Editing Appendix/Ramblings/HadleyAndImamuraSupplementaryDatabase (section)

==Specifics of Database==
The initial, axisymmetric toroidal configuration that is associated with each simulation is uniquely characterized by the following set of physical parameters:
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<table align="left" cellpadding="5" border="0" width="100%">
<tr><td align="left">

<table border="0" cellpadding="5" align="left">
<tr>
  <td align="center"><font size="+2">&bull;</font></td>
  <td align="left"><math>~n</math></td>
  <td align="center">&hellip;</td>
  <td align="left">Polytropic index</td>
</tr>
<tr>
  <td align="center"><font size="+2">&bull;</font></td>
  <td align="left"><math>~q</math></td>
  <td align="center">&hellip;</td>
  <td align="left">Power-law index characterizing angular velocity profile &#8212; but see the [[#Simple_Rotation_Profiles|relevant note regarding ''Simple Rotation Profiles'', below]]</td>
</tr>
<tr>
  <td align="center"><font size="+2">&bull;</font></td>
  <td align="left"><math>~M_*/M_\mathrm{disk}</math></td>
  <td align="center">&hellip;</td>
  <td align="left">Star-to-disk (''i.e.'', star-to-torus) mass ratio</td>
</tr>
<tr>
  <td align="center"><font size="+2">&bull;</font></td>
  <td align="left"><math>~R_-/R_+</math></td>
  <td align="center">&hellip;</td>
  <td align="left">Ratio of inner-to-outer edge of the torus, in its equatorial plane</td>
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</table>

</td></tr>
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In addition to specifying these four parameters, the Hadley &amp; Imamura collaboration also tag each evolution with an integer value of <math>~m</math>, to identify the nonaxisymmetric mode that was observed to be fastest growing.  For example, following the "star-disk models" link that appears at the top of the category list on the [http://pages.uoregon.edu/khadley/ home page of the ''Stardisks'' repository] will bring you to a page that provides active links to 32 separate model categories, such as:

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  <td align="right" colspan="1">&nbsp;</td>
  <td align="center" colspan="3"><b><font size="+1">Model Category</font></b></td>
  <td align="left">To go directly to the identified page &hellip;</td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="right"><math>~(n, q, m)</math></td>
  <td align="center"><math>~=</math></td>
  <td align="left"><math>~(1.5, 1.0, 1)</math></td>
  <td align="center">[https://www.khadley.com/research/star_disks/n1.5q1.0m1.html click this link]</td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="right"><math>~(n, q, m)</math></td>
  <td align="center"><math>~=</math></td>
  <td align="left"><math>~(1.5, 1.75, 3)</math></td>
  <td align="center">[https://www.khadley.com/research/star_disks/n1.5q1.75m3.html click this link]</td>
</tr>
<tr>
  <td align="right">Most relevant here &nbsp;&nbsp; <font size="+2" color="red">&#x27A0;</font></td>
  <td align="right"><math>~(n, q, m)</math></td>
  <td align="center"><math>~=</math></td>
  <td align="left"><math>~(1.5, 2.0, 2)</math></td>
  <td align="center">[https://www.khadley.com/research/star_disks/n1.5q2.0m2.html click this link]</td>
</tr>
<tr>
  <td align="right">&nbsp;</td>
  <td align="right"><math>~(n, q, m)</math></td>
  <td align="center"><math>~=</math></td>
  <td align="left"><math>~(2.5, 1.5, 2)</math></td>
  <td align="center">[https://www.khadley.com/research/star_disks/n2.5q1.5m2.html click this link]</td>
</tr>
</table>
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Clicking on any one of these category links will take you to a page containing a number of separate tables &#8212; each labeled with the relevant  <math>~M_*/M_\mathrm{disk}</math> value &#8212; and, within each table, a (sometimes large!) number of rows, each characterized (usually in the 2<sup>nd</sup> column) by the specific model's <math>~R_-/R_+</math>.  


For example, on the page associated with the model category that we have identified (see red arrow, above) as most relevant to our [[Apps/ImamuraHadleyCollaboration#Comparison_with_Results_from_the_Imamura_.26_Hadley_Collaboration|accompanying discussions]], we find that the 2nd row of the top-most table details various properties of a ''particular'' model evolution uniquely identified by the set of parameters, <math>~(n, q, M_*/M_\mathrm{disk}, R_-/R_+)</math> = <math>~(1.5, 2.0, 0.0, 0.1008)</math>.  And on the same page, the last row of the bottom-most table details various properties of another ''particular'' model evolution uniquely identified by the set of parameters, <math>~(n, q, M_*/M_\mathrm{disk}, R_-/R_+)</math> = <math>~(1.5, 2.0, 1000.0, 0.8002)</math>.  In both of these model evolutions, the fastest growing unstable mode was <math>~m = 2</math>.  


At the very bottom of every <math>~(n,q,m)</math> model category page is a list of &#8212; and a hypertext link to &#8212; ten "Image Tables."  Clicking on the "eqContour" link takes you to a page that contains a thumbnail image showing the meridional-plane density contours of the equilibrium model that served as the initial condition for every evolution catalogued under the chosen <math>~(n,q,m)</math> category.  The thumbnail images are laid out on a two-dimensional, rectangular lattice whose horizontal position is set by the specific model's mass ratio and whose vertical position is set by the model's geometric ratio, <math>R_-/R_+</math>.  

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<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><b><font size="+1">Figure 1</font></b><p></p>
<math>~(n, q, M_*/M_\mathrm{disk}, R_-/R_+) = ~(1.5, 2.0, 0.0, 0.1008)</math>
  </td>
</tr>
<tr>
  <td align="center">[[File:eqContourMass0.png|200px|Mass0, eqContour]]</td>
  <td align="center">[[File:LogAmpMass0.png|200px|Mass0, LogAmp]]</td>
  <td align="center">[[File:PhaseLocusMass0.png|200px|Mass0, PhaseLocus]]</td>
</tr>
<tr>
  <td align="center">Initial Meridional-plane Contours</td>
  <td align="center">Radial Profile of Unstable m = 2 Eigenfunction</td>
  <td align="center">Phase Locus of Unstable m = 2 Eigenfunction</td>
</tr>
</table>
</div>


The image displayed here, on the left in Figure 1, was extracted from coordinate position <math>~(M_*/M_\mathrm{disk}, R_-/R_+)</math> = <math>~(0.0,0.1)</math>, near the upper-left corner of the [http://pages.uoregon.edu/khadley/stardisks/n1.5q2.0m2_eqContour.html "eqContour" Image Table] associated with our above-identified "most relevant" model category, namely, <math>~(n,q,m)</math> = <math>~(1.5,2.0,2)</math>.  This is a very thick (toroidal-shaped), isolated self-gravitating disk; that is, there is no central star associated with this dynamical system.  The image displayed on the left in Figure 2, was extracted from coordinate position <math>~(M_*/M_\mathrm{disk}, R_-/R_+)</math> = <math>~(1000.0, 0.8)</math>, near the bottom-right corner of the same [https://www.khadley.com/research/star_disks/n1.5q2.0m2_eqContour.html "eqContour" Image Table].  This is a very slender and nearly massless disk/torus; it is orbiting a central (point-like) star whose mass is one thousand times larger than the mass of the disk.

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<table border="1" align="center" cellpadding="5">
<tr>
  <td align="center" colspan="3"><b><font size="+1">Figure 2</font></b><p></p>
<math>~(n, q, M_*/M_\mathrm{disk}, R_-/R_+) = ~(1.5, 2.0, 1000.0, 0.8002)</math>
  </td>
</tr>
<tr>
  <td align="center">[[File:eqContourMass1000.png|200px|Mass0, eqContour]]</td>
  <td align="center">[[File:LogAmpMass1000.png|200px|Mass0, LogAmp]]</td>
  <td align="center">[[File:PhaseLocusMass1000.png|200px|Mass0, PhaseLocus]]</td>
</tr>
<tr>
  <td align="center">Initial Meridional-plane Contours</td>
  <td align="center">Radial Profile of Unstable m = 2 Eigenfunction</td>
  <td align="center">Phase Locus of Unstable m = 2 Eigenfunction</td>
</tr>
</table>
</div>

In an analogous fashion, the images displayed here in the middle of Figure 1 and Figure 2 were extracted from the [http://pages.uoregon.edu/khadley/stardisks/n1.5q2.0m2_ee1.html <math>~|\delta\rho|/\rho(\varpi)</math> Image Table] associated with the <math>~(n,q,m)</math> = <math>~(1.5,2.0,2)</math> model category.  And the images displayed on the right in both figures were extracted from the [http://pages.uoregon.edu/khadley/stardisks/n1.5q2.0m2_ee2m2.html "<math>~\delta\rho</math> phase(''m'')" Image Table] associated with the <math>~(n,q,m)</math> = <math>~(1.5,2.0,2)</math> model category.