Editing SSC/Perturbations (section)

====Reconciliation====
It is instructive to explicitly demonstrate that the linearized "Euler + Poisson" equation that we derived and highlighted in our [[#Summary_2|brief summary subsection, above]], namely,  
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<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x \, ,
</math>
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conveys the same physics as [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor's (1957) linearized Euler equation] when applied to a spherically symmetric system,  namely,
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<math>\frac{\partial v_r}{\partial t}</math>
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<math>~=</math>
  </td>
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<math>~
\hat{\mathbf{e}}_r \cdot \mathbf{F}_1 - \frac{d}{d r_0} \biggl[ \frac{w}{\rho_0}  \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] \, .</math>
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<font color="maroon">Step #1:</font>  We recognize that, after linearization, <math>~\partial v_r/\partial t = d^2 r/dt^2</math>.  So, drawing on our [[#Euler_.2B_Poisson_Equations|earlier detailed handling of the "Euler + Poisson" equations]], we can make the replacement,

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<math>
\frac{\partial v_r}{\partial t}  
</math>
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<math>
\rightarrow
</math>
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<math>
- ~\omega^2 r_0 x~e^{i\omega t} \, .
</math>
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<font color="maroon">Step #2:</font>  As we have already recognized, swapping between our perturbation notation and Bonnor's leads to the replacement,

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<math>
w \biggl( \frac{dP}{d\rho}  \biggr)_0
</math>
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<math>
\rightarrow
</math>
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<math>
P_0 p e^{i\omega t} \, .
</math>
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Hence,

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<math>
\frac{d}{d r_0} \biggl[ \frac{w}{\rho_0}  \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr]
</math>
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<math>
\rightarrow
</math>
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<math>
\frac{d}{d r_0} \biggl[ \biggl( \frac{P_0 p}{\rho_0}\biggr) e^{i\omega t} \biggr]  
</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
  </td>
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<math>
e^{i\omega t} \biggl\{ \frac{p}{\rho_0} \frac{d P_0}{d r_0} 
+
P_0 p \frac{d}{d r_0} \biggl( \frac{1}{\rho_0}\biggr) 
+
\frac{P_0}{\rho_0} \frac{d p}{d r_0}  
\biggr\}  
</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>~=</math>
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<math>
e^{i\omega t} \biggl\{ \biggl[ \frac{1}{\rho_0} \frac{d P_0}{d r_0} \biggr] (p - d)
+
\frac{P_0}{\rho_0} \frac{d p}{d r_0}  
\biggr\}  
</math>
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&nbsp;
  </td>
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<math>~=</math>
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<math>
e^{i\omega t} \biggl[ (d - p) g_0 + \frac{P_0}{\rho_0} \frac{d p}{d r_0}  \biggr]  \, .
</math>
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where we have, again, used the relationship,
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<math>\nabla P_0 = \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_0 = \biggl( \frac{P_0 p}{\rho_o d} \biggr) \nabla \rho_0 </math>
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&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math> &nbsp; &nbsp; &nbsp; 
  </td>
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<math>~
\frac{d}{dr_0} \biggl( \frac{1}{\rho_0}\biggr) = - \frac{1}{\rho_0^2}\frac{d\rho_0}{dr_0} 
=- \biggl( \frac{d}{P_0 p} \biggr) \biggl[\frac{1}{\rho_0}\frac{dP_0}{dr_0} \biggr]  \, .</math>
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<font color="maroon">Step #3:</font>  Implementing these first two substitutions, Bonnor's linearized Euler equation becomes,
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<math>- ~\omega^2 r_0 x</math>
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<math>~=</math>
  </td>
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<math>~
e^{-i\omega t}\hat{\mathbf{e}}_r \cdot \mathbf{F}_1 - \biggl[ (d - p) g_0 + \frac{P_0}{\rho_0} \frac{d p}{d r_0}  \biggr] </math>
  </td>
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<math>\Rightarrow ~~~~ \frac{P_0}{\rho_0} \frac{d p}{d r_0}  </math>
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<math>~=</math>
  </td>
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<math>~
e^{-i\omega t}\hat{\mathbf{e}}_r \cdot \mathbf{F}_1 + (p - d) g_0 + \omega^2 r_0 x\, .</math>
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<font color="maroon">Step #4:</font>  In order to map Bonnor's <math>~\mathbf{F}_1</math> to our perturbation notation, we back up to expressions for the gravitational acceleration, as a whole, which establish that, for spherically symmetric systems,
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<math>~\hat{\mathbf{e}}_r \cdot \mathbf{F}</math>
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<math>~=</math>
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<math>~-~\frac{Gm}{r^2}</math>
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<math>~\Rightarrow~~~\hat{\mathbf{e}}_r \cdot \biggl[ \mathbf{F}_0 + \mathbf{F}_1 \biggr]</math>
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<math>~=</math>
  </td>
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<math>~-~\frac{Gm}{r^2}</math>
  </td>
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