Editing SSCpt2/IntroductorySummary (section)

====Isolated Configurations====
After supplementing the PGEs by specifying an [[SR#Supplemental_Relations|equation of state of the fluid]], the system of equations is usually solved by employing one of [[SSCpt2/SolutionStrategies#Solution_Strategies|three techniques]] to obtain a "detailed force-balanced" model that provides the radius, <math>R_\mathrm{eq}</math>, of the equilibrium configuration &#8212; given its mass, <math>M</math>, and central pressure, <math>P_c</math>, for example &#8212; as well as details regarding the internal radial profiles of the fluid density and fluid pressure.   If, for example, a [[SR#Barotropic_Structure|''polytropic'' equation of state]], 
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[[File:FreeNRGpressureRadiusIsothermal.png|250px|right|border|Whitworth's (1981) Isothermal Free-Energy Surface]]
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[[SSCpt1/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|Free-Energy Surface]]
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{{ Math/EQ_Polytrope01 }}
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is adopted, a detailed force-balanced model is fully described by the radially dependent function, <math>\Theta_H(\xi) = [\rho(\xi)/\rho_c]^{1/n}</math>, which is obtained by solving the
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<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
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{{Math/EQ_SSLaneEmden01}}
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As our various discussions illustrate (see the [[#TableOfContents|table of contents, below]]), simply varying the index, {{ Math/MP_PolytropicIndex }}, gives rise to equilibrium configurations that have a wide variety of internal structural profiles.  

If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[VE#Free_Energy_Expression|global free energy]].  An illustrative, undulating free-energy surface is displayed here, on the right; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.  

In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[VE#Scalar_Virial_Theorem|''scalar virial theorem'']].  Specifically, for [[SSC/Virial/Polytropes/Pt1#CentralPressure|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:
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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}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W}  
 \, ,</math>
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where all three of the dimensionless ''structural form factors'', <math>\mathfrak{f}_M</math>, <math>\mathfrak{f}_W</math>, and <math>\mathfrak{f}_A</math>, are unity, under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, and are otherwise generically ''of order unity'' for detailed force-balanced models having a wide range of internal structures.  Alternatively, if the parameter, {{ Math/MP_PolytropicConstant }} (which defines the specific entropy of fluid elements throughout the configuration), rather than the central pressure, is held fixed while searching for extrema in the free-energy, the [[SSC/Virial/Polytropes/Pt1#Mass-Radius_Relationship|virial equilibrium relation for isolated polytropes]] is,
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<math>M^{(n-1)/n} R_\mathrm{eq}^{(3-n)/n} \biggl( \frac{G}{K_\mathrm{n}} \biggr) </math>
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<math>=</math>
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<math>\frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W}   \biggl(\frac{3}{4\pi \mathfrak{f}_M}\biggr)^{1/n} \, .  
 </math>
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