Editing SSCpt2/IntroductorySummary (section)

====Bipolytropic Configurations====
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[[File:PlotSequencesBest02.png|300px|right|border|Virial Mass-Radius Relation]]
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Additional insight into the structural properties and evolution of stars can be gained by studying ''bipolytropes'' &#8212; also sometimes referred to as ''composite polytropes.''  These are models in which the configuration's "core" is described by a polytropic equation of state having one index &#8212; say, <math>~n_c</math> &#8212; and the configuration's "envelope" is described by a polytropic equation of state of a different index &#8212; say, <math>~n_e</math>.  We have found it particularly instructive to examine bipolytropes having <math>~(n_c, n_e) = (5, 1)\, ,</math> in part, because equilibrium models of such systems can be completely described analytically whether they are constructed via a detailed force-balance analysis or by identifying extrema in the free-energy function, that is, via a virial theorem analysis.  

Following the lead of {{ SC42full }} &#8212; see, also, {{ EFC98full }} &#8212; we have pieced these bipolytropic configurations together mathematically in such a way that the mean molecular weight, {{Math/MP_MeanMolecularWeight}}, of the fluid is allowed to change in a discontinuous fashion at the core-envelope interface.  As is illustrated in the figure, shown here on the right, a physically interesting equilibrium model "sequence" can be constructed by monotonically shifting the location of the core-envelope interface from the center of the configuration to its surface while holding fixed the value of the envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>.   Each curve shows how the relative mass of the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, correlates with the relative ''size'' of the core, as measured by the ratio of the radial position of the core-envelope interface to the equilibrium radius of the composite polytropic configuration, <math>~q \equiv r_i/R_\mathrm{eq}</math>.  As the figure illustrates, if the jump in the mean molecular weight is sufficiently extreme &#8212; specifically, if <math>~\mu_e/\mu_c < 1/3</math> for the bipolytropic configurations being considered here &#8212; there is a core mass, <math>~\nu_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, two equilibrium configurations having different core ''sizes'', <math>~q</math>, can be constructed for any system having a core mass, <math>~\nu < \nu_\mathrm{max}</math>.  The astrophysical significance of <math>~\nu_\mathrm{max}</math> was first identified in the early 1940s in bipolytropic configurations having <math>~(n_c, n_e) = (\infty, 3/2)</math>, and has been discussed extensively in the context of the evolutionary transition of stars from the main sequence to the giant branch.  It is usually referred to as the Sch&ouml;nberg-Chandrasekhar mass limit.