Editing SSC/Structure/BiPolytropes/Analytic00 (section)

=Summary=
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Given that we are only interested in bipolytropes having <math>~\gamma_e > \gamma_c</math>, the leading coefficient of the derived stability condition will be positive.  Hence, the bipolytropic configuration will be dynamically stable as long as,
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<math>
~\frac{( \gamma_e - \frac{4}{3})}{(\gamma_e - \gamma_c)}   f - \biggl[1 + \frac{5}{2} (g^2-1) \biggr] ~>~ 0 \, ,
</math>
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where,
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<math>~f(q,\rho_e/\rho_c)</math> 
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<math>~\equiv~</math>
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<math>1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2} - 1 \biggr)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \biggl(\frac{1}{q^5} - 1 \biggr) - \frac{5}{2}\biggl(\frac{1}{q^2} - 1 \biggr) \biggr] \, ,</math>
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<math>~g^2(q,\rho_e/\rho_c)</math> 
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<math>~\equiv~</math>
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<math>1  + 
\biggl(\frac{\rho_e}{\rho_0}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( 1-q \biggr) + 
\frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr]  \, .</math>
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It is also worth noting that the numerical value of the factor <math>~\Lambda</math> that appears in the leading coefficient can be obtained from the relation,
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<math>~\frac{1}{\Lambda}</math>
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&nbsp; <math>~=</math>&nbsp; 
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<math>\frac{5}{2}(g^2-1)  \, ,</math>
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hence, the term,
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<math>\biggl(1 + \frac{1}{\Lambda} \biggr)</math>
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&nbsp; <math>~=</math>&nbsp; 
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<math>
~\biggl[1+\frac{5}{2}(g^2-1)\biggr] = 1 + 5 \biggl(\frac{\rho_e}{\rho_c}\biggr) (1-q) + \frac{5}{2} \biggl(\frac{\rho_e}{\rho_c}\biggr)^2 \biggl( \frac{1}{q^2} - 3 + 2q \biggr) 
\, .</math>
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It is also worth remembering that, in the <math>(n_c, n_e) = (0, 0)</math> bipolytrope, the density drop at the interface is related to the two key parameters <math>~q</math> and <math>~\nu</math> via the expression,
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<math>\frac{\rho_e}{\rho_c} </math>
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&nbsp; <math>~=</math>&nbsp; 
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<math>
\frac{q^3(1-\nu)}{\nu(1-q^3)}
\, .</math>
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The "Dynamical Stability" figure below shows a large segment of the <math>~(q,\nu)</math> model parameter space.  The figure displays stability boundaries for four different choices of the core/envelope adiabatic pairs <math>~(\gamma_c, \gamma_e)</math>, as detailed in the accompanying "Stability Boundaries" table.  (The corresponding polytropic indexes are also listed, where, <math>~\gamma = 1+1/n</math>.)  Two bipolytrope model sequences are also drawn:  One shows how the core-to-total mass ratio, <math>~\nu</math>, varies with increasing <math>~q</math> when the density ratio at the core-envelope interface is <math>~\rho_e/\rho_c = 1/2</math> (purple crosses); the other shows the same information, but for <math>~\rho_e/\rho_c = 1/4</math> (purple asterisks).  For a given choice of adiabatic indexes, equilibrium configurations that lie below the relevant stability boundary, along either sequence, are stable while configurations that lie above the relevant boundary are dynamically unstable.

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Stability Boundaries
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<math>~n_c</math>
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<math>~n_e</math>
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<math>~\gamma_c</math>
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<math>~\gamma_e</math>
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Marker
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<math>~5</math>
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<math>~1</math>
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<math>~1.2</math>
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<math>~2</math>
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Light blue diamonds
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<math>~5</math>
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<math>~\frac{3}{2}</math>
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<math>~1.2</math>
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<math>~1.6667</math>
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Light green triangles
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<math>~20</math>
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<math>~1</math>
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<math>~1.05</math>
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<math>~2</math>
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Red squares
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<math>~20</math>
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<math>~\frac{3}{2}</math>
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<math>~1.05</math>
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<math>~1.6667</math>
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Orange circles
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&nbsp;
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[[Image:BipolytropeDynamicalStability2.png|400px]]
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It is important to emphasize that the stability analysis presented above is an approximation in the following sense:  The underlying equilibrium structure of each model is that of an <math>~(n_c, n_e) = (0, 0)</math> bipolytrope, that is, both the core and the envelope have uniform (but different) densities; the gravitational potential energy, <math>~W</math> and the thermal energy content of the component volumes, <math>~S_\mathrm{core}</math> and <math>~S_\mathrm{env}</math>, are determined exactly.  However, the relative stability of a configuration is determined by assuming that, when the bipolytropic configuration is perturbed from its equilibriums state, its core and envelope heat up (or cool) by evolving along <math>~(\gamma_c, \gamma_e)</math> adiabats.