Editing SSC/Stability/InstabilityOnsetOverview (section)

==Supporting Scratch Work==
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Note the following:
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<math>x_P\biggr|_{n=3}</math>
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<math>~=</math>
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<math>~
\frac{3(n-1)}{2n}\biggl[1 + \biggl(\frac{n-3}{n-1}\biggr) \biggl( \frac{1}{\xi \theta^{n}}\biggr) \frac{d\theta}{d\xi}\biggr] \, ;
</math>
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<math>~x_P\biggr|_{n=3}</math>
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<math>~=</math>
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<math>~
1 \, ;
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<math>~x_P\biggr|_{n=5}</math>
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<math>~=</math>
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<math>~
\frac{6}{5}\biggl[1 + \frac{1}{2}\biggl( \frac{1}{\xi \theta^{5}}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=5} 
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&nbsp;
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<math>~=</math>
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<math>~
\frac{6}{5} - \frac{3}{5\xi} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{5/2} \frac{\xi}{3} \biggl( 1 + \frac{\xi^2}{3} \biggr)^{-3/2} 
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&nbsp;
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<math>~=</math>
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<math>~
\frac{6}{5} - \frac{1}{5} \biggl( 1 + \frac{\xi^2}{3} \biggr)
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&nbsp;
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<math>~=</math>
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<math>~
1 - \frac{\xi^2}{15} \, ;
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<math>~x_P\biggr|_{n=1}</math>
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<math>~=</math>
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<math>~
-3 \biggl[ \biggl( \frac{1}{\xi \theta}\biggr) \frac{d\theta}{d\xi}\biggr]_{n=1} 
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&nbsp;
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<math>~=</math>
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<math>~
\frac{3}{\xi} \biggl( \frac{\xi}{\sin\xi}\biggr) \biggl[\frac{\sin\xi}{\xi^2} - \frac{\cos\xi}{\xi}  \biggr]
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&nbsp;
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<math>~=</math>
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<math>~
\frac{3}{\xi^2}\biggl[ 1- \xi \cot\xi  \biggr]  = 1 + \frac{\xi^2}{15} + \frac{2\xi^4}{315} + \frac{\xi^6}{1575} + \cdots \, .
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For isolated polytropes of finite extent, the conventionally adopted surface boundary condition is one that [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations|ensures finite-amplitude fluctuations]] at the surface, namely,
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<math>\frac{d\ln x}{d\ln r_0}</math>
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<math>=</math>
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<math>\frac{1}{\gamma_\mathrm{g}} \biggl( 4 - 3\gamma_\mathrm{g} + \frac{\omega^2R^3}{GM_\mathrm{tot}}\biggr) </math>
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&nbsp;
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<math>=</math>
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<math>- \alpha + \frac{1}{\gamma_\mathrm{g}} \biggl( \frac{3\omega^2}{4\pi G\rho_c \mathfrak{f}_M}\biggr) \, .</math>
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where, this last expression has been written in terms of the [[SSCpt1/Virial#Structural_Form_Factors|structural form factor]], <math>\mathfrak{f}_M = \bar\rho/\rho_c</math>.  Now, [[SSC/Virial/Polytropes#Role_of_Structural_Form_Factors|for polytropic configurations]], 
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<math>\mathfrak{f}_M</math>
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<math>=</math>
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<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} \, ,</math>
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in which case this conventional surface boundary condition becomes,
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<math>- \frac{d\ln x}{d\ln \xi}\biggr|_\mathrm{surface}</math>
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<math>=</math>
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<math>\alpha + \frac{\sigma_c^2}{6\gamma_\mathrm{g}} \biggl[\frac{\xi}{\theta^'}\biggr]_{\xi_1} \, .</math>
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<!--where,  <math>\sigma_c^2 \equiv 3\omega^2/(2\pi G \rho_c)</math>.  -->If we align the adiabatic exponent with the index of the polytrope via the expression, <math>\gamma_\mathrm{g} = (n+1)/n</math>, this becomes,
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<math>- \frac{d\ln x}{d\ln \xi}\biggr|_\mathrm{surface}</math>
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<math>=</math>
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<math>\biggl(\frac{3-n}{n+1}\biggr) + \frac{n\sigma_c^2}{6(n+1)} \biggl[\frac{\xi}{\theta^'}\biggr]_{\xi_1} \, .</math>
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[<font color="red">'''Note added on 1/4/2019:'''</font>  The numerator of the first term on the RHS of this last expression used to be "n-3".  This manuscript error has been fixed.]
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We have already demonstrated that this is the boundary condition used by [[SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29|Schwarzschild (1941)]] in his examination of radial oscillations in n = 3 polytropes.  Presumably this also matches up with the boundary condition employed by [[SSC/Stability/Polytropes#Boundary_Conditions|HRW66]].  Notice that when n = 0, this final expression reduces to <math>d\ln x/d\ln\xi = -3</math>, which is the surface boundary condition that we have been using to determine the eigenvectors of all pressure-truncated configurations.