Editing Apps/ImamuraHadleyCollaboration (section)

===Imamura &amp; Hadley Collaboration===
Motivated especially by the numerical study of unstable nonaxisymmetric modes 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 [[#See_Also|Imamura &amp; Hadley collaboration]], we have embarked on an effort to better understand what sparks the development of a wide range of complex eigenvectors in self-gravitating tori.  Of particular interest are "P-", "J-", and "E-modes," such as the ones whose eigenfunctions are illustrated here in Table 1.  (Click on a small image in the right-hand column to see a larger image.)  Our initial attempts to construct fits to these eigenvectors empirically have been described in [[Appendix/Ramblings/AzimuthalDistortions#Analyzing_Azimuthal_Distortions|a separate chapter]].  We begin, here, a more quantitative analysis of these structures.

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="7"><font size="+1"><b>Table 1:</b></font>&nbsp; J-, 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">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">Eigenfunction</td>
</tr>
<tr>
  <td align="center" colspan="6">Extracted from Table 2 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
  <td align="center" colspan="1">Extracted from Fig. 4 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</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">&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">[[File:ImamuraPaper2Fig4.png|175px|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">[[File:ImamuraPaper2Fig4ModelE3.png|175px|Model E3]]</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="1">Extracted from Fig. 3 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</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">[[File:ImamuraPaper2Fig3ModelP1.png|175px|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">[[File:ImamuraPaper2Fig3ModelP2.png|175px|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">&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">[[File:ImamuraPaper2Fig3ModelP4.png|175px|Model P4]]</td>
</tr>
<tr>
  <td align="center" colspan="6">Extracted from Table 1 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
  <td align="center" colspan="1">Extracted from Fig. 2 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
</tr>
<tr>
  <td align="center">J1</td>
  <td align="center"><math>~0.01</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, \tfrac{3}{2})</math></td>
  <td align="center"><math>~0.402</math></td>
  <td align="center"><math>~1.51</math></td>
  <td align="center"><math>~6.47</math></td>
  <td align="center">[[File:ImamuraPaper2Fig2ModelJ1B.png|175px|Model J1]]</td>
</tr>
<tr>
  <td align="left" colspan="7"><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>


[[File:ImamuraPaper2Fig4Eigenfunction.png|right|250px|Model E2 Radial Eigenfunction]]Here, for example, are a couple of questions guiding our study: 
*  Can we understand why the radial eigenfunction of, for example, model E2 &#8212; re-displayed here, on the right &#8212; exhibits a series of sharp dips whose spacing gets progressively smaller and smaller as the outer edge of the torus is approached?
*  Does the spatial structure of the unstable eigenvectors that appear in numerical simulations of geometrically thin, self-gravitating tori &#8212; such as model "P4" highlighted here in Table 1 &#8212; resemble the analytically defined eigenvector that is predicted by linear stability analyses to be unstable in ''massless'' Papaloizou-Pringle tori?