Editing Appendix/Ramblings/AzimuthalDistortions (section)

==Attempts to Match P-modes and E-modes==
Based especially on the analysis provided in [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H Paper I] and [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II] of the Imamura &amp; Hadley collaboration, the eigenvectors that have drawn our attention thus far can be categorized as J-modes (as discussed, for example, in &sect;3.2.1, Table 1 &amp; Figure 2 of Paper II) or I-modes.  Our empirically derived phase function needs to be altered in order to provide reasonable fits to the constant phase loci associated with P-modes and edge-modes (as discussed, for example in &sect;3.2.2, Table2, Figures 3 &amp; 4 of Paper II). We begin by constructing a table of equilibrium parameter values for the typical P- and E-mode models described in Table 2 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II].

===Setup===
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="10"><font size="+1"><b>Table 4:</b></font>&nbsp; P- and E-mode Model Parameters Highlighted in Paper II
<p></p>[http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H K. Z. Hadley, P. Fernandez, J. N. Imamura, E. Keever, R. Tumblin, &amp; W. Dumas (2014, ''Astrophysics and Space Science'', 353, 191-222)]</td>
</tr>
<tr>
  <td align="center" colspan="6">Extracted from Table 2 or Table 4 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
  <td align="center" colspan="3">Deduced Here</td>
  <td align="center" colspan="1">Extracted from Fig. 3 or Fig. 4 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
</tr>
<tr>
  <td align="center">Name</td>
  <td align="center"><math>~M_*/M_d</math></td>
  <td align="center"><math>~(n, q)</math><sup>&dagger;</sup></td>
  <td align="center"><math>~R_-/R_+</math></td>
  <td align="center"><math>~r_\mathrm{outer} \equiv \frac{R_+}{R_\mathrm{max}}</math></td>
  <td align="center"><math>~R_\mathrm{max}</math></td>
  <td align="center"><math>~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">Eigenfunction</td>
</tr>
<tr>
  <td align="center">E1</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.101</math></td>
  <td align="center"><math>~5.52</math></td>
  <td align="center"><math>~0.00613</math></td>
  <td align="center"><math>~0.0338</math></td>
  <td align="center"><math>~0.442</math></td>
  <td align="center"><math>~0.558</math></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center">E2</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.202</math></td>
  <td align="center"><math>~2.99</math></td>
  <td align="center"><math>~0.0229</math></td>
  <td align="center"><math>~0.0685</math></td>
  <td align="center"><math>~0396</math></td>
  <td align="center"><math>~0.604</math></td>
  <td align="center">[[File:ImamuraPaper2Fig4.png|150px|Model E2]]</td>
</tr>
<tr>
  <td align="center">E3</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.402</math></td>
  <td align="center"><math>~1.74</math></td>
  <td align="center"><math>~0.159</math></td>
  <td align="center"><math>~0.277</math></td>
  <td align="center"><math>~0.301</math></td>
  <td align="center"><math>~0.699</math></td>
  <td align="center">[[File:ImamuraPaper2Fig4ModelE3.png|150px|Model E3]]</td>
</tr>
<tr>
  <td align="center">P1</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.452</math></td>
  <td align="center"><math>~1.60</math></td>
  <td align="center"><math>~0.254</math></td>
  <td align="center"><math>~0.406</math></td>
  <td align="center"><math>~0.277</math></td>
  <td align="center"><math>~0.723</math></td>
  <td align="center">[[File:ImamuraPaper2Fig3ModelP1.png|150px|Model P1]]</td>
</tr>
<tr>
  <td align="center">P2</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.500</math></td>
  <td align="center"><math>~1.49</math></td>
  <td align="center"><math>~0.403</math></td>
  <td align="center"><math>~0.600</math></td>
  <td align="center"><math>~0.255</math></td>
  <td align="center"><math>~0.745</math></td>
  <td align="center">[[File:ImamuraPaper2Fig3ModelP2.png|150px|Model P2]]</td>
</tr>
<tr>
  <td align="center">P3</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.600</math></td>
  <td align="center"><math>~1.33</math></td>
  <td align="center"><math>~1.09</math></td>
  <td align="center"><math>~1.450</math></td>
  <td align="center"><math>~0.202</math></td>
  <td align="center"><math>~0.798</math></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center">P4</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.700</math></td>
  <td align="center"><math>~1.21</math></td>
  <td align="center"><math>~3.37</math></td>
  <td align="center"><math>~4.078</math></td>
  <td align="center"><math>~0.153</math></td>
  <td align="center"><math>~0.847</math></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="left" colspan="10"><sup>&dagger;</sup>In all three papers from the [[#See_Also|Imamura &amp; Hadley collaboration]], <math>~q = 2</math> means, "[[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|uniform specific angular momentum]]."</td>
</tr>
</table>

[[Appendix/Ramblings/ToHadleyAndImamura#Recognition_.233|As in our separate summary discussion]], our two-piece radial eigenfunction is defined by the expressions,

<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>
&nbsp; &nbsp; &hellip; &nbsp; &nbsp; 
<math>~\biggl[ f_\mathrm{blue}(\varpi) = 0</math> &nbsp;  &nbsp; otherwise<math>~\biggr]</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>
&nbsp; &nbsp; &hellip; &nbsp; &nbsp; 
<math>~\biggl[ f_\mathrm{green}(\varpi) = 0</math> &nbsp;  &nbsp; otherwise<math>~\biggr]</math>,
  </td>
</tr>
</table>
</div>
where, <math>~r_\mathrm{mid}|_\mathrm{green} \le r_\mathrm{mid}|_\mathrm{blue}</math>.  The expression that we are currently using for the radial eigenfunction is a sum of these two pieces, that is,

<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>~f_2(\varpi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f_\mathrm{green}(\varpi) + f_\mathrm{blue}(\varpi) \, .
</math>
  </td>
</tr>
</table>

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

Over the inner ("blue") region of the torus, we will continue to use the phase-function prescription that has been described [[Appendix/Ramblings/To_Hadley_and_Imamura#Constant_Phase_Loci|in our separate summary discussion]], namely, 

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

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" bgcolor="blue">&nbsp;</td>
  <td align="right">
<math>~m\phi_\mathrm{blue}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\pm \tan^{-1}[\aleph_m \cdot D_{1/2}(\varpi)] </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_- < \varpi < r_\mathrm{min} \, ,</math>
  </td>
</tr>
</table>

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

where, <math>~\aleph_m</math> is a constant, 
<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>~\biggl[ \frac{f_\ln(\varpi) - [f_\ln]_\mathrm{min}}{[f_\ln]_\mathrm{max} - [f_\ln]_\mathrm{min}} \biggr]^{1/2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~f_\ln(\varpi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\ln[ f_1(\varpi)] \, .</math>
  </td>
</tr>
</table>
</div>

and, <math>~r_\mathrm{min}</math> is the radial location of the minimum of the radial eigenfunction, <math>~f_\ln</math>.  Over the outer ("green") region of the torus, however, we now adopt a different prescription for the phase function, namely,

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

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" bgcolor="green">&nbsp;</td>
  <td align="right">
<math>~m\phi_\mathrm{green}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{A_0}\cdot \sin[\aleph_m \cdot D_{1/2}(\varpi)] </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; for &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>r_\mathrm{min} < \varpi < r_+ \, ,</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>
where, <math>~A_0</math> is a constant.  

===Application to Specific Models===
In what follows, we pick a model from Table 4, and extract from that model the locations of the inner and outer radii of the torus, namely, <math>~r_-</math> and <math>~r_\mathrm{outer}</math>.  Then, in an effort to construct an eigenvector that matches the behavior shown in the corresponding Paper II plots (as reprinted, here, in the last column of our Table 4) over the innermost region of the configuration &#8212; that is, the region around the innermost and most prominent minimum of the radial eigenfunction &#8212; we assign a value to <math>~r_\mathrm{blue}</math> that roughly corresponds to the radial location of the first (innermost) minimum, and we assign a value to <math>~r_+</math> that roughly corresponds to the radial location of the second minimum.  We set <math>~m = 2</math> for each model, because, following [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014)], we are focusing only on unstable P- and E-modes having "bar-like" structures; and, [[#Specific_Application_to_HI11.27s_Figure_16|as above]], when descritizing each model within 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} </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>
</table>
</div>

Then, for each model, we adjusted the values of five free parameters until a reasonably good fit to both the published radial eigenfunction and "constant phase locus" was obtained over the innermost region of the model.  The five adjustable parameters are:  <math>~p_\mathrm{blue}, ~r_\mathrm{green}, ~p_\mathrm{green}, ~\aleph_2,</math> and <math>~A_0</math>.  

<div align="center" id="Table5">
<table border="1" align="center" cellpadding="5" width="75%">
<tr><td align="center" colspan="5" bgcolor="yellow"><font size="+1"><b>Table 5:</b></font>&nbsp; Model E2<sup>&Dagger;</sup></td></tr>
<tr>
  <td align="center" colspan="4" bgcolor="white"><font size="+1"><b>Model Parameters</b></font></td>
  <td align="center" colspan="1" rowspan="1" bgcolor="white">Results from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II (Hadley et al. 2014)]  Simulation</td>
</tr>
<tr>
  <td align="center"><math>~r_-</math></td>
  <td align="center"><math>~r_+</math></td>
  <td align="center"><math>~r_\mathrm{outer}</math></td>
  <td align="center"><math>~\delta r</math></td>
  <td align="center" colspan="1" rowspan="6">
[[File:ImamuraPaper2Fig4.png|400px|right|Figure 4b from Paper II]]
  </td>
</tr>
<tr>
  <td align="center"><math>~0.60</math></td>
  <td align="center"><math>~2.0</math></td>
  <td align="center"><math>~3.0</math></td>
  <td align="center"><math>~0.007</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~r_\mathrm{blue}</math></td>
  <td align="center" bgcolor="lightblue"><math>~p_\mathrm{blue}</math></td>
  <td align="center" bgcolor="lightblue"><math>~r_\mathrm{green}</math></td>
  <td align="center" bgcolor="lightblue"><math>~p_\mathrm{green}</math></td>
</tr>
<tr>
  <td align="center"><math>~\$A\$68 = 1.006</math></td>
  <td align="center"><math>~1.2</math></td>
  <td align="center"><math>~\$A\$65 = 0.985</math></td>
  <td align="center"><math>~0.50</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~r_\mathrm{min}</math></td>
  <td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>
  <td align="center" colspan="2"><math>~[f_\ln]_\mathrm{max}</math></td>
  <td align="center" colspan="1" bgcolor="white">Our Empirically Constructed Eigenvector</td>
</tr>
<tr>
  <td align="center"><math>~\$A\$65 = 0.985</math></td>
  <td align="center"><math>~-3.49047</math></td>
  <td align="center" colspan="2"><math>~\ln[f_2(\$A\$1)]=+4.85166</math></td>
  <td align="center" colspan="1" rowspan="5">
[[File:ImamuraPaper2MyE2mirror.png|380px|right|Empirically constructed E2]]
  </td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~m</math></td>
  <td align="center" bgcolor="lightblue"><math>~\aleph_2</math></td>
  <td align="center" bgcolor="lightblue"><math>~A_0</math></td>
  <td align="center"><math>~\phi_0</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~\pi</math></td>
  <td align="center"><math>~0.25</math></td>
  <td align="center"><math>~\tfrac{9\pi}{16}</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="left" colspan="5">
<sup>&Dagger;</sup>Note the following:
* When the Paper II radial eigenfunction possesses more than one local minimum &#8212; and, in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians &#8212; we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.
* Our empirical model's 5 adjustable free parameters are highlighted in blue.
* Before plotting our empirically constructed phase function, the tabulated phase shift, <math>~\phi_0</math>, has been added to both <math>~\phi_\mathrm{blue}</math> and <math>~\phi_\mathrm{green}</math> in an effort to match the overall orientation of the "constant phase locus" published in Paper II.  (This is not considered to be an adjustable free parameter.)
  </td>
</tr>

</table>
</div>

<p></p>
<p></p>

<div align="center" id="Table6">
<table border="1" align="center" cellpadding="5" width="75%">
<tr><td align="center" colspan="5" bgcolor="yellow"><font size="+1"><b>Table 6:</b></font>&nbsp; Model E3<sup>&Dagger;</sup></td></tr>
<tr>
  <td align="center" colspan="4" bgcolor="white"><font size="+1"><b>Model Parameters</b></font></td>
  <td align="center" colspan="1" rowspan="1" bgcolor="white">Results from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II (Hadley et al. 2014)]  Simulation</td>
</tr>
<tr>
  <td align="center"><math>~r_-</math></td>
  <td align="center"><math>~r_+</math></td>
  <td align="center"><math>~r_\mathrm{outer}</math></td>
  <td align="center"><math>~\delta r</math></td>
  <td align="center" colspan="1" rowspan="6">
[[File:ImamuraPaper2Fig4ModelE3.png|400px|right|Figure 4a from Paper II]]
  </td>
</tr>
<tr>
  <td align="center"><math>~0.7</math></td>
  <td align="center"><math>~1.7</math></td>
  <td align="center"><math>~1.74</math></td>
  <td align="center"><math>~0.005</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center" bgcolor="white"><math>~r_\mathrm{blue}</math></td>
  <td align="center" bgcolor="lightblue"><math>~p_\mathrm{blue}</math></td>
  <td align="center" bgcolor="lightblue"><math>~r_\mathrm{green}</math></td>
  <td align="center" bgcolor="lightblue"><math>~p_\mathrm{green}</math></td>
</tr>
<tr>
  <td align="center"><math>~\$A\$77 = 1.085</math></td>
  <td align="center"><math>~1.2</math></td>
  <td align="center"><math>~\$A\$55 = 0.975</math></td>
  <td align="center"><math>~0.75</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~r_\mathrm{min}</math></td>
  <td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>
  <td align="center" colspan="2"><math>~[f_\ln]_\mathrm{max}</math></td>
  <td align="center" colspan="1" bgcolor="white">Our Empirically Constructed Eigenvector</td>
</tr>
<tr>
  <td align="center"><math>~\$A\$71 = 1.055</math></td>
  <td align="center"><math>~-1.34494</math></td>
  <td align="center" colspan="2"><math>~\ln[f_2(\$A\$1)]=+5.19688</math></td>
  <td align="center" colspan="1" rowspan="5">
[[File:ImamuraPaper2MyE3mirror.png|380px|right|Empirically constructed E3]]
  </td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~m</math></td>
  <td align="center" bgcolor="lightblue"><math>~\aleph_2</math></td>
  <td align="center" bgcolor="lightblue"><math>~A_0</math></td>
  <td align="center"><math>~\phi_0</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~\tfrac{11 \pi}{8}</math></td>
  <td align="center"><math>~3.0</math></td>
  <td align="center"><math>~\tfrac{2\pi}{5}</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="left" colspan="5">
<sup>&Dagger;</sup>Note the following:
* When the Paper II radial eigenfunction possesses more than one local minimum &#8212; and, in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians &#8212; we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.
* Our empirical model's 5 adjustable free parameters are highlighted in blue.
* The undesirable kink that appears in our empirically constructed radial eigenfunction (to the left of the function minimum) arises because the exponent, <math>~p_\mathrm{green}</math>, is a fraction that is less than unity.
* Before plotting our empirically constructed phase function, the tabulated phase shift, <math>~\phi_0</math>, has been added to both <math>~\phi_\mathrm{blue}</math> and <math>~\phi_\mathrm{green}</math> in an effort to match the overall orientation of the "constant phase locus" published in Paper II.  (This is not considered to be an adjustable free parameter.)
  </td>
</tr>

</table>
</div>

<!-- COMMENT OUT OLD TABLE 5
<table border="1" align="center" cellpadding="5">
<tr><td align="center" colspan="14"><font size="+1"><b>Table 5:</b></font>&nbsp; Empirical Fits<sup>&Dagger;</sup></td></tr>
<tr>
  <td align="center">Name</td>
  <td align="center"><math>~r_-</math></td>
  <td align="center"><math>~r_+</math></td>
  <td align="center"><math>~\delta r</math></td>
  <td align="center"><math>~r_\mathrm{blue}</math></td>
  <td align="center"><math>~p_\mathrm{blue}</math></td>
  <td align="center"><math>~r_\mathrm{green}</math></td>
  <td align="center"><math>~p_\mathrm{green}</math></td>
  <td align="center"><math>~r_\mathrm{min}</math></td>
  <td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>
  <td align="center"><math>~[f_\ln]_\mathrm{max}=\ln[f_2(\$A\$1)]</math></td>
  <td align="center"><math>~m</math></td>
  <td align="center"><math>~\aleph_2</math></td>
  <td align="center"><math>~A_0</math></td>
</tr>
<tr>
  <td align="center">E3</td>
  <td align="center"><math>~0.7</math></td>
  <td align="center"><math>~1.7</math></td>
  <td align="center"><math>~0.005</math></td>
  <td align="center"><math>~\$A\$77 = 1.085</math></td>
  <td align="center"><math>~1.2</math></td>
  <td align="center"><math>~\$A\$55 = 0.975</math></td>
  <td align="center"><math>~0.75</math></td>
  <td align="center"><math>~\$A\$71 = 1.055</math></td>
  <td align="center"><math>~-1.34494</math></td>
  <td align="center"><math>~+5.19688</math></td>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~\tfrac{11 \pi}{8}</math></td>
  <td align="center"><math>~3.0</math></td>
</tr>
<tr>
  <td align="center" colspan="14">
[[File:ImamuraPaper2Fig4ModelE3.png|400px|Model E3]]<p></p>
[[File:ImamuraPaper2MyE3.png|400px|My E3]]
  </td>
</tr>
<tr>
  <td align="left" colspan="14">
<sup>&Dagger;</sup>Note the following:
* When the Paper II radial eigenfunction possesses more than one local minimum &#8212; and, in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians &#8212; we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.
* The undesirable kink that appears in our empirically constructed radial eigenfunction, to the left of the function minimum, arises because the exponent, <math>~p_\mathrm{green}</math>, is a fraction that is less than unity.
  </td>
</tr>
</table>
END COMMENTING OUT OLD TABLE 5 -->

<p></p>
<p></p>

<div align="center" id="Table7">
<table border="1" align="center" cellpadding="5" width="75%">
<tr><td align="center" colspan="5" bgcolor="yellow"><font size="+1"><b>Table 7:</b></font>&nbsp; Model P1<sup>&Dagger;</sup></td></tr>
<tr>
  <td align="center" colspan="4" bgcolor="white"><font size="+1"><b>Model Parameters</b></font></td>
  <td align="center" colspan="1" rowspan="1" bgcolor="white">Results from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II (Hadley et al. 2014)]  Simulation</td>
</tr>
<tr>
  <td align="center"><math>~r_-</math></td>
  <td align="center"><math>~r_+</math></td>
  <td align="center"><math>~r_\mathrm{outer}</math></td>
  <td align="center"><math>~\delta r</math></td>
  <td align="center" colspan="1" rowspan="6">
[[File:ImamuraPaper2Fig3ModelP1.png|400px|right|Figure 3d from Paper II]]
  </td>
</tr>
<tr>
  <td align="center"><math>~0.723</math></td>
  <td align="center"><math>~1.35</math></td>
  <td align="center"><math>~1.6</math></td>
  <td align="center"><math>~0.003135</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center" bgcolor="white"><math>~r_\mathrm{blue}</math></td>
  <td align="center" bgcolor="lightblue"><math>~p_\mathrm{blue}</math></td>
  <td align="center" bgcolor="lightblue"><math>~r_\mathrm{green}</math></td>
  <td align="center" bgcolor="lightblue"><math>~p_\mathrm{green}</math></td>
</tr>
<tr>
  <td align="center"><math>~\$A\$100 = 1.00515</math></td>
  <td align="center"><math>~1.0</math></td>
  <td align="center"><math>~\$A\$96 = 0.99261</math></td>
  <td align="center"><math>~0.35</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~r_\mathrm{min}</math></td>
  <td align="center"><math>~[f_\ln]_\mathrm{min}</math></td>
  <td align="center" colspan="2"><math>~[f_\ln]_\mathrm{max}</math></td>
  <td align="center" colspan="1" bgcolor="white">Our Empirically Constructed Eigenvector</td>
</tr>
<tr>
  <td align="center"><math>~\$A\$96 = 0.99261</math></td>
  <td align="center"><math>~-3.06805</math></td>
  <td align="center" colspan="2"><math>~\ln[f_2(\$A\$1)]=+4.48864</math></td>
  <td align="center" colspan="1" rowspan="5">
[[File:ImamuraPaper2MyP1better.png|380px|right|Empirically constructed P1]]
  </td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="center"><math>~m</math></td>
  <td align="center" bgcolor="lightblue"><math>~\aleph_2</math></td>
  <td align="center" bgcolor="lightblue"><math>~A_0</math></td>
  <td align="center"><math>~\phi_0</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="center"><math>~2.55</math></td>
  <td align="center"><math>~0.094</math></td>
  <td align="center"><math>~\tfrac{\pi}{2}</math></td>
</tr>
<tr>
  <td bgcolor="gray" colspan="4">&nbsp;</td>
</tr>
<tr>
  <td align="left" colspan="5">
<sup>&Dagger;</sup>Note the following:
* When the Paper II radial eigenfunction possesses more than one local minimum &#8212; and, usually in tandem, the Paper II phase function swings through more than <math>~2\pi</math> radians &#8212; we have set the parameter, <math>~r_+</math>, to a value less than <math>~r_\mathrm{outer}</math> because our empirically constructed eigenfunction is only designed to accommodate a single (the innermost) function minimum.
* Our empirical model's 5 adjustable free parameters are highlighted in blue.
* Before plotting our empirically constructed phase function, the tabulated phase shift, <math>~\phi_0</math>, has been added to both <math>~\phi_\mathrm{blue}</math> and <math>~\phi_\mathrm{green}</math> in an effort to match the overall orientation of the "constant phase locus" published in Paper II.  (This is not considered to be an adjustable free parameter.)
  </td>
</tr>

</table>
</div>