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

=Entropy as a Step Function=

Useful Chapters:
<ul>
<li>[[Appendix/Mathematics/StepFunction|Appendix/Mathematics/Stepfunction]]</li>
<li>[[SSC/Structure/BiPolytropes/Analytic51Renormalize#From_Step-Function_Behavior_of_Specific_Entropy|Step-Function Behavior of Specific Entropy]]</li>
</ul>

==Review==
The unit &#8212; or, [https://en.wikipedia.org/wiki/Heaviside_step_function Heaviside] &#8212; step function, <math>H(x)</math>, is defined such that,

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="center">
<math>
H(x) =  
\begin{cases}
0; & ~~ x < 0 \\
1; & ~~ x > 0
\end{cases}
</math>
<p><br />
[<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], Part I, &sect;2.1 (p. 123), Eq. (2.1.6)
  </td>
  <td align="center" rowspan="2">
[[File:Heaviside01.png|300px|Heaviside Function]]
  </td>
</tr>
</table>
In evaluating this function at <math>x=0</math>, we will adopt the ''half-maximum convention'' and set <math>H(0) = \tfrac{1}{2}</math>.  As has been pointed out in, for example, [https://en.wikipedia.org/wiki/Heaviside_step_function a relevant Wikipedia discussion], the derivative of the unit step function is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dH(x)}{dx}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\delta(x)    \, ,</math>
  </td>
</tr>
</table>

where, <math>\delta(x)</math> is the Dirac Delta function.

==Perturbed Density==

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

<tr>
  <td align="right">
<math>\zeta</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{r}{r_i} - 1 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ [\rho^*]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>H(\zeta)\cdot \rho^*_\mathrm{env} + H(-\zeta) \cdot \rho^*_\mathrm{core} \, ;</math>
  </td>
</tr>
</table>
and, more generally after a perturbation, <math>\delta\rho(\zeta)</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho^* = [\rho^*]_\mathrm{eq} + \delta\rho</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>[\rho^*]_\mathrm{eq} \biggl\{1  + \frac{\delta\rho}{[\rho^*]_\mathrm{eq}}\biggr\} 
=
[\rho^*]_\mathrm{eq} \biggl\{1  + de^{i\omega t}\biggr\}
\, .</math>
  </td>
</tr>
</table>
Hence, in the [[SSC/Perturbations#Continuity_Equation|linearized version of the continuity equation]], we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>d e^{i \omega t}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\delta\rho}{[\rho^*]_\mathrm{eq}} 
\, .</math>
  </td>
</tr>
</table>

<font color="red"><b>CORE:</b></font>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-5/2}</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\gamma_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{6}{5} </math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\mathcal{S}_\mathrm{core} \equiv \frac{s(\gamma_g-1)}{\Re/\bar{\mu}}\biggr|_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln (5) \, . 
</math>
  </td>
</tr>
</table>

<font color="red"><b>ENVELOPE:</b></font>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho^*_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)]</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\gamma_g</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2 </math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
  <td align="right">
<math>~\mathcal{S}_\mathrm{env} \equiv \frac{s(\gamma_g-1)}{\Re/\bar{\mu}}\biggr|_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr] \, .
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>(\rho^*)^{\gamma_g} \cdot \exp\biggl[ \frac{s(\gamma_g-1)}{\Re/\bar{\mu}} \biggr]</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>[P^*]_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>H(\zeta)\cdot P^*_\mathrm{env} + H(-\zeta) \cdot P^*_\mathrm{core} \, ;</math>
  </td>
</tr>
</table>
and, more generally after a perturbation, <math>\delta P(\zeta)</math>,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P^* = [P^*]_\mathrm{eq} + \delta P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>[P^*]_\mathrm{eq} \biggl\{1  + \frac{\delta P}{[P^*]_\mathrm{eq}}\biggr\} 
=
[P^*]_\mathrm{eq} \biggl\{1  + pe^{i\omega t}\biggr\}
\, .</math>
  </td>
</tr>
</table>

Hence, in the [[SSC/Perturbations#Adiabatic_form_of_the_First_Law_of_Thermodynamics|linearized version of the first law of thermodynamics]], we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>p e^{i \omega t}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\delta P}{[P^*]_\mathrm{eq}} 
\, .</math>
  </td>
</tr>
</table>

==Obtaining Perturbed Density from Perturbed Pressure==
Given that, quite generally,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{P}{\rho^{\gamma_g}}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\exp\biggl[ \frac{s(\gamma_g - 1)}{\mathfrak{R}/\bar\mu}\biggr]\, ,
</math>
  </td>
</tr>
</table>

let's define the density-like quantity,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathcal{Q}^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
e^{-\mathcal{S/\gamma_g}}\biggl[P^*\biggr]^{1/\gamma_g}
\, ,</math>
  </td>
</tr>
</table>
in which case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho^*_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
H(\zeta)\cdot [ \mathcal{Q}^*_\mathrm{eq}]_\mathrm{env}
+ 
H(-\zeta) \cdot [ \mathcal{Q}^*_\mathrm{eq}]_\mathrm{core} 
\, .</math>
  </td>
</tr>
</table>
What happens if we perturb the pressure?  In either region (core or envelope),

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

<tr>
  <td align="right">
<math>\mathcal{Q}^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
e^{-\mathcal{S/\gamma_g}} [P^*_\mathrm{eq}]^{1/\gamma_g}\biggl[ 1 + \frac{\delta P}{P^*_\mathrm{eq}}\biggr]^{1/\gamma_g}
\approx
e^{-\mathcal{S/\gamma_g}} [P^*_\mathrm{eq}]^{1/\gamma_g}\biggl[ 1 + \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]
=
[Q^*_\mathrm{eq}]\biggl[ 1 + \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]
\, .</math>
  </td>
</tr>
</table>
As a result,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho^* = \rho^*_\mathrm{eq} + \delta \rho</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
H(\zeta)\cdot [ \mathcal{Q}^*]_\mathrm{env}
+ 
H(-\zeta) \cdot [ \mathcal{Q}^*]_\mathrm{core} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
H(\zeta)\cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ 1 + \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{env}
+ 
H(-\zeta) \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ 1 + \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{core} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
H(\zeta)\cdot [Q^*_\mathrm{eq}]_\mathrm{env}
+ 
H(-\zeta) \cdot [Q^*_\mathrm{eq}]_\mathrm{core} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+~
H(\zeta)\cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{env}
+ 
H(-\zeta) \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{core} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\rho^*_\mathrm{eq} +~
H(\zeta)\cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{env}
+ 
H(-\zeta) \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{core} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\delta\rho}{\rho^*_\mathrm{eq}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{H(\zeta)}{\rho^*_\mathrm{eq}} \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{env}
+ 
\frac{H(-\zeta)}{\rho^*_\mathrm{eq}} \cdot \biggl\{[Q^*_\mathrm{eq}]\biggl[ \frac{1}{\gamma_g} \cdot \frac{\delta P}{P^*_\mathrm{eq}}\biggr]\biggr\}_\mathrm{core} 
</math>
  </td>
</tr>
</table>

==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>

==Focus on Nonlinear Continuity Equation==

A spherical shell of ''core'' density, <math>\rho_0</math>, where the inner radius of the shell is <math>(r_0 - \Delta r)</math> and its outer radius is <math>(r_0+\Delta r)</math> has a ''shell'' mass given by the expression,

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

<tr>
  <td align="right">
<math>(\Delta m)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} \cdot \rho_0 \biggl[
(r_0+\Delta r)^3 - (r_0 - \Delta r)^3 
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} \cdot \rho_0 r_0^3 \biggl[
\biggl(1 + \frac{\Delta r}{r_0} \biggr)^3 - \biggl(1 - \frac{\Delta r}{r_0} \biggr)^3 
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{3}{4\pi \rho_0 r_0^3} \biggr](\Delta m)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 + \frac{2\Delta r}{r_0} + \biggl(\frac{\Delta r}{r_0}\biggr)^2\biggr] \biggl(1 + \frac{\Delta r}{r_0} \biggr) 
- 
\biggl[1 - \frac{2\Delta r}{r_0} + \biggl(\frac{\Delta r}{r_0}\biggr)^2\biggr] \biggl(1 - \frac{\Delta r}{r_0} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[1 + \frac{3\Delta r}{r_0} + 3\biggl(\frac{\Delta r}{r_0}\biggr)^2 + \biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr]  
- 
\biggl[1 - \frac{3\Delta r}{r_0} + 3\biggl(\frac{\Delta r}{r_0}\biggr)^2 - \biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{6\Delta r}{r_0} + 2\biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ (\Delta m)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{4\pi \rho_0 r_0^3}{3} \biggr] \biggl[ \frac{6\Delta r}{r_0} + 2\biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] \, . 
</math>
  </td>
</tr>
</table>

Similarly, as viewed from the perspective of the envelope, it has a shell mass of,

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

<tr>
  <td align="right">
<math>(\Delta m)_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} \cdot \biggl(\frac{\mu_e}{\mu_c}\biggr) \rho_0 \biggl[
(r_0+\Delta r)^3 - (r_0 - \Delta r)^3 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{\mu_e}{\mu_c}\biggr) 
\biggl[ \frac{4\pi \rho_0 r_0^3}{3} \biggr] \biggl[ \frac{6\Delta r}{r_0} + 2\biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] \, . 
</math>
  </td>
</tr>
</table>

Better yet, pick the two edges of the shell, <math>r_+</math> and <math>r_-</math>, and let <math>r_0 \equiv (r_+ + r_-)/2</math> and <math>\Delta r = (r_+ - r_-)/2</math>.  Given the value of <math>\rho_0</math>, the unperturbed mass in the shell is given by the above expression, that is,

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

<tr>
  <td align="right">
<math>(\Delta m)_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{4\pi \rho_0 r_0^3}{3} \biggr] \biggl[ \frac{6\Delta r}{r_0} + 2\biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr] \, . 
</math>
  </td>
</tr>
</table>

Now, let <math>r_+ \rightarrow (r_+)_0 + (\delta r_+), ~r_- \rightarrow (r_-)_0 + (\delta r_-), </math> and, <math>\rho_0 \rightarrow (\rho_0)_0 + (\delta \rho)</math>.  We then have,

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

<tr>
  <td align="right">
<math>2r_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[(r_+)_0 + (\delta r_+)] + [(r_-)_0 + (\delta r_-)] = 2(r_0)_0 + [\delta r_+ + \delta r_-] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>2\Delta r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[(r_+)_0 + (\delta r_+)] - [(r_-)_0 + (\delta r_-)] = 2(\Delta r)_0 + [\delta r_+ - \delta r_-] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\Delta r}{r_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2(\Delta r)_0 + [\delta r_+ - \delta r_-]}{ 2(r_0)_0 + [\delta r_+ + \delta r_-] } 
=
\frac{(\Delta r)_0}{(r_0)_0} \times 
\biggl\{1 + \biggl[ \frac{\delta r_+ - \delta r_-}{2(\Delta r)_0} \biggr]  \biggr\}
\biggl\{1 + \biggl[ \frac{\delta r_+ + \delta r_-}{2(r_0)_0} \biggr]  \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{(\Delta r)_0}{(r_0)_0} \times 
\biggl\{1 + \biggl[ \frac{\delta r_+ - \delta r_-}{2(\Delta r)_0} \biggr]  \biggr\}
\biggl\{1 - \biggl[ \frac{\delta r_+ + \delta r_-}{2(r_0)_0} \biggr]  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>\approx</math>
  </td>
  <td align="left">
<math>
\frac{(\Delta r)_0}{(r_0)_0} \times 
\biggl\{1 + \biggl[ \frac{\delta r_+ - \delta r_-}{2(\Delta r)_0} \biggr]  
- \biggl[ \frac{\delta r_+ + \delta r_-}{2(r_0)_0} \biggr]  \biggr\}
</math>
  </td>
</tr>
</table>
In order for the shell to have the same <math>\Delta m</math> in both the unperturbed and perturbed cases, the following relation must hold:

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

<tr>
  <td align="right">
<math>
\biggl[ \frac{8\pi (\rho_0)_0 (r_0)_0^3}{3} \biggr] \biggl\{ 
\frac{3(\Delta r)_0}{(r_0)_0} + \biggl[ \frac{(\Delta r)_0}{(r_0)_0}\biggr ]^3
\biggr\}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ \frac{\pi [(\rho_0)_0 + \delta\rho] [2r_0]^3}{2\cdot 3} \biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times 2\biggl\{ \frac{3\Delta r}{r_0} + \biggl(\frac{\Delta r}{r_0}\biggr)^3\biggr\} 
</math>
  </td>
</tr>
</table>