Editing SSC/Perturbations (section)

=====Implications=====
'''First Case:'''  If the surface pressure fluctuation is set to zero, we have just deduced that, at the surface of the configuration,
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<math>~3 x +  r_0 \frac{dx}{dr_0} =0 \, .</math>
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This implies that,
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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>~-3</math>
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<math>~\Rightarrow ~~~~ d\ln x</math>
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<math>~=</math>
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<math>~-3d\ln r_0</math>
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<math>~\Rightarrow ~~~~ \ln x </math>
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<math>~=</math>
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<math>~\ln C_0 - 3\ln r_0</math>
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<math>~\Rightarrow ~~~~ x </math>
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<math>~=</math>
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<math>~\frac{C_0}{r_0^3} \, .</math>
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It is customary to normalize the radial eigenfunction, <math>~x</math>, in such a way that it goes to unity at the surface.  Therefore, in order to satisfy this "first case" boundary condition, at the surface of the oscillating configuration, the eigenfunction must display the behavior,
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<math>~x = \biggl( \frac{R}{r_0}\biggr)^{3} \, .</math>
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'''Second Case:'''  If, instead, we insist that the ''first derivative'' of the surface pressure fluctuation be zero, then, as we have just deduced, 
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<math>~\biggl( 4 - 3\gamma_g + \frac{\omega^2 r_0}{g_0}\biggr) \frac{x}{\gamma_g}-r_0 \frac{dx}{dr_0}</math>
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<math>~=</math>
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<math>~0</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
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But, rearranging terms in the full ''[[#2ndOrderODE|linear adiabatic wave equation]]'', we see that, throughout the entire structure (including the surface),
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<math>~ 
\biggl(\frac{g_0 \rho_0 }{P_0 r_0} \biggr)\biggl[\biggl(4 - 3\gamma_\mathrm{g}  + \frac{\omega^2 r_0}{g_0} \biggr)  \frac{x}{\gamma_\mathrm{g}}
- r_0 \frac{dx}{dr_0}\biggr]</math>
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<math>~=</math>
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<math>~- \biggl[ \frac{d^2x}{dr_0^2} + \biggl(\frac{4}{r_0}\biggr)\frac{dx}{dr_0} \biggr] \, .</math>
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Since, according to the "second case" surface boundary condition, the term inside the square brackets on the left-hand side of this expression must be zero at the surface, it must also be true that the term inside the square brackets on the right-hand side is zero.  That is, the "second case" boundary condition will be satisfied if, 
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<math>~\frac{d^2x}{dr_0^2} + \biggl(\frac{4}{r_0}\biggr)\frac{dx}{dr_0} </math>
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<math>~=</math>
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<math>~0</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R </math>
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<math>~\Rightarrow ~~~~~\frac{1}{r_0^4} \cdot \frac{d}{dr_0}\biggl[ r_0^4\frac{dx}{dr_0} \biggr]</math>
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<math>~=</math>
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<math>~0</math>
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<math>~\Rightarrow ~~~~~r_0^4\frac{dx}{dr_0} </math>
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<math>~=</math>
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<math>~C_0</math>
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<math>~\Rightarrow ~~~~~dx </math>
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<math>~=</math>
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<math>~C_0 r_0^{-4} dr_0</math>
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<math>~\Rightarrow ~~~~~x </math>
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<math>~=</math>
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<math>~C_1 - \biggl(\frac{C_0}{3} \biggr) r_0^{-3} </math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
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Again, given that it is customary to normalize the radial eigenfunction, <math>~x</math>, such that it goes to unity at the surface, the eigenfunction must display the behavior,
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<math>~x = 1 + \frac{C_0}{3}\biggl(\frac{1}{R^3} - \frac{1}{r_0^3} \biggr) \, ,</math>
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at the surface of the oscillating configuration in order to satisfy this "second case" boundary condition.

'''Third Case:'''  If we follow the lead of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>] and choose to establish a surface boundary condition that effectively ignores the inertia of the configuration's atmosphere, then, as we have just determined,
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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>~- 3 + \frac{4}{\gamma_g} </math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
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As in the "first case" discussed above, this constraint leads to a power-law <math>~x(r_0)</math> behavior at the surface.  Specifically, this "third case" boundary condition &#8212; along with the convention that <math>~x \rightarrow 1</math> at the surface &#8212; demands an eigenfunction whose behavior at the surface is,
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<math>~x = \biggl( \frac{R}{r_0}\biggr)^{3-4/\gamma_g} \, .</math>
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