Editing Apps/PapaloizouPringle84 (section)

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


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<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;
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  <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,
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<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,
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<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,
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  <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,
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<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,

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<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, 

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<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>

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-->