Editing Appendix/Ramblings/AzimuthalDistortions (section)

==Discussion==

===Simpler Connection Between Radial and Phase Eigenfunctions===

When it is used as an argument of ATAN, the function, <math>~D(\varpi)</math>, [[#Constant_Phase_Loci|as defined above]] smoothly steers <math>~\phi_1</math> through a phase shift of approximately <math>~\pi</math> radians principally because the function itself smoothly varies between <math>~+1</math> and <math>~0</math> (then back again).  Let's play with some simpler expressions for this governing function that also vary smoothly between these limits.  

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

<tr>
  <td align="right">
<math>~\Delta_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~+0 ~~\rightarrow ~~ +1 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Delta_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~+1 ~~\rightarrow ~~ +0 \, .</math>
  </td>
</tr>
</table>
</div>

The argument of ATANH, [[#Radial_Eigenfunction|as presented above]], was obtained from this relatively simple expression via the shift,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1-2\Delta_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~+1 ~~\rightarrow ~~ -1 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~1-2\Delta_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~-1 ~~\rightarrow ~~ +1 \, .</math>
  </td>
</tr>
</table>
</div>

For the argument of <math>~D(\varpi)</math> function, we could therefore use,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1-\Delta_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_-}{r_\mathrm{mid}-r_-} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~+1 ~~\rightarrow ~~ 0 \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~1-\Delta_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi - r_+}{r_\mathrm{mid}-r_+} \biggr)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; has range of &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~0 ~~\rightarrow ~~ +1 \, .</math>
  </td>
</tr>
</table>
</div>


===Playing with the Radial Eigenfunction===
Up to this point, we've only considered radial eigenfunctions composed of two components (a "blue" inner component and a "green" outer component) that do not overlap.  Here we'll allow the two components to overlap by assigning different values of <math>~r_\mathrm{mid}</math> to the two separate components &#8212; more specifically, we'll allow <math>~r_\mathrm{mid}|_\mathrm{green} \le r_\mathrm{mid}|_\mathrm{blue}</math> &#8212; then add the two functions over the region of overlap.  Let's consider components of the following form:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~f_\mathrm{blue}(\varpi) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{blue} - \varpi}{\varpi - r_-} \biggr)^{p}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{blue} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~f_\mathrm{green}(\varpi) </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{green} - \varpi}{\varpi - r_+} \biggr)^{p}
</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{green} < \varpi < r_+ \, ,</math>
  </td>
</tr>
</table>
</div>
where, <math>~p</math> is a exponent yet to be specified.  So, for fixed values of the inner and outer radii of the torus, <math>~r_-</math> and <math>~r_+</math>, this two-component function has three adjustable variables.  They are, <math>~r_\mathrm{blue}</math>, <math>~r_\mathrm{green}</math>, and <math>~p</math>.

====Experimenting====
In Figure 5, <math>~r_\mathrm{blue}</math> and <math>~r_\mathrm{green}</math> are fixed, and <math>~p</math> is varied.

<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr>
  <th align="center"><font size="+1">Figure 5:</font>&nbsp; Variable exponent, <math>~p</math></th>
</tr>
<tr><td align="center">
<math>~r_- = 0.5</math>, <math>~r_+ = 1.5</math>
&nbsp; &nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~r_\mathrm{blue} = 1.25</math>, <math>~r_\mathrm{green}= 1.1</math>
</td></tr>
<tr><td align="center">
[[File:MontageAbrief.png|500px|Playing with radial eigenfunction]]
</td></tr>
<tr><td align="left">
In this example, the exponent, <math>~p</math>, is varied over the range, <math>~0.25 \le p \le 1.2</math>, as indicated by the numerical values shown in the upper-lefthand corner of each panel.
</td></tr>
</table>
</div>
Based on the [[#SquareRoot|above discussion]], I expected that the best match to the eigenfunctions found in [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11] would be <math>~p=0.5</math>, that is, a square-root.  However, as illustrated in Figure 5, this and other fractional exponents less than unity generate noncontinuous derivatives at the overlapping edges of our two-piece function.  Instead, a value of <math>~p = 1.2</math> seems to exhibit a more desired, smooth behavior.


In Figure 6, <math>~r_\mathrm{green}</math> and <math>~p</math> are fixed, and <math>~r_\mathrm{blue}</math> is varied.

<div align="center">
<table border="1" align="center" cellpadding="5" width="80%">
<tr>
  <th align="center"><font size="+1">Figure 6:</font>&nbsp; Variable <math>~r_\mathrm{blue}</math></th>
</tr>
<tr><td align="center">
<math>~r_- = 0.5</math>, <math>~r_+ = 1.5</math>
&nbsp; &nbsp; &nbsp; &nbsp; &hellip; &nbsp; &nbsp; &nbsp; &nbsp; 
<math>~p = 1.2</math>, <math>~r_\mathrm{green}= 0.9</math>
</td></tr>
<tr><td align="center">
[[File:MontageBbrief.png|500px|Playing with radial eigenfunction]]
</td></tr>
<tr><td align="left">
In this example, the "blue" edge is varied over the range, <math>~0.91 \le r_\mathrm{blue} \le 1.25</math>, as indicated by the numerical values shown in the upper-lefthand corner of each panel.
</td></tr>
</table>
</div>

The frames of Figure 6 illustrate the qualitative behavior we have been seeking.  Setting the exponent, <math>~p</math>, to a value greater than unity then varying one of the edges of the two-part eigenfunction provides a natural variation from "pointed" curves that look like adjoined arc-hyperbolic tangents to others that look more like a parabola.


====Trial Comparison with HI11====

In Figure 7, we show how a straightforward, smooth adjustment of one parameter &#8212; namely, <math>~r_\mathrm{blue}</math> &#8212; generates a series of eigenfunctions that nicely match the set of radial eigenfunctions that are displayed in Figure 16 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11].

<div align="center" id="Figure7">
<table border="1" cellpadding="3" align="center" width="60%">
<tr>
  <td align="center" colspan="2">
<b><font size="+1">Figure 7:</font></b> &nbsp;
Radial Eigenfunction Comparison
  </td>
</tr>
<tr>
  <td align="center">'''(a)''' &nbsp; Extracted from Figure 16 of [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H HI11]</td>
  <td align="center">'''(b)''' &nbsp; Our Empirically Constructed Function</td>
</tr>
<tr><td align="center" rowspan="4">
[[File:PaperI_Fig16.png|300px|Figure 16 from HI11]]
</td><td align="center">[[File:PhaseMovieFrameFirst.png|250px|Figure 16 from HI11]]
</td></tr>
<tr><td align="center">[[File:EigenfunctionMovie01.gif|250px|Figure 16 from HI11]]</td></tr>
<tr><td align="center">[[File:PhaseMovieFrameLast.png|250px|Figure 16 from HI11]]</td></tr>
<tr>
  <td align="left" colspan="1">
Here we set <math>~r_- = 0.6</math>, <math>~r_+ = 1.5</math>, <math>~r_\mathrm{blue}= 1.15</math>, and
<math>~p = 1.2</math>, then let the "green" edge vary over the range, <math>~0.605 \le r_\mathrm{green} \le 1.144</math>.  Via an animation, the middle panel illustrates the behavior of our empirically constructed eigenfunction over this entire range of values as indicated by the numerical value shown in the bottom-righthand corner of the panel; the top and bottom panels display the shape of our eigenfunction at the two extreme values of <math>~r_\mathrm{green}</math>.
  </td>
</tr>
</table>
</div>

===Relationship to Fourier Series Amplitude and Phase===
Recalling that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{i\alpha}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cos\alpha + i\sin\alpha \, ,</math>
  </td>
</tr>
</table>
</div>
we appreciate that the real part of the amplitude term &#8212; the term inside the curly braces of the expression found in [[#Figure1|Figure 1, above]] &#8212; can be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{A}_m \equiv \Re\{f_m e^{im\phi}\}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f_m \cos(m\phi_m) \, .</math>
  </td>
</tr>
</table>
</div>

When viewed in the context of a discrete Fourier series, we also recognize that the two functions, <math>~f_m</math> and <math>~\phi_m</math>, can be transformed into the pair of amplitude functions,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f_m ~~~\rightarrow ~~~ (a_m^2 + b_m^2)^{1/2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
and
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~m\phi_m ~~~\rightarrow ~~~ \tan^{-1}\biggl(-\frac{b_m}{a_m}\biggr) \, ;</math>
  </td>
</tr>
</table>
</div>
that is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_m ~=~ f_m\cos(m\phi_m)</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
and
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~b_m ~=~ -f_m\sin(m\phi_m) \, .</math>
  </td>
</tr>
</table>
</div>

Let's see where these expression lead us, given our above-adopted expressions for <math>~f_m</math> and <math>~\phi_m</math>.  Given,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f = e^{f_\ln}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{r_\mathrm{blue} - \varpi}{\varpi - r_-} \biggr)^{p} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m\phi_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan^{-1}[\aleph \cdot D(\varpi)] \, </math>
  </td>
</tr>
</table>
</div>
where,

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

<tr>
  <td align="right">
<math>~D(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{f_\ln(\varpi) - [f_\ln]_\mathrm{min}}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}} \, ,</math>
  </td>
</tr>
</table>
</div>
we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{r_\mathrm{blue} - \varpi}{\varpi - r_-} \biggr)^{p}
\cos\biggl[ \tan^{-1}\biggl( \aleph \cdot D \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{f_\ln} \biggl[ 1+(\aleph \cdot D )^2 \biggr]^{-1/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{f_\ln} \bigg\{ 1+\aleph^2_\mathrm{norm} \biggl( f_\ln(\varpi) - [f_\ln]_\mathrm{min} \biggr)^2 \biggr\}^{-1/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\aleph_\mathrm{norm} \equiv \frac{\aleph}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}} \, .</math>
</div>

===Another Improvement===

I've realized that the phase plot is much smoother &#8212; in particular, the derivatives of both segments appear to match at their point of intersection &#8212; if we use <math>~D^{1/2}</math>, instead of simply <math>~D</math>, as the argument of the arctangent function.  So, we define,
<div align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~D_{1/2}(\varpi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\sqrt{D(\varpi)} = \biggl[ \frac{f_\ln(\varpi) - [f_\ln]_\mathrm{min}}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}} \biggr]^{1/2} \, ,</math>
  </td>
</tr>
</table>

</div>
and henceforth will set,

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~m\phi_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \tan^{-1}[\aleph \cdot D_{1/2}(\varpi)] \, .</math>
  </td>
</tr>
</table>

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

With this new argument for the arctangent function, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{r_\mathrm{blue} - \varpi}{\varpi - r_-} \biggr)^{p}
\cos\biggl[ \tan^{-1}\biggl( \aleph \cdot D_{1/2} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{f_\ln} \biggl[ 1+(\aleph \cdot D_{1/2} )^2 \biggr]^{-1/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{f_\ln} \bigg\{ 1+\aleph^2_\mathrm{norm} \biggl( f_\ln(\varpi) - [f_\ln]_\mathrm{min} \biggr) \biggr\}^{-1/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{f_\ln} \bigg\{ C_0 +\aleph^2_\mathrm{norm} f_\ln(\varpi) \biggr\}^{-1/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~C_0 \equiv 1 - \aleph_\mathrm{norm}^2 [f_\ln]_\mathrm{min} \, ,</math>
</div>
and,
<div align="center">
<math>~\aleph_\mathrm{norm} \equiv \frac{\aleph^2}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}} \, .</math>
</div>

===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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