Editing SSC/Stability/BiPolytropes/RedGiantToPN/Pt4 (section)

===Finite-Difference Expressions===

The discussion  in this subsection is guided by our [[SSC/Stability/Polytropes/Pt3#Numerical_Integration_from_the_Center,_Outward|previous attempt at numerical integration]].

Here, we focus on the LAWE that is relevant to the envelope, namely,
<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_0^2} + \biggl[4 - (n+1) Q_1 \biggr] \frac{1}{r_0} \cdot\frac{dx}{dr_0} 
- \biggl[(n+1)\alpha Q_1 \biggr]  \frac{x}{r_0^2} 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{d^2x}{dr_0^2} + \biggl[4 - 2 Q_1 \biggr] \frac{1}{r_0} \cdot\frac{dx}{dr_0} 
- \biggl[2 Q_1 \biggr]  \frac{x}{r_0^2} 
\, ,
</math>
  </td>
</tr>
</table>
<span id="FD">where we have plugged</span> in the values, <math>(n,\alpha) = (1, 1)</math>.  Using the [[Appendix/Ramblings/NumericallyDeterminedEigenvectors#General_Approach|general finite-difference approach described separately]], we make the substitutions,

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

<tr>
  <td align="right">
<math>\biggl[\frac{dx}{dr_0}\biggr]_i</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{x_+ - x_-}{2 \Delta_r}   \, ;
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and,&nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>\biggl[\frac{d^2x}{dr_0^2}\biggr]_i</math>
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>\frac{x_+ - 2x_i + x_-}{\Delta_r^2} \, ;</math>
  </td>
</tr>
</table>
which will provide an approximate expression for <math>x_+ \equiv x_{i+1}</math>, given the values of <math>x_- \equiv x_{i-1}</math> and <math>x_i</math>.

<font color="orange"><b>A:</b></font> &nbsp; &nbsp; Pick <math>\xi_\mathrm{int}</math>; this will give analytic expressions for <math>\eta_\mathrm{int}</math>, <math>B</math>, and for <math>\eta_\mathrm{surf}</math>, as well as analytic expressions for <math>(r_0)_\mathrm{int}</math> and <math>(r_0)_\mathrm{surf}</math>.

<font color="orange"><b>B:</b></font> &nbsp; &nbsp; Divide the radial coordinate grid into 99 spherical shells <math>\Rightarrow~ \Delta_r = [(r_0)_\mathrm{surf} - (r_0)_\mathrm{int}]/99.</math>  Then tabulate 100 values of <math>(r_0)_i, \eta_i, (Q_1)_i = [1 -  \eta\cot(\eta-B) ]_i</math>.

Generally speaking, after multiplying through by <math>r_0^2</math>, the finite-difference  representation of the envelope's 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>
r_0^2\biggl[\frac{x_+ - 2x_i + x_-}{\Delta_r^2}\biggr] 
+ \biggl[4 - 2 Q_1 \biggr] r_0 \biggl[\frac{x_+ - x_-}{2 \Delta_r}\biggr] 
- \biggl[2 Q_1 \biggr]  x_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_+ \biggl\{ \frac{r_0^2}{\Delta_r^2} + (4-2Q_1)\frac{r_0}{2 \Delta_r} \biggr\}
+
x_i \biggl\{- \frac{2r_0^2}{\Delta_r^2} - 2Q_1  \biggr\}
+
x_- \biggl\{ \frac{r_0^2}{\Delta_r^2} - (4-2Q_1) \frac{r_0}{2 \Delta_r} \biggr\}
</math>
  </td>
</tr>
</table>
Multiplying through by <math>(\Delta_r^2/r_0^2)</math> and solving for <math>x_+</math> gives,

<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_+ \biggl\{ 1 + (4-2Q_1)\frac{\Delta_r}{2r_0} \biggr\}
-
2x_i \biggl\{1 + Q_1\biggl( \frac{\Delta_r^2}{r_0^2} \biggr)  \biggr\}
+
x_- \biggl\{ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ \biggl\{ 1 + (4-2Q_1)\frac{\Delta_r}{2r_0} \biggr\}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2x_i \biggl\{1 + Q_1\biggl( \frac{\Delta_r^2}{r_0^2} \biggr)  \biggr\}
-
x_- \biggl\{ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ x_+ </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{~
2x_i \biggl[1 + Q_1\biggl( \frac{\Delta_r^2}{r_0^2} \biggr)  \biggr]
-
x_- \biggl[ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr]
~\biggr\}~\biggl[ 1 + (4-2Q_1)\frac{\Delta_r}{2r_0} \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
Now, at the interface &#8212; as viewed from the perspective of both the core and the envelope &#8212; we know the value of <math>x_i =x_\mathrm{int}</math>, but we don't know the value of <math>x_-</math> as viewed from the envelope.  However &#8212; [[#STEPS|see <font color="maroon">STEP #4</font> below]] &#8212; we know analytically the value of the first derivative at the interface as viewed from the perspective of the envelope, namely, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ \frac{dx}{dr_0}\biggr]_\mathrm{int}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{x_\mathrm{int}}{r_0} \cdot \biggl\{ \frac{d\ln x}{d\ln r_0}\biggr|_\mathrm{int} \biggr\}_\mathrm{env}</math>
  </td>
</tr>
</table>
Therefore, from the [[#FD|above-specified finite-difference representation]] of the first derivative, we deduce that,
<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>
x_+ - 2\Delta_r \cdot \biggl[ \frac{dx}{dr_0}\biggr]_\mathrm{int}
</math>
  </td>
</tr>
</table>
Hence, at the interface &#8212; and only ''at'' the interface &#8212; the finite-difference representation of the envelope's LAWE can be written as,

<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_+ \biggl[ 1 + (4-2Q_1)\frac{\Delta_r}{2r_0} \biggr]
-
2x_i \biggl[1 + Q_1\biggl( \frac{\Delta_r^2}{r_0^2} \biggr)  \biggr]
+
\biggl\{x_+ - 2\Delta_r \cdot \biggl[ \frac{dx}{dr_0}\biggr]_\mathrm{int} \biggr\}\cdot \biggl[ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_+ \biggl[ 1 + (4-2Q_1)\frac{\Delta_r}{2r_0} \biggr]
-
2x_i \biggl[1 + Q_1\biggl( \frac{\Delta_r^2}{r_0^2} \biggr)  \biggr]
+
x_+  \biggl[ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr]
- 2\Delta_r \cdot \biggl[ \frac{dx}{dr_0}\biggr]_\mathrm{int} \cdot \biggl[ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
x_+ \biggl[ 1 + (4-2Q_1)\frac{\Delta_r}{2r_0} \biggr]
+
x_+  \biggl[ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2x_i \biggl[1 + Q_1\biggl( \frac{\Delta_r^2}{r_0^2} \biggr)  \biggr]
+ 2\Delta_r \cdot \biggl[ \frac{dx}{dr_0}\biggr]_\mathrm{int} \cdot \biggl[ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~
x_+ 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
x_i \biggl[1 + Q_1\biggl( \frac{\Delta_r^2}{r_0^2} \biggr)  \biggr]
+ \Delta_r \cdot \biggl[ \frac{dx}{dr_0}\biggr]_\mathrm{int} \cdot \biggl[ 1 - (4-2Q_1) \frac{\Delta_r}{2r_0} \biggr]
\, .
</math>
  </td>
</tr>
</table>