Editing Appendix/Ramblings/HadleyAndImamuraSupplementaryDatabase (section)

==Relationship to Studies of Slender, Massless Tori==

In an [[Apps/ImamuraHadleyCollaboration#Characteristics_of_Unstable_Eigenvectors_in_Self-Gravitating_Tori|accompanying chapter]], we have discussed the "characteristics of unstable eigenvectors in self-gravitating tori" &#8212; such as the ones presented by the [[#See_Also|Hadley &amp; Imamura collaboration]] &#8212; in the context of analytic studies of normal modes of oscillation in tori that &hellip;
<ul>
<li>have <math>~n = \tfrac{3}{2}</math> &#8212; although this can be readily generalized;</li>
<li>have uniform specific angular momentum &#8212; that is, have <math>~q = 2.0</math>;</li>
<li>are ''massless'', that is, have <math>~M_*/M_\mathrm{disk} = \infty</math>;</li>
<li>are ''slender'' &#8212; that is, they have geometric ratios, <math>~R_-/R_+</math> near unity.</li>
</ul>

As a group, these configurations define a subset of a much larger set of ''massless'' tori whose (initial) axisymmetric, equilibrium structures are completely definable in terms of analytic expressions.  As has been discussed in an [[Apps/PapaloizouPringleTori#Massless_Polytropic_Tori|accompanying chapter]], the structural and stability properties of these configurations were first brought to the attention of the astrophysics community, in the context of Newtonian systems, by [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou &amp; Pringle] (1984, MNRAS, 208, 721-750).  Hence, they are frequently referred to as Papaloizou-Pringle tori &#8212; or, PP tori, for short.  


[[Apps/PapaloizouPringleTori#Boundary_Conditions|As we have pointed out]], the relative thickness of a PP torus can be expressed in terms of the dimensionless Bernoulli constant, <math>~C_B^'</math> &#8212; a physical parameter highlighted by [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou &amp; Pringle (1984)] &#8212; as follows:
<div align="center">
<math>
~\frac{R_{-}}{R_{+}}  =  \frac{ 1 - \sqrt{1-2C_\mathrm{B}^'} }{1 + \sqrt{1-2C_\mathrm{B}^'} } .
</math>
</div>

And in our review of [[Apps/PapaloizouPringleTori#Model_as_Described_by_Kojima|Kojima's (1986)]] work, we showed that the square of the Mach number at the cross-sectional center (subscript "0") of the torus can also be expressed in terms of this dimensionless Bernoulli constant.  Specifically,

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

<tr>
  <td align="right">
<math>~\mathfrak{M}_0^2 \equiv \frac{(v_\varphi|_0)^2}{(c_s|_0)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2n [ 1- 2C_B^' ]^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ [1 - 2C_B^'] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2n}{\mathfrak{M}_0^2} \, . </math>
  </td>
</tr>
</table>
</div>

Now, instead of specifying the system's Mach number in his ''analytic'' examination of the stability of PP tori, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] defines the dimensionless order parameter,

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

<tr>
  <td align="right">
<math>~\beta^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{2n}{\mathfrak{M}_0^2} \, .</math>
  </td>
</tr>

</table>
</div>
Putting these algebraic expressions together, it is clear that in massless PP tori,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_-}{R_+}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1-\beta}{1 + \beta} \, ;</math>
  </td>
</tr>
</table>
</div>
or, inverting this expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \beta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1 - R_-/R_+}{1 + R_-/R_+} \, .</math>
  </td>
</tr>
</table>
</div>

With this information in hand, we expect that results from the stability analysis performed by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] on PP tori having <math>~M_*/M_\mathrm{disk} = \infty</math> and <math>~\beta \ll 1</math> should most appropriately be compared to models from the Hadley &amp; Imamura collaboration that have <math>~M_*/M_\mathrm{disk} \gg 1</math> and <math>~R_-/R_+ \approx 1</math>.  Numerical simulation results from the model illustrated, above, in our Figure 1 are not likely to compare well with the analytic stability analysis presented by Blaes because the torus is very thick <math>~(R_-/R_+ = 0.1)</math> and massive <math>~(M_*/M_\mathrm{disk} = 0)</math>.  On the other hand, the model illustrated, above, in our Figure 2 presents a likely candidate for comparison, as its parameter values are,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{M_*}{M_\mathrm{disk}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1000 \, ,</math>
  </td>
</tr>
<tr><td colspan="3" align="center">and,</td></tr>
<tr>
  <td align="right">
<math>~\frac{R_-}{R_+} = 0.8</math>
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;<math>~\Rightarrow</math>&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>~\beta \approx 0.11 \, .</math>
  </td>
</tr>
</table>
</div>

Panels B and C of Figure 3 [[Apps/ImamuraHadleyCollaboration#Figure3|in an accompanying chapter]] presents this direct comparison.