Editing SSC/Perturbations (section)

=====Ignore Atmospheric Inertia=====
Here we echo the discussion presented in &sect;38.1 of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>], where an alternative boundary condition at the surface of our spherically symmetric, oscillating configuration is recommended; text drawn verbatim from this reference is shown here in green.  <font color="green">We simplify the atmosphere by assuming its mass <math>~m_a</math> to be comprised in a thin layer at <math>~r_0 = R(t)</math>, which follows the changing <math>~R</math> during the oscillations and provides the outer boundary condition at each moment by its weight</font>.  This amounts to ignoring the inertia of this very thin, nearly massless layer and is accomplished, in practice, by setting to zero the second time-derivative on the left-hand side of the [[#PGE:Euler|above "Euler + Poisson Equation"]]. Hence, the pressure, <math>~P_b</math>, at the base of the outermost (atmospheric) layer of the oscillating configuration is described, at all times, by,
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<math>~\cancelto{0}{\frac{d^2 r}{dt^2}}</math>
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<math>~\approx</math>
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<math>~- 4\pi r^2 \frac{dP}{dM_r} - \frac{GM_r}{r^2}</math>
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<math>~\Rightarrow ~~~~~ \frac{GM_\mathrm{tot}}{4\pi R^4}</math>
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<math>~\approx</math>
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<math>~- \frac{dP}{dM_r} </math>
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&nbsp;
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<math>~\approx</math>
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<math>~- \frac{\Delta P}{\Delta M_r} = - \frac{(0 - P_b)}{m_a} = \frac{P_b}{m_a} \, .</math>
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Appreciating that <math>~R</math> and <math>~P_b</math> are the only time-varying quantities in this expression, perturbing then linearizing the expression gives (at <math>~r_0 = R</math>),
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<math>~(P_b)_0[1 + p e^{i\omega t}]</math>
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<math>~=</math>
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<math>~\frac{GM_\mathrm{tot} m_a}{4\pi R_0^4}\biggl[ 1 + x e^{i\omega t} \biggr]^{-4}</math>
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<math>~\Rightarrow ~~~~~ 1 + p e^{i\omega t}</math>
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<math>~\approx</math>
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<math>~1 -4 x e^{i\omega t} </math>
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<math>~\Rightarrow ~~~~~ p + 4x</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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This is identical to the boundary condition presented as equation (38.12) in [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>].  Combining this condition with the [[#Summary_Set_of_Linearized_Equations|above "linearized adiabatic form of the first law of thermodynamics"]] allows us to write, as well,
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<math>~ \gamma_g d + 4x</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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and, in combination with the  [[#Summary_Set_of_Linearized_Equations|above "linearized continuity equation"]], to also conclude that,
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<math>~ r_0 \frac{dx}{dr_0}</math>
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<math>~=</math>
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<math>~-3x + \frac{4x}{\gamma_g}</math>
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<math>~ \Rightarrow ~~~~~ \biggl( \frac{r_0}{x} \biggr) \frac{dx}{dr_0}</math>
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<math>~=</math>
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<math>~\frac{1}{\gamma_g} (4-3\gamma_g )</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>~r_0 = R \, .</math>
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This matches the boundary condition, as presented in equation (38.13) of [<b>[[Appendix/References#KW94|<font color="red">KW94</font>]]</b>].