Editing SSC/Virial/PolytropesSummary (section)

====Summary====

The algebraic free-energy function associated with pressure-truncated <math>~n=4</math> polytropes is,
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<math>
\mathfrak{G}^*\biggr|_{n=4} = 
-3\mathcal{A} \chi^{-1} +~ 4\mathcal{B} \chi^{-3/4} +~ \mathcal{D}\chi^3 \, ,
</math>
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and the corresponding ''renormalized'' free-energy function is,
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<math>
\mathfrak{G}^{**}\biggr|_{n=4} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} = 
-3 \Chi^{-1} +~ 4\Chi^{-3/4} +~ \Pi_\mathrm{ad}\Chi^3 \, .
</math>
</div>
As has been demonstrated, above, the two equilibrium states that are supported by the same external pressure of, <math>~P_e/P_\mathrm{norm} = 1.71 \times 10^4</math>, are associated with extrema found in the following free-energy curves:  The ''unstable'' equilibrium appears as a relative ''maximum'' in the free-energy curves having the coefficient values,
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<math>~\Pi_\mathrm{ad} = 0.01</math> &nbsp;&nbsp;&nbsp;&nbsp; or &nbsp;&nbsp;&nbsp;&nbsp;
<math>(\mathcal{A}, \mathcal{B}, \mathcal{D}) = (1, 2.682, 7.16\times 10^4) \, .</math>
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The ''stable'' equilibrium appears as a relative ''minimum'' in the free-energy curves having the coefficient values,
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<math>~\Pi_\mathrm{ad} = 0.02369</math> &nbsp;&nbsp;&nbsp;&nbsp; or &nbsp;&nbsp;&nbsp;&nbsp;
<math>(\mathcal{A}, \mathcal{B}, \mathcal{D}) = (1, 2.542, 7.16\times 10^4) \, .</math>
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Configurations Sharing the Same External Pressure
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'''<font color="maroon">ASIDE:</font>'''  In retrospect, it is obvious that pairs of truncated equilibrium configurations of a given polytropic index that are bounded by the same external pressure &#8212; and, hence, that may share a ''physical'' evolutionary connection &#8212; will share the same value of Horedt's dimensionless pressure,
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<math>
~p_a 
</math>
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<math>~\equiv</math>
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<math>
\tilde\theta^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)}  \, .
</math>
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The implication is that a single free-energy curve with ''constant'' coefficients cannot connect the two equilibrium states.  There are certainly two separate equilibrium states that can be supported by the specified external pressure, but these two states exhibit somewhat different values of the structural form factors, which leads to different values of the coefficient, <math>~\mathcal{B}</math>.   The righthand plot in the following figure shows how <math>~\mathcal{B}</math> varies with the applied external pressure in <math>~n=4</math> polytropes.  


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Variation of Various Physical Parameters along the Sequence of Pressure-Truncated <math>~n=4</math> Polytropes

[Structural data obtained from the table provided on p. 399 of [http://adsabs.harvard.edu/abs/1986Ap%26SS.126..357H Horedt (1986, ApJS, vol. 126)]]

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[[File:SecondN4Parameters.png|750px|Parameters for n = 4 Embedded Polytropes]]
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'''<font color="maroon">Left:</font>'''  This log-log plot displays the variation with applied external pressure, <math>~p_a</math> (increasing to the right along the horizontal axis), of the renormalized pressure, <math>~\Pi_\mathrm{ad}</math> (light blue diamonds), the renormalized equilibrium radius, <math>~\Chi_\mathrm{ad}</math> (light green triangles), and the key physical parameter, <math>~\eta_\mathrm{ad}</math> (maroon circles).  As the diagram illustrates, each parameter is double-valued, demonstrating that, for any choice of the dimensionless external pressure (as long as the pressure is less than a well-defined limiting value), there are two available equilibrium states.  Along all three curves, parameter values associated with the ''stable'' equilibrium are traced by the ''upper'' portion of the curve.  The red vertical line has been drawn at the value of <math>~{p_a} = 0.176</math>, corresponding to the external pressure <math>~(P_e/P_\mathrm{norm} = 1.71\times 10^4)</math> examined in the above two tables.  This red line intersects the <math>~\Pi(p_a)</math> curve at <math>~\Pi = 0.01</math> (unstable state examined above) and at <math>~\Pi = 0.02369</math> (stable state examined above).

'''<font color="maroon">Right:</font>'''  This plot (linear scale on both axes) shows how <math>~\mathcal{B}</math> (curve outlined by light blue diamonds) varies with the applied external pressure, <math>~P_e/P_\mathrm{norm}</math>, in <math>~n=4</math> polytropes.  The curve bends back on itself, showing that at any value of <math>~P_e</math>, below some limiting value, two equilibrium configurations exist and they have different values of <math>~\mathcal{B}</math>.  The vertical red line identifies the value of the external pressure <math>~(P_e/P_\mathrm{norm} = 1.71\times 10^4)</math> that has been used as an example in the above two tables to illustrate how a pair of ''physically associated'' equilibrium states can be identified.  This red line intersects the displayed curve at <math>~\mathcal{B} = 2.682</math> (unstable state examined above) and at <math>~\mathcal{B} = 2.542</math> (stable state examined above).
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