Editing Appendix/Ramblings/51BiPolytropeStability/BetterInterfacePt2 (section)

===Interface===

<font color="darkgreen">CORE:</font> &nbsp; When <math>J = (i - 1)</math> (where <math>i</math> means interface), we can obtain the fractional displacements at the interface, <math>x_i</math> and <math>p_i</math>, via the expressions,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_{i}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{i-2} - 2\Delta\tilde{r} \biggl\{
\frac{1}{\tilde{r}}\biggl[ 
3x + \frac{p}{\gamma_g}\biggr]
\biggr\}_{i-1} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>p_{i}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{i-2} + 2\Delta\tilde{r}
\biggl\{
\frac{\tilde{\rho}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}
\biggl[ (4x + p)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] 
\biggr\}_{i-1}\, .
</math>
  </td>
</tr>
</table>
Then, setting <math>J = i</math>, the pair of radial derivatives '''at the interface''' and '''<font color="darkgreen">as viewed from the perspective of the core</font>''' is given by the expressions,

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

<tr>
  <td align="right"><math>\biggl(\frac{dx}{d \tilde{r}}\biggr)_i \biggr|_\mathrm{core}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{1}{\tilde{r}_i}\biggl[ 
3x_i + \frac{p_i}{6/5}\biggr] \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\biggl(\frac{dp}{d \tilde{r}}\biggr)_i\biggr|_\mathrm{core} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{(\tilde{\rho}_i)_\mathrm{core}}{\tilde{P}_i} \cdot \frac{\tilde{M}_\mathrm{core}}{\tilde{r}_i^2}
\biggl[ (4x_i + p_i)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}_i^3 }{\tilde{M}_\mathrm{core}}\biggr) x_i \biggr]\, . 
</math>
  </td>
</tr>
</table>

It is important to recognize that, throughout the core, <math>(dx/d\tilde{r})</math> has been evaluated by setting <math>\gamma_g = 6/5</math>. If we continue to use this value of <math>\gamma_g</math> at the interface, we are determining the slope ''as viewed from the perspective of the core''.  


<font color="darkgreen">ENVELOPE:</font> &nbsp; On the other hand, ''as viewed from the perspective of the envelope'', all parameters used to determine <math>(dx/d\tilde{r})_i</math> at the interface (and throughout the entire envelope) are the same ''except'' <math>\gamma_g</math>, which equals 2 instead of 6/5.  Specifically at the interface, we have,

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

<tr>
  <td align="right"><math>\biggl(\frac{dx}{d \tilde{r}}\biggr)_i \biggr|_\mathrm{env}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
- \frac{1}{\tilde{r}_i}\biggl[ 
3x_i + \frac{p_i}{2}\biggr] \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>\biggl(\frac{dp}{d \tilde{r}}\biggr)_i\biggr|_\mathrm{env} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{(\tilde{\rho}_i)_\mathrm{env}}{\tilde{P}_i} \cdot \frac{\tilde{M}_\mathrm{core}}{\tilde{r}_i^2}
\biggl[ (4x_i + p_i)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}_i^3 }{\tilde{M}_\mathrm{core}}\biggr) x_i \biggr]\, . 
</math>
  </td>
</tr>
</table>

(See, for example, our [[SSC/Stability/BiPolytropes#Interface_Conditions|related discussion]].)  Hence, we appreciate that there is a discontinuous change in the value of this slope at the interface.  We note as well &#8212; <font color="red">for the first time (8/17/2023)!</font> &#8212; that there must also be a discontinuous jump in the slope of the "pressure perturbation."  All of the variables used to evaluate <math>(dp/d\tilde{r})_i</math> are the same irrespective of your core/envelope point of view ''except'' the leading density term.  As viewed from the perspective of the core, <math>(\tilde{\rho}_i)|_\mathrm{core} = m_\mathrm{surf}^5 (\mu_e/\mu_c)^{-10} \theta_i^5</math> whereas, from the perspective of the envelope, <math>(\tilde{\rho}_i)|_\mathrm{env} = m_\mathrm{surf}^5 (\mu_e/\mu_c)^{-9} \theta_i^5\phi_i</math>.  Appreciating that <math>\phi_i = 1</math>, this means that the slope of the "pressure perturbation" is a factor of <math>\mu_e/\mu_c</math> smaller as viewed from the perspective of the envelope.

Then the value of the fractional radial displacement and the value of the pressure perturbation at the first zone outside of the interface are obtained by setting <math>J = i</math>.  That is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_{i+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
x_{i-1} - 2\Delta\tilde{r} \biggl\{
\frac{1}{\tilde{r}}\biggl[ 
3x + \frac{p}{2}\biggr]
\biggr\}_i \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>p_{i+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
p_{i-1} + 2\Delta\tilde{r}
\biggl\{
\frac{(\tilde{\rho}_i)|_\mathrm{env}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}
\biggl[ (4x + p)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] 
\biggr\}_i\, .
</math>
  </td>
</tr>
</table>
But, as written, these two expressions are unacceptable because the values just inside the interface, <math>x_{i-1}</math> and <math>p_{i-1}</math>, are not known '''as viewed from the perspective of the envelope.''' However, we can fix this by drawing from the "average" expressions as replacements, namely,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>x_{i}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}(x_{i-1} + x_{i+1}) ~~~ \Rightarrow~~~ x_{i-1} = (2x_i - x_{i+1})
\, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>p_{i}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{2}(p_{i-1} + p_{i+1}) ~~~ \Rightarrow~~~ p_{i-1} = (2p_i - p_{i+1})
\, ,
</math>
  </td>
</tr>
</table>
in which case we have,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right"><math>2x_{i+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2x_{i} - 2\Delta\tilde{r} \biggl\{
\frac{1}{\tilde{r}}\biggl[ 
3x + \frac{p}{2}\biggr]
\biggr\}_i \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;</td>
  <td align="right"><math>2p_{i+1}</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
2p_{i} + 2\Delta\tilde{r}
\biggl\{
\frac{(\tilde{\rho}_i)|_\mathrm{env}}{\tilde{P}} \cdot \frac{\tilde{M}_r}{\tilde{r}^2}
\biggl[ (4x + p)
+
\sigma_c^2 \biggl(\frac{2\pi}{3} \cdot \frac{\tilde{\rho}_c \tilde{r}^3 }{\tilde{M}_r}\biggr) x \biggr] 
\biggr\}_i\, .
</math>
  </td>
</tr>
</table>