Editing SSC/StabilityEulerianPerspective (section)

=Scratch Work=


{{ SGFworkInProgress }}

==Part 1==

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

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  <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">

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  <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">

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  <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>


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<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.

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<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>.


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<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,


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<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>


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<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>
</table>
</div>


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<math>~
\biggl(\frac{dP}{d\rho}\biggr)_0 = \biggl( \frac{\gamma_g P}{\rho}\biggr)_0
</math>
</div>

==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;


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<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>



* Term involving <math>~\Phi^'</math>:


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  <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>


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



<div align="center">
<math>~
- \frac{\rho^'}{\rho_0} \frac{\partial P_0}{\partial r} = \frac{1}{\rho_0} \frac{\partial P_0}{\partial r} \biggl[ \frac{\partial}{\partial r}(\rho_0 \delta 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{\partial }{\partial r}\biggl\{ \gamma_g P_0 \biggl[ \frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \delta r \biggr) \biggr]  \biggr\} 
+ \rho_0 \delta r \biggl[ \sigma^2  - \frac{1}{\rho_0}\frac{\partial P_0}{\partial r} \biggl(\frac{4}{r} \biggr)\biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0 
</math>
  </td>
</tr>
</table>
</div>


<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/1958HDP....51..353L P. Ledoux &amp; Th. Walraven (1958)]
  </th>
<tr>
  <td>
[[File:LedouxWalraven1958Eq57.23.png|600px|center|Ledoux &amp; Walraven (1958)]]
  </td>
</tr>
</table>
</div>


<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 [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf S. Rosseland (1969)]
  </th>
<tr>
  <td>
[[File:RosselandEq2.18.png|600px|center|Rosseland (1969)]]
  </td>
</tr>
</table>
</div>



<div align="center">
<math>~\Downarrow</math>
</div>

==Part 4==



<div align="center">
<p>&nbsp;</p>
<p>&nbsp;</p>
<font size="+10">&#8629;</font>
<p>&nbsp;</p>
<p>&nbsp;</p>
<font size="+50">&#9166;</font>
<p>&nbsp;</p>
<p>&nbsp;</p>

<font size="+2"><math>~\Rightarrow</math></font>
</div>



<div align="center">
<math>\frac{d^2 v}{dt^2} = - \frac{1}{\rho} \frac{d}{dt} \biggl[\nabla_r P\biggr] + \frac{\nabla_r P}{\rho^2} \frac{d\rho}{dt}
- \frac{d}{dt}\biggl[\nabla_r \Phi\biggr]</math>
</div>



<div align="center">
<math>
~g \equiv \nabla_r \Phi = \frac{GM_r}{r^2}
</math>
</div>




<div align="center">
<math>
~g_0 = -\frac{1}{\rho_0} \nabla_r P_0~~~~~~~~\frac{d}{dt}\biggl[\nabla_r \Phi \biggr]
</math>
</div>



<div align="center">
<math>
~\frac{\nabla_r P}{\rho} \biggl[ \frac{1}{\rho} \frac{d\rho}{dt}\biggr] = -\frac{\nabla_r P}{\rho} \biggl[\nabla\cdot\vec{v}\biggr] 
= -\frac{\nabla_r P}{\rho} \biggl[\frac{\partial v_r}{\partial r} + \frac{2v_r}{r}\biggr]
= \biggl[ \frac{dv_r}{dt} + g \biggr] \biggl[\frac{\partial v_r}{\partial r} + \frac{2v_r}{r}\biggr]
</math>
</div>


For any scalar variable, <math>~q(\vec{r},t)</math>, the relationship between a ''Lagrangian'' (total) and ''Eulerian'' (partial) time-derivative is,

<div align="center">
<math>
~\frac{dq}{dt} = \frac{\partial q}{\partial t} + \vec{v}\cdot \nabla q \, .
</math>
</div>



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

<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl[ \nabla_r P \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial t}\biggl[ \nabla_r P \biggr] + v_r \frac{\partial }{\partial r} \biggl[ \nabla_r P \biggr]
= \frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) + \frac{\partial }{\partial r} \biggl[ v_r (\nabla_r P )\biggr] -  \nabla_r P \biggl( \frac{\partial v_r}{\partial r} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) 
+ \frac{\partial }{\partial r} \biggl[ v_r (\nabla_r P )\biggr] 
-  \nabla_r P \biggl( \frac{\partial v_r}{\partial r} \biggr) </math>
  </td>
</tr>
</table>
</div>



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

<tr>
  <td align="right">
<math>~\frac{d}{dt}\biggl[ \nabla_r P \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial }{\partial r}\biggl( \frac{\partial P}{\partial t} \biggr) 
+ \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  - \frac{\partial P}{\partial t} \biggr] 
- \rho \biggl[  \frac{\nabla_r P}{\rho} \biggr]\frac{\partial v_r}{\partial r} 
= \frac{\partial }{\partial r} \biggl[ \frac{dP}{dt}  \biggr] 
+ \rho \biggl[  \frac{dv_r}{dt} + g \biggr]\frac{\partial v_r}{\partial r}
</math>
  </td>
</tr>
</table>
</div>

<div align="center"><math>~dP/dt</math></div>

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