Editing Apps/PapaloizouPringle84 (section)

=====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>