Editing SSC/Perturbations (section)

=====Ensure Finite-Amplitude Fluctuations=====
Here we follow the discussion provided by [http://adsabs.harvard.edu/abs/1967IAUS...28....3C J. P. Cox (1967)]; specifically, the relevant discussion begins in the middle of p. 21, in association with Cox's equation (3.7).  Text drawn directly from [http://adsabs.harvard.edu/abs/1967IAUS...28....3C J. P. Cox (1967)] is presented here in green.  


<font color="green">At the surface <math>~(r_0 = R)</math> of our</font> oscillating, spherically symmetric configuration <font color="green">we must require, in general, that all relative pulsation variables &hellip; be finite.  The specific surface condition can be obtained most generally from the</font> [[#Summary_Set_of_Linearized_Equations|above derived]],
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<font color="#770000">'''Linearized Euler + Poisson Equations'''</font><br />
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x .
</math>
</div>
Multiplying through by <math>~(R\rho_0/P_0)</math> and remembering that 
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<math>g_0 = - \frac{1}{\rho_0} \frac{dP_0}{dr_0} \, ,</math>
</div>
this relation assumes the form of equation (3.7) in [http://adsabs.harvard.edu/abs/1967IAUS...28....3C J. P. Cox (1967)], namely,
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<math>~R \frac{dp}{dr_0}</math>
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<math>~=</math>
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<math>~- \frac{R}{P_0} \frac{dP_0}{dr_0}\biggl[(4x + p) + \frac{\omega^2 r_0 x}{g_0} \biggr] </math>
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</tr>

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&nbsp;
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<math>~=</math>
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<math>~\frac{R}{\lambda_p}\biggl[(4x + p) + \frac{\omega^2 r_0 x}{g_0} \biggr] \, ,</math>
  </td>
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</table>
</div>
where, as is highlighted by Cox in association with his equation (2.42'),
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<math>~\lambda_p \equiv -\biggl(\frac{d\ln P_0}{dr_0} \biggr)^{-1} \, ,</math>
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<font color="green">is the (equilibrium) pressure scale height</font> of the configuration.  (Note that, the [https://en.wikipedia.org/wiki/Scale_height pressure scale height] is often represented by the variable, <math>~H</math>, instead of <math>~\lambda_p</math>.)


<font color="green">Since <math>~R/\lambda_p \gg 1</math> at the photosphere for most stars (and <math>~R/\lambda_p = \infty</math> if <math>~P/\rho</math> is assumed to vanish at the surface), a reasonable surface boundary condition would be</font> to force the terms inside the square brackets on the right-hand side of this expression to sum to zero, that is, for the pressure fluctuation to obey the relation,
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<math>~p</math>
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<math>~=</math>
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<math>~-\biggl(4 + \frac{\omega^2 r_0 }{g_0} \biggr)x </math>
  </td>
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&nbsp;
  </td>
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<math>~=</math>
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<math>~-\biggl( 4 + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) x</math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
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<font color="green">This boundary condition prevents <math>~dp/dr_0</math> from having a large (or infinite) value at <math>~r_0 = R</math> and also requires that <math>~p</math> be finite at <math>~r_0 = R</math> even if <math>~P_0 = 0</math> here.</font>  This is the surface boundary condition specified in two key review articles on this subject &#8212; one by [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C R. F. Christy (1965]; see [[#Review_Article_by_Christy_.281966.29|discussion below]]) and another, almost a decade later, by [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974]; see [[#Review_Article_by_Cox_.281974.29|additional discussion below]]).  

<span id="ChristyCox">
[http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C Christy (1965)] and [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)] also point out that, by calling upon the [[#PGE:AdiabaticFirstLaw|above "Linearized Adiabatic Form of the First Law of Thermodynamics"]] to replace the fractional pressure variation, <math>~p</math>, in favor of the fractional density variation, <math>~d</math>; then using the [[#Continuity|above "Linearized Equation of Continuity"]] to replace <math>~d</math> in favor of the fractional radial displacement, <math>~x</math>, the same boundary condition may be written as,</span>

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<math>~- \gamma_g d</math>
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<math>~=</math>
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<math>~\biggl( 4 + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) x</math>
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<math>~\Rightarrow ~~~~~ 3 x +  r_0 \frac{dx}{dr_0}</math>
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<math>~=</math>
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<math>~\biggl( 4 + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \frac{x}{\gamma_g}</math>
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<math>~\Rightarrow ~~~~~ r_0 \frac{d\ln x}{dr_0}</math>
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<math>~=</math>
  </td>
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<math>~\frac{1}{\gamma_g} \biggl( 4 - 3\gamma_g + \frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) </math>&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, ,</math>
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</table>
</div>

which, as [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)] summarizes (see following his equation 3.9), <font color="green">gives the logarithmic slope <math>~d\ln x/dr_0</math> of the relative pulsation amplitude <math>~x</math> at <math>~r_0 = R</math> in terms of <math>~\gamma_g</math> and the dimensionless frequency <math>~\omega^2R^3/GM_\mathrm{tot}</math> and assures that both <math>~x</math> and <math>~dx/dr_0</math> are finite at <math>~r_0 = R</math>.</font>  This is the boundary condition <font color="green">conventionally used in connection with the adiabatic wave equation</font>.