Editing SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt3

__FORCETOC__
=BiPolytrope with (n<sub>c</sub>, n<sub>e</sub>) = (5, 1) Renormalized=
<table border="1" align="center" width="100%" colspan="8">
<tr>
  <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize|Part I: &nbsp; (5, 1) Analytic Renormalize]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt2|Part II:&nbsp; Envelope]]<br />&nbsp;</td>
  <td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt3|III:&nbsp; Interface Pressure Gradient]]<br />&nbsp;</td>
</tr>
</table>


==Examine Pressure Gradient at the Interface==
Determine the interface-pressure-gradient from two different perspectives:  (1) Look at the behavior of the pressure as determined when the hydrostatic-balance models have been constructed; and (2) Look at the behavior of the specific entropy at the interface.

===From Hydrostatic Balance===

====Pressure Gradient at Core Interface====
[[SSC/Structure/BiPolytropes/Analytic51|Step 4: Throughout the core]] &hellip; we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P^* \equiv \frac{P}{[ K_c \rho_0^{6/5}  ]}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-3} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{dP^*}{d\xi} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2\xi \biggl( 1 + \frac{1}{3}\xi^2 \biggr)^{-4} 
\, ;
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r^* \equiv \biggl[\frac{G\rho_0^{4/5}}{K_c}\biggr]r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\xi}{r^*} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{2\pi}{3}\biggr)^{1 / 2} \, .
</math>
  </td>
</tr>
</table>
Hence, from the perspective of the core, at the interface <math>(\xi_i)</math> the radial pressure derivative is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dP^*}{dr^*} \biggr|_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- 2\xi_i \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-4} \biggl(\frac{2\pi}{3}\biggr)^{1 / 2}
\, .
</math>
  </td>
</tr>
</table>

Note, as well, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[ \frac{r^*}{P^*} \cdot \frac{dP^*}{dr^*} \biggr]_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- 2\xi_i \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-4} \biggl(\frac{2\pi}{3}\biggr)^{1 / 2}
\biggl[ \biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi_i \biggr]\biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{3}
</math>
  </td>
</tr>

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


====Pressure Gradient at Envelope Interface====

[[SSC/Structure/BiPolytropes/Analytic51|Step 8: Throughout the envelope]] &hellip; we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>P^* \equiv \frac{P}{[ K_c \rho_0^{6/5}  ]}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\theta_i^6 \phi^2 
=
A^2\biggl[ 1 + \frac{\xi_i^2}{3} \biggr]^{-3} \biggl[\frac{\sin(\eta-B)}{\eta}\biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{dP^*}{d\eta} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2A^2\biggl[ 1 + \frac{\xi_i^2}{3} \biggr]^{-3} \biggl[\frac{\sin(\eta-B)}{\eta}\biggr]\biggl\{
\frac{\cos(\eta-B)}{\eta} - \frac{\sin(\eta-B)}{\eta^2}
\biggr\} 
\, ;
</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r^* \equiv \biggl[\frac{G\rho_0^{4/5}}{K_c}\biggr]r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{\eta}{r^*} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{2}_i (2\pi)^{1/2} \, .
</math>
  </td>
</tr>
</table>
Hence, from the perspective of the envelope, at the interface <math>(\eta_i)</math> the radial pressure derivative is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dP^*}{dr^*} \biggr|_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr) (2^3\pi)^{1/2}A^2\biggl[ 1 + \frac{\xi_i^2}{3} \biggr]^{-4} \biggl[\frac{\sin(\eta_i-B)}{\eta_i^2}\biggr]\biggl\{
\eta_i\cos(\eta_i-B) - \sin(\eta_i-B)
\biggr\}  \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\eta_i}{\sin(\eta_i-B)} \, ,
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\eta_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
3^{1/2} \xi_i \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{2}\, .
</math>
  </td>
</tr>
</table>
Simplifying this last expression a bit, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dP^*}{dr^*} \biggr|_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr) (2^3\pi)^{1/2}\biggl[ 1 + \frac{\xi_i^2}{3} \biggr]^{-4} \biggl[
\eta_i\cot(\eta_i-B) - 1
\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-2 \xi_i \biggl(\frac{2\pi}{3}\biggr)^{1/2}\biggl( 1 + \frac{\xi_i^2}{3} \biggr)^{-4} \biggl\{
1 - \eta_i\cot(\eta_i-B)
\biggr\} \biggl( \frac{\mu_e}{\mu_c} \biggr) \frac{\sqrt{3}}{\xi_i}  \, .
</math>
  </td>
</tr>
</table>

Note, as well, that,

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

<tr>
  <td align="right">
<math>\biggl[ \frac{r^*}{P^*} \cdot \frac{dP^*}{dr^*} \biggr]_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr) (2^3\pi)^{1/2}A^2\biggl[ 1 + \frac{\xi_i^2}{3} \biggr]^{-4} \biggl[\frac{\sin(\eta_i-B)}{\eta_i^2}\biggr]\biggl\{
\eta_i\cos(\eta_i-B) - \sin(\eta_i-B)
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
\times \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \theta^{-2}_i (2\pi)^{-1/2}\eta_i 
\biggl\{ A^2\biggl[ 1 + \frac{\xi_i^2}{3} \biggr]^{-3} \biggl[\frac{\sin(\eta_i-B)}{\eta_i}\biggr]^2 \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2\biggl[ 1 + \frac{\xi_i^2}{3} \biggr]^{-1} \biggl\{
\eta_i\cot(\eta_i-B) - 1
\biggr\} \theta^{-2}_i \eta_i
</math>
  </td>
</tr>

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

====Pressure Gradient Summary====

Whether viewed from the perspective of the core or the envelope, we have shown that the pressure at the interface is the same.  However, at the interface, the first derivative (or the logarithmic derivative) of the pressure as viewed from the envelope is "larger" than what is viewed from the perspective of the core by the following factor:

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

<tr>
  <td align="right">
factor
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{
1 - \eta_i\cot(\eta_i-B)
\biggr\}   \frac{\sqrt{3}}{\xi_i}\biggl( \frac{\mu_e}{\mu_c}\biggr) 
\, .
</math>
  </td>
</tr>
</table>
We note, for later use, that ''averaging'' these two pressure-gradients at the interface gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{1}{2} \biggl\{
\biggl[ \frac{r^*}{P^*} \cdot \frac{dP^*}{dr^*} \biggr]_\mathrm{core}
+
\biggl[ \frac{r^*}{P^*} \cdot \frac{dP^*}{dr^*} \biggr]_\mathrm{env}
\biggr\}_i
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \xi_i \biggl( 1 + \frac{1}{3}\xi_i^2 \biggr)^{-1} \biggl\{
\xi_i 
+\biggl[
1 - \eta_i\cot(\eta_i-B)
\biggr] 3^{1/2}\biggl( \frac{\mu_e}{\mu_c}\biggr) 
\biggr\}
\, .
</math>
  </td>
</tr>
</table>

===From Step-Function Behavior of Specific Entropy===

====Strategically Incorporate Step Function====
As we have discussed separately, a useful expression for the specific entropy of any individual Lagrangian fluid element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{s}{\mathfrak{R}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{\mu (\gamma-1)} \cdot \ln \biggl(\frac{P}{\rho^\gamma}\biggr)
\, .
</math>
  </td>
</tr>
</table>
How does <math>s/\mathfrak{R}</math> vary as a function of the Lagrangian mass shell (or Lagrangian radial coordinate)?  In the case of a spherical bipolytropic configuration: <math>s = s_c</math> (a constant) throughout the core; <math>s = s_e</math> (another constant) throughout the envelope; and a [[Appendix/Mathematics/StepFunction|unit step function]],

<table border="0" align="center" cellpadding="8">
<tr>
  <td align="center">
<math>
H(\zeta) =  
\begin{cases}
0; & ~~ \zeta < 0 \\
1; & ~~ \zeta > 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>
</tr>
</table>

can be introduced to accomplish the instantaneous jump from <math>s_c</math> to <math>s_e</math> at the core/envelope interface.  Specifically, after defining,

<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>
</table>

we obtain the correct physical description of the variation of specific entropy with mass shell, <math>m</math>, via the expression,

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

<tr>
  <td align="right">
<math>s(\zeta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>s_c + H(\zeta)\cdot (s_e - s_c) \, .</math>
  </td>
</tr>
</table>
Adopting the [[Appendix/Mathematics/StepFunction|''half-maximum convention'']] &#8212; which states that <math>H(\zeta = 0) = \tfrac{1}{2}</math> &#8212; we acknowledge that the functional value of the specific entropy at the interface is, <math>s_i = \tfrac{1}{2}(s_c + s_e)</math>.  Also, from our [[Appendix/Mathematics/StepFunction|accompanying brief discussion of the behavior of the unit step function]], we appreciate that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dH(\zeta)}{d\zeta}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\delta(\zeta)    \, ,</math>
  </td>
</tr>
</table>
where, <math>\delta(\zeta)</math> is the Dirac delta function.  We conclude, therefore, that precisely at the interface,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{ds(\zeta)}{d\zeta}\biggr|_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[(s_e - s_c) \cdot \delta(\zeta) \biggr]_i = (s_e - s_c)   \, .</math>
  </td>
</tr>
</table>
Generally speaking, the two parameters, <math>\mu, \gamma</math>, and the mass density, <math>\rho</math>, also will exhibit a step-function behavior at the interface of each equilibrium bipolytrope.  The following table summarizes how we model the radial variation of these quantities.

<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" rowspan="2">Quantity</td>
  <td align="center" rowspan="2">Functional Behavior</td>
  <td align="center" colspan="3">At Interface</td>
</tr>

<tr>
  <td align="center">Value</td>
  <td align="center">Derivative wrt <math>\zeta</math></td>
  <td align="center">Derivative wrt <math>r</math></td>
</tr>

<tr>
  <td align="center">Specific Entropy</td>
  <td align="center"><math>s(\zeta) = s_c + H(\zeta)\cdot (s_e - s_c)</math></td>
  <td align="center"><math>s_i \equiv s(0) = \tfrac{1}{2}(s_c + s_e)</math></td>
  <td align="center"><math>\frac{ds(\zeta)}{d\zeta}\biggr|_i = (s_e - s_c)</math></td>
  <td align="center"><math>\frac{ds}{dr}\biggr|_i = (s_e - s_c)/r_i</math></td>
</tr>

<tr>
  <td align="center">Mean Molecular Weight</td>
  <td align="center"><math>\mu(\zeta) = \mu_c + H(\zeta)\cdot (\mu_e - \mu_c)</math></td>
  <td align="center"><math>\mu_i \equiv \mu(0) = \tfrac{1}{2}(\mu_c + \mu_e)</math></td>
  <td align="center"><math>\frac{d\mu(\zeta)}{d\zeta}\biggr|_i = (\mu_e - \mu_c)</math></td>
  <td align="center"><math>\frac{d\mu}{dr}\biggr|_i = (\mu_e - \mu_c)/r_i</math></td>
</tr>

<tr>
  <td align="center">Ratio of Specific Heats</td>
  <td align="center"><math>\gamma(\zeta) = \gamma_c + H(\zeta)\cdot (\gamma_e - \gamma_c)</math></td>
  <td align="center"><math>\gamma_i \equiv \gamma(0) = \tfrac{1}{2}(\gamma_c + \gamma_e)</math></td>
  <td align="center"><math>\frac{d\gamma(\zeta)}{d\zeta}\biggr|_i = (\gamma_e - \gamma_c)</math></td>
  <td align="center"><math>\frac{d\gamma}{dr}\biggr|_i = (\gamma_e - \gamma_c)/r_i</math></td>
</tr>
</table>

The step-function that arises in a proper description of the density distribution must be handled with a bit more care.  Throughout the core,
<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; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>
r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{3}{2\pi}\biggr)^{1 / 2} \xi \, ;</math>
  </td>
</tr>
</table>
and throughout the envelope,
<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 \biggl[ \frac{\eta_i}{\sin(\eta_i - B)}\biggr]
\biggl[ \frac{\sin(\eta - B)}{\eta}\biggr] \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>
r^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\mu_e}{\mu_c}\biggr)^{-1} \theta_i^{-2} (2\pi)^{-1 / 2}\eta \, .</math>
  </td>
</tr>
</table>
The complete functional expression for the normalized mass density can therefore be written as,

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

<tr>
  <td align="right">
<math>\rho^*(\zeta)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho^*_\mathrm{core}(\zeta) + H(\zeta)\cdot \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr] \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d}{dr^*} \biggl[\rho^*(\zeta)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\rho^*_\mathrm{core}}{dr^*} 
+ \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr] \frac{1}{r_i} \frac{dH(\zeta)}{d\zeta}
+ H(\zeta)\cdot \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} - \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr] 
\, .</math>
  </td>
</tr>
</table>

Sanity check:
<ul>
  <li><math>\zeta < 0:</math> &nbsp;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d}{dr^*} \biggl[\rho^*(\zeta)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\rho^*_\mathrm{core}}{dr^*} 
+ \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr] \frac{1}{r_i} \cancelto{0}{\frac{dH(\zeta)}{d\zeta} }
+ \cancelto{0}{H(\zeta)}\cdot \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} - \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr] 
=
\frac{d\rho^*_\mathrm{core}}{dr^*}
\, .</math>
  </td>
</tr>
</table>
  </li>

  <li><math>\zeta > 0:</math> &nbsp;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d}{dr^*} \biggl[\rho^*(\zeta)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{d\rho^*_\mathrm{core}}{dr^*} 
+ \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr] \frac{1}{r_i} \cancelto{0}{\frac{dH(\zeta)}{d\zeta} }
+ \cancelto{1}{H(\zeta)}\cdot \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} - \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr] 
=
\frac{d\rho^*_\mathrm{env}}{dr^*}
\, .</math>
  </td>
</tr>
</table>
  </li>
  <li><math>\zeta = 0:</math> &nbsp;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d}{dr^*} \biggl[\rho^*(\zeta)\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{d\rho^*_\mathrm{core}}{dr^*} \biggr]_i
+ \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr]_i \frac{1}{r_i} \cancelto{1}{\frac{dH(\zeta)}{d\zeta} }
+ \cancelto{1/2}{H(\zeta)}\cdot \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} - \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr]_i 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2} \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} + \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr]_i 
+ \frac{1}{r_i} \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr]_i 
\, .</math>
  </td>
</tr>
</table>
  </li>
</ul>

====Now Take Radial Derivative of Pressure====
Solving for <math>P</math> in the expression for specific entropy, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\ln \biggl[\frac{P^*}{(\rho^*)^\gamma}\biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\mu (\gamma-1)s}{\mathfrak{R}}
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
<math>\frac{dP^*}{dr^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\gamma(\rho^*)^{\gamma-1} \biggl\{ \exp\biggl[ \frac{\mu (\gamma-1)s}{\mathfrak{R}} \biggr]\biggr\} \frac{d\rho^*}{dr^*}
+
(\rho^*)^{\gamma} \biggl\{ \exp\biggl[ \frac{\mu (\gamma-1)s}{\mathfrak{R}} \biggr]\biggr\} \frac{d}{dr^*}\biggl[
\frac{\mu (\gamma-1)s}{\mathfrak{R}}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{1}{P^*} \cdot \frac{dP^*}{dr^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\gamma}{\rho^*} \frac{d\rho^*}{dr^*}
+
\frac{d}{dr^*}\biggl[
\frac{\mu (\gamma-1)s}{\mathfrak{R}}
\biggr]
\, .
</math>
  </td>
</tr>
</table>

We will need to recognize that, unless we are sitting exactly at the interface &#8212; that is, unless <math> \zeta= 0</math> precisely &#8212; 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{d}{dr^*}\biggl[\frac{\mu (\gamma-1)s}{\mathfrak{R}}\biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
0\, .
</math>
  </td>
</tr>
</table>

Hence, for two of the separate physical regimes &hellip;

<ul>
  <li>
<math>\zeta < 0:</math>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{dP^*}{dr^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\gamma(\rho^*)^{\gamma-1} \biggl\{ \exp\biggl[ \frac{\mu (\gamma-1)s}{\mathfrak{R}} \biggr]\biggr\} \frac{d\rho^*}{dr^*}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{1}{P^*_\mathrm{core}} \cdot \frac{dP^*_\mathrm{core}}{dr^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\gamma_c}{\rho^*_\mathrm{core}} \cdot \frac{d\rho^*_\mathrm{core}}{dr^*}
</math>
  </td>
</tr>
</table>
  </li>

  <li>
<math>\zeta > 0:</math>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{1}{P^*_\mathrm{env}} \cdot \frac{dP^*_\mathrm{env}}{dr^*}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\gamma_e}{\rho^*_\mathrm{env}} \cdot \frac{d\rho^*_\mathrm{env}}{dr^*}
</math>
  </td>
</tr>
</table>
  </li>
</ul>

However, at the interface where <math>\zeta = 0</math> precisely, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{\mathfrak{Z}_0}{r_i} \equiv
\biggl\{\frac{d}{dr^*}\biggl[\frac{\mu (\gamma-1)s}{\mathfrak{R}}\biggr]\biggr\}_{\zeta=0}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\mathfrak{R}}
\biggl\{(\gamma-1)s\frac{d\mu}{dr^*}
+
\mu  s\frac{d\gamma}{dr^*}
+
\mu (\gamma-1)\frac{ds}{dr^*}

\biggr\}_{\zeta=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\mathfrak{R}}
\biggl\{(\gamma-1)s\frac{d\mu}{dr^*}
+
\mu  s\frac{d\gamma}{dr^*}
+
\mu (\gamma-1)\frac{ds}{dr^*}

\biggr\}_{\zeta=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\mathfrak{R}}\biggl[\tfrac{1}{2}(\mu_c + \mu_e) \biggr] \biggl[ \tfrac{1}{2}(\gamma_c + \gamma_e)-1 \biggr]
\biggl[\tfrac{1}{2}(s_c + s_e) \biggr]
\biggl\{\biggl[\frac{2}{(\mu_c + \mu_e) }\biggr]\frac{d\mu}{dr^*}
+
\biggl[ \frac{2}{(\gamma_c + \gamma_e)-1} \biggr]\frac{d\gamma}{dr^*}
+
\biggl[\frac{2}{(s_c + s_e)} \biggr] \frac{ds}{dr^*}
\biggr\}_{\zeta=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{\mathfrak{R}}\biggl[\tfrac{1}{2}(\mu_c + \mu_e) \biggr] \biggl[ \tfrac{1}{2}(\gamma_c + \gamma_e)-1 \biggr]
\biggl[\tfrac{1}{2}(s_c + s_e) \biggr]
\biggl\{\frac{2(\mu_e - \mu_c)}{(\mu_c + \mu_e) }
+
\frac{2(\gamma_e - \gamma_c)}{(\gamma_c + \gamma_e)-1} 
+
\frac{2(s_e - s_c)}{(s_c + s_e)}  
\biggr\}\frac{1}{r_i^*} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2^2 \mathfrak{R}}
\biggl\{ (\mu_e - \mu_c)[ (\gamma_c + \gamma_e)-1 ](s_c + s_e)
+
(\gamma_e - \gamma_c)(\mu_c + \mu_e) (s_c + s_e) 
+
(s_e - s_c)(\mu_c + \mu_e)[ (\gamma_c + \gamma_e)-1 ]  
\biggr\}\frac{1}{r_i^*} \, .
</math>
  </td>
</tr>
</table>

At the interface, then, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl[\frac{1}{P^*} \cdot \frac{dP^*}{dr^*} \biggr]_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{
\frac{\gamma}{\rho^*} \frac{d\rho^*}{dr^*}
+
\frac{d}{dr^*}\biggl[
\frac{\mu (\gamma-1)s}{\mathfrak{R}}
\biggr]\biggr\}_{\zeta=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\mathfrak{Z}_0}{r_i} +
\biggl\{
\frac{\gamma}{\rho^*} 
\biggr\}_{\zeta=0} \cdot
\biggl\{
\frac{1}{2} \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} + \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr]_i 
+ \frac{1}{r_i} \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr]_i 
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Finally, we see that,

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

<tr>
  <td align="right">
<math>\biggl\{
\frac{\gamma}{\rho^*} 
\biggr\}_{\zeta=0}
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\tfrac{1}{2}(\gamma_c + \gamma_e) \biggl\{
\rho^*_\mathrm{core}(\zeta) + H(\zeta)\cdot \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr]
\biggr\}^{-1}_{\zeta=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\tfrac{1}{2}(\gamma_c + \gamma_e) \biggl\{\frac{1}{2} \biggl[
\rho^*_\mathrm{core}(\zeta) +  \rho^*_\mathrm{env}(\zeta)  \biggr]
\biggr\}^{-1}_{\zeta=0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(\gamma_c + \gamma_e) \biggl[
\rho^*_\mathrm{core}(\zeta) +  \rho^*_\mathrm{env}(\zeta)  \biggr]_i^{-1}
\, ,
</math>
  </td>
</tr>
</table>

so, at the interface,

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

<tr>
  <td align="right">
<math>\biggl[\frac{1}{P^*} \cdot \frac{dP^*}{dr^*} \biggr]_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{\mathfrak{Z}_0}{r_i} +
\biggl\{
\frac{1}{2} \biggl[ \frac{d\rho^*_\mathrm{env}}{dr^*} + \frac{d\rho^*_\mathrm{core}}{dr^* }\biggr]_i 
+ \frac{1}{r_i} \biggl[ \rho^*_\mathrm{env}(\zeta) - \rho^*_\mathrm{core}(\zeta) \biggr]_i 
\biggr\} 
(\gamma_c + \gamma_e) \biggl[
\rho^*_\mathrm{core}(\zeta) +  \rho^*_\mathrm{env}(\zeta)  \biggr]_i^{-1} \, .
</math>
  </td>
</tr>
</table>

=Try Again=

Additional studies of radial oscillations in models that lie along the "51 Renormalized" sequences can be [[SSC/Structure/BiPolytropes/51RenormaizePart2|found here]].

==Example BiPolytrope Sequence 0.3100==
For the case of <math>(n_c, n_e) = (5, 1)</math> and <math>\mu_e/\mu_c = 0.3100</math>, we consider here the examination of models with three relatively significant values of the core/envelope interface:
<ul>
  <li>
<math>(\xi_i, \bar\rho/\rho_c, q, \nu) \approx (2.06061, 1.1931E+02, 0.16296, 0.13754)</math>: Approximate location along the sequence of the model with the maximum fractional core radius.
  </li>
  <li>
<math>(\xi_i, \bar\rho/\rho_c, q, \nu) \approx (2.69697, 3.0676E+02, 0.15819, 0.19161)</math>: Approximate location along the sequence of the onset of fundamental-mode instability.
  </li>
  <li>
<math>(\xi_i, \bar\rho/\rho_c, q, \nu) \approx (9.0149598, 1.1664E+06, 0.075502255, 0.337217006)</math>: Exact location along the sequence of the model with the maximum fractional core mass.
  </li>
</ul>

=See Also=
<ul>
  <li>Prasad, C. (1953), Proc. Natn. Inst. Sci. India, Vol. 19, 739, ''Radial Oscillations of a Composite Model.''</li>
  <li>Singh, Manmohan, (1969), Proc. Natn. Inst. Sci., India, Part A, Vol. 35, pp. 586 - 589, ''Radial Oscillations of Composite Polytropes &#8212; Part I</li>
  <li>[https://www.osti.gov/biblio/4178929-radial-oscillations-composite-polytropes-part-ii Singh, Manmohan, (1969)], Proc. Nat. Inst. Sci., India, Part A, Vol. 35, pp. 703 - 708, ''Radial Oscillations of Composite Polytropes &#8212; Part II</li>
  <li>[https://ui.adsabs.harvard.edu/abs/2021A%26AT...32..371K/abstract Kumar, S., Saini, S., Singh, K. K., Bhatt, V., &amp; Vashishta, L. (2021)], Astronomical &amp; Astrophysical Transactions, Vol. 32, Issue 4, pp. 371-382, ''Radial Pulsations of distorted Polytropes of Non-Uniform Density.''</li>
</ul>

{{ SGFfooter }}