Editing SSC/FreeEnergy/PolytropesEmbedded/Pt2 (section)

==Case M Free-Energy Surface==

It is useful to rewrite the free-energy function in terms of dimensionless parameters.  Here we need to pick normalizations for energy, radius, and pressure that are expressed in terms of the gravitational constant, <math>~G</math>, and the two fixed parameters, <math>~K</math> and <math>~M</math>.  We have chosen to use,
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<math>~x</math>
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<math>~\equiv</math>
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<math>~\frac{R}{R_0}</math>
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<math>~R_\mathrm{norm}</math>
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<math>~\equiv</math>
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<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
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<math>~P_\mathrm{norm}</math>
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<math>~\equiv</math>
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<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)}  \, ,</math>
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which, as is detailed in an [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|accompanying discussion]], are similar but not identical to the normalizations used by [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and by [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)].  The self-consistent energy normalization is,

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<math>~E_\mathrm{norm}</math>
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<math>~\equiv</math>
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<math>~P_\mathrm{norm} R^3_\mathrm{norm} \, .</math>
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As we have [[SSCpt1/Virial#Gathering_it_all_Together|demonstrated elsewhere]], after implementing these normalizations, the expression that describes the "Case M" free-energy surface is,
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<math>
\mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} = 
-3\mathcal{A} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} +~ n\mathcal{B} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n} 
+~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \, ,
</math>
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Given the polytropic index, <math>~n</math>, we expect to obtain a different "Case M" free-energy surface for each choice of the dimensionless truncation radius, <math>~\tilde\xi</math>; this choice will imply corresponding values for <math>~\tilde\theta</math> and <math>~\tilde\theta^'</math> and, hence also, corresponding (constant) values of the coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>.

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<math>~\frac{P_e}{P_\mathrm{norm}}</math>
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<math>~=</math>
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<math>~\frac{1}{20\pi x_\mathrm{eq}^4} \biggl[ 15\biggl(\frac{3}{4\pi}\biggr)^{1/n} x_\mathrm{eq}^{(n-3)/n} - 3\biggr]</math>
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<math>~[x_\mathrm{eq}]_\mathrm{crit}</math>
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<math>~=</math>
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<math>~\biggl[ \frac{4n}{15(n+1)} \biggl( \frac{4\pi}{3} \biggr)^{1/n} \biggr]^{n/(n-3)}</math>
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<math>~[x_\mathrm{eq}]_\mathrm{turn}</math>
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<math>~=</math>
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<math>~\biggl[ \frac{4n}{15(n+1)} \biggl( \frac{4\pi}{3} \biggr)^{1/n} \biggr]^{n/(n-3)}</math>
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<math>~\frac{P_\mathrm{max}}{P_\mathrm{norm}}</math>
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<math>~=</math>
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<math>~\frac{1}{20\pi} \biggl( \frac{n-3}{n+1} \biggr) \biggl[ \frac{15(n+1)}{4n}\biggl(\frac{3}{4\pi}\biggr)^{1/n} \biggr]^{4n/(n-3)}</math>
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