Editing SSC/PerspectiveReconciliation (section)

==Introduction==
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<font size="-1">[[H_BookTiledMenu#Stability_Analysis|<b>(poor attempt at)<br />Reconciliation</b>]]</font>
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===Eulerian Approach===
In a [[SSC/StabilityEulerianPerspective#Stability_of_Spherically_Symmetric_Configurations_.28Eulerian_Perspective.29|related discussion]], we have shown that, from a standard Eulerian perspective, the perturbation and linearization of the [[PGE#Principal_Governing_Equations|principal governing equations]] leads to an equation of continuity (EOC), equation of motion (EOM), and Poisson equation of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \nabla\cdot \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial}{\partial t}\biggl( \frac{\rho_1}{\rho_0} \biggr) + \vec{v} \cdot \frac{\nabla\rho_0}{\rho_0} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\partial \vec{v}}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \nabla\Phi_1 - \nabla\biggl[ \frac{\rho_1}{\rho_0}\biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nabla^2\Phi_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G \rho_1 \, .
</math>
  </td>
</tr>
</table>
</div>
The standard <font color="red">''Eulerian''</font> approach to combining these expressions is to <font color="red">take the time-derivative of the EOC, take the divergence of the EOM (and combine with the Poisson equation), then add the two</font> to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial^2}{\partial t^2}\biggl( \frac{\rho_1}{\rho_0} \biggr) + \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial \vec{v}}{\partial t} 
- 4\pi G \rho_1 - \nabla^2 \biggl[ \frac{\rho_1}{\rho_0}\biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .
</math>
  </td>
</tr>
</table>
</div>

After defining the ''fractional Eulerian density variation'' as,
<div align="center">
<math>~s \equiv \frac{\rho_1}{\rho_0} \, ,</math>
</div>
this becomes what we have [[SSC/StabilityEulerianPerspective#EulerianWaveEquation|referred to elsewhere as the Eulerian Wave Equation]].

===Lagrangian Approach===
In [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|another related discussion]], we have shown that, from a standard Lagrangian perspective, the perturbation and linearization of the principal governing equations leads to an equation of continuity (EOC), and equation of motion (EOM) ''already combined'' with the Poisson equation of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 3x - r_0 \biggl(\frac{dx}{dr_0}\biggr) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{\gamma P_0}{\rho_0} \biggl[ \frac{d}{dr_0}\biggl(d \biggr) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\omega^2 r_0 x + g_0\biggl[ 4x + \gamma d \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
The standard <font color="red">''Lagranian''</font> approach to combining these expressions is to <font color="red">take the radial-derivative of the EOC, then substitute the EOC's expression for <math>~d</math> as well as its radial derivative into the EOM</font> to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{\gamma P_0}{\rho_0} \biggl[ 4\biggl(\frac{dx}{dr_0}\biggr) + r_0 \frac{d^2x}{dr_0^2} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\omega^2 r_0 x + g_0\biggl\{ 4x - \gamma \biggl[ 3x + r_0 \biggl(\frac{dx}{dr_0}\biggr) \biggr] \biggr\} \, ,</math>
  </td>
</tr>
</table>
</div>
which, after rearrangement of terms, gives what we have [[SSC/Perturbations#2ndOrderODE|referred to elsewhere as the Adiabatic Wave (or Radial Pulsation) Equation]].

===Rosseland's Approach===
[[SSC/PerspectiveReconciliation#Rosseland.27s_Derivation_of_the_Wave_Equation|Rosseland's approach]] is to <font color="red">take the total time-derivative of the EOM &#8212; even before linearizing any of the governing equations &#8212; then combine it strategically with the EOC and (integrated) Poisson equation</font> to obtain,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2v_r}{dt^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] 
+ \biggl\{ - \frac{d}{dt}\biggl( \frac{\partial\Phi}{\partial r}\biggr) + \biggl( \frac{\partial \Phi}{\partial r}\biggr) \frac{2v_r}{r} \biggr\}  
+ \frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\rho} \frac{\partial }{\partial r} \biggl[ \gamma_g P \nabla\cdot \vec{v}\biggr] + \frac{4gv_r}{r}  
+ \cancelto{\mathrm{small}}{\frac{2v_r}{r} \biggl[ \frac{dv_r}{dt} \biggr]} \, .
</math>
  </td>
</tr>
</table>
</div>

By recognizing that <math>~\vec{v} = 0</math> in the initial equilibrium state and, therefore, that after perturbing the system, <math>~v_r</math> is a "small quantity," this equation is already linearized because every term contains one factor of <math>~v_r</math>.  Actually, as indicated, the last term on the right-hand-side can be dropped because it is of order <math>~(v_r)^2</math>.


===Approach by Ledoux and Walraven===
Note: It also might prove valuable to look at the arguments presented in &sect;2.6 of [https://archive.org/details/AllerStellarStructure Ledoux's Chapter 10, pp. 499-574 of ''Stellar Structure'' (1965)]

[http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)] discuss linearization of the principal governing equations and stellar pulsation primarily from an Eulerian perspective.  Focusing on &sect;57 (pp. 455 - 458) of their ''Handbuch der Physik'' article &#8212; which falls under the major heading, "Radial oscillations of a gaseous sphere under its own gravitation" &#8212; we note, first that they use <math>~\delta r</math> to denote the radial displacement and use primes to identify all ''Eulerian'' perturbations.  Then, in separating out the spatial and time dependences, they use the notation (see their equation 57.14),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta r</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\delta r e^{i\sigma t} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~v^'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~i \sigma \delta r e^{i\sigma t} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho^'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\rho^'(r) e^{i\sigma t} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p^'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~p^'(r) e^{i\sigma t} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Phi^'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\Phi^'(r) e^{i\sigma t} \, .</math>
  </td>
</tr>
</table>
</div>
 
According to their derivations, the linearized equation of continuity (EOC), equation of motion (EOM), and Poisson equation &#8212; see, repectively, their equations 57.15, 57.16 (with nonadiabatic effects set to zero), and 57.21 &#8212; are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho^' + \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho \delta r \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~- \sigma^2 \delta r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\partial \Phi^'}{\partial r} + \frac{\rho^'}{\rho^2} \frac{\partial p}{\partial r} - \frac{1}{\rho}\frac{\partial p^'}{\partial r} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\nabla^2 \Phi^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho^' \, .</math>
  </td>
</tr>
</table>
</div>

Next, with the aim of writing each of the three terms on the right-hand-side of the EOM in terms of the displacement, <math>~\delta r</math>, Ledoux and Walraven do the following: (1) By combining the EOC with the Poisson equation they obtain the perturbation in the gravitational acceleration, namely (see their equation 57.22),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{r^2} \frac{\partial}{\partial r} \biggl( r^2 \frac{\partial\Phi^'}{\partial r} \biggr) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( 4\pi G r^2 \rho \delta r \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ \frac{\partial\Phi^'}{\partial r}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4\pi G \rho \delta r \, .</math>
  </td>
</tr>
</table>
</div>

(2) The pressure fluctuation is obtained by using the adiabatic form of the 1<sup>st</sup> law of thermodynamics to relate <math>~P</math> to <math>~\rho</math>, replacing the ''Lagrangian'' time derivatives with their ''Eulerian'' counterparts, then using the linearized EOC to provide an expression for the density fluctuation in terms of the radial displacement.  Specifically (see their equations 56.15 and 57.19 with nonadiabatic terms set to zero),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dP}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\gamma P}{\rho}\biggr) \frac{d\rho}{dt}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ \frac{\partial p^'}{\partial t} + v^' \frac{\partial p}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\gamma P}{\rho}\biggr) \biggl[ \frac{\partial \rho^'}{\partial t} + v^' \frac{\partial \rho}{\partial r} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ p^'+ \delta r \frac{\partial p}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\gamma P}{\rho}\biggr) \biggl[ \rho^' + \delta r \frac{\partial \rho}{\partial r} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\gamma P}{\rho}\biggr) \biggl[ - \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho \delta r \biggr) + \delta r \frac{\partial \rho}{\partial r} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma P \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr] </math>
  </td>
</tr>
</table>
</div>
With these substitutions, the linearized EOM (tentatively) takes the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma^2 \rho\delta r - \rho \frac{\partial \Phi^'}{\partial r} - \frac{\partial p^'}{\partial r} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{\rho^'}{\rho} \frac{\partial p}{\partial r}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~~~ \rho \delta r \biggl[ \sigma^2  + 4\pi G \rho \biggr] 
+ \frac{\partial }{\partial r}\biggl\{ \gamma P \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr]  
+\delta r \frac{\partial p}{\partial r} \biggr\} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho \delta r \biggr)\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~~~ \rho \delta r \biggl[ \sigma^2  + 4\pi G \rho \biggr] 
+ \frac{\partial }{\partial r}\biggl\{ \gamma P \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr]  \biggr\} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho \delta r \biggr)\biggr]
- \frac{\partial}{\partial r} \biggl[ \delta r \frac{\partial p}{\partial r}  \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{2\rho \delta r}{r} + \frac{\partial}{\partial r}(\rho \delta r)\biggr]
+ \biggl\{  \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl(\frac{2\rho\delta r}{r} \biggr)
- \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 \delta r \frac{\partial p}{\partial r} \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{\rho} \frac{\partial p}{\partial r} \biggl(\frac{4\rho\delta r}{r} \biggr) 
+\frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{\partial}{\partial r}(\rho \delta r)\biggr]
- \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 \delta r \frac{\partial p}{\partial r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow ~~~~~ \rho \delta r \biggl[ \sigma^2  + \frac{4Gm(r)}{r^3}\biggr] 
+ \frac{\partial }{\partial r}\biggl\{ \gamma P \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr]  \biggr\} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\rho\delta r(4\pi G \rho) 
+\frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{\partial}{\partial r}(\rho \delta r)\biggr]
- \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 \delta r \frac{\partial p}{\partial r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\rho\delta r \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl( r^2 \frac{\partial\Phi}{\partial r}\biggr) \biggr] 
+\frac{1}{\rho} \frac{\partial p}{\partial r} \biggl[  \frac{\partial}{\partial r}(\rho \delta r)\biggr]
- \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 (\rho \delta r )\frac{1}{\rho}\frac{\partial p}{\partial r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\rho\delta r \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl( r^2 \frac{\partial\Phi}{\partial r}\biggr) \biggr] 
- \frac{\partial \Phi}{\partial r} \biggl[  \frac{\partial}{\partial r}(\rho \delta r)\biggr]
+ \frac{1}{r^2}\frac{\partial}{\partial r}\biggl[ r^2 (\rho \delta r )\frac{\partial \Phi}{\partial r} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
0 \, .
</math>
  </td>
</tr>
</table>
</div>
This matches equation (57.23) of Ledoux &amp; Walraven, if all nonadiabatic terms are set to zero.