Editing SSC/Structure/OtherAnalyticModels (section)

===Specific Entropy Distribution===

From an [[PGE/FirstLawOfThermodynamics#EntropyLL75|accompanying discussion]] of specific entropy distributions, <math>s</math>, we realize that to within an additive constant <math>(s_0)</math>,
<div align="center" id="LL75">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>s - s_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>c_P \ln \biggl(\frac{P^{1/\gamma_g}}{\rho} \biggr)\, .</math>
  </td>
</tr>
</table>
[<b>[[Appendix/References#LL75|<font color="red">LL75</font>]]</b>], &sect;80, Eq. (80.12)
</div>
Or (see a [[Appendix/Ramblings/PatrickMotl#Tying_Expressions_into_H_Book_Context|related discussion]]), given that 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~c_P </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\gamma_g}{(\gamma_g-1)} \biggl( \frac{\Re}{\bar\mu} \biggr) \, ,</math>
  </td>
</tr>
</table>
we can also write,

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

<tr>
  <td align="right">
<math>\frac{(s - s_0)}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{\gamma_g}{(\gamma_g-1)} \biggl\{ 
\frac{1}{\gamma_g}\ln P - \ln\rho
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{(\gamma_g-1)} \biggl\{ 
\ln P - \gamma_g\ln\rho
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{(\gamma_g-1)} \biggl\{ 
\ln P - \gamma_g\ln\rho - \ln(\gamma_g-1)
\biggr\} + \frac{\ln(\gamma_g-1)}{(\gamma_g - 1)}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(\gamma_g-1)}\ln \biggl[ \frac{P}{(\gamma_g-1)\rho^{\gamma_g}} \biggr]
+ \frac{\ln(\gamma_g-1)}{(\gamma_g - 1)} \, .</math>
  </td>
</tr>
</table>
Notice that if we set the constant,

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

<tr>
  <td align="right">
<math>\frac{s_0}{\Re/\bar{\mu}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
- \frac{\ln(\gamma_g-1)}{(\gamma_g - 1)} \, ,
</math>
  </td>
</tr>
</table>

we obtain the same expression for the entropy distribution as we used in our [[Appendix/Ramblings/PatrickMotl#Tying_Expressions_into_H_Book_Context|discussions with Patrick Motl]], namely,

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

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

Shifting this expression for the specific entropy by another constant (not written out explicitly here) leads us to the more "normalized" expression,

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

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~ (\gamma_g - 1)\exp\biggl[\frac{(\gamma_g - 1)s}{\Re/\bar{\mu}} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{P/P_c}{(\rho/\rho_c)^{\gamma_g}} \biggr] 
</math>
  </td>
</tr>
</table>
Plugging in the expressions for the pressure and density distributions that are relevant to the {{ Prasad49 }} model having a "parabolic" density distribution, then gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(\gamma_g-1)\exp\biggl[\frac{(\gamma_g - 1)s}{\Re/\bar{\mu}} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\biggl[1-\biggl(\frac{r}{R}\biggr)^2\biggr]^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r}{R}\biggr)^2\biggr] 
\biggr\}
\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr]^{-\gamma_g} 
</math>
  </td>
</tr>

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