Editing SSC/VirialStability (section)

==Expressions for Mass==
Inside the core, the expression for the mass interior to any radius, <math>0 \le \xi \le 1</math>, is,
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<math>M_\xi = \frac{4\pi}{3} \rho_c r_i^3 \xi^3</math> .
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The expression for the mass interior to any position within the envelope, <math>1 \le \xi \le \xi_s</math>, is,
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<math>M_\xi = \frac{4\pi}{3} r_i^3 \biggl[\rho_c  + \rho_e(\xi^3 - 1) \biggr]</math> .
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Hence, the mass of the core, the mass of the envelope, and the total mass are, respectively,
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<math>
M_\mathrm{core} = \frac{4\pi}{3} \rho_c r_i^3
</math> ;

<math>
M_\mathrm{env} = \frac{4\pi}{3} r_i^3 \biggl[\rho_e (\xi_s^3 - 1) \biggr] 
</math> ;

<math>
M_\mathrm{tot} = \frac{4\pi}{3} r_i^3 \biggl[\rho_c  + \rho_e(\xi_s^3 - 1) \biggr] 
</math> .
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Following the work of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)] &#8212; see [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|our accompanying discussion]]  &#8212; we are seeking equilibrium configurations in the <math>\nu - q</math> plane where,
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<math>\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} </math> &nbsp; &nbsp; and &nbsp; &nbsp;
<math>q \equiv \frac{r_i}{R} = \frac{1}{\xi_s}</math>.
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So we can combine the above expressions to obtain,
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<math>\frac{\rho_e}{\rho_c} =  \frac{M_\mathrm{env}}{M_\mathrm{core}} (\xi_s^3 - 1)^{-1} 
= \biggl[ \frac{1-\nu}{\nu}\biggr] (\xi_s^3 - 1)^{-1}
= \frac{q^3}{\nu}\biggl( \frac{1 - \nu}{1- q^3} \biggr) \, ,
</math>
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or,
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<math>\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^3} - 1\biggr) \biggr]^{-1}</math> .
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It is worth noting that exactly the same result arises from an examination of the analytically definable, structural properties of [[SSC/Structure/BiPolytropes/Analytic51#Parameter_Values|bipolytropes having <math>n_c = 5</math> and <math>n_e = 1</math>]].  That is, the ratio of the average density in the envelope to the average density in the core is,
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<math>
\frac{\bar{\rho}_e}{\bar{\rho}_c} = 
\frac{(M^*_\mathrm{tot} - M^*_\mathrm{core}) / [(R^*)^3 - (r^*_\mathrm{core})^3]}{M^*_\mathrm{core} / (r^*_\mathrm{core})^3 } =
\frac{q^3 (1- \nu)}{\nu (1-q^3)} \, .
</math>
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This is, of course, at it should be.
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It is worth noting that, because <math>\bar\rho \equiv 3M_\mathrm{tot}/(4\pi R^3) \,</math>, we can write,
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<math>\frac{\rho_c}{\bar\rho} = \frac{\nu}{q^3} \, ,</math> &nbsp;&nbsp;&nbsp;and &nbsp;&nbsp;&nbsp; 
<math>\frac{\rho_e}{\bar\rho} = \frac{1-\nu}{1-q^3} \, ,</math>
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which is consistent with the above expression for the ratio, <math>\rho_e/\rho_c \, .</math>  The following figure shows how <math>\nu \,</math> varies with <math>q \,</math> for various choices of the mass density ratio, <math>\rho_e/\rho_c \,</math>.  It illustrates that, for a given core-to-total mass ratio, <math>\nu \,</math>, the relative location of the interface radius, <math>q \,</math>, can vary between zero and one, but each value of <math>q \,</math> reflects a different ratio of envelope-to-core mass density.

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[[File:NuVersusQ.png|600px|center|Nu versus Q]]
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