Editing SSC/StabilityEulerianPerspective (section)

====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 [[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.  
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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)]]
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  <td>
[[File:Jeans1902Eq12.png|400px|center|Jeans (1902)]]
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  <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>
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</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.
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Wave Equation for Self-Gravitating Fluids as Derived and Presented by [http://adsabs.harvard.edu/abs/1928asco.book.....J Jeans (1928)]
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  <td>
[[File:JamesJeans1928Eq314.6.png|400px|center|Jeans (1928)]]
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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>
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=====W. B. Bonnor (1957)=====
Above, in our opening layout of the governing equations and supplemental relations, we pointed out that, 
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<math>~\nabla P_0</math>
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  <td align="center">
<math>~=</math>
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<math>~\biggl( \frac{dP}{d\rho} \biggr)_0 \nabla \rho_0 \, .</math>
  </td>
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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,
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  <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>
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  <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>
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  <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>
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</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.

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  <th align="center" colspan="3">
Bonnor's [http://adsabs.harvard.edu/abs/1957MNRAS.117..104B (1957, MNRAS, 117, 104)] Derivation
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  <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>
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  <td align="center">
[[File:Bonnor1957Eq2.1.png|250px|center|Bonnor's (1957) Equation 2.1]]
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[[File:Bonnor1957Eq2.7.png|250px|center|Bonnor's (1957) Equation 2.7]]
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  <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>
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</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%">
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  <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>
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  <td>
[[File:Bonnor1957Eq2.10.png|400px|center|Bonnor (1957)]]
  </td>
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</div>