Editing Apps/PapaloizouPringle84 (section)

==Rewritten Velocity Components==

===PP84===

Again following the lead of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84], we let <math>~W^'</math> represent the (normalized) perturbation in the fluid entropy, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W^' </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P^'}{\rho_0(\sigma + m{\dot\varphi}_0)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\frac{\partial}{\partial\varpi}\biggl(\frac{P^'}{\rho_0} \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial}{\partial\varpi} \biggl[ W^'(\sigma + m{\dot\varphi}_0 )\biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\sigma + m{\dot\varphi}_0 )\frac{\partial W^'}{\partial\varpi}  + mW^'\frac{\partial  {\dot\varphi}_0 }{\partial\varpi} </math>
  </td>
</tr>
</table>
</div>
in which case the three linearized components of the Euler equation may be rewritten as,

<div align="center">
<table border="0" cellpadding="8" align="center">

<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' <math>\varpi</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~{\dot\varpi}^'
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
i \biggl[ \frac{\partial W^'}{\partial\varpi}  + \frac{mW^'}{(\sigma + m{\dot\varphi}_0)}\frac{\partial  {\dot\varphi}_0 }{\partial\varpi}  
- \frac{2{\dot\varphi}_0 (\varpi {\dot\varphi}^' )}{(\sigma + m{\dot\varphi}_0)}   \biggr]
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' <math>\varphi</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^')  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ mW^'}{\varpi}  + i~ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]  \, ;
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">''' ''Linearized'' <math>~z</math> Component of Euler Equation'''</font></td></tr>
<tr>
  <td align="right">
<math>~
~{\dot{z}}^'  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
i~\frac{\partial W^'}{\partial z} \, . 
</math>
  </td>
</tr>
</table>

</div>

Using the second of these three relations to provide an expression for <math>~(\varpi {\dot\varphi}^')</math> in terms of <math>~W^'</math> and <math>~{\dot\varpi}^'</math>, and plugging this expression into the first relation allows us to solve for the radial component of the perturbed velocity in terms of <math>~W^'</math> and its radial derivative.  Specifically, we obtain,

<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>~{\dot\varpi}^' 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \frac{\partial W^'}{\partial \varpi} + i~\frac{mW^'}{(\sigma + m{\dot\varphi}_0)} \biggl[ \frac{\kappa^2}{2\varpi {\dot\varphi}_0} - \frac{2 {\dot\varphi}_0 }{\varpi}\biggr]
- i~ \frac{2 {\dot\varphi}_0 }{(\sigma + m{\dot\varphi}_0)} 
\biggl[ -  \frac{ mW^'}{\varpi}  + i~ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl( \frac{ \kappa^2 \varpi }{ 2{\dot\varphi}_0 } \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \frac{\partial W^'}{\partial \varpi} 
+ i~\frac{mW^'}{(\sigma + m{\dot\varphi}_0)} \biggl[ \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr]  
+ \biggl[  \frac{2 {\dot\varphi}_0 }{(\sigma + m{\dot\varphi}_0)} \biggr]\biggl[ \frac{{\dot\varpi}^'}{\varpi(\sigma + m{\dot\varphi}_0)}\biggl( \frac{ \kappa^2 \varpi }{ 2{\dot\varphi}_0 } \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \biggl[ \frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0} \biggr) \frac{ mW^'}{\bar\sigma}  \biggr]
+ \biggl[ {\dot\varpi}^'\biggl( \frac{ \kappa^2 }{  {\bar\sigma}^2 } \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ {\dot\varpi}^' ({\bar\sigma}^2 - \kappa^2 )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) mW^' \bar\sigma   \biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>

where, adopting notation from PP84,

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

<tr>
  <td align="right">
<math>~\kappa^2 \equiv \frac{2{\dot\varphi}_0}{\varpi} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~{\bar\sigma} \equiv (\sigma + m{\dot\varphi}_0) \, .</math>
  </td>
</tr>
</table>
</div>

This means, as well, that,

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

<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^')  ({\bar\sigma}^2 - \kappa^2 )
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ mW^'}{\varpi} ({\bar\sigma}^2 - \kappa^2 ) 
- \frac{ 1 }{\varpi \bar\sigma }\biggl[ \frac{\kappa^2 \varpi }{ 2{\dot\varphi}_0 }  \biggr]  
\biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{2 {\dot\varphi}_0}{\varpi} + \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) mW^' \bar\sigma   \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  +  \frac{ m\kappa^2W^'}{\varpi} 
- \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{2 {\dot\varphi}_0}{\varpi} + \frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>


In summary, the three components of the perturbed velocity are:


<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">
Cylindrical-Coordinate Components of the Perturbed Velocity from PP84
  </th>
</tr>
<tr><td>

<table border="0" cellpadding="8" align="center">

<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varpi</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~ {\dot\varpi}^' ({\bar\sigma}^2 - \kappa^2 )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \biggl[ {\bar\sigma}^2~\frac{\partial W^'}{\partial \varpi} 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) mW^' \bar\sigma   \biggr] \, ,
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varphi</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^')  ({\bar\sigma}^2 - \kappa^2 )
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  \frac{ m{\bar\sigma}^2 W^'}{\varpi}  - \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }   \biggl[ ~\frac{\partial W^'}{\partial \varpi} 
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{mW^' }{\bar\sigma }  \biggr] \, .
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~
~{\dot{z}}^'  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
i~\frac{\partial W^'}{\partial z} \, . 
</math>
  </td>
</tr>
</table>

where, the square of the epicyclic frequency,

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

<tr>
  <td align="right">
<math>~\kappa^2 \equiv \frac{2{\dot\varphi}_0}{\varpi} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~{\bar\sigma} \equiv (\sigma + m{\dot\varphi}_0) </math>
  </td>
</tr>
</table>

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


These three velocity-component expressions  match, respectively, equations (3.14), (3.15), and (3.16) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84].

===GGN86===
In &sect;2.2 of their paper, [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G P. Goldreich, J. Goodman, and R. Narayan (1986, MNRAS, 221, 339)] &#8212; hereafter, GGN86 &#8212; also present expressions for the three components of the perturbed velocity.  Here we seek to identify key differences in approach but, ultimately, highlight the degree of agreement between the GGN86 and the [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] analyses.

====Preamble====
First, let's make the substitution,
<div align="center">
<math>Q_{JT} \equiv (\sigma + m{\dot\varphi}_0) W^' \, ,</math>
</div>
in which case,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial W^'}{\partial\varpi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\sigma + m{\dot\varphi}_0)^{-1} \frac{\partial  Q_{JT} }{\partial \varpi}
- Q_{JT} (\sigma + m{\dot\varphi}_0)^{-2} m\frac{\partial {\dot\varphi}_0}{\partial \varpi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\sigma + m{\dot\varphi}_0)^{-2} \biggl[ (\sigma + m{\dot\varphi}_0)\frac{\partial  Q_{JT} }{\partial \varpi}
- m Q_{JT} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Then we can rewrite the radial component of the perturbed velocity as,

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

<tr>
  <td align="right">
<math>~i~ {\dot\varpi}^' ( \kappa^2 - {\bar\sigma}^2  )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ (\sigma + m{\dot\varphi}_0)\frac{\partial  Q_{JT} }{\partial \varpi}
- m Q_{JT} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr] 
+\biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) m Q_{JT}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\sigma + m{\dot\varphi}_0)\frac{\partial  Q_{JT} }{\partial \varpi}
+m Q_{JT} \biggl[ \biggl( \frac{\kappa^2}{2\varpi {\dot\varphi}_0}  \biggr) 
-\biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr)  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\sigma + m{\dot\varphi}_0)\frac{\partial  Q_{JT} }{\partial \varpi}
+m Q_{JT} \biggl\{ \frac{1}{\varpi^2} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]
-\biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr)  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\sigma + m{\dot\varphi}_0)\frac{\partial  Q_{JT} }{\partial \varpi}
+ (2\dot\varphi_0)\frac{m Q_{JT}}{\varpi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~{\dot\varpi}^' ( \kappa^2 - {\bar\sigma}^2  )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- i~\bar\sigma \frac{\partial  Q_{JT} }{\partial \varpi}
- (2i\dot\varphi_0)\frac{m Q_{JT}}{\varpi}  \, .
</math>
  </td>
</tr>
</table>
</div>

Similarly, we can rewrite the azimuthal component of the perturbed velocity as,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^')  ( \kappa^2 - {\bar\sigma}^2)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{ m{\bar\sigma}^2 }{\varpi}(\sigma + m{\dot\varphi}_0)^{-1}Q_{JT}   + \frac{\kappa^2 {\bar\sigma} }{ 2{\dot\varphi}_0 }  
 \biggl\{ ~(\sigma + m{\dot\varphi}_0)^{-2} \biggl[ (\sigma + m{\dot\varphi}_0)\frac{\partial  Q_{JT} }{\partial \varpi}
- m Q_{JT} \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \biggr]
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{m}{\bar\sigma } (\sigma + m{\dot\varphi}_0)^{-1}Q_{JT} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{ m{\bar\sigma} }{\varpi} Q_{JT}   + \frac{\kappa^2 }{ 2{\dot\varphi}_0 }  
 \biggl\{ ~\biggl[ \frac{\partial  Q_{JT} }{\partial \varpi}
- \biggl( \frac{\partial {\dot\varphi}_0}{\partial \varpi} \biggr) \frac{m}{\bar\sigma} Q_{JT} \biggr]
+\biggl(\frac{\partial {\dot\varphi}_0}{\partial\varpi} \biggr) \frac{m}{\bar\sigma } Q_{JT} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl( \frac{\kappa^2 }{ 2{\dot\varphi}_0 }  \biggr)
\frac{\partial  Q_{JT} }{\partial \varpi}  +  \bar\sigma \frac{mQ_{JT} }{\varpi} \, .
</math>
  </td>
</tr>
</table>
</div>

Finally, the vertical component of the perturbed velocity becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
~{\dot{z}}^'  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
i~(\sigma + m{\dot\varphi}_0)^{-1} \frac{\partial Q_{JT}}{\partial z} \, . 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~i~\bar\sigma {\dot{z}}^'  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{\partial Q_{JT}}{\partial z} \, . 
</math>
  </td>
</tr>
</table>
</div>

====Nod to Oort Constants and Simple Rotation Profiles====
Acknowledging the galactic dynamics community's familiarity with the so-called [https://en.wikipedia.org/wiki/Oort_constants ''Oort constants''], and in anticipation of our [[#Velocity_Components|review of the GGN86 derivation]] that follows, we define the following two "Oort functions":
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{1}{2}\biggl[ {\dot\varphi}_0 - \frac{\partial}{\partial \varpi}\biggl( \varpi {\dot\varphi}_0 \biggr) \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{2}\biggl[ {\dot\varphi}_0 + \frac{\partial}{\partial \varpi}\biggl( \varpi {\dot\varphi}_0 \biggr) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
 
Given these definitions, we note that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~B - A </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~{\dot\varphi}_0 \, ;</math>
  </td>
</tr>
</table>
</div>
and, given the definition of the square of the epicyclic frequency, above, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\kappa^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4{\dot\varphi}_0 B = 4B (B - A) \, .</math>
  </td>
</tr>
</table>
</div>

In line with [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|our own discussion of ''simple rotation profiles'']] (but note that, in [[AxisymmetricConfigurations/SolutionStrategies#SRPtable|that chapter]], the variable we use for the power-law exponent is different from theirs), GGN86 adopt a generalized power-law rotation profile of the form (see their equation 2.1),
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\dot\varphi}_0(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \Omega_0 \biggl( \frac{\varpi}{\varpi_0} \biggr)^{-q} \, ,</math>
  </td>
</tr>
</table>
</div>
in which case we also have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial}{\partial \varpi} \biggl( \varpi {\dot\varphi}_0 \biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{\partial}{\partial \varpi} \biggl[ \Omega_0 \varpi_0^{q} \varpi^{1-q}\biggr] = (1-q){\dot\varphi}_0 \, .</math>
  </td>
</tr>
</table>
</div>

Given this ''particular'' adopted profile, it is therefore clear that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~A_\mathrm{GGN}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{1}{2}\biggl[ {\dot\varphi}_0 - (1-q) {\dot\varphi}_0\biggr] = - \frac{q}{2} {\dot\varphi}_0  \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B_\mathrm{GGN}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \frac{1}{2}\biggl[ {\dot\varphi}_0 + (1-q) {\dot\varphi}_0\biggr] = \frac{1}{2} (2-q) {\dot\varphi}_0  \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\kappa^2_\mathrm{GGN}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4{\dot\varphi}_0 \biggl[ \frac{1}{2} (2-q) {\dot\varphi}_0 \biggr] = 2(2-q){\dot\varphi}_0^2  \, .</math>
  </td>
</tr>
</table>
</div>
These three expressions are in line with GGN86 equations (2.4), (2.6), and (2.24), respectively.

<!--
{{LSU_WorkInProgress}}
-->

====Velocity Components====

=====Direct Comparison of Derived Equations=====
The left panel of the following equation-table presents, once again, [[#Preamble|our above rewrite]] of the three components of the perturbed velocity derived in [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] after we have replaced <math>~\kappa^2</math> (on the right-hand side of the <math>~\varphi</math> component) with its expression in terms of the "Oort function", <math>~B</math>.  For comparison, the right panel of the same equation-table shows the analogous perturbed velocity-component expressions derived by [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G GGN86] (see their equations 2.21 - 2.25).

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">
Rewrite of the Components of the Perturbed Velocity from PP84
  </th>
  <th align="center">
Perturbed Velocity Components from &sect;2.2 of [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G GGN86]
  </th>
</tr>
<tr><td>

<table border="0" cellpadding="8" align="center">

<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varpi</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~{\dot\varpi}^' ( \kappa^2 - {\bar\sigma}^2  )</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- i~\bar\sigma \frac{\partial  Q_{JT} }{\partial \varpi}
- (2i\dot\varphi_0)\frac{m Q_{JT}}{\varpi}  \, ,
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>\varphi</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^')  ( \kappa^2 - {\bar\sigma}^2)
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2B \frac{\partial  Q_{JT} }{\partial \varpi}  +  \bar\sigma \frac{mQ_{JT} }{\varpi} \, ,
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~\Rightarrow~~~~i~\bar\sigma {\dot{z}}^'  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{\partial Q_{JT}}{\partial z} \, . 
</math>
  </td>
</tr>
</table>

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

<table border="0" cellpadding="8" align="center">

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~x \equiv (\varpi - \varpi_0)</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~ u ( \kappa^2 - \sigma^2_\mathrm{GGN})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \biggl[ \sigma_\mathrm{GGN}~\frac{\partial Q}{\partial x} 
- 2\Omega_0 k Q \biggr] \, ,
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~y \equiv (\varpi_0 \varphi)</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~ v ( \kappa^2 - \sigma^2_\mathrm{GGN})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2B~\frac{\partial Q}{\partial x}  - \sigma_\mathrm{GGN} k Q  \, ,
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~
~-i~\sigma_\mathrm{GGN}w
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{\partial Q}{\partial z} \, . 
</math>
  </td>
</tr>
</table>

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

These sets of expressions are identical if we adopt the following three variable mappings,

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

<tr>
  <td align="right">
<math>~Q_{JT}\equiv\bar\sigma W^'</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~Q \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~-~\bar\sigma </math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~\sigma_\mathrm{GGN} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~m/\varpi</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~k \, ,</math>
  </td>
</tr>
</table>
</div>

and recognize that the appropriate association between the variable names that has been used for the three perturbed velocity-components is:

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

<tr>
  <td align="right">
<math>~{\dot\varpi}^'</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~u \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~(\varpi {\dot\varphi}^') </math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~v \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~{\dot{z}}^'</math>
  </td>
  <td align="center">
<math>~\leftrightarrow</math>
  </td>
  <td align="left">
<math>~w \, .</math>
  </td>
</tr>
</table>
</div>

=====Checking Self-Consistency=====


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

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- \frac{1}{2}\biggl[ {\dot\varphi}_0 - \frac{\partial}{\partial \varpi}\biggl( \varpi {\dot\varphi}_0 \biggr) \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~ -2A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \varpi \frac{\partial {\dot\varphi}_0}{\partial \varpi}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~ \frac{\partial {\dot\varphi}_0}{\partial \varpi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2A}{\varpi}</math>
  </td>
</tr>

</table>
</div>

Now, expand the function, <math>~{\dot\varphi}_0(\varpi)</math> in a Taylor series &hellip;
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~{\dot\varphi}_0(\varpi) </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\Omega_0 + (\varpi - \varpi_0)\frac{\partial {\dot\varphi}_0}{\partial\varpi}\biggr|_{\varpi_0}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\Omega_0 + (\varpi - \varpi_0)\frac{2A}{\varpi_0}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\bar\sigma \equiv (\sigma + m{\dot\varphi}_0)</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\sigma + \biggl[m\Omega_0 + \frac{2mA}{\varpi_0}(\varpi - \varpi_0)\biggr]</math>
  </td>
</tr>
</table>
</div>

Now, from equations (2.18) and (2.15) of GGN86, along with their definition of the independent variable, <math>~x</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \sigma_\mathrm{GGN}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \omega_\mathrm{GGN} + 2Akx</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \omega_\mathrm{GGN} + \frac{2mA}{\varpi_0} (\varpi-\varpi_0) \, .</math>
  </td>
</tr>
</table>
</div>
Hence, we can understand the desired mapping, <math>\bar\sigma \leftrightarrow - \sigma_\mathrm{GGN}</math>, if we acknowledge the more fundamental mapping,
<div align="center">
<math>~\omega_\mathrm{GGN} ~~ \leftrightarrow ~~ - (\sigma+m\Omega_0) \, .</math>
</div>

Adopting [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] <math>~y_1</math> and <math>~y_2</math> notation, which we have discussed in a [[Appendix/Ramblings/AzimuthalDistortions#Adopted_Notation|separate but closely related chapter]], we therefore have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~y_1 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\mathrm{Re}(\omega_\mathrm{GGN})}{\Omega_0} - m </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{1}{\Omega_0} \biggr) \mathrm{Re}\biggl[ - (\sigma+m\Omega_0) \biggr] - m </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~y_2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\mathrm{Im}(\omega_\mathrm{GGN})}{\Omega_0} \, ,</math>
  </td>
</tr>
</table>
</div>



<!--
********************


Let's see if the deduced variable mappings make sense in the context of, for example, the [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G GGN86] derivation.  If we make the specified substitution for <math>~k</math>, and insert the (above derived) power-law expression for the "Oort function", <math>~A</math>, into the GGN86 definition of <math>~\sigma_\mathrm{GGN}</math> &#8212; see their equation (2.18) &#8212; we find,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma_\mathrm{GGN}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\omega - 2Akx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\omega - 2Akx \, .
</math>
  </td>
</tr>
</table>
</div>

Following the prescribed variable mapping, this should be compared with the (negative of the) PP84 definition of <math>~\bar\sigma</math>, namely,

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

<tr>
  <td align="right">
<math>~- \bar\sigma</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-\sigma - m{\dot\varphi}_0 \, .</math>
  </td>
</tr>
</table>
</div>

<table border="1" cellpadding="8" align="center">
<tr>
  <th align="center">
Perturbed Velocity Components from &sect;2.2 of [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G GGN86]
  </th>
</tr>
<tr><td>

<table border="0" cellpadding="8" align="center">

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~x</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~ u ( \kappa^2 - \sigma^2_\mathrm{GGN})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~i \biggl[ \sigma_\mathrm{GGN}~\frac{\partial Q}{\partial x} 
- 2\Omega_0 k Q \biggr] \, ,
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~y</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~ v ( \kappa^2 - \sigma^2_\mathrm{GGN})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2B~\frac{\partial Q}{\partial x}  - \sigma_\mathrm{GGN} k Q  \, ,
</math>
  </td>
</tr>

<tr><td align="center" colspan="3"><font color="#770000">'''<math>~z</math> Component'''</font></td></tr>
<tr>
  <td align="right">
<math>~
~w
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- i \biggl(\frac{1}{\sigma_\mathrm{GGN}}\biggr) \frac{\partial Q}{\partial z} \, . 
</math>
  </td>
</tr>
</table>

where, 

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

<tr>
  <td align="right">
<math>~\kappa^2 \equiv \frac{2{\dot\varphi}_0}{\varpi} \biggl[ \frac{\partial (\varpi^2\dot\varphi_0)}{\partial\varpi} \biggr]</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~{\bar\sigma} \equiv (\sigma + m{\dot\varphi}_0) </math>
  </td>
</tr>
</table>

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

Likely transformations:
<ul>
  <li><math>~\sigma_\mathrm{GGN} \equiv \biggl( \omega - \frac{2Amx}{\varpi_0} \biggr) ~~~ \leftrightarrow ~~~ \bar\sigma \equiv (\sigma_\mathrm{Blaes} + m{\dot\varphi}_0 )</math>  </li>
  <li><math>\frac{Q}{\sigma_\mathrm{GGN}} ~~~ \leftrightarrow ~~~ - W^'</math>  </li>
</ul>

****************
-->