Editing SSC/Virial/PolytropesSummary (section)

===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 
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<math>\frac{4\pi (5-n)}{3}\biggl[\frac{P_e R_\mathrm{eq}^4}{G M_\mathrm{limit}^2} \biggr]_\mathrm{Horedt} </math>
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<math>~=</math>
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<math>~\eta_\mathrm{ad} \, .</math>
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Hence, the pair of parametric equations obtained via a solution of the detailed force-balanced equations satisfy our, slightly rearranged, 
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<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,
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<math>~\Chi_\mathrm{ad} = \frac{r_a}{r_n}</math>
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&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp;
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<math>~\Pi_\mathrm{ad} = \frac{p_a}{p_n} \, .</math>
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Specifically in terms of the coefficients in the free-energy expression,
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<math>~r_n^{n-3}</math>
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<math>~\equiv</math>
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<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>
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and,
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<math>~p_n^{n-3}</math>
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<math>~\equiv</math>
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<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>
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while, in terms of the structural form factors,

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<math>~r_n^{n-3}</math>
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<math>~\equiv</math>
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<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>
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and,
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<math>~p_n^{n-3}</math>
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<math>~\equiv</math>
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<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>
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<!--  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,
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<math>\Pi_\mathrm{ad} \Chi_\mathrm{ad}^4</math>
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<math>~=</math>
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<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>
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&nbsp;
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<math>~=</math>
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<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>
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While the righthand side becomes,
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<math>
~\Chi_\mathrm{ad}^{(n-3)/n} - 1
</math>
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<math>~=</math>
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<math>~
\biggl[ \frac{\mathcal{B}}{\mathcal{A}} \biggr]
\biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^{(n-3)/n} -1 \, .
</math>
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Together, then, we have,
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<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>
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<math>~=</math>
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<math>~
\mathcal{B} \biggl(  \frac{R_\mathrm{eq}}{R_\mathrm{norm} } \biggr)_\mathrm{Horedt}^{(n-3)/n} -\mathcal{A}
</math>
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<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>
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<math>~=</math>
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<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>
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Or, simplifying,
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<math>~
\frac{1}{3(n+1)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} p_a r_a^4
</math>
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<math>~=</math>
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<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>
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<math>~\Rightarrow~~~ \frac{1}{3} p_a r_a^4
</math>
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<math>~=</math>
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<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>
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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,

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<math>~p_a r_a^4
</math>
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<math>~=</math>
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<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>
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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.

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