Editing SSC/Structure/BiPolytropes/51RenormaizePart2 (section)

====Behavior at the Interface====

It is worth pointing out that the second derivative of the pressure (with respect to <math>\tilde{M}_r</math>) exhibits a discontinuous jump at the interface.  Specifically,

The smooth, solid curves in <b>Figure C1</b> (blue for the core and green for the envelope) show the analytically prescribed behavior of the quantity, <math>\tilde{M}_r/\tilde{r}^2</math> as a function of <math>\tilde{M}_r</math> throughout the unperturbed <b>Model C</b>.  These curves intersect at the core/envelope interface (marked by the vertical, black dashed line), which means that the quantity, <math>\tilde{M}_r/\tilde{r}^2</math> has the same value whether viewed from the perspective of the core or from the perspective of the envelope.  But, as the figure illustrates, the curves exhibit different slopes at the interface.

Quite generally we can write,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3}\frac{d\tilde{r}}{d\tilde{M}_r} 
=
\frac{1}{\tilde{r}^2}\biggl\{ 1 - 2 \cdot \frac{d\ln \tilde{r}}{d\ln \tilde{M}_r}
\biggr\}
\, .
</math>
  </td>
</tr>
</table>

This means that, for the core,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3} \biggl[ \frac{d\xi}{d\tilde{M}_r} \cdot \frac{d\tilde{r}}{d\xi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3} 
\biggl[ \mathcal{m}_\mathrm{surf} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} 
\biggl( \frac{\pi }{2\cdot 3} \biggr)^{1/2} 
\biggl\{ 
\frac{1}{3\xi^2 }\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{5/2} 
\biggr\} 
 \biggr] 
\cdot 
\biggl[ \mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2}  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2}\biggl\{1 
-
2\tilde{M}_r 
\biggl[ \mathcal{m}_\mathrm{surf} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} 
\biggl( \frac{\pi }{2\cdot 3} \biggr)^{1/2} 
\frac{1}{3\xi^3 }\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{5/2} 
 \biggr] 
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3\tilde{r}^2}\biggl( 1 - \frac{2}{3}\xi^2 \biggr) \, .
</math>
  </td>
</tr>
</table>
Specifically at the interface (from the perspective of the core),
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]\biggr\}_{i,\mathrm{core}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3\tilde{r}_i^2}\biggl( 1 - \frac{2}{3}\xi_i^2 \biggr) \, .
</math>
  </td>
</tr>
</table>


<!--
----


<math>
\mathcal{m}_\mathrm{surf} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} 
\biggl( \frac{\pi }{2\cdot 3} \biggr)^{1/2} 
\biggl\{ 
\frac{1}{3\xi^2 }\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{5/2} 
\biggr\} 
</math><br />
<math>\mathcal{m}_\mathrm{surf}^{-2} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{4} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi</math><br />
<math>
\mathcal{m}_\mathrm{surf}^{-1} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{2} 
\biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} \biggl[ \xi^3 \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3/2} \biggr]
</math>

----

<math>A \mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
\biggl\{ 
\eta\sin(\eta-B)
\biggr\} 
</math><br />
<math>\mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2}\eta </math><br />
<math>\mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
A\biggl[ \sin(\eta-B) - \eta\cos(\eta-B) \biggr] \, ,</math>
----

-->

And for the envelope,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3} \biggl[ \frac{d\eta}{d\tilde{M}_r} \cdot \frac{d\tilde{r}}{d\eta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} 
-
\frac{2\tilde{M}_r}{\tilde{r}^3} \biggl[ 
A \mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
\biggl\{ 
\eta\sin(\eta-B)
\biggr\} 
\biggr]^{-1} \cdot 
\biggl[ \mathcal{m}_\mathrm{surf}^{-2} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{3} \theta^{-2}_i (2\pi)^{-1/2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} \biggl\{1
-
\frac{2\tilde{M}_r}{\eta} \biggl[ 
A \mathcal{m}_\mathrm{surf}^{-1}~ \theta^{-1}_i \biggl( \frac{2}{\pi} \biggr)^{1/2} 
\eta\sin(\eta-B)
\biggr]^{-1} 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2} \biggl\{1
-
\frac{2}{\eta^2} 
\biggl[ 1 - \eta\cot(\eta-B) \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>

Now, pulling from [[SSC/Structure/BiPolytropes/Analytic51#Step_6:_Envelope_Solution|our original derivation]], we appreciate that,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\eta_i \cot(\eta_i - B) 
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\eta_i \Lambda_i
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3} \theta_i^2 \xi_i  \biggr] 
\frac{\xi_i}{\sqrt{3}} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ 1 - \theta_i^2 \xi_i^2 \biggl(\frac{\mu_e}{\mu_c}\biggr)\biggr]
</math>
  </td>
</tr>
</table>

Hence, at the interface (from the perspective of the envelope) we find,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>\biggl\{
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
\biggr\}_{i, \mathrm{env}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
\frac{2}{\eta_i^2} 
\biggl[ 1 - \eta_i \cot(\eta_i-B) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
\frac{2}{\eta_i^2} 
\biggl[ \theta_i^2 \xi_i^2 \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
2 \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3} \theta_i^2 \xi_i \biggr]^{-2} 
\biggl[ \theta_i^2 \xi_i^2 \biggl(\frac{\mu_e}{\mu_c}\biggr) \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
\frac{2}{3} 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\biggl(1+\frac{1}{3}\xi_i^2\biggr) 
\biggr\} \, .
</math>
  </td>
</tr>
</table>

<!--
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\theta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(1+\frac{1}{3}\xi_i^2\biggr)^{-1 / 2} \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\eta_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>\biggl(\frac{\mu_e}{\mu_c}\biggr)~ \sqrt{3} \theta_i^2 \xi_i \, ,</math></td>
</tr>

<tr>
  <td align="right"><math>\Lambda_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>
\frac{\xi_i}{\sqrt{3}} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\frac{1}{\theta_i^2 \xi_i^2} - 1\biggr] 
\, ,</math></td>
</tr>
</table>
-->

<table border="1" align="center" cellpadding="8" width="80%">
<tr>
  <td align="center"><b>Figure C2</b></td>
</tr>
<tr>
  <td align="center">[[File:ModelC Fig2.png|750px|Model C Slopes at Interface]]</td>
</tr>

<tr>
  <td align="left">
A pair of line-segments with arrowheads has been added to Figure C1:  
<ul>
  <li>The red arrow is tangent to the solid blue curve at the core/envelope interface; its slope is,
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]\biggr\}_{i,\mathrm{core}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3\tilde{r}_i^2}\biggl( 1 - \frac{2}{3}\xi_i^2 \biggr) = -9.177751 \times 10^{4}\, .
</math>
  </td>
</tr>
</table>
  </li>
  <li>The green arrow is tangent to the solid green curve at the core/envelope interface; its slope is, 
<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]\biggr\}_{i,\mathrm{env}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}_i^2} \biggl\{1
-
\frac{2}{3} 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\biggl(1+\frac{1}{3}\xi_i^2\biggr) 
\biggr\} =-4.552725\times 10^5\, .
</math>
  </td>
</tr>
</table>
  </li> 
</ul>
They illustrate that the ''slope'' of the function, <math>\tilde{M}_r/\tilde{r}^2</math>, has a discontinuous jump at the interface.  Given that, 
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>\frac{d\tilde{P}}{d\tilde{M}_r} </math></td>
  <td align="center"><math>=</math></td>
  <td align="left"><math>- \frac{\tilde{M}_r}{\tilde{r}^2} \, ,</math></td>
</tr>
</table>
in the <b>Model C</b> equilibrium configuration, this also illustrates that this bipolytropic model has a discontinuous jump in the ''second-derivative'' of the pressure (with respect to <math>\tilde{M}_r</math>) at the core/envelope interface.
  </td>
</tr>
</table>


Note that this discontinuity will disappear in a model for which the two slopes have the same value, that is, when,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{1}{3}\biggl( 1 - \frac{2}{3}\xi_i^2 \biggr)
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{1
-
\frac{2}{3} 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\biggl(1+\frac{1}{3}\xi_i^2\biggr) 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ 1 - \frac{2}{3}\xi_i^2
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3
-
2 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\biggl(1+\frac{1}{3}\xi_i^2\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \xi_i^2
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}
+
\biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}\xi_i^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl(\frac{\mu_e}{\mu_c}\biggr)\xi_i^2
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 
+
\xi_i^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~ \biggl(\frac{\mu_e}{\mu_c} - 1\biggr)\xi_i^2
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
3 \, .
</math>
  </td>
</tr>
</table>
This will never occur in this bipolytropic model.

<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
<div align="center"><b>Evaluation of the Logarithmic Derivative</b><br /><math>d\ln \tilde{r}/d\ln \tilde{M}_r</math></div>
At the [[#Behavior_at_the_Interface|beginning of this subsection]], we demonstrated that, quite generally,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\frac{d}{d\tilde{M}_r} \biggl[ \frac{\tilde{M}_r}{\tilde{r}^2} \biggr]
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{\tilde{r}^2}\biggl\{ 1 - 2 \cdot \frac{d\ln \tilde{r}}{d\ln \tilde{M}_r}
\biggr\}
\, .
</math>
  </td>
</tr>
</table>
It is therefore the case that, at the interface and from the perspective of the core,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d\ln \tilde{r}}{d\ln \tilde{M}_r} \biggr\}_{i, \mathrm{core}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3}\biggl(1 + \frac{\xi_i^2}{3}\biggr) \, ;
</math>
  </td>
</tr>
</table>
while, at the interface but from the perspective of the envelope,

<table border="0" align="center" cellpadding="8">

<tr>
  <td align="right">
<math>
\biggl\{ \frac{d\ln \tilde{r}}{d\ln \tilde{M}_r} \biggr\}_{i, \mathrm{env}}
</math>
  </td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\frac{1}{3}\biggl(1 + \frac{\xi_i^2}{3}\biggr)\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} 
\, .
</math>
  </td>
</tr>
</table>

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