Editing Apps/ImamuraHadleyCollaboration (section)

====Comparison with Results from the Imamura &amp; Hadley Collaboration====

=====Visual Comparison=====
With the above information in hand, we can now directly compare the eigenfunction of the unstable, ''m'' = 2 mode discovered and defined analytically by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] with the eigenfunction of unstable, ''m'' = 2 modes found by the [[#See_Also|Imamura &amp; Hadley collaboration]] while using numerical hydrodynamic techniques to simulate the evolution of similarly slim tori.  Panel A of Figure 3 displays a pair of plots, extracted from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014; Paper II)], that present information regarding the ''m'' = 2 mode, nonaxisymmetric structure that developed in the equatorial plane of their model P4.  The model P4 disk/torus is geometrically slim <math>~(\beta \approx 0.18)</math> and its mass, although not zero, is only 1% of the mass of the central star <math>~(M_*/M_\mathrm{disk} = 10^2)</math>.   Panel B of Figure 3 displays a similar pair of plots from a separate, [[Appendix/Ramblings/HadleyAndImamuraSupplementaryDatabase#Supplementary_Dataset_Generated_by_Hadley_.26_Imamura_Collaboration|unpublished Hadley et al. model evolution]]; in this model, the disk/torus mass is only 0.1% that of the central star <math>~(M_*/M_\mathrm{disk} = 10^3)</math>.  In both panels, the blue curves show the geometric structure of non-axisymmetric, equatorial-plane density fluctuations, <math>~|\rho^'/\rho_0|_2</math>, and the red curves show the structure of the "enthalpy + gravity" perturbation, <math>~\mathcal{W}</math>:  The left-hand plot displays log<sub>10</sub> of (the modulus of) the amplitude versus radius; while the right-hand plot displays the unstable mode's constant phase locus.  In each of these plots, the radial coordinate is, <math>~\chi \equiv \varpi/\varpi_0</math>.  


Panel C of Figure 3 presents similar plots that we have generated to show the ''m'' = 2 mode, non-axisymmetric, equatorial-plane density and enthalpy fluctuations predicted by the [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] analysis.  More specifically, on the left, we have plotted (blue curve) <math>~\log_{10}|g_{0,0,2}|</math> versus <math>~\chi</math>, for <math>~\beta = 0.12</math> and a normalization coefficient, <math>~C = 4</math>; and (red curve) <math>~\log_{10}\mathcal{W}</math> versus <math>~\chi</math>, where we have used a normalization coefficient, <math>~C = 2</math>.  (See the third column of Table 5 for example data values.)  On the right, we present an equatorial-plane plot of (see columns five and six of Table 5) the constant phase locus, <math>~\tfrac{1}{2}\alpha_{0,0,2}</math> versus <math>~\chi</math>, and &#8212; reflecting the ''m'' = 2 mode structure &#8212; its twin phase locus, shifted in entirety by <math>~\pi</math> radians. [Note that, in the right-hand plot we have flipped the sign of <math>~\varphi_\mathrm{max}</math> in order to match the sign convention exhibited by the constant phase locus plots presented by the [[#See_Also|Imamura &amp; Hadley collaboration]].]

 
<div align="center" id="Figure3">
<table border="1" align="center" cellpadding="5">
<tr><th><font size="+1">Figure 3:  Comparison</font></th></tr>
<tr><th><font size="+1">Panel A: &nbsp;&nbsp; Model P4 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014; Paper II)];</font><p></p>
<math>~M_*/M_\mathrm{disk} = 100</math></th></tr>
<tr><td align="right">
[[File:Montage01Apart1.png|500px|center|Imamura &amp; Hadley collaboration]]
</td></tr>
<tr><th><font size="+1">Panel B: &nbsp;&nbsp; [[Appendix/Ramblings/HadleyAndImamuraSupplementaryDatabase#Supplementary_Dataset_Generated_by_Hadley_.26_Imamura_Collaboration|Unpublished model]] from Imamura &amp; Hadley collaboration;</font><p></p>
<math>~M_*/M_\mathrm{disk} = 1000</math></th></tr>
<tr><td align="right">
[[File:Montage01Apart3.png|500px|center|Imamura &amp; Hadley collaboration]]
</td></tr>
<tr><th><font size="+1">Panel C: &nbsp;&nbsp; [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] Analytic Solution with</font> <math>~(\beta, m, C) = (0.12, 2, 0.25)</math>;<p></p>
<math>~M_*/M_\mathrm{disk} = \infty</math></th></tr>
<tr><td align="right">
[[File:BlaesAnalyticCombined3.png|500px|center|Analytic Solution from Blaes(1985)]]
</td></tr>
</table>
</div>

=====Details Regarding Radial Coordinate Specification=====
We should emphasize that values of <math>~|\Delta|</math>, <math>~|g_{0,0,2}|</math>, and <math>~\alpha_{0,0,2}</math> that have been tabulated in Table 5 and that have been used to generate the two Panel C plots, were determined from analytic expressions that are functions of the parameter, <math>~\eta</math>, not explicitly functions of the radial coordinate, <math>~\chi</math>.  How did we determine <math>~\chi</math> from <math>~\eta</math> ?  Noting that, in the equatorial plane,
<div align="center">
<math>\chi = 1 - x\cos\theta \, ,</math>
</div>
with <math>~\cos\theta = \pm 1</math>, we ''could'' have used the [[#Normal_Modes_in_Slender_Tori|slender torus approximation]], <math>~x \approx \eta\beta</math>, to generate the algebraic mapping,
<div align="center">
<math>\chi \approx 1 \pm \eta\beta \, .</math>
</div>
This would have produced amplitude curves with reflection symmetry about the torus center <math>~(\chi = 1)</math>, and a "constant phase locus" exhibiting symmetry after double-reflection &#8212; reflection about the phase angle, - &pi;/2, as well as about <math>~(\chi = 1)</math>.  Instead, here we have adopted [[#Establishing_the_Simpler_Eigenvalue_Problem|the more accurate and more realistic, asymmetric relation]] between <math>~x</math> and <math>~\eta</math>, namely,
<div align="center">
<math>~x^2 \pm 2x^3 = (\beta\eta)^2 \, .</math>
</div>

In an <!-- [[Appendix/Ramblings/PPTori#Cubic_Equation_Solution|accompanying discussion]]--> [[Appendix/Ramblings/PPToriPt1A#Cubic_Equation_Solution|accompanying discussion]], we show that the relevant roots of this cubic equation give, from the inner edge of the torus to the pressure/density maximum,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{6}\biggl\{1 + 2\cos\biggl[\tfrac{1}{3}\cos^{-1}(1-54\beta^2\eta^2) + \frac{2\pi}{3}  \biggr]
\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
while, from the pressure/density maximum to the outer edge of the torus,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{6}\biggl\{1 + 2\cos\biggl[\tfrac{1}{3}\cos^{-1}(1-54\beta^2\eta^2) - \frac{2\pi}{3}  \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
Sample values of <math>~x_\mathrm{inner}</math> and <math>~x_\mathrm{outer}</math> are given in the last two columns of Table 5, assuming <math>~\beta = 0.12</math>.  In either case, the desired dimensionless radial coordinate is then obtained from the expression,
<div align="center">
<math>~\chi = 1 + x_\mathrm{inner/outer} \, .</math>
</div>

=====Discussion=====
Figure 3 reveals a remarkably strong resemblance between the eigenfunctions that have been generated using analytic expressions from the [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] analysis, and the curves that have emerged from the [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014)]  numerical simulations.  There are certainly differences in detail among the corresponding curves.  But rather than attributing this to shortcomings in the linear perturbation technique used by Blaes, or errors in the hydrodynamic scheme employed by the Imamura &amp; Hadley collaboration, we suggest that the variations seen are real, and principally due to the effects of self-gravity.  When moving from panel C, to panel B, to panel A: 
*  The red "enthalpy + gravity" curve, <math>~\mathcal{W}(\chi)</math>, maintains its overall shape but the central dip becomes progressively more pronounced.  This is presumably because the contribution to this function by the perturbation in the gravitational potential, <math>~\delta\Phi</math>, becomes larger (in an absolute sense) as the star-to-disk mass ratio decreases.
*  The blue "density fluctuation" curve becomes very slightly more "rounded."
*  The blue "constant phase locus" maintains its overall shape, but its end-to-end length gets progressively shorter.  In this regard, the Blaes85 analysis tells us that, for slim ''massless'' PP tori &#8212; that is, in the limit of <math>~M_*/M_\mathrm{disk} = \infty</math> &#8212; the total angular extent of the "constant phase locus" is <font size="+1">(</font>evaluated, here, for <math>~n = \tfrac{3}{2}</math> and <math>~m=2</math><font size="+1">)</font>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Delta \varphi|_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{m}\biggl\{ \pi - 2\tan^{-1}\biggl[ \frac{3}{8(n+1)} \biggr]^{1/2} \biggr\} = 1.20129~\mathrm{radians} 
= 68.83~\mathrm{degrees} \, .
</math>
  </td>
</tr>
</table>
</div>