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

==Set of Linearized Equations==

Borrowing from [[SSC/Perturbations#Summary_Set_of_Linearized_Equations|an accompanying discussion]], we have &hellip;

<div align="center">
<table border="1" cellpadding="10">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
r_0 \frac{dx}{dr_0} = - 3 x - d ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler + Poisson Equations'''</font></span><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>
p = \gamma_\mathrm{g} d  \, .
</math>
</td></tr>
</table>
</div>

Rearranging terms in the "Linearized Euler + Poisson Equations" as follows &hellip;

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

<tr>
  <td align="right">
<math>\frac{1}{\rho_0} \cdot \frac{dp}{dr_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{P_0}\biggl[ (4x + p)g_0 + \omega^2 r_0 x \biggr]
</math>
  </td>
</tr>
</table>

we realize that the expression on the RHS has the same value at the interface, whether you're viewing the equation from the point of view of the core or the envelope; and we recognize as well that <math>\rho_0</math> is a simple step function at the interface.  Hence, letting a prime indicate differentiation with respect to <math>r_0</math>, we can write,

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

<tr>
  <td align="right">
<math>
H(\zeta)\cdot (p')_\mathrm{env}
+
H(-\zeta)\cdot (p')_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{[\rho_0]_\mathrm{core}}{P_0}\biggl[ (4x + p)g_0 + \omega^2 r_0 x \biggr]
\biggl\{
H(\zeta)\cdot \biggl[ \frac{\mu_e}{\mu_c} \biggr]
+
H(-\zeta) \biggr\} \, .
</math>
  </td>
</tr>
</table>
Analogously, the "Linearized Equation of Continuity" can be rewritten as,

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

<tr>
  <td align="right">
<math>
H(\zeta)\cdot (x')_\mathrm{env}
+
H(-\zeta)\cdot (x')_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{3x}{r_0} 
-
\frac{1}{r_0}\biggl\{
H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env}
+
H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\}
\, . 
</math>
  </td>
</tr>
</table>

Now, given that <math>\zeta = (r_0/r_i - 1)</math>, we see that,

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

<tr>
  <td align="right">
<math>
\frac{dH(\zeta)}{dr_0} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\partial H(\zeta)}{\partial \zeta} \cdot \frac{d\zeta}{dr_0}
=
r_o^{-1} \delta(\zeta) \, ;

</math>
  </td>
</tr>
</table>
and,

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

<tr>
  <td align="right">
<math>
\frac{dH(-\zeta)}{dr_0} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{\partial H(\zeta)}{\partial \zeta} \cdot \frac{d\zeta}{dr_0}
=
-r_o^{-1} \delta(\zeta) \, .

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

Hence, differentiation of the "Linearized Equation of Continuity" gives,

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

<tr>
  <td align="right">
<math>
\frac{d}{dr_0}\biggl[H(\zeta)\cdot (x')_\mathrm{env}\biggr]
+
\frac{d}{dr_0}\biggl[H(-\zeta)\cdot (x')_\mathrm{core}\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr]
-
\frac{1}{r_0} \frac{d}{dr_0}\biggl\{
H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env}
+
H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\}

+
\frac{1}{r_0^2}\biggl\{
H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env}
+
H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\}
\, . 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
(x')_\mathrm{env}\frac{d}{dr_0}\biggl[H(\zeta)\biggr] + H(\zeta)\cdot (x'')_\mathrm{env}
+
(x')_\mathrm{core}\frac{d}{dr_0}\biggl[H(-\zeta)\biggr] + H(-\zeta)\cdot (x'')_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr]
+
\frac{1}{r_0^2}\biggl\{
H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env}
+
H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\}
\, . 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env}\frac{d}{dr_0}\biggl\{H(\zeta) \biggr\}
-
\frac{1}{r_0} \biggl\{
H(\zeta)\cdot \biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{env}
\biggr\}

-
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \frac{d}{dr_0}\biggl\{H(-\zeta) \biggr\}
-
\frac{1}{r_0} \biggl\{H(-\zeta)\biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{core} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~
(x')_\mathrm{env} \biggl[\frac{\delta(\zeta)}{r_0}\biggr] + H(\zeta)\cdot (x'')_\mathrm{env}
-
(x')_\mathrm{core}\biggl[\frac{\delta(\zeta)}{r_0}\biggr] + H(-\zeta)\cdot (x'')_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr]
+
\frac{1}{r_0^2}\biggl\{
H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env}
+
H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\}
\, . 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\}
-
\frac{1}{r_0} \biggl\{
H(\zeta)\cdot \biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{env}
\biggr\}

+
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\}
-
\frac{1}{r_0} \biggl\{H(-\zeta)\biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{core} \biggr\}
</math>
  </td>
</tr>
</table>

===From the Perspective of the Core===

When <math>r_0/r_i \le 1</math> &#8212; that is, from the perspective of the core while including the interface,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
-
(x')_\mathrm{core}\biggl[\frac{\delta(\zeta)}{r_0}\biggr] + H(-\zeta)\cdot (x'')_\mathrm{core}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr]
+
\frac{1}{r_0^2}\biggl\{
H(-\zeta)\biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggr\}
\, . 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\}

+
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\}
-
\frac{1}{r_0} \biggl\{H(-\zeta)\biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{core} \biggr\}
</math>
  </td>
</tr>
</table>

And examining only the interface, where <math>\delta(\zeta) = 1</math> while <math>H(-\zeta) = 0</math>,

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

<tr>
  <td align="right">
<math>
-
(x')_\mathrm{core}\biggl[\frac{\delta(\zeta)}{r_0}\biggr] 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr]
-
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\}

+
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ 
- (x')_\mathrm{core} 
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\biggl[ 3x' - \frac{3x}{r_0}\biggr]
-
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} 

+
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} 
</math>
  </td>
</tr>
</table>

===From the Perspective of the Envelope===

When <math>r_0/r_i \ge 1</math> &#8212; that is, from the perspective of the envelope while including the interface,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
(x')_\mathrm{env} \biggl[\frac{\delta(\zeta)}{r_0}\biggr] + H(\zeta)\cdot (x'')_\mathrm{env}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 
\biggl[ \frac{3}{r_0} \cdot x' - \frac{3x}{r_0^2}\biggr]
+
\frac{1}{r_0^2}\biggl\{
H(\zeta)\cdot \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env}
\biggr\}
\, . 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
-
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{env} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\}
-
\frac{1}{r_0} \biggl\{
H(\zeta)\cdot \biggl[ \frac{p'}{\gamma_g} \biggr]_\mathrm{env}
\biggr\}

+
\frac{1}{r_0} \biggl[ \frac{p}{\gamma_g} \biggr]_\mathrm{core} \biggl\{\frac{\delta(\zeta)}{r_0} \biggr\}
</math>
  </td>
</tr>
</table>