Editing SSC/Perturbations (section)

===Bonnor (1957)===

In a paper titled, "Jeans' Formula for Gravitational Instability," [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor (1957, MNRAS, 117, 104)] carried out a linear perturbation analysis, preferring to examine the development of Eulerian fluctuations in the matter density rather than the development of Lagrangian position displacements.  Here we show the relationship between his approach to a perturbation analysis and the one that we have focused on, above.  

====Linearized Equations on a Static Background====
First, we examine how [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor's (1957)] ''linearized'' Euler equation (2.7) was derived from the nonlinear Euler equation, numbered (2.1) in his paper.

<div align="center">
<table border="2" cellpadding="10">
<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>
</table>
</div>

As has been made clear in our [[PGE/ConservingMomentum#Eulerian_Representation|introductory discussions]], the

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

{{Math/EQ_Euler02}}
</div>

can be counted among the principal set of equations that govern the dynamics of self-gravitating fluids.  Accepting that he uses the boldface variable <math>~\mathbf{u}</math> instead of <math>~\vec{v}</math> to represent the fluid velocity, we see that the lefthand side of Bonnor's equation (2.1) exactly matches the lefthand side of the Euler equation, as we have presented it.  The term on the righthand side of Bonnor's equation (2.1) that involves a gradient in the gas pressure also matches ours.  What remains is to recognize that, in Bonnor's paper,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{F}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla\Phi \, .</math>
  </td>
</tr>
</table>
</div>
This is confirmed by Bonnor's equation (2.6), which presents another one of our identified set of [[PGE|principal governing equations]], namely the
<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span>

{{Math/EQ_Poisson01}}
</div>
in terms of the vector, <math>~\mathbf{F}</math>, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla\cdot\mathbf{F}</math>
  </td>
  <td align="center">
<math>~= - \nabla^2\Phi =</math>
  </td>
  <td align="left">
<math>~-4\pi G \rho \, .</math>
  </td>
</tr>
</table>
</div>

As we have done in our [[#The_Eigenvalue_Problem|development of the eigenvalue problem]], [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor (1957)] began the process of ''linearizing'' the Euler equation by writing each physical variable in terms of its initial, unperturbed value (denoted by subscript "0") plus a "small quantity."  For example, he wrote,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathbf{u}_0 + \mathbf{u} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathbf{F}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathbf{F}_0 + \mathbf{F}_1 </math> &nbsp; &nbsp; &hellip; and we, furthermore, will write &hellip; &nbsp; &nbsp; <math>~\mathbf{F}_0 = -\nabla \Phi_0</math> &nbsp; and &nbsp; <math>~\mathbf{F}_1 = - \nabla \Phi_1 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_0 + w \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~P_0 + q \, .</math>
  </td>
</tr>
</table>
</div>
[For future reference, notice that the perturbation variable names that we introduced for density and pressure &#8212; [[#The_Eigenvalue_Problem|see above]] &#8212; are different from the ones used by Bonner.  They are related via the expressions, <math>(\rho_0 d e^{i\omega t}) \leftrightarrow w</math> and <math>(P_0 p e^{i\omega t}) \leftrightarrow q</math>.]  He also initially assumed, as have we, that the unperturbed system is in hydrostatic balance, so,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathbf{u}_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathbf{F}_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\rho_0} \nabla P_0 \, .</math>
  </td>
</tr>
</table>
</div>

Notice that &#8212; confusion notwithstanding &#8212; Bonnor did not affix a subscript to the variable being used to represent the velocity perturbation, at least not initially.  So, after setting <math>~\mathbf{u}_0 = 0</math>, the ''generic'' velocity vector, <math>~\mathbf{u}</math>, on the lefthand side of his equation (2.1) becomes the ''small in magnitude'' velocity perturbation, <math>~\mathbf{u}</math>, on the lefthand side of his equation (2.7); and, in his equation (2.7), the <math>~(\mathbf{u}\cdot \nabla)\mathbf{u}</math> term disappears altogether because it involves the product of two quantities that are both ''small in magnitude.''  Although details of the derivation are not presented in Bonnor's (1957) paper, it is reasonable to assume that he took the following steps in ''linearizing'' the righthand side of equation (2.1):
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
RHS of equation (2.1)
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\mathbf{F}_0 + \mathbf{F}_1 - \frac{1}{(\rho_0 + w)} \nabla(P_0 + q) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathbf{F}_0 + \mathbf{F}_1 - \frac{1}{\rho_0} \biggl(1 + \frac{w}{\rho_0} \biggr)^{-1}\nabla(P_0 + q) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\mathbf{F}_0 + \mathbf{F}_1 - \frac{1}{\rho_0} \biggl(1 - \frac{w}{\rho_0} \biggr)\nabla(P_0 + q) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \mathbf{F}_0 - \frac{1}{\rho_0}\nabla P_0 \biggr] 
+ \mathbf{F}_1 - \frac{1}{\rho_0} \nabla q 
+ \biggl(\frac{w}{\rho_0^2} \biggr)\nabla P_0
+ \biggl(\frac{w}{\rho_0^2} \biggr)\nabla q</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathbf{F}_1 - \frac{1}{\rho_0} \nabla q 
+ \biggl(\frac{w}{\rho_0^2} \biggr)\nabla P_0 \, .</math>
  </td>
</tr>
</table>
</div>

As is shown in the following framed image, if the discussion is restricted only to fluctuations in the radial coordinate direction of a spherically symmetric configuration, in which case <math>~\nabla \rightarrow \partial/\partial r</math>, this expression exactly matches the righthand side of the linearized Euler equation, derived and presented as equation (12) by James H. Jeans in 1902 [http://adsabs.harvard.edu/abs/1902RSPTA.199....1J (Philosophical Transactions of the royal Society of London. Series A, 199, 1)].
<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|450px|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 [using [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor's (1957)] variable notation] is clear after accepting the following variable mappings:
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>~\frac{\partial V'}{\partial r}</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~\mathbf{F}_1= - \nabla \Phi_1</math></td>
</tr>
<tr>
  <td align="right"><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>~q</math></td>
</tr>
<tr>
  <td align="right"><math>~\rho'</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~w</math></td>
</tr>
</table>
The lefthand side of equation (12) from Jeans (1902) also matches the lefthand side of Bonnor's (1957) linearized Euler equation, although this may not be immediately apparent because the variable "u" has been assigned different meanings in the two publications.  In Bonnor's paper,  <math>~u = \hat{\mathbf{e}}_r \cdot \mathbf{u}</math> is the perturbed velocity in the radial-coordinate direction; while, in the paper by Jeans, <math>~u</math> is the radial displacement itself.  Hence, 
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right"><math>~\frac{\partial u}{\partial t} \biggr|_\mathrm{Jeans}</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~u \biggr|_\mathrm{Bonner}</math></td>
</tr>
<tr>
  <td align="right"><math>~\Rightarrow~~~\frac{\partial^2 u}{\partial t^2} \biggr|_\mathrm{Jeans}</math></td>
  <td align="center">&nbsp; &nbsp; <math>~~~ \rightarrow ~~</math>&nbsp; &nbsp; </td>
  <td align="left"><math>~\frac{\partial u}{\partial t} \biggr|_\mathrm{Bonner}</math></td>
</tr>
</table>
  </td>
</tr>
</table>
</div>



Now, in order to morph this last expression into the expression found on the righthand side of Bonnor's equation (2.7), as reprinted above, we need to draw upon the result obtained, above, from [[#Adiabatic_form_of_the_First_Law_of_Thermodynamics|linearizing the adiabatic form of the First Law of Thermodynamics]].  After shifting to Bonnor's variable notation (as clarified in earlier remarks), the relevant result is,
<div align="center">
<math>\frac{q}{P_0} \approx \gamma_g \frac{w}{\rho_0} \, .</math>
</div>
In addition, we appreciate that,
<div align="center">
<math>\gamma_g = \biggl( \frac{d\ln P}{d\ln \rho} \biggr)_0 = \frac{\rho_0}{P_0}\cdot \biggl( \frac{dP}{d\rho} \biggr)_0 \, .</math>
</div>
Hence (see also Bonnor's equation 3.7), 
<div align="center">
<math>q \approx w \biggl( \frac{dP}{d\rho} \biggr)_0 \, ,</math>
</div>
which allows us to write the righthand side of the linearized Euler equation as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
RHS of equation (2.1)
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathbf{F}_1 - \frac{1}{\rho_0} \biggl\{ \nabla \biggl[ w \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] 
- \biggl(\frac{w}{\rho_0} \biggr)\nabla P_0\biggr\} \, .</math>
  </td>
</tr>
</table>
</div>
But, given the adopted barotropic equation of state, we can also write,
<div align="center">
<math>\nabla P_0 = \biggl(\frac{dP}{d\rho} \biggr)_0 \nabla\rho_0 \, ,</math>
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
RHS of equation (2.1)
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\mathbf{F}_1 - \biggl\{\frac{1}{\rho_0}  \nabla \biggl[ w \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] 
- \biggl(\frac{w}{\rho_0^2} \biggr)\biggl(\frac{dP}{d\rho} \biggr)_0 \nabla\rho_0\biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{F}_1 - \biggl\{ \frac{1}{\rho_0} \nabla \biggl[ w \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] 
+ w\biggl(\frac{dP}{d\rho} \biggr)_0 \nabla\biggl(\frac{1}{\rho_0} \biggr) \biggr\} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathbf{F}_1 - \nabla \biggl[ \frac{w}{\rho_0}  \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
This precisely matches the righthand side of the linearized Euler equation derived and presented as equation (2.7) by Bonnor &#8212; see the reprinted equation, above.

====Reconciliation====
It is instructive to explicitly demonstrate that the linearized "Euler + Poisson" equation that we derived and highlighted in our [[#Summary_2|brief summary subsection, above]], namely,  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>
\frac{P_0}{\rho_0} \frac{dp}{dr_0}  = (4x + p)g_0 + \omega^2 r_0 x \, ,
</math>
  </td>
</tr>
</table>
</div>

conveys the same physics as [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B Bonnor's (1957) linearized Euler equation] when applied to a spherically symmetric system,  namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\partial v_r}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{\mathbf{e}}_r \cdot \mathbf{F}_1 - \frac{d}{d r_0} \biggl[ \frac{w}{\rho_0}  \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

<font color="maroon">Step #1:</font>  We recognize that, after linearization, <math>~\partial v_r/\partial t = d^2 r/dt^2</math>.  So, drawing on our [[#Euler_.2B_Poisson_Equations|earlier detailed handling of the "Euler + Poisson" equations]], we can make the replacement,

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{\partial v_r}{\partial t}  
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
- ~\omega^2 r_0 x~e^{i\omega t} \, .
</math>
  </td>
</tr>
</table>

<font color="maroon">Step #2:</font>  As we have already recognized, swapping between our perturbation notation and Bonnor's leads to the replacement,

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
w \biggl( \frac{dP}{d\rho}  \biggr)_0
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
P_0 p e^{i\omega t} \, .
</math>
  </td>
</tr>
</table>
Hence,

<table align="center" border="0" cellpadding="5">
<tr>
  <td align="right">
<math>
\frac{d}{d r_0} \biggl[ \frac{w}{\rho_0}  \biggl( \frac{dP}{d\rho} \biggr)_0 \biggr]
</math>
  </td>
  <td align="center">
<math>
\rightarrow
</math>
  </td>
  <td align="left">
<math>
\frac{d}{d r_0} \biggl[ \biggl( \frac{P_0 p}{\rho_0}\biggr) e^{i\omega t} \biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
e^{i\omega t} \biggl\{ \frac{p}{\rho_0} \frac{d P_0}{d r_0} 
+
P_0 p \frac{d}{d r_0} \biggl( \frac{1}{\rho_0}\biggr) 
+
\frac{P_0}{\rho_0} \frac{d p}{d r_0}  
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
e^{i\omega t} \biggl\{ \biggl[ \frac{1}{\rho_0} \frac{d P_0}{d r_0} \biggr] (p - d)
+
\frac{P_0}{\rho_0} \frac{d p}{d r_0}  
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
e^{i\omega t} \biggl[ (d - p) g_0 + \frac{P_0}{\rho_0} \frac{d p}{d r_0}  \biggr]  \, .
</math>
  </td>
</tr>
</table>
where we have, again, used the relationship,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\nabla P_0 = \biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_0 = \biggl( \frac{P_0 p}{\rho_o d} \biggr) \nabla \rho_0 </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow~~~~</math> &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~
\frac{d}{dr_0} \biggl( \frac{1}{\rho_0}\biggr) = - \frac{1}{\rho_0^2}\frac{d\rho_0}{dr_0} 
=- \biggl( \frac{d}{P_0 p} \biggr) \biggl[\frac{1}{\rho_0}\frac{dP_0}{dr_0} \biggr]  \, .</math>
  </td>
</tr>
</table>
</div>

<font color="maroon">Step #3:</font>  Implementing these first two substitutions, Bonnor's linearized Euler equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>- ~\omega^2 r_0 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{-i\omega t}\hat{\mathbf{e}}_r \cdot \mathbf{F}_1 - \biggl[ (d - p) g_0 + \frac{P_0}{\rho_0} \frac{d p}{d r_0}  \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~~ \frac{P_0}{\rho_0} \frac{d p}{d r_0}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{-i\omega t}\hat{\mathbf{e}}_r \cdot \mathbf{F}_1 + (p - d) g_0 + \omega^2 r_0 x\, .</math>
  </td>
</tr>
</table>
</div>

<font color="maroon">Step #4:</font>  In order to map Bonnor's <math>~\mathbf{F}_1</math> to our perturbation notation, we back up to expressions for the gravitational acceleration, as a whole, which establish that, for spherically symmetric systems,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\hat{\mathbf{e}}_r \cdot \mathbf{F}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{Gm}{r^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\hat{\mathbf{e}}_r \cdot \biggl[ \mathbf{F}_0 + \mathbf{F}_1 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{Gm}{r^2}</math>
  </td>
</tr>
</table>
</div>