Editing SSC/PerspectiveReconciliation (section)

=Reconciling Eulerian versus Lagrangian Perspectives=

<table border="1" cellpadding="8" align="center" width="90%">
<tr>
  <th align="center" colspan="3">Adopting the Perturbation Notation<sup>&dagger;</sup> of [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)]</th>
</tr>
<tr>
  <th align="center">''Eulerian''</th><th align="center">''Lagrangian''</th>
</tr>
<tr>
<td align="center" width="50%">
  
<!-- EULERIAN-->
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>~\hat{e}_r r</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\hat{e}_r [r_0 + \delta r(r) e^{i\sigma t}] </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\hat{e}_r v</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\hat{e}_r [0 + v_r^'(r) e^{i\sigma t}] = \hat{e}_r[i\sigma \delta r(r) e^{i\sigma t} ]</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\rho_0 + \rho^'(r) e^{i\sigma t} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~P_0 + P^'(r) e^{i\sigma t} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\Phi_0 + \Phi^'(r) e^{i\sigma t} </math>
  </td>
</tr>
</table>

</td>
<td align="center" width="50%">
  
<!-- LAGRANGIAN-->
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>~\hat{e}_r r</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\hat{e}_r [r_0 + \delta r(m)  e^{i\sigma t}] </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\hat{e}_r v</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\hat{e}_r [0 + \delta v_r(m)  e^{i\sigma t}] = \hat{e}_r[ i\sigma \delta r(m)  e^{i\sigma t}]</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\rho_0 + \delta \rho(m) e^{i\sigma t} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~P_0 + \delta P(m) e^{i\sigma t} </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~\Phi_0 + \delta \Phi(m) e^{i\sigma t} </math>
  </td>
</tr>
</table>

</td>
</tr>
<tr>
  <td align="left" colspan="2"><sup>&dagger;</sup>See especially the following clarifying comments from, respectively, &sect;43 (p. 432) and &sect;55 (p. 452) of [http://adsabs.harvard.edu/abs/1958HDP....51..353L Ledoux &amp; Walraven (1958)]: 
* The time derivatives of Eulerian coordinates have no meaning; hence, the (components of) <math>~\vec{r}</math> must be treated as Lagrangian coordinates in the definition,

<div align="center"><math>~\frac{d\vec{r}}{dt} = \vec{v}(\vec{r},t) \, .</math></div>

* Consistent with this understanding, we shall always denote a ''perturbation'' of <math>~\mathbf{r}</math> by <math>~\delta\mathbf{r}</math> but for all dependent variables <math>~\psi</math> we shall distinguish between the local or Eulerian perturbations <math>~\psi^'</math> and the Lagrangian perturbations <math>~\delta\psi</math> taken in following the motion.
</td>
</tr>
</table>


==Introduction==
{| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 125px; width: 125px; background-color:white;" |
<font size="-1">[[H_BookTiledMenu#Stability_Analysis|<b>(poor attempt at)<br />Reconciliation</b>]]</font>
|}
===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.

==Separate Analyses of Homogeneous Sphere==
[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 22 June 2015:  Last night I realized that a key to understanding how to reconcile the Eulerian and Lagrangian perspectives was an analysis of the eigenvalue problem for the homogeneous sphere.]]In an [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|accompanying discussion]], we have reviewed [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S T. E. Sterne's (1937, MNRAS, 97, 582)] study of radial pulsation modes in the homogeneous sphere.  He solved the eigenvalue problem as defined by the

<div align="center" id="LagragianApproach">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>

and, hence, as established via a [[SSC/Perturbations#Consistent_Lagrangian_Formulation|Lagrangian formulation of the problem]].  The eigenvectors and eigenvalues that Sterne derived for the first two or three radial modes have also appeared &#8212; usually in the context of separate, re-derivations &#8212; in other publications:  See, for example, &sect;38.2 (pp. 402-403) of [[Appendix/References#KW94|[<font color="red">KW94</font>]]].

After finishing that review, we became aware that a separate study of radial pulsation modes in the homogeneous sphere has been published by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf S. Rosseland (1969)]
In his book titled, ''The Pulsation Theory of Variable Stars'' (see, specifically his &sect; 3.2, beginning on p. 27).  Rosseland solved an eigenvalue problem as defined by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial r} \biggl( \gamma P_0 \nabla\cdot \vec{\xi}\biggr) + \biggl( \omega^2 + \frac{4g_0}{r}\biggr) \rho_0 \xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>
(see his equation 2.23 on p. 20, with the adiabatic condition being enforced by setting the right-hand-side equal to zero), where,
<div align="center">
<math>~\vec\xi = \mathbf{\hat{e}}_r \xi(r) \, .</math>
</div>

Rosseland derived this expression in an earlier section of his book via an Eulerian formulation of the problem.  
Realizing that, for a spherically symmetric system,
<div align="center">
<math>\nabla\cdot \vec\xi = \frac{1}{r^2}\frac{\partial}{\partial r}\biggl(r^2 \xi\biggr) = \frac{\partial \xi}{\partial r} + \frac{2\xi}{r} \, ,</math>
</div>
as is demonstrated in [[SSC/Structure/OtherAnalyticModels#Eulerian_Approach|and accompanying discussion]], this relation can be rewritten in the more familiar form of a 2<sup>nd</sup>-order ODE, namely,

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

<tr>
  <td align="left">
<math>~P_0 \frac{\partial^2 \xi}{\partial r^2}
+ \biggl[ \frac{2P_0}{r}- \rho_0 g_0  \biggr] \frac{\partial \xi}{\partial r}
+ \biggl[ \biggl( \frac{\omega^2\rho_c}{\gamma}  + \frac{4\rho_c g_0}{\gamma r}\biggr) \biggl(\frac{\rho_0}{\rho_c}\biggr)
- \biggl(\frac{2\rho_c g_0 }{r}\biggr)\biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2P_0}{r^2} \biggr] \xi 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="right">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>

Here we will repeat the setup and solution of both eigenvalue problems in an effort to reconcile differences.  As we have [[SSC/Stability/UniformDensity#Properties_of_the_Equilibrium_Configuration|explained elsewhere]], an equilibrium, homogeneous, self-gravitating sphere has the following structural properties:
<div align="center">
<math>\frac{\rho_0}{\rho_c} = 1 \, ,</math>
</div>
<div align="center">
<math>\frac{P_0}{P_c} = 1 - \chi_0^2 \, ,</math>
</div>
<div align="center">
<math> 
\frac{g_0}{g_\mathrm{SSC}} = 2\chi_0  \, ,
</math><br />
</div>
where,
<div align="center">
<math>P_c = \frac{3G}{8\pi}\biggl( \frac{M^2}{R^4} \biggr) </math>
&nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  and &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\rho_c = \frac{3M}{4\pi R^3} \, ,</math>
</div>
and the characteristic gravitational acceleration is defined as,
<div align="center">
<math> 
g_\mathrm{SSC} \equiv \frac{P_c}{R \rho_c} \, .
</math>
</div>
In addition, it will be useful to recognize that the square of the characteristic time for dynamical oscillations in spherically symmetric configurations (SSC) is,
<div align="center">
<math> 
\tau_\mathrm{SSC}^2 \equiv \frac{R^2 \rho_c}{P_c}  = \frac{2R^3}{G M} \, .
</math><br />
</div>


==Lagrangian Reformulation==

Quite generally, we can rewrite the ''Lagrangian-formulated'' wave equation as,
<div align="center">
<math>
\biggl(\frac{P_0}{P_c}\biggr)\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4}{\chi_0}\biggl(\frac{P_0}{P_c}\biggr) 
- \biggl(\frac{\rho_0}{\rho_c}\biggr)  \biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \biggr] \frac{dx}{d\chi_0} 
+ \biggl(\frac{\rho_0}{\rho_c}\biggr) \biggl(\frac{1}{\gamma_\mathrm{g}} \biggr)\biggl[\tau_\mathrm{SSC}^2 \omega^2 + (4 - 3\gamma_\mathrm{g})\biggl(\frac{g_0}{g_\mathrm{SSC}}\biggr) \frac{1}{\chi_0} \biggr]  x = 0 .
</math><br />
</div>
Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, <math>~n=1</math> polytrope are plugged into it, as is demonstrated in an [[SSC/Stability/Polytropes#MurphyFiedler1985b|accompanying discussion]], we obtain exactly the same 2<sup>nd</sup>-order ODE as published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985)].  For an homogeneous sphere, in particular, this expression can be rewritten as follows.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 )\frac{d^2x}{d\chi_0^2} + \biggl[\frac{4(1-\chi_0^2)}{\chi_0} 
-  2\chi_0 \biggr] \frac{dx}{d\chi_0} 
+ \frac{1}{\gamma_\mathrm{g}} \biggl[2\biggl(\frac{\omega^2 R^3}{GM}\biggr) + 2(4 - 3\gamma_\mathrm{g}) \biggr]  x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 )\frac{d^2x}{d\chi_0^2} + \frac{1}{\chi_0}\biggl[4 -  6\chi_0^2 \biggr] \frac{dx}{d\chi_0} + \mathfrak{F}  x \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\mathfrak{F}  \equiv \frac{2}{\gamma_\mathrm{g}} \biggl[\biggl(\frac{\omega^2 R^3}{GM}\biggr) + (4 - 3\gamma_\mathrm{g}) \biggr]  \, .</math>
</div>
This expression precisely matches equation (2) of [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)].

Drawing from [[SSC/Stability/UniformDensity#Sterne.27s_Presentation|Sterne's presentation]], the following table details the eigenfunctions for the four lowest radial modes that satisfy this wave equation.

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr><th colspan="4">Sterne's (1937) Eigenfunctions for Homogeneous Sphere</th></tr>
<tr>
  <td align="center">Mode</td>
  <td align="center" rowspan="2">Eigenvector</td>
  <td align="center">Square of Eigenfrequency:<p></p><math>~3\omega^2/(4\pi G\rho)</math></td>
  <td align="center" rowspan="6">[[File:Sterne1937SolutionPlot1.png|350px|center|Sterne (1937)]]</td>
</tr>
<tr>
  <td align="center"><math>~j</math></td>
  <td align="center"><math>~\gamma[3+j(2j+5)] - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~0</math></td>
  <td align="left"><math>~x = 1</math></td>
  <td align="center"><math>~3\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~1</math></td>
  <td align="left"><math>~x = 1 -\frac{7}{5} \chi_0^2</math></td>
  <td align="center"><math>~10\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="left"><math>~x = 1 -\frac{18}{5} \chi_0^2 + \frac{99}{35} \chi_0^4</math></td>
  <td align="center"><math>~21\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~3</math></td>
  <td align="left"><math>~x = 1 -\frac{33}{5} \chi_0^2 + \frac{429}{35} \chi_0^4 - \frac{143}{21} \chi_0^6</math></td>
  <td align="center"><math>~36\gamma-4</math></td>
</tr>
</table>
</div>

==Eulerian Reformulation==

Using the same characteristic time scale and gravitational acceleration, we can similarly rewrite the ''Eulerian-formulated'' expression as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="left">
<math>~
\biggl(\frac{P_0}{P_c}\biggr) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \biggl[ \frac{2}{\chi_0}\biggl(\frac{P_0}{P_c}\biggr)  - \frac{g_0 }{g_\mathrm{SSC}}\biggl(\frac{\rho_0}{\rho_c}\biggr) \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl\{ \biggl[\frac{\omega^2\tau_\mathrm{SSC}^2}{\gamma}  + \frac{2}{\chi_0 } \biggl(\frac{2}{\gamma } - 1\biggr)\frac{g_0}{g_\mathrm{SSC}}\biggr] 
\biggl(\frac{\rho_0}{\rho_c}\biggr) - \frac{2}{\chi_0^2} \biggl(\frac{P_0}{P_c}\biggr)  \biggr\} \xi
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="right">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
Note that we are convinced that this expression is error-free because, for example, when the structural properties of an equilibrium, [[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|"linear stellar model"]] are plugged into it, we obtain exactly the same 2<sup>nd</sup>-order ODE as published by [http://adsabs.harvard.edu/abs/1967ApJ...148..305S R. Stothers &amp; J. A. Frogel (1967, ApJ, 148, 305)] &#8212; see their equation (2).  For an homogeneous sphere, in particular, this expression can be rewritten as follows.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \biggl[ \frac{2( 1-\chi_0^2 )}{\chi_0}  - 2\chi_0\biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl\{ \frac{1}{\gamma}\biggl[2\biggl( \frac{\omega^2 R^3}{GM} \biggr) + 4\biggl(2 - \gamma\biggr) \biggr] 
- \frac{2}{\chi_0^2} ( 1-\chi_0^2 )  \biggr\} \xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \frac{1}{\chi_0}\biggl[ 2 - 4\chi_0^2\biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl\{ \frac{2}{\gamma}\biggl[\biggl( \frac{\omega^2 R^3}{GM} \biggr) + \biggl(4 - 3\gamma\biggr) +\gamma\biggr] 
- \frac{2}{\chi_0^2} ( 1-\chi_0^2 )  \biggr\} \xi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \frac{1}{\chi_0}\biggl[ 4 - 6\chi_0^2 - 2(1-\chi_0^2) \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl\{ \mathfrak{F} + \biggl(4 - \frac{2}{\chi_0^2} \biggr) \biggr\} \xi \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \frac{1}{\chi_0}\biggl[ 4 - 6\chi_0^2 \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \mathfrak{F} \xi 
- \frac{1}{\chi_0}\biggl[ 2(1-\chi_0^2) \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl(4 - \frac{2}{\chi_0^2} \biggr) \xi 
</math>
  </td>
</tr>
</table>
</div>


Drawing from Rosseland's presentation (see his p. 29), the following table details the eigenfunctions for the three lowest radial modes that satisfy this wave equation; a virtually identical summary can be found in &sect;6.5 (see especially Figure 6.1  on p. 140) of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>].

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="5">
Rosseland's (1964) Eigenfunctions for Homogeneous Sphere<p></p>
Figure in the right-most column extracted from p. 29 of [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)]<p></p>
"''The Pulsation Theory of Variable Stars''" (New York:  Dover Publications, Inc.)
  </th>
</tr>
<tr>
  <td align="center">Mode</td>
  <td align="center" colspan="2">Eigenvector</td>
  <td align="center">Square of Eigenfrequency:<p></p><math>~3\sigma^2/(4\pi G\rho)</math></td>
  <td align="center" rowspan="5">[[File:RosselandModesFigure3.png|350px|center|Rosseland (1937)]]</td>
</tr>
<tr>
  <td align="center"><math>~m</math></td>
  <td align="center">As Published</td><td align="center">Rewritten</td>
  <td align="center"><math>~\frac{m}{2}(m+1)\gamma - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="left"><math>~\xi = -2\chi_0</math></td>
  <td align="left"><math>~-\frac{1}{2}\biggl( \frac{\xi}{\chi_0}\biggr) = 1</math></td>
  <td align="center"><math>~3\gamma - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~4</math></td>
  <td align="left"><math>~\xi = -\frac{20}{3}\chi_0 + \frac{28}{3}\chi_0^3</math></td>
  <td align="left"><math>~-\frac{3}{20}\biggl( \frac{\xi}{\chi_0}\biggr) = 1- \frac{7}{5}\chi_0^2</math></td>
  <td align="center"><math>~10\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~6</math></td>
  <td align="left"><math>~\xi = -14\chi_0 + \frac{252}{5} \chi_0^3 - \frac{198}{5} \chi_0^5</math></td>
  <td align="left"><math>~-\frac{1}{14}\biggl( \frac{\xi}{\chi_0}\biggr) = 1 - \frac{18}{5} \chi_0^2 + \frac{99}{35} \chi_0^4</math></td>
  <td align="center"><math>~21\gamma-4</math></td>
</tr>
</table>
</div>

==Reconciliation by Rosseland==
[https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1969)] points out that these two different formulations can be reconciled by adopting the following dependent coordinate substitution (see his equation 3.11):
<div align="center">
<math>~x \rightarrow \frac{\xi}{\chi_0} \, .</math>
</div>
This means that the derivatives that appear in the above ''Lagrangian formulation'' should be replaced by the expressions,
<div align="center">
<math>~\frac{dx}{d\chi_0} \rightarrow \biggl(\frac{1}{\chi_0}\biggr)\frac{d\xi}{d\chi_0} - \frac{\xi}{\chi_0^2} \, ,</math>
</div>
and,
<div align="center">
<math>~\frac{d^2x}{d\chi_0^2} \rightarrow \biggl(\frac{1}{\chi_0}\biggr)\frac{d^2\xi}{d\chi_0^2} - \biggl( \frac{2}{\chi_0^2} \biggr) \frac{d\xi}{d\chi_0} 
+ \frac{2\xi}{\chi_0^3} \, .</math>
</div>
Doing this, we have,

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

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1-\chi_0^2 )\biggl[ \biggl(\frac{1}{\chi_0}\biggr)\frac{d^2\xi}{d\chi_0^2} - \biggl( \frac{2}{\chi_0^2} \biggr) \frac{d\xi}{d\chi_0} 
+ \frac{2\xi}{\chi_0^3}  \biggr] + \frac{1}{\chi_0}\biggl[4 -  6\chi_0^2 \biggr] \biggl[ \biggl(\frac{1}{\chi_0}\biggr)\frac{d\xi}{d\chi_0} 
- \frac{\xi}{\chi_0^2} \biggr] + \mathfrak{F} \biggl( \frac{\xi}{\chi_0} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{1}{\chi_0}\biggr) \biggl\{
( 1-\chi_0^2 )\biggl[ \frac{d^2\xi}{d\chi_0^2} - \biggl( \frac{2}{\chi_0} \biggr) \frac{d\xi}{d\chi_0} 
+ \frac{2\xi}{\chi_0^2}  \biggr] + \biggl[4 -  6\chi_0^2 \biggr] \biggl[ \biggl(\frac{1}{\chi_0}\biggr)\frac{d\xi}{d\chi_0} 
- \frac{\xi}{\chi_0^2} \biggr] + \mathfrak{F} \xi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{1}{\chi_0}\biggr) \biggl\{
( 1-\chi_0^2 )\frac{d^2\xi}{d\chi_0^2} + \frac{1}{\chi_0} \biggl[4 -  6\chi_0^2 \biggr] \frac{d\xi}{d\chi_0} + \mathfrak{F} \xi 
-\biggl( \frac{2}{\chi_0} \biggr)  ( 1-\chi_0^2 )\frac{d\xi}{d\chi_0}  + \biggl[4\chi_0^2 -2 \biggr] \frac{\xi}{\chi_0^2}  \biggr\}  \, ,
</math>
  </td>
</tr>
</table>
</div>
which, indeed, matches the expression derived via Rosseland's ''Eulerian formulation''.