Editing SSC/Virial/Polytropes (section)

==Relationship to Detailed Force Balance Solution==

Let's plug these form-factors into our expressions for the dimensionless equilibrium radius, <math>~\chi_\mathrm{ad}</math>, and dimensionless surface pressure, <math>~\Pi_\mathrm{ad}</math>, that have been derived from the identification of extrema in the free-energy function and see how they compare to the dimensionless radius, <math>~r_a \equiv R_\mathrm{eq}/R_\mathrm{Horedt}</math>, and dimensionless pressure, <math>~p_a \equiv P_e/P_\mathrm{Horedt}</math>, given by the detailed force-balance models provided by {{ Horedt70 }}.  Note that expressions for <math>~r_a</math> and <math>~p_a</math> are given in our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|accompanying discussion of embedded polytropic spheres]] and that the conversion from the {{ Horedt70 }} scaling parameters to our normalization parameters, <math>~R_\mathrm{Horedt}/R_\mathrm{norm}</math> and <math>~P_\mathrm{Horedt}/P_\mathrm{norm}</math>, can be found in our [[SSCpt1/Virial#Normalizations|introductory discussion of the virial equilibrium of spherical configurations]].

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

<tr>
  <td align="right">
<math>~\chi_\mathrm{ad} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{ad}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)
\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_a \biggl[ \biggl(\frac{\gamma - 1}{\gamma}\biggr) (4\pi)^{\gamma-1}\biggr]^{1/(4-3\gamma)}
\biggl[ 5 \biggl( \frac{4\pi}{3} \biggr)^{1-\gamma_g} \frac{\mathfrak{f}_A}{\mathfrak{f}_W \mathfrak{f}_M^{\gamma_g-2}}
\biggr]^{1/(4-3\gamma_g)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_a \biggl[ 5 \cdot 3^{\gamma-1} \biggl(\frac{\gamma - 1}{\gamma}\biggr) 
\frac{\mathfrak{f}_A \mathfrak{f}_M^{2-\gamma_g}}{\mathfrak{f}_W }
\biggr]^{1/(4-3\gamma_g)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
r_a \biggl[ \frac{5 \cdot 3^{1/n} }{n+1} \cdot \frac{\mathfrak{f}_A \mathfrak{f}_M^{(n-1)/n}}{\mathfrak{f}_W } \biggr]^{n/(n-3)}
= r_a \biggl[ 3 \mathfrak{f}_M^{n-1} \biggl( \frac{5 }{n+1} \cdot \frac{\mathfrak{f}_A }{\mathfrak{f}_W }\biggr)^n \biggr]^{1/(n-3)}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} \equiv \frac{P_\mathrm{e}}{P_\mathrm{ad}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{P_\mathrm{e}}{P_\mathrm{Horedt}} \biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)
\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p_a \biggl[ \biggl( \frac{\gamma}{\gamma-1}\biggr)^{3\gamma} (4\pi)^{-\gamma} \biggr]^{1/(4-3\gamma)}
\biggl[ \mathfrak{f}_A^{-4} \biggl( \frac{4\pi}{3\cdot  5^3} \cdot \frac{\mathfrak{f}_W^3}{\mathfrak{f}_M^2} \biggr)^{\gamma_g} 
\biggr]^{1/(4-3\gamma_g)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p_a \mathfrak{f}_A^{-4/(4-3\gamma)} \biggl[ \biggl( \frac{\gamma}{\gamma-1}\biggr)^{3}  
\biggl( \frac{1}{3\cdot  5^3} \cdot \frac{\mathfrak{f}_W^3}{\mathfrak{f}_M^2} \biggr)\biggr]^{\gamma/(4-3\gamma)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~p_a \mathfrak{f}_A^{-4n/(n-3)} \biggl[ \frac{(n+1)^3}{3\cdot  5^3} \cdot \frac{\mathfrak{f}_W^3}{\mathfrak{f}_M^2} \biggr]^{(n+1)/(n-3)}
</math>
  </td>
</tr>
</table>
</div>

Now we insert the form-factor expressions [[SSC/Virial/Polytropes#Summary|from above]], to obtain,

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

<tr>
  <td align="right">
<math>~\chi_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
r_a \biggl[ 3 \biggl( \frac{5 }{n+1} \biggr)^n \biggr]^{1/(n-3)}  
\biggl[ \frac{\mathfrak{f}_A^n \mathfrak{f}_M^{n-1} }{\mathfrak{f}_W^n } \biggr]^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
r_a \biggl[ 3 \biggl( \frac{5 }{n+1} \biggr)^n \biggr]^{1/(n-3)}  
\biggl\{ 
\biggl[ \frac{3(n+1) }{(5-n)} ~( \Theta^' )^2 \biggr]^n
\biggl[ \frac{3^2\cdot 5}{5-n} \biggl( \frac{\Theta^'}{\xi} \biggr)^2 \biggr]^{-n}
\biggl[ - \frac{3\Theta^'}{\xi} \biggr]^{n-1}
\biggr\}^{1/(n-3)}_{\xi_1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
r_a \biggl\{ 3^n \biggl( \frac{5 }{n+1} \biggr)^n \biggl[ \frac{3(n+1) }{(5-n)} \biggr]^n \biggl[ \frac{5-n}{3^2\cdot 5}  \biggr]^{n}
\biggr\}^{1/(n-3)}  
\biggl\{ \xi^{2n}
\biggl[ - \frac{\Theta^'}{\xi} \biggr]^{n-1}
\biggr\}^{1/(n-3)}_{\xi_1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
r_a \biggl[ \xi^{n+1} ( - \Theta^' )^{n-1} \biggr]^{1/(n-3)}_{\xi_1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~p_a \mathfrak{f}_A^{-4n/(n-3)} \biggl[ \frac{(n+1)^3}{3\cdot  5^3} \cdot \frac{\mathfrak{f}_W^3}{\mathfrak{f}_M^2} \biggr]^{(n+1)/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p_a \biggl[ \frac{3(n+1) }{(5-n)} ~( \Theta^' )^2 \biggr]_{\tilde\xi}^{-4n/(n-3)} 
\biggl\{ 
\frac{(n+1)^3}{3\cdot  5^3} \cdot 
\biggl[\frac{3^2\cdot 5}{5-n} \biggl( \frac{\Theta^'}{\xi} \biggr)^2\biggr]^3 
\biggl[  - \frac{3\Theta^'}{\xi} \biggr]^{-2}
\biggr\}^{(n+1)/(n-3)}_{\xi_1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~p_a \biggl\{ \biggl[ \frac{5-n}{3(n+1)} \biggr]^{4n} \biggl[ \frac{(n+1)^3 \cdot 3^6 \cdot 5^3}{3^3 \cdot 5^3 (5-n)^3} \biggr]^{(n+1)} \biggr\}^{1/(n-3)}
\biggl[ ( \Theta^' )^{-8n} \biggl( \frac{\Theta^'}{\xi} \biggr)^{4(n+1)}
\biggr]^{1/(n-3)}_{\xi_1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~p_a \biggl[ \frac{5-n}{3(n+1)} \biggr] \biggl[\xi^{n+1} ( \Theta^' )^{n-1} \biggr]^{-4/(n-3)}_{\xi_1}
</math>
  </td>
</tr>
</table>
</div>

Notice that the bracketed term that is to be evaluated at <math>\xi_1</math> is identical in both expressions.  After [[SSC/Virial/Polytropes#Renormalization|renormalization, as derived above]], the statement of virial equilibrium for embedded polytropes is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\chi_\mathrm{ad}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\Pi_\mathrm{ad}} \biggl[ \chi_\mathrm{ad}^{4-3\gamma_g} - 1 \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>
or, setting <math>~\gamma_g = (n+1)/n</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} \chi_\mathrm{ad}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_\mathrm{ad}^{(n-3)/n} - 1  \, .</math>
  </td>
</tr>
</table>
</div>
This relation can now be written in terms of the {{ Horedt70 }} dimensionless radius and pressure.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
~p_a \biggl[ \frac{5-n}{3(n+1)} \biggr] \biggl[\xi^{n+1} ( \Theta^' )^{n-1} \biggr]^{-4/(n-3)}_{\xi_1}
\cdot r_a^4 \biggl[ \xi^{n+1} ( - \Theta^' )^{n-1} \biggr]^{4/(n-3)}_{\xi_1} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ r_a^{(n-3)/n} \biggl[ \xi^{n+1} ( - \Theta^' )^{n-1} \biggr]^{1/n}_{\xi_1}  - 1</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>
\Rightarrow ~~~~ p_a r_a^4 \biggl[ \frac{5-n}{3(n+1)} \biggr]  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ r_a^{(n-3)/n} \biggl[ \xi^{n+1} ( - \Theta^' )^{n-1} \biggr]^{1/n}_{\xi_1}  - 1 \, ,</math>
  </td>
</tr>
</table>
</div>
where, from our separate [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|summary of Horedt's presentation]], 
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~r_a 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
= \biggl[ \tilde\xi^{n+1} (-\tilde\theta^')^{n-1} \biggr]^{-1/(n-3)}\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~p_a 
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the virial relation becomes,

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

<tr>
  <td align="right">
<math>
 ~1
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl\{ \frac{ [ \xi^{n+1} ( - \Theta^' )^{n-1} ]_{\xi_1} }{ [ \tilde\xi^{n+1} (-\tilde\theta^')^{n-1} ] } 
 \biggr\}^{1/n}
- \biggl[ \frac{5-n}{3(n+1)} \biggr] \tilde\theta_n^{n+1} \biggl\{ \frac{( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}}{ 
[ \tilde\xi^{n+1} (-\tilde\theta^')^{n-1} ]^{4/(n-3)} } \biggr\}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{ \xi_1^{n+1} ( - \Theta^' )^{n-1}_{\xi_1} }{ \tilde\xi^{n+1} (-\tilde\theta^')^{n-1}  } 
 \biggr]^{1/n}
- \biggl[ \frac{5-n}{3(n+1)} \biggr] \tilde\theta_n^{n+1} \biggl\{ \frac{ \tilde\xi^{4n+4} ( -\tilde\theta' )^{2n+2}}{ 
\tilde\xi^{4n+4} (-\tilde\theta^')^{4n-4}  } \biggr\}^{1/(n-3)}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ \frac{ \xi_1^{n+1} ( - \Theta^' )^{n-1}_{\xi_1} }{ \tilde\xi^{n+1} (-\tilde\theta^')^{n-1}  } 
 \biggr]^{1/n}
- \biggl[ \frac{5-n}{3(n+1)} \biggr] \frac{ \tilde\theta_n^{n+1} }{ ( -\tilde\theta' )^{2} } \, .</math>
  </td>
</tr>
</table>
</div>
This needs to be checked, perhaps with specific applications to the cases <math>~n=1</math> and <math>~n=5</math>.