Editing SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2 (section)

===Discrete Determination of Bipolytropic Envelope===
Here we focus on the ''specific'' <math>(n_c, n_e) = (5, 1)</math> equilibrium model sequence that has <math>(\mu_e/\mu_c) = 0.31</math>; and along this sequence, we attempt to analyze the dynamical stability of "model <b>A</b>" [[#Model_Pairings|from above]], which sits along the sequence at the maximum-core-mass turning point for which &hellip;
<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="8"><math>(n_c, n_e) = (5, 1)</math> and <math>(\mu_e/\mu_c) = 0.31</math></td>
</tr>
<tr>
  <td align="center">Model</td>
  <td align="center"><math>A</math></td>
  <td align="center"><math>B</math></td>
  <td align="center"><math>\xi_i</math></td>
  <td align="center"><math>\theta_i = (1 + \xi_i^2/3)^{-1 / 2}</math></td>
  <td align="center"><math>\eta_i = 3^{1 / 2}\biggl(\frac{\mu_e}{\mu_c}\biggr) \xi_i \theta_i^2</math></td>
  <td align="center"><math>\eta_s = B + \pi</math></td>
  <td align="center"><math>\mathcal{m}_\mathrm{surf}=\biggl( \frac{2}{\pi}\biggr)^{1 / 2}\frac{A\eta_s}{\theta_i}</math></td>
</tr>
<tr>
  <td align="center"><b>A</b></td>
  <td align="center">0.200812422</td>
  <td align="center">- 0.859270052</td>
  <td align="center">9.0149598</td>
  <td align="center">0.188679805</td>
  <td align="center">0.17232050</td>
  <td align="center">2.28232260</td>
  <td align="center">1.9381270</td>
</tr>
</table>


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3">Key Parameter-Parameter Ratios</td>
</tr>
<tr>
  <td align="center"><math>\frac{\xi}{\tilde{r}} = \biggl(\frac{2\pi}{3}\biggr)^{1 / 2} \mathcal{m}_\mathrm{surf}^2 \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-4}</math></td>
  <td align="center"><math>\frac{\eta}{\tilde{r}} = \mathcal{m}_\mathrm{surf}^{2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-3} \theta^{2}_i (2\pi)^{1/2}</math></td>
  <td align="center"><math>\frac{\eta}{\xi}</math></td>
</tr>
<tr>
  <td align="center">588.6362811</td>
  <td align="center">11.25175286</td>
  <td align="center">0.019114950</td>
</tr>
</table>

As [[#Core|presented above]], when <math>\sigma_c^2 = 0</math>, the eigenfunction for the core that we have deduced via the B-KB74 conjecture appears to be well represented by the expressions,

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

<tr>
  <td align="right">
<math>x_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>C_0\biggl[ 1 - \frac{\xi^2}{15} \biggr] \, ,</math> &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; <math>\frac{dx}{d\xi}\biggr|_\mathrm{core} = - \frac{2 C_0 \xi}{15}\, ,</math> &nbsp; &nbsp; &nbsp; with, &nbsp; &nbsp; &nbsp; <math>C_0 = - 0.0011 \, ,</math>
  </td>
</tr>
</table>
over the radial-parameter range, <math>0 \le \xi \le \xi_i\, .</math>


<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="5">At the Core/Envelope Interface<br /><font size="-1">(as viewed from the core)</font></td>
</tr>
<tr>
  <td align="center"><math>x_i</math></td>
  <td align="center"><math>\frac{dx}{d\xi}\biggr|_i</math></td>
  <td align="center"><math>\tilde{r}_i</math></td>
  <td align="center"><math>\frac{dx}{d\tilde{r}}\biggr|_i</math></td>
  <td align="center"><math>\biggl[\frac{d\ln x}{d\ln \xi}\biggr]_i = \biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_i</math></td>
</tr>
<tr>
  <td align="center">+ 0.004859763</td>
  <td align="center">+ 0.001322194</td>
  <td align="center">0.015314992</td>
  <td align="center">+ 0.778291359</td>
  <td align="center">+ 2.4526969</td>
</tr>
</table>


Copying from our [[#Envelope|earlier discussion of the envelope]] for "model <b>A</b>", the range of the radial parameter 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]], <font color="red">we expect</font> that 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>

[[#Discrete_Form_of_LAWE|As above]], we will integrate the discrete LAWE ''outward'' using the finite-difference expression,

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

<tr>
  <td align="right">
<math>x_+ \overbrace{\biggl[2\phi_i +\frac{4\Delta_\eta \phi_i}{\eta_i} - \Delta_\eta (n+1)(- \phi^')_i\biggr]}^{\mathrm{TERM1}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_- \overbrace{\biggl[\frac{4\Delta_\eta \phi_i}{\eta_i} - \Delta_\eta (n+1)(- \phi^')_i - 2\phi_i\biggr]}^{\mathrm{TERM2}} 
+  x_i \overbrace{\biggl\{4\phi_i - \frac{\Delta_\eta^2(n+1)}{3}\biggl[ \frac{\sigma_c^2}{\gamma_g}  - 
2\alpha \biggl(- \frac{3\phi^'}{\eta}\biggr)_i\biggr]  \biggr\}}^{\mathrm{TERM3}} \, .</math>
  </td>
</tr>
</table>
When we started the integration at the center of the configuration, we kickstarted the process by, first, setting <math>x_1 = 1</math>; then, [[#Determining_Discrete_Representation_of_Eigenfunction|second, setting]],

<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_\eta^2}{60} \biggr] \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; where, &nbsp; &nbsp; &nbsp; </td>
  <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\biggr] \, .</math>
  </td>
</tr>
</table>
Having obtained <math>x_1 \rightarrow x_-</math> and <math>x_2 \rightarrow x_i</math>, we then used the finite-difference expression to calculate <math>x_+ \rightarrow x_3</math>, as well as all subsequent "<math>x_+</math>" values, all the way to the surface.  

Here, instead, we want to start the ''envelope'' integration at the core/envelope interface as follows:
<ol>
  <li>The displacement function for the core gives us the value of the displacement function, <math>x_i = + 0.004859763</math>, at <math>\xi_i = 9.0149598</math>, that is, at <math>\eta_i = 0.17232050</math>; we recognize that this value of <math>x_i</math> (at the interface) also furnishes the value of <math>x_i</math> in the first integration step of the finite-difference expressions.</li>
  <li>We will then "guess" the slope of the envelope's displacement function, <math>q \equiv [dx/d\eta]_i</math>, at the interface.</li>
  <li>Our discrete representation of this first derivative permits us to write,

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

<tr>
  <td align="right">
<math>
q
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2\Delta_\eta}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ x_-
</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
x_+ - 2q\Delta_\eta \, .
</math>
  </td>
</tr>
</table>
Inserting this expression into the finite-difference approximation to the LAWE gives <b>for the first integration step only!</b>

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

<tr>
  <td align="right">
<math>x_+ \cdot \mathrm{TERM1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(x_+ - 2q\Delta_\eta) \cdot \mathrm{TERM2} 
+  x_i \cdot \mathrm{TERM3}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \cdot [\mathrm{TERM1} - \mathrm{TERM2}]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i \cdot \mathrm{TERM3}
- 2q\Delta_\eta \cdot \mathrm{TERM2} \, .
</math>
  </td>
</tr>
</table>

  </li>
</ol>
<font color="red">NOTE:</font> &nbsp; Judging by the behavior of the B-KB74 generated displacement function, at the interface we expect the slope, <math>[d\ln x/d\ln \tilde{r}]_\mathrm{env}</math>, from the envelope's perspective to be shallower than the slope, <math>[d\ln x/d\ln \tilde{r}]_\mathrm{core}</math>, from the core's perspective.  That is to say, we expect to "guess" values of <math>q</math> such that at the 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[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
- \biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{env}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{d\ln x}{d\ln \eta}\biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{\eta}{x} \cdot \frac{dx}{d \eta}\biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl[\frac{dx}{d \eta}\biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\frac{x_i}{\eta_i} \cdot \biggl[\frac{d\ln x}{d\ln \tilde{r}}\biggr]_\mathrm{core}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~q</math>
  </td>
  <td align="center">
<math><</math>
  </td>
  <td align="left">
<math>
\biggl[\frac{+ 0.004859763}{+ 0.1723205}\biggr] \cdot 2.4526969 = +0.06917068 \, .
</math>
  </td>
</tr>
</table>