Editing SSC/Stability/BiPolytrope00CompareApproaches (section)

===Physical Properties===

[[File:CommentButton02.png|right|100px|This adopted parameter notation pays tribute to the notation that was introduced by Chandrasekhar and his collaborators in the early 1940s in papers associated with the discovery of the Sch&ouml;nberg-Chandrasekhar mass limit.]]Aside from specifying its radius, <math>~R</math>, and total mass, <math>~M_\mathrm{tot}</math>, there are three particularly interesting ''dimensionless'' parameters that characterize the internal structure of a bipolytrope having <math>~(n_c,n_e) = (0,0)</math>.  They are, the radial location of the core/envelope interface,
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<math>~q \equiv \frac{r_i}{R} \, ;</math>
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the ratio of the density of the envelope material to the density of the core, <math>~0 \le \rho_e/\rho_c \le 1</math>; and the fraction of the total mass that is contained in the core,
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<math>~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} \, .</math>
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Identifying a unique bipolytropic configuration requires the specification of two of these three dimensionless parameters; the third parameter is, then, necessarily determined via what we will refer to as the,

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<font color="#770000">'''Primary Algebraic Constraint'''</font><br />
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<math>~\frac{\rho_e}{\rho_c} </math>
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<math>=</math>
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<math>~\frac{q^3(1-\nu)}{\nu(1-q^3)} \, .</math>
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It is also relatively straightforward to appreciate that, in dimensional units, the value of the central density is,
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<math>~\rho_c</math>
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<math>~=</math>
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<math>~\frac{3M_\mathrm{tot}}{4\pi G R^3} \cdot \frac{\nu}{q^3} \, .</math>
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[[SSC/Structure/BiPolytropes/Analytic00#gdefinition|Our study of equilibrium configurations has shown]] that once, for example, the pair of parameters, <math>~q</math> and <math>~\rho_e/\rho_c</math>, has been specified, other properties of the associated equilibrium configuration can be succinctly expressed in terms of the function,
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<math>~g^2</math>
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<math>~\equiv</math>
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<math>
1  + \biggl(\frac{\rho_e}{\rho_c}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( 1-q \biggr) + 
\frac{\rho_e}{\rho_c} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \, .
</math>
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For example, the central pressure is given by the expression,
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<math>~P_c</math>
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&nbsp; <math>~=</math>&nbsp; 
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<math>\biggl( \frac{3}{2^3\pi} \biggr) \frac{\nu^2 g^2}{q^4} \biggl[ \frac{GM_\mathrm{tot}^2}{R^4} \biggr] \, .</math>
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