Editing SSC/Structure/PolytropesEmbedded (section)

===Whitworth's Presentation===

In &sect;5 of his paper, {{ Whitworth81}} &#8212; hereafter, {{ Whitworth81hereafter }} &#8212; also presents the set of parametric 
equations that define what the equilibrium radius, <math>R_\mathrm{eq}</math>, is of an embedded polytrope for a certain imposed external pressure, <math>P_\mathrm{e}</math>, namely,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~R_\mathrm{eq}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
~R_\mathrm{rf} \biggl\{ \frac{4\eta}{5|\eta-1|} \biggl(\frac{\xi}{3} \biggr)^\eta 
\biggl|\frac{d\theta_n}{d\xi} \biggr|^{(2-\eta)} \biggr\}_{\xi_e}^{1/(3\eta - 4)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(3-n)}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{4(n+1)}{5} \biggr]^{n} \biggl(\frac{\xi_e}{3} \biggr)^{(n+1)} 
\biggl|\frac{d\theta_n}{d\xi} \biggr|^{(n-1)}_{\xi_e} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~P_\mathrm{e}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
P_\mathrm{rf} \biggl\{  2^{-8/\eta} \biggl(\frac{5|\eta-1|}{\eta} \biggr)^3 \biggl(\frac{3}{\xi} \biggr)^4 
\biggl|\frac{d\theta_n}{d\xi} \biggr|^{-2} \biggr\}_{\xi_e}^{\eta/(3\eta - 4)} \theta_n^{\eta/(\eta-1)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ \biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr)^{(3-n)}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
2^{-8n}\biggl\{  \biggl(\frac{5}{n+1} \biggr)^3 \biggl(\frac{3}{\xi} \biggr)^4 \theta_n^{(3-n)}
\biggl|\frac{d\theta_n}{d\xi} \biggr|^{-2} \biggr\}_{\xi_e}^{(n+1)} \, ,
</math>
  </td>
</tr>

</table>
</div>
where, in order to obtain the second line of the two relations we have used the substitution, <math>\eta \rightarrow (1+1/n)</math>, and, as is detailed in an [[SSC/Structure/PolytropesASIDE1|accompanying ASIDE]], {{ Whitworth81hereafter }} "referenced" <math>P_\mathrm{e}</math> and <math>R_\mathrm{eq}</math> to, respectively,

<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
P_\mathrm{rf}^{(4-3\eta)}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
2^{-2(4+\eta)} \biggl( \frac{3^4 \cdot 5^3}{\pi} \biggr)^\eta \biggl[ \frac{K_n^4}{G^{3\eta} M^{2\eta} } \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ P_\mathrm{rf}^{(n-3)}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
2^{-2(5n+1)} \biggl( \frac{3^4 \cdot 5^3}{\pi} \biggr)^{(n+1)} \biggl[ \frac{K_n^{4n}}{G^{3(n+1)} M^{2(n+1)} } \biggr] \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~R_\mathrm{rf}^\eta 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\frac{2^2}{K_n} \biggl(\frac{GM}{3\cdot 5}\biggr)^\eta P_\mathrm{rf}^{(1-\eta)}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ R_\mathrm{rf}^{(n+1)} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2^2}{K_n} \biggr)^{n} \biggl(\frac{GM}{3\cdot 5}\biggr)^{(n+1)} P_\mathrm{rf}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~~ R_\mathrm{rf}^{(3-n)} 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2^2}{K_n} \biggr)^{n(3-n)/(n+1)} \biggl(\frac{GM}{3\cdot 5}\biggr)^{(3-n)} P_\mathrm{rf}^{(n-3)/(n+1)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{2^2}{K_n} \biggr)^{n(3-n)/(n+1)}  \biggl(\frac{GM}{3\cdot 5}\biggr)^{(3-n)} 
\biggl\{2^{-2(5n+1)} \biggl( \frac{3^4 \cdot 5^3}{\pi} \biggr)^{(n+1)} \biggl[ \frac{K_n^{4n}}{G^{3(n+1)} M^{2(n+1)} } \biggr]
\biggr\}^{1/(n+1)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>K_n^{n} ( 2^2 )^{-(n+1)} 
\biggl(\frac{GM}{3\cdot 5}\biggr)^{(3-n)} 
\biggl( \frac{3^4 \cdot 5^3}{\pi} \biggr) \biggl[ \frac{1}{G^{3} M^{2} } \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
2^{-2(n+1)} \pi^{-1} 3^{n+1} \cdot 5^{n} K_n^n G^{-n} M^{1-n}
</math>
  </td>
</tr>
</table>
</div>
Via these normalizations, {{ Whitworth81hereafter }} &#8212; as did {{ Horedt70hereafter }} &#8212; chose to express <math>R_\mathrm{eq}</math> and <math>P_\mathrm{e}</math> in terms of {{Math/MP_PolytropicConstant}} and the system's total mass, <math>M</math>.

To convert from Whitworth's expressions, which use one set of normalization parameters <math>(R_\mathrm{rf},P_\mathrm{rf})</math>, to Horedt's expressions, which use a somewhat different set of normalization parameters &#8212; identified here as <math>(R_\mathrm{Horedt},P_\mathrm{Horedt})</math> &#8212; one simply needs to make use of the relations,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
\biggl( \frac{R_\mathrm{rf}}{R_\mathrm{Horedt}} \biggr)^{(3-n)}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
3^{(n+1)} \biggl[ \frac{5}{2^2 (n+1)} \biggr]^{n} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\biggl( \frac{P_\mathrm{rf}}{P_\mathrm{Horedt}} \biggr)^{(3-n)}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
2^{8n} \biggl[ \frac{(n+1)^3}{3^4 \cdot 5^3} \biggr]^{(n+1)} \, ,
</math>
  </td>
</tr>
</table>
</div>