Editing SSCpt2/IntroductorySummary (section)

====Pressure-Truncated Configurations====
If the physical system under consideration &#8212; such as a protostellar gas cloud &#8212; is not isolated but is, instead, surrounded and ''truncated'' by a hot, tenuous medium that exerts on the system a confining external pressure, <math>P_e</math>, the configuration's equilibrium parameters will be related via the expression,

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<math>\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
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<math>=</math>
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<math>\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W}  \biggl[ \mathfrak{f}_A - \frac{P_e}{P_c} \biggr]
 \, ;</math>
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or, fixing {{ Math/MP_PolytropicConstant }} instead of <math>P_c</math>, the [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_2|relevant virial equilibrium expression]] is,
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<math>P_e </math>
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<math>=</math>
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<math>K_\mathrm{n} \mathfrak{f}_A \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \cdot \frac{1}{\mathfrak{f}_M} \biggr)^{1 + 1/n}  
- \frac{\mathfrak{f}_W}{5} \biggl(\frac{3GM^2}{4\pi R_\mathrm{eq}^4} \cdot \frac{1}{\mathfrak{f}_M^2} \biggr) \, .</math>
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It is a virial expression specifically of this form <math>(</math>with <math>n = \infty</math> and <math>\mathfrak{f}_M = \mathfrak{f}_W = \mathfrak{f}_A = 1)</math> that identifies extrema (e.g., valleys or ridges) in the rainbow-colored free-energy surface, <math>\mathfrak{G}^*(R_\mathrm{eq}, P_e)</math>, [[SSCpt1/Virial/PolytropesEmbeddedOutline#ASIDE:__Isothermal_Configurations|displayed above]].  As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if <math>P_e</math> is greater than some limiting value, <math>P_\mathrm{crit}</math>; the equilibrium configuration associated with the limiting condition, <math>P_e = P_\mathrm{crit}</math>, is marked by a red dot on the above-displayed free-energy surface.  The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert spheres]].

After rearranging terms, for any specified values of the parameters <math>P_e</math> and {{ Math/MP_PolytropicConstant }}, this virial equilibrium expression can also be viewed as a [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_3|mass-radius relation of the form]],
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[[File:MassRadiusVirialLabeled.png|250px|right|border|Virial Mass-Radius Relation]]
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<math>a R_\mathrm{eq}^4 - b M^{(n+1)/n} R_\mathrm{eq}^{(n-3)/n} + c M^2</math>
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<math>=</math>
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<math>0\, ,</math>
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where the constants,
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<math>a \equiv \frac{4\pi}{3} \cdot  P_e \, ,</math> &nbsp;  &nbsp; &nbsp;<math>b \equiv  \biggl( \frac{3}{4\pi} \biggr)^{1/n}   \cdot K_\mathrm{n} \mathfrak{f}_A  \mathfrak{f}_M^{-(n+1)/n} \, ,</math> &nbsp;  &nbsp; &nbsp; and  &nbsp;  &nbsp; &nbsp;<math>c \equiv   \frac{G\mathfrak{f}_W}{5\mathfrak{f}_M^2}  \, .</math>
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Using this virial equilibrium relation (and, for illustration purposes, assuming a = b = c = 1), the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation|curves drawn in the figure that is displayed here, on the right]], show how the equilibrium radius of an embedded, pressure-truncated polytropic sphere varies with mass for seven different adopted polytropic indexes.  In direct analogy with the critical pressure that is associated with Bonnor-Ebert spheres, for systems having <math>~n \ge 3</math>, there is a mass, <math>M_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, when <math>n > 3</math>, multiple equilibrium configurations having different radii can be constructed for any system having a mass, <math>M < M_\mathrm{max}</math>.

A [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|detailed force-balance analysis of the structure of embedded, pressure-truncated polytropic configurations]] reveals, for each choice of {{ Math/MP_PolytropicIndex }}, a mass-radius relationship that is qualitatively similar to the one deduced from a virial equilibrium analysis.  However, the resulting mass-radius relationship is invariably different in detail and quantitatively more correct than the one prescribed by the virial theorem.  At its foundation are models whose internal structural profile &#8212; of, for example, the fluid pressure and fluid density &#8212; is not uniform but, rather, is precisely that which is required to achieve detailed force balance throughout.  Hence, we appreciate that even as <math>P_e</math> and {{ Math/MP_PolytropicConstant }} are held fixed, in essence the structural form factors must vary from model to model along the more precise "detailed force-balanced" equilibrium ''sequence.''