Editing Appendix/Ramblings/AzimuthalDistortions (section)

===Specific Application to HI11's Figure 16 ===

Next, let's see how well we can match the eigenmode structures presented in Figure 16 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] by varying only a single parameter in our expression for <math>~f_\ln(\varpi)</math>.  Given the radial locations of the inner and outer edges of the torus (normalized to the location of the density maximum; see the last two columns of our Table 1), <math>~r_- = 0.611</math> and <math>~r_+ = 1.490</math>, we set up an Excel spreadsheet with 199 radial zones spanning this range of radii.  

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9"><font size="+1"><b>Table 1:</b></font>&nbsp; [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] Model Parameters</td>
</tr>
<tr>
  <td align="center" colspan="7">Extracted from Table 2 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]</td>
  <td align="center" colspan="2">Deduced Here</td>
</tr>
<tr>
  <td align="center"><math>~T/|W|</math></td>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~q</math></td>
  <td align="center"><math>~R_\mathrm{max}</math></td>
  <td align="center"><math>~R_+</math></td>
  <td align="center"><math>~R_-/R_+</math></td>
  <td align="center"><math>~\epsilon \equiv \frac{R_\mathrm{max}-R_-}{R_\mathrm{max}}</math></td>
  <td align="center"><math>~r_- \equiv \frac{R_-}{R_\mathrm{max}}</math></td>
  <td align="center"><math>~r_+ \equiv \frac{R_+}{R_\mathrm{max}}</math></td>
</tr>
<tr>
  <td align="center"><math>~0.2729</math></td>
  <td align="center"><math>~\tfrac{3}{2}</math></td>
  <td align="center"><math>~\tfrac{3}{2}</math></td>
  <td align="center"><math>~6.744</math></td>
  <td align="center"><math>~10.051</math></td>
  <td align="center"><math>~0.410</math></td>
  <td align="center"><math>~0.388</math></td>
  <td align="center"><math>~0.611</math></td>
  <td align="center"><math>~1.490</math></td>
</tr>
</table>


Specifically, in Excel we set, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta r</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{r_+ - r_-}{200} = 0.004395 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\$A\$N</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r_- + N \delta r</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~p</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1.2 </math>
  </td>
</tr>
</table>
</div>

Table 2 tabulates the value of <math>~f_\mathrm{blue}</math> over a range of radii for the specific case when we set <math>~r_\mathrm{blue} = 1.107635</math>. 

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="4">
<font size="+1"><b>Table 2:</b></font><p></p>
Left-hand (blue) segment of eigenfunction
  </td>
</tr>
<tr>
  <td align="center" colspan="4">
<math>~r_\mathrm{blue} = \$A\$113 = 1.107635</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">Example Excel Grid Zone</td>
  <td align="center" colspan="2">Function Evaluated</td>
</tr>
<tr>
  <td align="center"><math>~N</math></td>
  <td align="center"><math>~\varpi</math></td>
  <td align="center"><math>~f_\mathrm{blue}</math></td>
  <td align="center"><math>~\ln(f_\mathrm{blue})</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="right">0.615395</td>
  <td align="right">287.78</td>
  <td align="right">5.6622</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="right">0.61979</td>
  <td align="right">123.92</td>
  <td align="right">4.8197</td>
</tr>
<tr>
  <td align="center">32</td>
  <td align="right">0.75164</td>
  <td align="right">3.04791</td>
  <td align="right">1.1145</td>
</tr>
<tr>
  <td align="center">50</td>
  <td align="right">0.83075</td>
  <td align="right">1.3195</td>
  <td align="right">0.27733</td>
</tr>
<tr>
  <td align="center">62</td>
  <td align="right">0.88349</td>
  <td align="right">0.79107</td>
  <td align="right">-0.23437</td>
</tr>
<tr>
  <td align="center">82</td>
  <td align="right">0.97139</td>
  <td align="right">0.31121</td>
  <td align="right">-1.1673</td>
</tr>
<tr>
  <td align="center">92</td>
  <td align="right">1.01534</td>
  <td align="right">0.16987</td>
  <td align="right">-1.7727</td>
</tr>
<tr>
  <td align="center">102</td>
  <td align="right">1.05929</td>
  <td align="right">0.06908</td>
  <td align="right">-2.6725</td>
</tr>
<tr>
  <td align="center">112</td>
  <td align="right">1.10324</td>
  <td align="right">0.003475</td>
  <td align="right">-5.66220</td>
</tr>
</table>
</div>


Then, for two separate choices of <math>~r_\mathrm{green}</math> &#8212; specifically, 1.05929 and 0.61979 &#8212; Table 3 lists values of <math>~f_\mathrm{green}</math> at various radial positions across the HI11 torus.  Given values of <math>~f_\mathrm{blue}</math> and <math>~f_\mathrm{green}</math> at each radial position, Table 3 also lists corresponding values of <math>~f_\ln</math>, <math>~D_{1/2}</math>, and <math>~\phi_1</math>.  Notice that the values of <math>~[f_\ln]_\mathrm{min}</math> (highlighted in yellow) and <math>~[f_\ln]_\mathrm{max}</math> (highlighted in orange) that are used in the calculation of <math>~D_{1/2}</math> depend on the choice of <math>~r_\mathrm{green}</math>.


<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="9">
<font size="+1"><b>Table 3:</b></font><p></p>
Right-hand (green) segment, and total eigenfunction
  </td>
</tr>
<tr>
  <td align="center" colspan="1" rowspan="2"><math>~r_\mathrm{green}</math></td>
  <td align="center" colspan="2">Example Excel Grid Zone</td>
  <td align="center" colspan="6">Functions Evaluated<sup>&dagger;</sup></td>
<tr>
  <td align="center"><math>~N</math></td>
  <td align="center"><math>~\varpi</math></td>
  <td align="center"><math>~f_\mathrm{green}</math></td>
  <td align="center"><math>~f_\mathrm{blue}</math></td>
  <td align="center" colspan="2"><math>~f_\ln \equiv \ln(f_\mathrm{green}+f_\mathrm{blue})</math></td>
  <td align="center"><math>~D_{1/2}</math></td>
  <td align="center"><math>~(\phi_1/\pi)</math></td>
</tr>
<tr>
  <td align="center" rowspan="6">
    <math>~\$A\$102 = 1.05929</math><p>&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
[[File:M02cropped.png|200px|2nd frame of movie]]
  </td>
  <td align="center">199</td>
  <td align="right">1.485605</td>
  <td align="right">242.17</td>
  <td align="right">0.0</td>
  <td align="center">+5.490</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.9897</td>
  <td align="center">-0.4212</td>
</tr>
<tr>
  <td align="center">151</td>
  <td align="right">1.274645</td>
  <td align="right">1.00000</td>
  <td align="right">0.0</td>
  <td align="center">0.000</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.5750</td>
  <td align="center">-0.3695</td>
</tr>
<tr>
  <td align="center">112</td>
  <td align="right">1.10324</td>
  <td align="right">0.073556</td>
  <td align="right">0.003475</td>
  <td align="center">-2.564</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.1662</td>
  <td align="center">-0.1868</td>
</tr>
<tr>
  <td align="center">106</td>
  <td align="right"><font color="red"><b>1.07687</b></font></td>
  <td align="right">0.022632</td>
  <td align="right">0.038349</td>
  <td align="center" bgcolor="yellow">-2.797</td>
  <td align="center"><math>~\leftarrow~~[f_\ln]_\mathrm{min}</math></td>
  <td align="center">0.0000</td>
  <td align="center">0.0000</td>
</tr>
<tr>
  <td align="center">103</td>
  <td align="right">1.063685</td>
  <td align="right">0.004129</td>
  <td align="right">0.060897</td>
  <td align="center">-2.733</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.0871</td>
  <td align="center">+0.1068</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="right">0.615395</td>
  <td align="right">0.0</td>
  <td align="right">287.78</td>
  <td align="center" bgcolor="orange">+5.662</td>
  <td align="center"><math>~\leftarrow~~[f_\ln]_\mathrm{max}</math></td>
  <td align="center">1.0000</td>
  <td align="center">+0.4220</td>
</tr>
<tr>
  <td align="center" colspan="9" bgcolor="purple">&nbsp;</td>
</tr>
<tr>
  <td align="center" rowspan="7">
    <math>~\$A\$2 = 0.61979</math><p>&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
[[File:M07cropped.png|200px|2nd frame of movie]]
  </td>
  <td align="center">199</td>
  <td align="right">1.485605</td>
  <td align="right">566.71</td>
  <td align="right">0.0</td>
  <td align="center" bgcolor="orange">+6.340</td>
  <td align="center"><math>~\leftarrow~~[f_\ln]_\mathrm{max}</math></td>
  <td align="center">1.0000</td>
  <td align="center">-0.4220</td>
</tr>
<tr>
  <td align="center">151</td>
  <td align="right">1.274645</td>
  <td align="right">3.79829</td>
  <td align="right">0.0</td>
  <td align="center">+1.335</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.4673</td>
  <td align="center">-0.3436</td>
</tr>
<tr>
  <td align="center">112</td>
  <td align="right">1.10324</td>
  <td align="right">1.30705</td>
  <td align="right">0.003475</td>
  <td align="center">+0.270</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.2285</td>
  <td align="center">-0.2357</td>
</tr>
<tr>
  <td align="center">106</td>
  <td align="right">1.07687</td>
  <td align="right">1.12898</td>
  <td align="right">0.038349</td>
  <td align="center">+0.155</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.1848</td>
  <td align="center">-0.2026</td>
</tr>
<tr>
  <td align="center">103</td>
  <td align="right">1.063685</td>
  <td align="right">1.04969</td>
  <td align="right">0.060897</td>
  <td align="center">+0.105</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.1623</td>
  <td align="center">-0.1833</td>
</tr>
<tr>
  <td align="center">83</td>
  <td align="right"><font color="red"><b>0.975785</b></font></td>
  <td align="right">0.64322</td>
  <td align="right">0.29488</td>
  <td align="center" bgcolor="yellow">-0.064</td>
  <td align="center"><math>~\leftarrow~~[f_\ln]_\mathrm{min}</math></td>
  <td align="center">0.0000</td>
  <td align="center">0.0000</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="right">0.615395</td>
  <td align="right">0.0</td>
  <td align="right">287.78</td>
  <td align="center">+5.662</td>
  <td align="center">&nbsp;</td>
  <td align="center">0.9456</td>
  <td align="center">+0.4177</td>
</tr>
<tr>
  <td align="left" colspan="9">
<sup>&dagger;</sup>Throughout this table, the phase angle, <math>~\phi_1(\varpi)</math>, has been calculated assuming that <math>~\aleph = 4</math>, and it has been assigned a positive (negative) value if <math>~\varpi</math> is inside (outside) the radial location (identified in red) of the minimum function value, <math>~[f_\ln]_\mathrm{min}</math>.
  </td>
</tr>

</table>
</div>


The miniaturized image displayed near the top of the left-most column of Table 3 contains a pair of plots associated with our choice of the parameter, <math>~r_\mathrm{green} = 1.05929</math>:  On the left is a semi-log plot of the eigenfunction, <math>~f_\ln</math> versus <math>~\varpi</math> &#8212; actually, <math>~f_\mathrm{log10}</math> versus <math>~\varpi</math> &#8212; and on the right is a plot in polar coordinates of <math>~\phi_1</math> versus <math>~\varpi</math>, that is, a "constant phase locus" plot.  In an analogous manner, the miniaturized image displayed near the bottom of the left-most column of Table 3 contains a semi-log plot of the radial eigenfunction and a polar plot of the "constant phase locus" associated with our choice of the parameter, <math>~r_\mathrm{green} = 0.61979</math>.  The pair of plots found in each of these miniaturized images also appear within a single frame of the animation that is displayed in Figure 8, below.  Each frame of the animation is stamped with the numerical value corresponding to the radial coordinate, <math>~r_\mathrm{min}</math>, at which the eigenfunction has its minimum (see the red numbers in Table 3).  In the case of <math>~r_\mathrm{green} = 1.05929</math>, we find <math>~r_\mathrm{min} = 1.077</math>; and in the case of <math>~r_\mathrm{green} = 0.61979</math>, we find <math>~r_\mathrm{min} = 0.976</math>.


<div align="center">
<table border="1">
<tr>
  <td align="center" colspan="3">
<b><font size="+1">Figure 8:</font></b> &nbsp;
Radial and Azimuthal Eigenfunction Comparison
  </td>
</tr>
<tr>
  <td align="center" colspan="2">'''(a)''' &nbsp; Our Empirically Constructed Function</td>
  <td align="center">'''(b)''' &nbsp; Extracted from Figure 16 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]</td>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center">Specified<p></p><math>~r_\mathrm{green}</math><p></p><hr></td>
  <td align="center">Resulting<p></p><math>~r_\mathrm{min}</math><p></p><hr></td>
</tr>
<tr>
  <td align="center">1.10324</td>
  <td align="center">1.103</td>
</tr>
<tr>
  <td align="center">1.05929</td>
  <td align="center">1.077</td>
</tr>
<tr>
  <td align="center">1.01534</td>
  <td align="center">1.059</td>
</tr>
<tr>
  <td align="center">0.97139</td>
  <td align="center">1.042</td>
</tr>
<tr>
  <td align="center">0.88349</td>
  <td align="center">1.020</td>
</tr>
<tr>
  <td align="center">0.75164</td>
  <td align="center">0.998</td>
</tr>
<tr>
  <td align="center">0.61979</td>
  <td align="center">0.976</td>
</tr>
</table>

</td>
<td align="center">
[[File:HI11Fig16Animate.gif|350px|Figure 16 from HI11]]
</td>
<td align="center">
[[File:HI11_Fig16ThreeQuarters.png|400px|Figure 16 from HI11]]
</td>
</tr>
</table>
</div>

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