Editing SSC/PerspectiveReconciliation (section)

==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>