Editing SSC/Structure/BiPolytropes/Analytic00 (section)

===Second Derivative of Free Energy===
The relative stability of an equilibrium configuration can be determined by the sign of the second derivative of the free energy evaluated at the equilibrium radius; if the second derivative is negative, the system is dynamically unstable.  The second derivative of the free energy is,
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<math>\frac{d^2\mathfrak{G}}{dR^2} \biggr|_{R_\mathrm{eq}}</math>
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<math>~=~</math>
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<math>\biggl[ -2A_\mathrm{grav}R^{-3} - 2(2-3\gamma_c)C_\mathrm{core} R^{1-3\gamma_c} - 
2(2-3\gamma_e)C_\mathrm{env} R^{1-3\gamma_e} \biggr]_\mathrm{eq} </math>
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&nbsp;
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<math>~=~</math>
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<math>2R^{-2}_\mathrm{eq} \biggl[ W - (2-3\gamma_c)C_\mathrm{core} R^{3-3\gamma_c} - 
(2-3\gamma_e)C_\mathrm{env} R^{3-3\gamma_e} \biggr]_\mathrm{eq}  \, .</math>
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From the equilibrium condition derived above, we know that,
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<math>- C_\mathrm{env} R^{3-3\gamma_e}</math>
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<math>~=~</math>
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<math>C_\mathrm{core} R^{3-3\gamma_c} + \frac{W}{2} \, .</math>
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Hence,
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<math>\frac{1}{2} R^{2}_\mathrm{eq} \frac{d^2\mathfrak{G}}{dR^2} \biggr|_{R_\mathrm{eq}}</math>
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<math>~=~</math>
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<math>W_\mathrm{eq} - (2-3\gamma_c)C_\mathrm{core} R_\mathrm{eq}^{3-3\gamma_c} + 
(2-3\gamma_e) \biggl[ C_\mathrm{core} R^{3-3\gamma_c} + \frac{W}{2}\biggr]_\mathrm{eq} </math>
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&nbsp;
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<math>~=~</math>
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<math>\biggl( 2 - \frac{3}{2}\gamma_e \biggr) W_\mathrm{eq} + 3 (\gamma_c - \gamma_e)C_\mathrm{core} R_\mathrm{eq}^{3-3\gamma_c}  \, .</math>
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But, from our above discussion, we can also write,
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<math>~C_\mathrm{core}</math>
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<math>~=~</math>
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<math>2\pi q^3 P_i R_\mathrm{eq}^{3\gamma_c} (1+\Lambda) \, .</math>
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Hence,
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<math>\biggl[ \frac{R^{2}_\mathrm{eq}}{2} \biggr] \frac{d^2\mathfrak{G}}{dR^2} \biggr|_{R_\mathrm{eq}}</math>
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<math>~=~</math>
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<math>6\pi (\gamma_c - \gamma_e)q^3 P_i R_\mathrm{eq}^{3} (1+\Lambda) + \biggl( 2 - \frac{3}{2}\gamma_e \biggr) W_\mathrm{eq} 
  </math>
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<math>\Rightarrow ~~~~ \biggl[ \frac{1}{P_iR_\mathrm{eq}} \biggr] \frac{d^2\mathfrak{G}}{dR^2} \biggr|_{R_\mathrm{eq}}</math>
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<math>~=~</math>
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<math>
12\pi (\gamma_c - \gamma_e)q^3 (1+\Lambda)  - \frac{1}{P_iR^3_\mathrm{eq}} ( 4 - 3\gamma_e) 
\biggl[  \frac{3GM_\mathrm{tot}^2}{5 R_\mathrm{eq}} \biggl(\frac{\nu^2}{q}\biggr) f \biggr]
</math>
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&nbsp;
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<math>~=~</math>
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<math>
~12\pi (\gamma_c - \gamma_e)q^3 (1+\Lambda)  - 4\pi ( 4 - 3\gamma_e) q^3 \Lambda f 
</math>
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<math>\Rightarrow ~~~~ \frac{d^2\mathfrak{G}}{dR^2} \biggr|_{R_\mathrm{eq}}</math>
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<math>~=~</math>
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<math>
~12\pi P_iR_\mathrm{eq}\biggl[ (\gamma_c - \gamma_e)q^3 (1+\Lambda)  - \biggl( \frac{4}{3} - \gamma_e \biggr) q^3 \Lambda f \biggr] 
</math>
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&nbsp;
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<math>~=~</math>
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<math>
~12\pi P_iR_\mathrm{eq} (\gamma_e - \gamma_c) q^3 \Lambda \biggl[ 
\frac{( \gamma_e - \frac{4}{3})}{(\gamma_e - \gamma_c)}   f 
- \biggl(1+\frac{1}{\Lambda} \biggr)  \biggr] 
</math>
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&nbsp;
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<math>~=~</math>
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<math>
~12\pi P_iR_\mathrm{eq} (\gamma_e - \gamma_c) q^3 \Lambda \biggl[ 
\frac{( \gamma_e - \frac{4}{3})}{(\gamma_e - \gamma_c)}   f 
-1 -\frac{5}{2} (g^2-1)  \biggr] \, .
</math>
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