View source for User:Tohline/SSC/Stability Eulerian Perspective

<!-- __FORCETOC__ will force the creation of a Table of Contents -->
<!-- __NOTOC__ will force TOC off -->
=Stability of Spherically Symmetric Configurations (Eulerian Perspective)=
{{LSU_HBook_header}}

A standard technique that is used throughout astrophysics to test the stability of self-gravitating fluids involves ''perturbing'' physical variables away from their initial (usually equilibrium) values then ''linearizing'' each of the principal governing equations before seeking solutions describing the time-dependent behavior of the variables that simultaneously satisfy all of the equations.  When the effects of the fluid's self gravity are ignored and this analysis technique is applied to an initially homogeneous medium, the combined set of ''linearized'' governing equations generates a [http://en.wikipedia.org/wiki/Wave_equation wave equation] that governs the propagation of sound waves.  Here we build on [[User:Tohline/SSC/SoundWaves#Sound_Waves|our separate, introductory discussion of sound waves]] and apply standard perturbation &amp; linearization techniques to spherically symmetric, inhomogeneous and self-gravitating fluids.  We will assume that the reader has read this separate introductory discussion and, in particular, understands how the linear wave equation that governs the propagation of sound waves is derived from the set of nonlinear, principal governing equations.

==Assembling the Key Relations==

===Governing Equations and Supplemental Relations===
We begin with the set of [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] that provides the foundation for all of our discussions in this H_Book, namely, the

<div align="center">
<span id="ConservingMass:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Continuity Equation,

{{User:Tohline/Math/EQ_Continuity02}}


<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,

{{User:Tohline/Math/EQ_Euler02}}


<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

{{User:Tohline/Math/EQ_FirstLaw02}} .


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{User:Tohline/Math/EQ_Poisson01}}
</div>

As was done in our [[User:Tohline/SSC/SoundWaves#Governing_Equations_and_Supplemental_Relations|separate, introductory discussion of sound waves]], we will assume that we are dealing with an ideal gas and supplement this set of equations with a barotropic (polytropic) equation of state,
<div align="center">

<math>~P = K\rho^{\gamma_\mathrm{g}}</math>&nbsp; &nbsp; &hellip; with &hellip; &nbsp; &nbsp; 
<math>\gamma_\mathrm{g} \equiv \frac{d\ln P_0}{d\ln \rho_0} = \frac{\rho_0}{P_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \, ,</math>
</div>
which will ensure that the adiabatic form of the first law of thermodynamics is satisfied.  When we develop the linearized Euler equation, below, it will be useful to recognize that, assuming <math>~\gamma_\mathrm{g}</math> is uniform throughout the fluid, we can rewrite this last expression as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\nabla P_0}{P_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{\rho_0}{P_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] \frac{\nabla \rho_0}{\rho_0}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \nabla P_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_0 \, .</math>
  </td>
</tr>
</table>
</div>

 

===Perturbation then Linearization of Equations===

In this ''Eulerian'' analysis, we are investigating how conditions vary with time at a fixed point in space, <math>~\vec{r}</math>.  By analogy with [[User:Tohline/SSC/SoundWaves#Perturbation_then_Linearization_of_Equations|our separate introductory  analysis of sound waves]], we will write the four  primary variables in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0(\vec{r}) + \rho_1(\vec{r},t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cancelto{0}{\vec{v}_0} + \vec{v}_1(\vec{r},t) = \vec{v}(\vec{r},t) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0(\vec{r}) + P_1(\vec{r},t)  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Phi_0(\vec{r}) + \Phi_1(\vec{r},t)  \, ,</math>
  </td>
</tr>
</table>
</div>
where quantities with subscript "0" are initial values &#8212; independent of time, but not necessarily spatially uniform, and usually specified via some choice of an initial equilibrium configuration &#8212; and quantities with subscript "1" denote variations away from the initial state, which are assumed to be small in amplitude &#8212; for example, <math>~|\rho_1/\rho_0 | \ll 1</math> and <math>~| P_1/P_0 | \ll 1</math>.  As indicated, we will assume that the fluid configuration is initially stationary <math>~(\mathrm{i.e.,}~\vec{v}_0 = 0)</math> and, for simplicity, will not append the subscript "1" to the velocity perturbation.  It is to be understood, however, that the velocity, {{User:Tohline/Math/VAR_VelocityVector01}}, is small also where, ultimately, this will mean <math>~|\vec{v}| \ll c_s</math>.  In what follows, by definition, <math>~P_1</math>, <math>~\rho_1</math>, <math>~\Phi_1</math>, and <math>~\vec{v}</math> are considered to be of first order in smallness, while products of these quantities are of second (or even higher) order in smallness.

====Continuity Equation====
Substituting the expression for <math>~\rho</math> into the lefthand side of the continuity equation and neglecting small quantities of the second order, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~~\frac{\partial}{\partial t} (\rho_0 + \rho_1) + \nabla\cdot [(\rho_0 + \rho_1)\vec{v}]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cancelto{0}{\frac{\partial \rho_0}{\partial t}} + \frac{\partial \rho_1}{\partial t}  
+ \nabla\cdot (\rho_0 \vec{v}) + \nabla\cdot\cancelto{\mathrm{small}}{(\rho_1\vec{v} )}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 \, , 
</math>
  </td>
</tr>
</table>
</div>
where, in the first line, the first term on the righthand side has been set to zero because <math>~\rho_0</math> is independent of time.  Hence, we have the desired,
<div align="center">
<font color="#770000">'''Linearized Continuity Equation'''</font>
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
</div>
<!--
Note that, if we were to assume that the initial configuration is homogeneous, then we could set <math>~\nabla\rho_0 = 0</math> and drop the last term on the righthand side of this expression, retrieving the linearized continuity equation used in [[User:Tohline/SSC/SoundWaves#Perturbation_then_Linearization_of_Equations|our introductory discussion of sound waves]].
-->

====Euler Equation====
Next, we turn to the Euler equation and note that the term,
<div align="center">
<table border="0" cellpadding="5">
<tr><td align="center">
<math>(\vec{v} \cdot \nabla)\vec{v} \, ,</math>
</td></tr>
</table>
</div>
may be altogether neglected because it is of second order in smallness.   Substituting the expressions for <math>~\rho</math>, <math>~P</math>, and <math>~\Phi</math> into the righthand side of the Euler equation and neglecting small quantities of the second order, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \frac{1}{(\rho_0 + \rho_1)} \nabla (P_0 + P_1) - \nabla(\Phi_0 + \Phi_1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\rho_0} \biggl( 1 + \frac{\rho_1}{\rho_0} \biggr)^{-1} \biggl[ \nabla P_0 + \nabla P_1\biggr] - \nabla(\Phi_0 + \Phi_1)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\frac{1}{\rho_0} \biggl[ 1 - \frac{\rho_1}{\rho_0} + \cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2} + \cdots \biggr] 
\biggl[ \nabla P_0 + \nabla P_1\biggr] - \nabla(\Phi_0 + \Phi_1)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \cancelto{0}{\biggl[ \frac{1}{\rho_0} \nabla P_0 + \nabla \Phi_0 \biggr]}
-\frac{1}{\rho_0} \nabla P_1 - \nabla\Phi_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0
+ \frac{1}{\rho_0^2} \cancelto{\mathrm{small}}{(\rho_1 \nabla P_1)}  \, ,
</math>
  </td>
</tr>
</table>
</div>
where, the binomial theorem has been used to obtain the expression on the righthand side of the second line and, in the last line, the sum of the first pair of terms has been set to zero because the initial configuration is assumed to be in equilibrium.  Combining these simplification steps, we have the,
<div align="center">
<font color="#770000">'''Linearized Euler Equation'''</font><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\frac{\partial \vec{v}}{\partial t}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0 - \nabla\Phi_1 \, .
</math>
  </td>
</tr>
</table>
</div>
<!--
Ultimately, as emphasized in [[User:Tohline/Appendix/References#LL75|LL75]], <FONT COLOR="#007700">the condition that the</FONT> linearized governing equations <FONT COLOR="#007700">should be applicable to the propagation of sound waves is that the velocity of the fluid particles in the wave should be small compared with the velocity of sound</FONT>, that is, <math>~|\vec{v}| \ll c_s</math>.
-->

====First Law of Thermodynamics====
In a similar fashion, perturbing the variables in the barotropic equation of state leads to,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
P_0 + P_1  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K (\rho_0 + \rho_1)^{\gamma_\mathrm{g}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ P_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K\rho_0^{\gamma_\mathrm{g}} \biggl(1 + \frac{\rho_1}{\rho_0} \biggr)^{\gamma_\mathrm{g}} - K\rho_0^{\gamma_\mathrm{g}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
K\rho_0^{\gamma_\mathrm{g}} \biggl[1 + 
\gamma_\mathrm{g}\biggl(\frac{\rho_1}{\rho_0} \biggr) + \frac{\gamma_\mathrm{g}(\gamma_\mathrm{g}-1)}{2} 
\cancelto{\mathrm{small}}{\biggl(\frac{\rho_1}{\rho_0} \biggr)^2}
+ \cdots \biggr] - K\rho_0^{\gamma_\mathrm{g}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\gamma_\mathrm{g} \biggl( \frac{P_0}{\rho_0} \biggr) \rho_1 \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, as in [[User:Tohline/SSC/SoundWaves#Perturbation_then_Linearization_of_Equations|our separate introductory discussion of sound waves]], we have the,
<div align="center">
<font color="#770000">'''Linearized Equation of State'''</font><br />
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\, .
</math>
  </td>
</tr>
</table>
</div>

====Poisson Equation====
Finally, plugging the "perturbed" expressions for <math>~\Phi</math> and <math>~\rho</math> into the Poisson equation &#8212; which, by its very nature, is a linear equation &#8212; we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla^2 (\Phi_0 + \Phi_1)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\pi G (\rho_0 + \rho_1)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \cancelto{0}{\nabla^2 \Phi_0 - 4\pi G \rho_0} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\nabla^2 \Phi_1 - 4\pi G \rho_1 \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in the second line, the sum of the pair of terms on the lefthand side has been set to zero because it is a self-contained representation of the Poisson equation for the initial unperturbed medium.  This gives us the desired,
<div align="center">
<font color="#770000">'''Linearized Poisson Equation'''</font><br />
<table border="0" cellpadding="5" align="center">

<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>

===Summary and Combinations===
In summary, the following four linearized equations govern the time-dependent physical relationship between the four perturbation amplitudes <math>~P_1(\vec{r},t)</math>, <math>~\rho_1(\vec{r},t)</math>, <math>~\Phi_1(\vec{r},t)</math> and <math>~\vec{v}(\vec{r},t)</math> in self-gravitating fluids:

<div align="center">
<table border="1" cellpadding="15">
<tr><td align="center">
<font color="#770000">'''Linearized'''</font><br />
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
\frac{\partial \rho_1}{\partial t}  + \rho_0\nabla\cdot \vec{v} + \vec{v}\cdot \nabla\rho_0 = 0 ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:Euler"><font color="#770000">'''Euler Equation'''</font></span><br />
<math>
~\frac{\partial \vec{v}}{\partial t}  = - \nabla\Phi_1 - \frac{1}{\rho_0} \nabla P_1 + \frac{\rho_1}{\rho_0^2} \nabla P_0  \, ,
</math><br />

<font color="#770000">'''Linearized'''</font><br />
<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

<math>
P_1 = \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\, ,
</math>

<font color="#770000">'''Linearized'''</font><br />
<font color="#770000">'''Poisson Equation'''</font><br />

<math>
\nabla^2 \Phi_1 = 4\pi G \rho_1\, .
</math>
</td></tr>
</table>
</div>

This set of linearized governing equations is more general than the [[User:Tohline/SSC/SoundWaves#Summary|set of equations that has traditionally been used to describe the propagation of sound waves]].  The equations governing sound-wave propagation can be retrieved by choosing an initially homogeneous medium &#8212; in which case, <math>~\nabla\rho_0 = 0</math> and <math>~\nabla P_0 = 0</math> &#8212; and by ignoring the fluid's self gravity, that is, by setting <math>~\Phi_1 = 0</math>.

As is explicitly demonstrated in [[User:Tohline/SSC/Stability_Eulerian_Perspective#W._B._Bonnor_.281957.29|our discussion of Bonnor's (1957) research]], below, the linearized Euler equation can be combined with the linearized adiabatic form of the first law of thermodynamics to give,
<div align="center">
<table border="0" cellpadding="5" align="center">

<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>
</table>
</div>
Taking the divergence of this relation, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial }{\partial t} \nabla\cdot \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla^2 \Phi_1 - \nabla^2\biggl[\frac{\rho_1}{\rho_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- 4\pi G \rho_1 - \nabla^2\biggl[\frac{\rho_1}{\rho_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where, in order to obtain this last expression, we have used the linearized Poisson equation to replace <math>~\Phi_1</math> in favor of <math>~\rho_1</math>.  Alternatively, taking the time derivative of the linearized continuity equation gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial^2 \rho_1}{\partial t^2}  + \rho_0 \frac{\partial}{\partial t}\nabla\cdot \vec{v} 
+ \nabla\rho_0 \cdot \frac{\partial \vec{v}}{\partial t} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{\partial}{\partial t}\nabla\cdot \vec{v} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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} \, .
</math>
  </td>
</tr>
</table>
</div>
Combining these last two expressions, then, gives us a
<div align="center" id="EulerianWaveEquation">
<table border="1" cellpadding="8">
<tr><td align="center">
<table border="0" cellpadding="1" align="center">
<tr>
  <td align="center" colspan="3">
<font color="#770000">'''Wave Equation for Self-Gravitating Fluids'''</font>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\frac{\partial^2 s }{\partial t^2}   +  \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial \vec{v}}{\partial t} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>4\pi G \rho_0 s + \nabla^2\biggl[s \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] </math>
  </td>
</tr>
</table>
</td></tr>
</table>
</div>
that describes the time-variation at any point in space of the ''fractional'' density fluctuation,
<div align="center">
<math>s \equiv \frac{\rho_1}{\rho_0} \, ,</math>
</div>
in a self-gravitating, barotropic fluid.  For purposes of comparison with the [[User:Tohline/SSC/SoundWaves#Wave_Equation_Derivation|wave equation that governs the behavior of sound waves]], it is worth rewriting this expression in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{\partial^2 s }{\partial t^2}   - c_s^2 \nabla^2 s 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>4\pi G \rho_0 s - \frac{\nabla\rho_0}{\rho_0} \cdot \frac{\partial \vec{v}}{\partial t}   \, ,</math>
  </td>
</tr>
</table>
</div>
where, as before,
<div align="center">
<math>
c_s \equiv \sqrt{\biggl( \frac{dP}{d\rho} \biggr)_0 } \, .
</math>
</div>

====Comparison with Classic Research Publications====

=====James Jeans (1902 &amp; 1928)=====

James H. Jeans [http://adsabs.harvard.edu/abs/1902RSPTA.199....1J (1902, Philosophical Transactions of the royal Society of London. Series A, 199, 1)] used precisely this type of perturbation and linearization analysis when he first derived what is now commonly referred to as the ''Jeans Instability.'' <!-- &#8212; quoting from p. 112 of [[User:Tohline/Appendix/References#Shu92|Shu92]], "&hellip; [the Jeans criterion for gravitational stability] constitutes perhaps the most frequently cited result of instability theory in all of astronomy." -->   For example, if our discussion is restricted only to fluctuations in the radial coordinate direction of a spherically symmetric configuration &#8212; in which case <math>~\nabla \rightarrow \partial/\partial r</math> and <math>\vec{v} \rightarrow \hat\mathbf{e}_r \cdot \vec{v} = v_r</math> &#8212; our expression for the linearized Euler equation exactly matches equation (12) from  Jeans (1902), which, for purposes of illustration, is displayed in the following framed image.  
<div align="center">
<table border="2" cellpadding="10" width="75%">
<tr>
  <th align="center">
Linearized Euler Equation as Derived and Presented by [http://adsabs.harvard.edu/abs/1902RSPTA.199....1J Jeans (1902)]
[[File:Jeans1902Title.png|350px|center|Jeans (1902)]]
  </th>
<tr>
  <td>
[[File:Jeans1902Eq12.png|400px|center|Jeans (1902)]]
  </td>
</tr>
<tr>
  <td align="left">
The correspondence between the righthand-sides of equation (12) from [http://adsabs.harvard.edu/abs/1902RSPTA.199....1J Jeans (1902)] and our derived expression for the linearized Euler equation is clear after accepting the following variable mappings:
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>~V'</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~- \Phi_1 \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">&nbsp; &nbsp; &nbsp; &nbsp; <math>~\varpi_0</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~P_0</math></td>
</tr>
<tr>
  <td align="right"><math>~\varpi'</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~P_1 \, ;</math> &nbsp; &nbsp; &nbsp; &nbsp;</td>
  <td align="right">&nbsp; &nbsp; &nbsp; &nbsp; <math>~\rho'</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~\rho_1</math></td>
</tr>
</table>
The lefthand side of equation (12) from Jeans (1902) also matches the lefthand side of our linearized Euler equation, although this may not be immediately apparent.  In the paper by Jeans, <math>~u</math> is not a component of the velocity vector but is, rather, the radial displacement of a fluid element.  Hence, 
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>~\frac{\partial u}{\partial t}</math></td>
  <td align="center">&nbsp; <math>~~~ \rightarrow ~~</math> &nbsp; </td>
  <td align="left"><math>~v_r</math>&nbsp; &nbsp; &nbsp; &nbsp; </td>
  <td align="right"><math>~\Rightarrow~~~\frac{\partial^2 u}{\partial t^2} </math></td>
  <td align="center">&nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; </td>
  <td align="left"><math>~\frac{\partial v_r}{\partial t} </math></td>
</tr>
</table>
  </td>
</tr>
</table>
</div>

A broader discussion of gravitational instability in the context of the formation of "great nebulae" (''i.e.,'' galaxies) and stars appears in Chapter XIII &#8212; specifically, pp. 337-342 &#8212; of the book by [http://adsabs.harvard.edu/abs/1928asco.book.....J Jeans (1928)] titled, "Astronomy and Cosmogony."  The governing wave equation for self-gravitating fluids that we have derived, above, appears as equation (314.6) in this published discussion by Jeans (1928), although the term on the lefthand side involving <math>~\nabla\rho_0</math> does not appear, presumably because Jeans assumed that the initial, unperturbed medium was homogeneous.  In an effort to facilitate comparison with our derived expression, equation (314.6) from Jeans (1928) has been reprinted here as a framed image.
<div align="center">
<table border="2" cellpadding="10" width="65%">
<tr>
  <th align="center">
Wave Equation for Self-Gravitating Fluids as Derived and Presented by [http://adsabs.harvard.edu/abs/1928asco.book.....J Jeans (1928)]
  </th>
<tr>
  <td>
[[File:JamesJeans1928Eq314.6.png|400px|center|Jeans (1928)]]
  </td>
</tr>
<tr>
  <td align="left">
Note that Jeans (1928) uses <math>~\rho</math>, without the subscript "0", to represent the initial density, and <math>~\gamma</math>, rather than "G", for the Newtonian gravitational constant.
  </td>
</tr>
</table>
</div>

=====W. B. Bonnor (1957)=====
Above, in our opening layout of the governing equations and supplemental relations, we pointed out that, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla P_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_0 \, .</math>
  </td>
</tr>
</table>
</div>
If we make this substitution in our linearized Euler equation, and also use the linearized first law of thermodynamics to replace <math>~P_1</math> in favor of <math>~\rho_1</math>, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<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 - \frac{1}{\rho_0} \nabla\biggl[ \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\biggr] 
+ \frac{\rho_1}{\rho_0^2} \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_0  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla\Phi_1 - \frac{1}{\rho_0} \nabla\biggl[ \biggl( \frac{dP}{d\rho} \biggr)_0 \rho_1\biggr] 
- \rho_1 \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \biggl(\frac{1}{\rho_0} \biggr)  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </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>
</table>
</div>

This is the version of the linearized Euler equation that was derived by [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor (1957, MNRAS, 117, 104)] in the section of his paper that addresses the growth of Newtonian, self-gravitating fluctuations on a static (cosmological) background.  The two equation images reproduced in the following outlined box document Bonnor's (1957) initial expression (his equation 2.1) for the nonlinear Euler equation and his derived expression (equation 2.7) for the linearized Euler equation.  After allowing for the identified variable mappings, Bonnor's two expressions precisely match, respectively, the form of the nonlinear Euler equation that is included among our set of principal governing equations, and this last derived form of our linearized Euler equation.

<div align="center">
<table border="2" cellpadding="10" width="75%">
<tr>
  <th align="center" colspan="3">
Bonnor's [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B (1957, MNRAS, 117, 104)] Derivation
  </th>
<tr>
  <td align="center">
Original ''nonlinear'' Euler Equation
  </td>
  <td align="center" rowspan="2">
&nbsp; &nbsp; &nbsp; <math>~\rightarrow</math> &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="center">
''Linearized'' Euler Equation
  </td>
</tr>
<tr>
  <td align="center">
[[File:Bonnor1957Eq2.1.png|250px|center|Bonnor's (1957) Equation 2.1]]
  </td>
  <td align="center">
[[File:Bonnor1957Eq2.7.png|250px|center|Bonnor's (1957) Equation 2.7]]
  </td>
</tr>
<tr>
  <td align="left" colspan="3">
The correspondence between these two equations from [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor (1957)] and our derived expressions is clear after accepting the following variable mappings:
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>~\mathbf{u}</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~\vec{v}</math></td>
</tr>
<tr>
  <td align="right"><math>~\mathbf{F}</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~- \nabla\Phi</math></td>
</tr>
<tr>
  <td align="right"><math>~w</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~\rho_1</math></td>
</tr>
</table>
  </td>
</tr>
</table>
</div>
Bonnor then proceeded to combine the full set of linearized governing equations, in the manner we have detailed above, into a wave equation that is appropriately modified to handle self-gravitating fluids. In an effort to facilitate comparison with our derived expression, Bonnor's (1957) modified wave equation (2.10) has been reprinted here as a framed image.
<div align="center">
<table border="2" cellpadding="10" width="65%">
<tr>
  <th align="center">
Wave Equation for Self-Gravitating Fluids as Derived and Presented by [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor (1957)]
  </th>
<tr>
  <td>
[[File:Bonnor1957Eq2.10.png|400px|center|Bonnor (1957)]]
  </td>
</tr>
</table>
</div>

=See Also=

* Part II of ''Spherically Symmetric Configurations'': [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Stability]]
* [http://en.wikipedia.org/wiki/Wave_equation Wave Equation]
* [http://www.astronomy.ohio-state.edu/~dhw/A825/notes6.pdf Sound Waves and Gravitational Instability] &#8212; class notes provided online by David H. Weinberg (The Ohio State University)

=Scratch Work=


{{LSU_WorkInProgress}}

==Part 1==

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

<tr>
  <td align="right">
<math>~\frac{\partial\vec{v}}{\partial t} + \cancelto{\mathrm{small}}{(\vec{v}\cdot \nabla) \vec{v} }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{\rho} \nabla P - \nabla \Phi</math>
  </td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g \equiv \frac{d\Phi}{dr}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{GM_r}{r^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\rho_0}\frac{dP_0}{dr_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-g_0</math>
  </td>
</tr>
</table>
</div>

<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>~\gamma_g P \nabla \cdot \vec{v}</math>
  </td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\ln P}{d\ln\rho}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\gamma_g</math>
  </td>
</tr>
</table>
</div>


<div align="center">
<table border="1" cellpadding="15">
<tr><td align="center">

<font color="#770000" size="+1">'''Linearized'''</font><br />


<span>Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

<math>
P^' = \biggl( \frac{\gamma_g P}{\rho} \biggr)_0 \rho^'
</math>


<span><font color="#770000">'''Equation of Continuity'''</font></span><br />
<math>
\frac{\partial (f_\sigma \rho^')} {\partial t}  + \nabla\cdot [\hat{e}_r  (\rho_0 f_\sigma v_r^')] = 0 
</math><br />


<span><font color="#770000">'''Euler Equation'''</font></span><br />
<math>
~\frac{\partial [\hat{e}_r  (f_\sigma v_r^')]}{\partial t}  = \hat{e}_r f_\sigma \biggl\{ - \nabla_r\Phi^' - \frac{1}{\rho_0} \nabla_r P^' + \frac{\rho^'}{\rho_0^2} \nabla_r P_0 \biggr\} 
</math><br />


<font color="#770000">'''Poisson Equation'''</font><br />

<math>
\nabla^2 \Phi^' = 4\pi G \rho^'
</math><br />
<br />
<br />


</td></tr>
</table>
</div>

Replace <math>~P^'</math> in favor of <math>~\rho^'</math> in Euler equation.

<div align="center">
<math>
~\frac{\partial \vec{v}}{\partial t}  = - \nabla\Phi^' - \nabla \biggl[ \frac{\rho^'}{\rho_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] 
</math>
</div>



Take the divergence of this entire expression, then use linearized Poisson equation to replace <math>~\Phi^'</math> in favor of <math>~\rho^'</math>.


<div align="center">
<math>
~\frac{\partial }{\partial t}\nabla\cdot \vec{v}  = - 4\pi G \rho^' - \nabla^2 \biggl[ \frac{\rho^'}{\rho_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] 
</math>
</div>


Rearrange terms in the linearized equation of continuity, then take the partial time-derivative of the entire expression.


<div align="center">
<math>~\frac{\partial }{\partial t}\nabla\cdot \vec{v}  = - \frac{\partial^2}{\partial t^2}\biggl(\frac{\rho^'}{\rho_0} \biggr) - \vec{v}\cdot \frac{\nabla\rho_0}{\rho_0}
</math>
</div>

Subtract the step #2 expression from the step #3 expression.


<div align="center">
<math>~\frac{\partial^2}{\partial t^2}\biggl(\frac{\rho^'}{\rho_0} \biggr) + \vec{v}\cdot \frac{\nabla\rho_0}{\rho_0}
~= 4\pi G \rho^' + \nabla^2 \biggl[ \frac{\rho^'}{\rho_0} \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] ~~~~\Downarrow ~~~~~\Leftarrow
</math>
</div>



<div align="center">
<math>~\frac{d\ln P}{dt} = \gamma_g \frac{d\ln\rho}{dt} ~~~~~\Rightarrow ~~~~~\frac{dP}{dt} = \biggl(\frac{\gamma_g P}{\rho}\biggr) \frac{d\rho}{dt}</math>
</div>

 

If homentropic as well, then,


<div align="center">
<math>~\frac{d\ln P}{dr} = \gamma_g \frac{d\ln\rho}{dr} ~~~~~\Leftrightarrow ~~~~~\nabla P = \biggl(\frac{\gamma_g P}{\rho}\biggr) \nabla\rho</math>
</div>



<div align="center">
<math>~ - \frac{1}{\rho_0} \nabla_r P^' + \frac{1}{\rho_0^2} \biggl[ \rho^' \biggr] \biggl[\nabla_r P_0\biggr] 
= - \frac{1}{\rho_0} \nabla_r P^' + \frac{1}{\rho_0^2} \biggl[ \biggl( \frac{\rho}{\gamma_g P}\biggr)_0 P^' \biggr] 
\biggl[\biggl( \frac{\gamma_g P}{\rho} \biggr)_0 \nabla_r \rho_0\biggr] = - \nabla_r \biggl( \frac{P^'}{\rho_0}\biggr)
</math>
</div>



<div align="center">
<math>~= \hat{e}_r f_\sigma \biggl\{ - \nabla_r\Phi^' - \nabla_r \biggl[\frac{\rho^'}{\rho_0}  \biggl( \frac{\gamma_g P}{\rho}\biggr)_0 \biggr] \biggr\}</math>
</div>



<div align="center">
<math>~
\frac{\partial}{\partial t} \nabla \cdot [\hat{e}_r(f_\sigma v_r^')]
= - f_\sigma \biggl\{ 4\pi G \rho^' + \nabla^2 \biggl[\frac{\rho^'}{\rho_0}  \biggl( \frac{\gamma_g P}{\rho}\biggr)_0 \biggr] \biggr\}
</math>
</div>



<div align="center">
<math>~
\frac{\partial}{\partial t} \nabla \cdot [\hat{e}_r(f_\sigma v_r^')]
= - \frac{\partial^2 }{\partial t^2} \biggl(\frac{f_\sigma \rho^'}{\rho_0}\biggr)
- \biggl[\frac{\nabla_r \rho_0}{\rho_0} \biggr] \frac{\partial (f_\sigma v_r^') }{\partial t}
</math>
</div>



<div align="center">
<math>~
\frac{\partial^2 }{\partial t^2} \biggl(\frac{f_\sigma \rho^'}{\rho_0}\biggr)
+ \biggl[\frac{\nabla_r \rho_0}{\rho_0} \biggr] \frac{\partial (f_\sigma v_r^') }{\partial t}
=  f_\sigma \biggl\{ 4\pi G \rho^' + \nabla^2 \biggl[\frac{\rho^'}{\rho_0}  \biggl( \frac{\gamma_g P}{\rho}\biggr)_0 \biggr] \biggr\}
</math>
</div>



<div align="center">
<math>~
\frac{\partial^2 s}{\partial t^2} 
+ \biggl[\frac{\nabla_r \rho_0}{\rho_0} \biggr] \frac{\partial (f_\sigma v_r^') }{\partial t}
=  4\pi G \rho_0 s + \nabla^2 \biggl[s \biggl( \frac{\gamma_g P}{\rho}\biggr)_0 \biggr] 
</math>
</div>



where:  <math>~s \equiv \biggl(\frac{f_\sigma \rho^'}{\rho_0}\biggr)</math>

==Part 2==

&nbsp;
<p></p>
<font color="green" size="+3">&#x2460;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2461;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2462;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2463;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2464;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2465;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2466;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2467;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2468;</font>&nbsp;&nbsp;
<font color="green" size="+3">&#x2469;</font>&nbsp;&nbsp;


<div align="center">
<table border="1" cellpadding="15">
<tr><td align="center">

<font color="#770000" size="+1">'''Linearized'''<br />
(explicit time-dependence removed)</font><br />


<span><font color="#770000">'''Equation of Continuity'''</font></span><br />

<math>
\rho^' = - \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \rho_0 \delta r \biggr)  
</math><br />


<span><font color="#770000">'''Euler Equation'''</font></span><br />

<math>
~\sigma^2 \rho_0 \delta r  = \rho_0 \frac{\partial \Phi^'}{\partial r} + \frac{\partial P^'}{\partial r} - \frac{\rho^'}{\rho_0} \frac{\partial P_0}{\partial r} 
</math><br />


<font color="#770000">'''Poisson Equation'''</font><br />

<math>
\nabla^2 \Phi^' = 4\pi G \rho^'
</math><br />


<span>Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

<math>
P^' = \biggl( \frac{\gamma_g P}{\rho} \biggr)_0 \rho^'
</math>

</td></tr>
</table>
</div>


<div align="center">
<math>
\frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \frac{\partial\Phi^'}{\partial r}\biggr) = - \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 4\pi G \rho_0 \delta r \biggr)
~~~\Rightarrow~~~ \frac{\partial\Phi^'}{\partial r} = -4\pi G \rho_0 \delta r = - \delta r \nabla^2 \Phi_0
</math>
</div>


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

<tr>
  <td align="right">
<math>~\rho_0 \frac{\partial \Phi^'}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ (\rho_0 \delta r) \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl(\frac{r^2}{\rho_0} \frac{\partial P_0}{\partial r}\biggr)  \biggr]
= \frac{1}{r^2} \frac{\partial}{\partial r} \biggl[ (r^2 \delta r) \frac{\partial P_0}{\partial r} \biggr]
- \frac{1}{\rho_0} \frac{\partial P_0}{\partial r} \frac{\partial (\rho_0 \delta r)}{\partial r}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial r} \biggl[ (\delta r) \frac{\partial P_0}{\partial r} \biggr]
+ \frac{1}{\rho_0}\frac{\partial P_0}{\partial r} \biggl( \frac{2\rho_0 \delta r}{r} \biggr)
- \frac{1}{\rho_0} \frac{\partial P_0}{\partial r} \frac{\partial (\rho_0 \delta r)}{\partial r} \, ;</math>
  </td>
</tr>
</table>
</div>

<div align="center">
<math>~\frac{\partial \Phi_0}{\partial r} = - \frac{1}{\rho_0} \frac{\partial P_0}{\partial r}</math>
</div>




<div align="center">
<math>- \frac{1}{\rho_0} \frac{\partial P_0}{\partial r} \biggl[ \rho^' \biggr]
= + \frac{1}{\rho_0} \frac{\partial P_0}{\partial r} \biggl[ \frac{1}{r^2} \frac{\partial}{\partial r} \biggl(r^2 \rho_0 \delta r\biggr) \biggr]
= \frac{1}{\rho_0} \frac{\partial P_0}{\partial r} \biggl[ \frac{\partial (\rho_0 \delta r )}{\partial r}  + \frac{2\rho_0 \delta r}{r} \biggr]
</math>
</div>



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

<tr>
  <td align="right">
<math>~- \frac{1}{\rho_0} \frac{\partial P_0}{\partial r} \biggl[ \rho^' \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \frac{1}{\rho_0} \frac{\partial P_0}{\partial r} \biggl[ \frac{1}{r^2} \frac{\partial}{\partial r} \biggl(r^2 \rho_0 \delta 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_0} \frac{\partial P_0}{\partial r} \biggl[ \frac{\partial (\rho_0 \delta r )}{\partial r}  + \frac{2\rho_0 \delta r}{r} \biggr] \, ;</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{\partial P^'}{\partial r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{\partial}{\partial r} \biggl\{ -\gamma_g P_0 \biggl[\frac{1}{r^2} \frac{\partial}{\partial r}(r^2 \delta r)  \biggr] \biggr\}
- \frac{\partial}{\partial r}\biggl[ (\delta r) \frac{\partial P_0}{\partial r} \biggr] \, .</math>
  </td>
</tr>
</table>
</div>


<div align="center">
<math>~\frac{\partial P}{\partial t} + \vec{v}\cdot \nabla P = \biggl( \frac{\gamma_g P}{\rho}\biggr) \biggl[\frac{\partial \rho}{\partial t} + \vec{v}\cdot \nabla \rho \biggr]</math>
</div>



<div align="center">
<math>~\frac{\partial (f_\sigma P^')}{\partial t} + f_\sigma v_r^'\nabla_r P_0
= \biggl( \frac{\gamma_g P}{\rho}\biggr)_0 \biggl[\frac{\partial (f_\sigma\rho^')}{\partial t} + f_\sigma v_r^'\nabla_r \rho_0 \biggr]</math>
</div>



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

<tr>
  <td align="right">
<math>~P^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\gamma_g P_0 \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl(r^2 \delta r \biggr)  \biggr] - 
\delta r \biggl( \frac{\gamma_g P }{\rho}\biggr)_0 \frac{\partial\rho_0}{\partial r} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\gamma_g P_0 \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl(r^2 \delta r \biggr)  \biggr] - 
\delta r \frac{\partial P_0}{\partial r} </math>
  </td>
</tr>
</table>
</div>



<div align="center">
<math>~P^'
= -\gamma_g P_0 \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl(r^2 \delta r \biggr)  \biggr] - 
\delta r \biggl( \frac{\gamma_g P }{\rho}\biggr)_0 \frac{\partial\rho_0}{\partial r}
= -\gamma_g P_0 \biggl[\frac{1}{r^2} \frac{\partial}{\partial r} \biggl(r^2 \delta r \biggr)  \biggr] - 
\delta r \frac{\partial P_0}{\partial r} </math>
</div>

==Part 3==


<div align="center">
<math>~
- \frac{1}{\rho_0}\nabla_r P^' + \frac{\rho^'}{\rho_0^2} \nabla_r P_0 = 
- \frac{1}{\rho_0}\nabla_r \biggl[ \biggl( \frac{\gamma_g P_0}{\rho_0}\biggr) \rho^' \biggr] + \frac{\rho^'}{\rho_0^2} \biggl( \frac{\gamma_g P_0}{\rho_0}\biggr) \nabla_r \rho_0 =
- \nabla_r \biggl[ \biggl( \frac{\gamma_g P_0}{\rho_0}\biggr) \frac{\rho^'}{\rho_0} \biggr]
</math>
</div>



<div align="center">
<math>~\Rightarrow ~~~
~\frac{\partial [\hat{e}_r  (f_\sigma v_r^')]}{\partial t}  = \hat{e}_r f_\sigma \biggl\{ - \nabla_r\Phi^' - \nabla_r \biggl[ \biggl( \frac{\gamma_g P_0}{\rho_0}\biggr) \frac{\rho^'}{\rho_0} \biggr] \biggr\} 
</math>
</div>



{{LSU_HBook_footer}}