Editing SSC/Virial/PolytropesSummary (section)

==Relationship to Detailed Force-Balanced Models==
===Structural Form Factors===

In our [[SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying detailed analysis]], we demonstrate that the expressions given above for the free-energy function and the virial theorem are correct in sufficiently strict detail that they can be used to precisely match &#8212; and assist in understanding &#8212; the equilibrium of embedded polytropes whose structures have been determined from the set of detailed force-balance equations.  In order to draw this association, it is only necessary to realize that, very broadly, the constant coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, in the above [[SSC/Virial/PolytropesSummary#FreeEnergyExpression|algebraic free-energy expression]] are expressible in terms of three [[SSCpt1/Virial#Structural_Form_Factors|structural form factors]], <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_W</math>, and <math>~\tilde\mathfrak{f}_A</math>, as follows:
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2 \cdot \tilde\mathfrak{f}_W \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} 
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  
\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A 
= \frac{4\pi}{3} 
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3(n+1)/n} \biggr]_\mathrm{eq} 
\cdot \tilde\mathfrak{f}_A \, ;
</math>
  </td>
</tr>
</table>
</div>
and that, specifically in the context of spherically symmetric, pressure-truncated polytropes, we can write &hellip;

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl[ - \frac{3\tilde\theta^'}{\tilde\xi} \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \tilde\theta^' \biggr]^2  + \tilde\theta^{n+1} \, .
</math>
  </td>
</tr>
</table>
</div>


<font color="red"><b>
January 13, 2015:
</b></font> 
As is noted in our [[SSC/Virial/PolytropesEmbeddedOutline#Third_Effort|accompanying outline of work]], I no longer believe that <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math> have the same expressions as in isolated polytropes.  Hence, all of the material that follows is suspect and needs to be reworked.

{{ SGFworkInProgress }}

After plugging these nontrivial expressions for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> into the righthand sides of the above equations for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> and, simultaneously, using Horedt's detailed force-balanced expressions for <math>~r_a</math> and <math>~p_a</math> to specify, respectively, <math>~\chi_\mathrm{eq}</math> and <math>~P_e/P_\mathrm{norm}</math> in these same equations, we find that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_\mathrm{ad} (1 + \eta_\mathrm{ad})^{-4n/(n-3)} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1 + \eta_\mathrm{ad})^{n/(n-3)} \, ,</math>
  </td>
</tr>
</table>
</div>
where the newly identified, key physical parameter, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{(5-n) \tilde\theta^{n+1}}{3(n+1) (\tilde\theta^')^2} \, .</math>
  </td>
</tr>
</table>
</div>

It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[SSC/Virial/PolytropesSummary#ConciseVirial|virial theorem presented above]].

===Physical Meaning of Parameter &eta;<sub>ad</sub>===

As defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of the two terms that are summed together in the definition of the structural form factor, <math>\tilde\mathfrak{f}_A</math>.  It is worth pointing out what physical quantities are associated with these two terms.  

At any radial location within a polytropic configuration, the [[SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden function]], <math>\theta</math>, is defined in terms of a ratio of the local density to the configuration's central density, specifically,
<div align="center">
<math>\theta \equiv \biggl(\frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
</div>
Remembering that, at any location within the configuration, the pressure is related to the density via the polytropic equation of state,
<div align="center">
<math>P = K\rho^{(n+1)/n} \, ,</math>
</div>
we see that,
<div align="center">
<math>\frac{P}{P_c} = \theta^{n+1} \, .</math>
</div>
Hence, the quantity, <math>\tilde\theta^{n+1}</math>, which appears as the second term in our definition of <math>\tilde\mathfrak{f}_A</math>, is the ratio, <math>(P/P_c)_{\tilde\xi}</math>, evaluated at the surface of the truncated polytropic sphere.  But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,
<div align="center">
<math>\tilde\theta^{n+1} = \frac{P_e}{P_c}  \, .</math>
</div>

In our [[SSC/Virial/Polytropes#Strategy2|accompanying detailed analysis]], we have employed the virial theorem expression to demonstrate that the first term in our definition of <math>~\tilde\mathfrak{f}_A</math> provides a measure the configuration's normalized central pressure.  Specifically, we show that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{4\pi}{3} \biggr) \frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>[3 (n+1) (\tilde\theta^')^2]^{-1} \, .</math>
  </td>
</tr>
</table>
</div>
We conclude, therefore, that quite generally,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>(5-n) \tilde\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4}  + (5-n) \frac{P_e}{P_c} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{4\pi} \biggr) \frac{G M_\mathrm{limit}^2}{P_c R_\mathrm{eq}^4} \biggl[1  + \eta_\mathrm{ad} \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
and that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\eta_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{4\pi (5-n)}{3} \biggr] \frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \, .</math>
  </td>
</tr>
</table>
</div>

===Desired Pressure-Radius Relation===
It is now clear from our review, above, of [[User:Tohline/SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution|Horedt's detailed force-balanced solution]], that 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{4\pi (5-n)}{3}\biggl[\frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta_\mathrm{ad} \, .</math>
  </td>
</tr>
</table>
</div>
Hence, the pair of parametric equations obtained via a solution of the detailed force-balanced equations satisfy our, slightly rearranged, 
<div align="center" id="ConciseVirial3">
<font color="#770000">'''Algebraic Expression of the Virial Theorem'''</font><br />

<math>
\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4 = \Chi_\mathrm{ad}^{(n-3)/n} - 1 \, .
</math>
</div>
More to the point, it is now clear that this virial theorem expression provides the direct relationship between the configuration's dimensionless equilibrium radius as defined by Horedt, <math>~r_a</math>, and the dimensionless applied external pressure as defined by Horedt, <math>~p_a</math>, that was not apparent from the original pair of parametric relations.  Horedt's parameters, <math>~r_a</math> and <math>~p_a</math>, can be directly associated to our parameters, <math>~\Chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math>, via two new normalizations,  <math>~r_n</math> and <math>~p_n</math>, defined through the relations,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Chi_\mathrm{ad} = \frac{r_a}{r_n}</math>
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>~\Pi_\mathrm{ad} = \frac{p_a}{p_n} \, .</math>
  </td>
</tr>
</table>
</div>
Specifically in terms of the coefficients in the free-energy expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_n^{n-3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{(n+1)^n}{4\pi} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^n
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{-n} \, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p_n^{n-3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{3^{n-3}}{(4\pi)^4 (n+1)^{3(n+1)}} \biggl[ \mathcal{A} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \biggr]^{-3(n+1)}
\biggl[ \mathcal{B} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} \biggr]^{4n} \, ;
</math>
  </td>
</tr>
</table>
</div>
while, in terms of the structural form factors,

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

<tr>
  <td align="right">
<math>~r_n^{n-3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1}{3} \biggl[ \frac{(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr]^n \mathfrak{f}_M^{1-n}
\, ,
</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p_n^{n-3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{1}{(4\pi)^8} \biggl[ \frac{3\cdot 5^3}{(n+1)^3} \cdot \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W^3} \biggr]^{n+1} \mathfrak{f}_A^{4n}
\, .
</math>
  </td>
</tr>
</table>
</div>


<!--  THE FOLLOWING DERIVATION IS CORRECT IN DETAIL, BUT NOT PARTICULARLY USEFUL

Let's plug Horedt's expressions into the virial relation and see how it reduces without inserting specific expressions for the free-energy coefficients, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>.  The lefthand side becomes,
<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>~
\frac{4\pi}{3} \biggl[ \frac{\mathcal{A}^{3(n+1)}}{\mathcal{B}^{4n}} \biggr]^{1/(n-3)} \biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]^{4n/(n-3)}
\biggl(  \frac{P_e}{P_\mathrm{norm} } \biggr)_\mathrm{Horedt} \biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^4 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4\pi}{3} \biggl[ \frac{1}{\mathcal{A}} \biggr] 
\biggl(  \frac{P_e}{P_\mathrm{norm} } \biggr)_\mathrm{Horedt} \biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^4 \, .
</math>
  </td>
</tr>
</table>
</div>
While the righthand side becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
~\Chi_\mathrm{ad}^{(n-3)/n} - 1
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]
\biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^{(n-3)/n} -1 \, .
</math>
  </td>
</tr>

</table>
</div>
Together, then, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\frac{4\pi}{3} 
\biggl(  \frac{P_e}{P_\mathrm{norm} } \biggr)_\mathrm{Horedt} \biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^4 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathcal{B} \biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^{(n-3)/n} -\mathcal{A}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~
\frac{4\pi}{3} \biggl\{ p_a
\biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{(n+1)/(n-3)} \biggr\}
\biggl\{ r_a
\biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)}
\biggr\}^4 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathcal{B} \biggl\{ r_a
\biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)}
\biggr\}^{(n-3)/n} -\mathcal{A}
</math>
  </td>
</tr>
</table>
</div>
Or, simplifying,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{1}{3(n+1)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} p_a r_a^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_a^{(n-3)/n}
\mathcal{B} \biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/n}
-\mathcal{A}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{1}{3} p_a r_a^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_a^{(n-3)/n}
\mathcal{B} ( 4\pi )^{1/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-(n+1)/n} 
-\mathcal{A} (n+1) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \, .
</math>
  </td>
</tr>
</table>
</div>
Derived from the virial theorem, this expression shows, in the most general case, how the equilibrium radius identified by Horedt, <math>~r_a</math>, relates to the dimensionless external pressure, <math>~p_a</math>, as defined by Horedt.  It is somewhat unsatisfactory that this algebraic <math>~p_a - r_a</math> relationship explicitly involves <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, two of the coefficients found in the free-energy expression.  Unsatisfactory as it may be, its broad applicability can be straightforwardly demonstrated.  After plugging in the expressions given above for <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math> in terms of the structural form factors, to obtain,

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

<tr>
  <td align="right">
<math>~p_a r_a^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_a^{(n-3)/n} \biggl[3^{(n+1)/n}  \biggr] \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}}
-\biggl[ \frac{3(n+1)}{5} \biggr] \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,
</math>
  </td>
</tr>
</table>
</div>
one need only plug in Horedt's expressions for <math>~r_a</math> and <math>~p_a</math>, and our expressions for the three structural form factors &#8212; all given in terms of <math>~\tilde\theta</math>, <math>~\tilde\theta^'</math>, and <math>~\tilde\xi</math> &#8212; to see that the lefthand side equals the righthand side in precise detail.

END OF BLOCKED-OUT SUBSECTION -->