eta - B_x)}{\sin^2(\eta - B_x)}
\biggr]
- \frac{2}{\eta}\biggl[
\frac{\cos(\eta - B_x)}{\sin(\eta - B_x)}
+
\frac{\cos^3(\eta - B_x)}{\sin^3(\eta - B_x)}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{6}{\eta^4} - \frac{2}{\eta^2}\biggl[1 + \cot^2(\eta - B_x) \biggr]
- \frac{2\cot(\eta-B_x)}{\eta^3}
- \frac{2}{\eta}\biggl[
\cot(\eta - B_x)
+
\cot^3(\eta - B_x)
\biggr]\, .
</math>
  </td>
</tr>
</table>
These match the expressions for <math>dx_P/d\eta</math> and <math>d^2x_P/d\eta^2</math> that we separately derived in a [[Appendix/Ramblings/BiPolytrope51AnalyticStability#Attempt_4B|subsection labeled ''Attempt_4B'' of an accompanying discussion labeled]].  Plugging these three relations into the LAWE, then multiplying through by <math>\eta^4</math>, gives,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>
\frac{6}{\eta^4} - \frac{2}{\eta^2}\biggl[1 + \cot^2(\eta - B_x) \biggr]
- \frac{2\cot(\eta-B_x)}{\eta^3}
- \frac{2}{\eta}\biggl[
\cot(\eta - B_x)
+
\cot^3(\eta - B_x)
\biggr]
</math>
</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{2}{\eta}\biggl\{ 1 + \eta\cot(\eta - B) \biggr\} \biggl[ -\frac{2}{\eta^3 }  
+ 
\frac{1}{\eta} 
+ 
\frac{\cot(\eta - B_x)}{\eta^2} 
+ 
\frac{\cot^2(\eta - B_x)}{\eta}
 \biggr] 
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl\{
2 \biggl[ 1 - \eta\cot(\eta - B) \biggr]  \frac{1}{\eta^2}
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) \biggl[ \frac{\eta}{ A\sin(\eta-B)}\biggr]
\biggr\} \biggl[ \frac{1}{\eta^2 }  - \frac{\cot(\eta - B_x)}{\eta}\biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
6 - 2\eta^2 \biggl[1 + \cot^2(\eta - B_x) \biggr]
- 2\eta \cot(\eta-B_x)
- 2\eta^3 \biggl[
\cot(\eta - B_x)
+
\cot^3(\eta - B_x)
\biggr]
</math>
</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\biggl\{ 1 + \eta\cot(\eta - B) \biggr\} \biggl[ -2  
+ 
\eta^2 
+ 
\eta\cot(\eta - B_x) 
+ 
\eta^2 \cot^2(\eta - B_x)
 \biggr] 
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ \biggl\{
2 \biggl[ 1 - \eta\cot(\eta - B) \biggr] 
- 2 \biggl( \frac{\sigma_c^2}{12} \biggr) \biggl[ \frac{\eta^3}{ A\sin(\eta-B)}\biggr]
\biggr\} \biggl[ 1  - \eta \cot(\eta - B_x)\biggr]
</math>
</td>
</tr>
</table>

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\Rightarrow ~~~
\biggl( \frac{\sigma_c^2}{6} \biggr) \biggl[ \frac{\eta^3}{ A\sin(\eta-B)}\biggr]
\biggl[ 1  - \eta \cot(\eta - B_x)\biggr]
</math>
</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 1 + \eta\cot(\eta - B) \biggr] \biggl[ 4  
- 
2\eta^2 
- 
2\eta\cot(\eta - B_x) 
- 
2\eta^2 \cot^2(\eta - B_x)
 \biggr] 
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 
2 \biggl[ 1 - \eta\cot(\eta - B) \biggr] 
\biggl[ 1  - \eta \cot(\eta - B_x)\biggr]

-6 + 2\eta^2 \biggl[1 + \cot^2(\eta - B_x) \biggr]
+ 2\eta \cot(\eta-B_x)
+ 2\eta^3 \biggl[
\cot(\eta - B_x)
+
\cot^3(\eta - B_x)
\biggr]
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 4 - 2\eta^2 - 2\eta\cot(\eta - B_x) - 2\eta^2 \cot^2(\eta - B_x)\biggr] 
+
\eta\cot(\eta - B) 
\biggl[ 4 - 2\eta^2 - 2\eta\cot(\eta - B_x) - 2\eta^2 \cot^2(\eta - B_x)\biggr] 

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
+ 
\biggl[ 2 - 2\eta\cot(\eta - B) - 2\eta \cot(\eta - B_x) + 2\eta^2 \cot(\eta - B)\cot(\eta - B_x)\biggr]

-6 + 2\eta^2 + 2\eta^2 \cot^2(\eta - B_x) 
+ 2\eta \cot(\eta-B_x)
+ 2\eta^3 \cot(\eta - B_x) 
+ 2\eta^3 \cot^3(\eta - B_x)
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\eta\cot(\eta - B_x) - 2\eta^2 \cot^2(\eta - B_x) 
+
4\eta\cot(\eta - B) - 2\eta^3\cot(\eta - B) - 2\eta^2 \cot(\eta - B)\cot(\eta - B_x) - 2\eta^3\cot(\eta - B) \cot^2(\eta - B_x)

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">&nbsp;</td>
  <td align="left">
<math>
- 2\eta\cot(\eta - B) - 2\eta \cot(\eta - B_x) + 2\eta^2 \cot(\eta - B)\cot(\eta - B_x)
+ 2\eta^2 \cot^2(\eta - B_x) 
+ 2\eta \cot(\eta-B_x)
+ 2\eta^3 \cot(\eta - B_x) 
+ 2\eta^3 \cot^3(\eta - B_x)
</math>
</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- 2\eta^3 \biggl[\cot(\eta - B)  +\cot(\eta - B) \cot^2(\eta - B_x)
- \cot(\eta - B_x) - \cot^3(\eta - B_x) \biggr]
+ 2\eta \biggl[\cot(\eta - B) - \cot(\eta - B_x) \biggr]
</math>
</td>
</tr>
</table>

Notice that if we set <math>B_x = B</math>, the RHS of this LAWE expression goes to zero.  This [thankfully] is as expected.

=See Also=
<ul>
  <li>Prasad, C. (1953), Proc. Natn. Inst. Sci. India, Vol. 19, 739, ''Radial Oscillations of a Composite Model.''</li>
  <li>Singh, Manmohan, (1969), Proc. Natn. Inst. Sci., India, Part A, Vol. 35, pp. 586 - 589, ''Radial Oscillations of Composite Polytropes &#8212; Part I</li>
  <li>[https://www.osti.gov/biblio/4178929-radial-oscillations-composite-polytropes-part-ii Singh, Manmohan, (1969)], Proc. Nat. Inst. Sci., India, Part A, Vol. 35, pp. 703 - 708, ''Radial Oscillations of Composite Polytropes &#8212; Part II</li>
  <li>[https://ui.adsabs.harvard.edu/abs/2021A%26AT...32..371K/abstract Kumar, S., Saini, S., Singh, K. K., Bhatt, V., &amp; Vashishta, L. (2021)], Astronomical &amp; Astrophysical Transactions, Vol. 32, Issue 4, pp. 371-382, ''Radial Pulsations of distorted Polytropes of Non-Uniform Density.''</li>
</ul>

{{ SGFfooter }}