Editing SSC/PerspectiveReconciliation (section)

=Linearizing the Key Relations=

<table border="3" align="center" cellpadding="10">
<tr>
  <th align="center" colspan="2"><font color="darkgreen" size="+1">Continuity Equation</font></th>
</tr>
<tr>
  <th align="center" width="50%">Lagrangian Perspective</th>
  <th align="center" width="50%">Eulerian Perspective</th>
</tr>
<tr>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\rho}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \rho \nabla\cdot\vec{v}</math>
  </td>
</tr>
</table>
</div>

  </td>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial\rho}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \rho \nabla\cdot\vec{v} - \vec{v}\cdot \nabla\rho</math>
  </td>
</tr>
</table>
</div>

  </td>
</tr>
<tr>
  <th align="center" colspan="2">Spherically Symmetric Initial Configurations &amp; Purely Radial Perturbations</th>
</tr>
<tr>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d\rho}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr)</math>
  </td>
</tr>
</table>
</div>

  </td>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial\rho}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr) - v_r \frac{\partial \rho}{\partial r}</math>
  </td>
</tr>
</table>
</div>

  </td>
</tr>
<!-- EXPLAIN PERTURBATION VARIABLE CHOICES -->
<tr>
  <td align="left" colspan="2">
In an interval of time, <math>~dt = \partial t</math>, a fluid element initially at position <math>~r_0</math> moves to position, <math>~r = r_0 + r_1 = r_0(1 + \xi)</math>.  [For later reference, note that <math>~\xi</math> can be a function of <math>~r_0</math> as well as of <math>~t</math>.]  On the righthand side of the expression, the radial coordinate will be handled as follows:  From the Lagrangian perspective, <math>~r \rightarrow r_0 (1+ \xi)</math>, while from the Eulerian perspective, we want to stay at the original coordinate location, so <math>~r \rightarrow r_0</math>.  From both perspectives, 
<div align="center"><math>~v_r = \frac{\partial ( r_0 \xi )}{\partial t} = r_0 \frac{\partial \xi}{\partial t}  \, .</math></div>
Riding with the fluid element (Lagrangian perspective), <math>~\rho \rightarrow (\rho_0 + \rho_L) = \rho_0(1+s_L)</math>, while at a fixed coordinate location (Eulerian perspective), <math>~\rho \rightarrow (\rho_0 + \rho_E) = \rho_0(1 + s_E)</math>.  Finally, in maintaining a ''Lagrangian'' perspective, we will need to ensure that the same element of mass is being tracked as we "ride along" with the fluid element to its new position.  For radial perturbations associated with a spherically symmetric configuration, this means that the differential mass in each spherical shell, <math>~dm = 4\pi r^2 \rho dr</math>, must remain constant; that is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~4\pi r_0^2 \rho_0 dr_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi r^2 \rho dr</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi [r_0(1+\xi)]^2 \rho_0(1+s_L) dr</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi r_0^2 \rho_0 \biggl(1+2\xi + \cancelto{\scriptstyle\text{small}}{\xi^2} + \cdots \biggr) (1+s_L) dr</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~4\pi r_0^2 \rho_0 \biggl(1+2\xi + s_L + 2\cancelto{\scriptstyle\mathrm{small}}{\xi s_L} \biggr) dr</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{d}{dr}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~(1+2\xi + s_L ) \frac{d}{dr_0} \, .</math>
  </td>
</tr>
</table>
</div>
  </td>
</tr>

<tr>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl[\rho_0(1+s_L)\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \biggl\{ \frac{\rho_0(1+\cancelto{}{s_L})}{[r_0(1+\cancelto{}{\xi})]^2} \biggr\}(1+2\cancelto{}{\xi s_L}) \frac{\partial}{\partial r_0} 
\biggl\{ [r_0(1+\cancelto{}{\xi})]^2 v_r \biggr\}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{d s_L}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[\frac{2v_r}{r_0} + \frac{\partial v_r}{\partial r_0} \biggr] - \frac{1}{\rho_0} \frac{d\rho_0}{dt}</math>
  </td>
</tr>
</table>
</div>

  </td>
  <td align="center">

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\partial}{\partial t}\biggl[\rho_0(1+s_E)\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\rho_0(1+\cancelto{\mathrm{small}}{s_E})}{r_0^2} \frac{\partial}{\partial r_0} \biggl( r_0^2 v_r \biggr) 
- v_r \frac{\partial [\rho_0(1+\cancelto{\mathrm{small}}{s_E})] }{\partial r_0}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{\partial s_E}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[\frac{2v_r}{r_0} + \frac{\partial v_r}{\partial r_0} \biggr] 
- \frac{v_r}{\rho_0} \frac{\partial \rho_0 }{\partial r_0}</math>
  </td>
</tr>
</table>
</div>

  </td>
</tr>
<tr>
  <td align="left" colspan="2">
Note:  The last term that appears on the righthand side of the two expressions appears to be different.  But if, as we are assuming here, <math>~\rho_0</math> has no explicit time dependence but ''may'' be considered to be a function of the radial coordinate, <math>~r_0</math>, then the two terms are the same.  This is because, quite generically for any scalar function <math>~q</math>, the ''total'' time-derivative (Lagrangian perspective) differs from the ''partial'' time-derivative (Eulerian perspective) via the expression, <math>dq/dt - \partial q /\partial t = \vec{v}\cdot \nabla q</math>.  In our case, <math>~\partial \ln \rho_0/\partial t = 0</math>, so <math>~d\ln\rho_0/dt = \vec{v}\cdot \nabla \ln \rho_0</math>.
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
<math>~s_L ~~\rightarrow~~ \Delta_L(r_0) e^{i\omega t}</math>
&nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  &hellip; and &hellip; &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  
<math>~s_E ~~\rightarrow~~ \Delta_E(r_0) e^{i\omega t}</math>

<math>~\xi ~~\rightarrow~~ x(r_0) e^{i\omega t}</math>
&nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  <math>\Rightarrow</math> &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  
<math>~v_r ~~\rightarrow~~ (i\omega)r_0 x(r_0) e^{i\omega t}</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~e^{i\omega t} \biggl[ (i\omega)\Delta_L + \frac{d\Delta_L}{dt} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- e^{i\omega t} \biggl[ 2(i\omega)x + (i\omega)x +  (i\omega)r_0 \frac{\partial x}{\partial r_0} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~r_0 \frac{\partial x}{\partial r_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Delta_L -3 x - \frac{1}{(i\omega)} \biggl[ v_r \frac{\partial\Delta_L}{\partial r_0} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Delta_L - x \biggl[3 + \cancelto{\scriptstyle\mathrm{small}}{\frac{\partial\Delta_L}{\partial \ln r_0} }\biggr]</math>
  </td>
</tr>
</table>
</div>
  </td>
  <td align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{i\omega t} (i\omega)\Delta_E </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- e^{i\omega t} \biggl[ 2(i\omega)x + (i\omega)x +  (i\omega)r_0 \frac{\partial x}{\partial r_0} \biggr]
- \biggl[\frac{1}{\rho_0} \frac{\partial \rho_0}{\partial r_0} \biggr](i\omega)r_0 x e^{i\omega t}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~r_0 \frac{\partial x}{\partial r_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \Delta_E - x \biggl[3 + \frac{\partial \ln\rho_0}{\partial \ln r_0} \biggr] </math>
  </td>
</tr>
</table>
</div>
  </td>
</tr>
</table>