Editing Apps/ImamuraHadleyCollaboration (section)

===Our Additional Analysis===

====Generic Formulation====
At any point in time, the fractional density fluctuation associated with any azimuthal mode, <math>~m</math>, can be represented by the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\rho^'}{\rho_0} \biggr]_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~g_m(\eta,\theta)e^{im\varphi}  \, .</math>
  </td>
</tr>
</table>
</div>
In general, we must assume that the function, <math>~g_m</math>, has both a real and an imaginary component, that is, we should assume that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_m(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{A}(\eta,\theta) + i\mathcal{B}(\eta,\theta) \, ,</math>
  </td>
</tr>
</table>
</div>
in which case the square of the modulus of this function is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~|g_m|^2 \equiv g_m \cdot g^*_m </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathcal{A}^2 + \mathcal{B}^2 \, .</math>
  </td>
</tr>
</table>
</div>
We can rewrite this complex function in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~g_m(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~|g_m(\eta,\theta)|e^{i[\alpha(\eta,\theta)]} \, ,</math>
  </td>
</tr>
</table>
</div>
if the angle, <math>~\alpha(\eta,\theta)</math> is defined such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cos\alpha = \frac{\mathcal{A}}{\sqrt{\mathcal{A}^2 + \mathcal{B}^2}}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\sin\alpha = \frac{\mathcal{B}}{\sqrt{\mathcal{A}^2 + \mathcal{B}^2}}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \alpha</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl(\frac{\mathcal{B}}{\mathcal{A}}\biggr) 
= \tan^{-1}\biggl[ \frac{\mathrm{Im}(g_m)}{\mathrm{Re}(g_m)} \biggr] 
\, .</math>
  </td>
</tr>
</table>
</div>

<span id="DensityEigenfunction">Hence, the spatial structure of the eigenfunction can be rewritten as,</span>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\rho^'}{\rho_0} \biggr]_m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~|g_m(\eta,\theta)| e^{i[\alpha(\eta,\theta)+m\varphi]}  \, .</math>
  </td>
</tr>
</table>
</div>

From this representation we can see that, at each spatial location, <math>~(\eta,\theta)</math>, the phase angle(s) at which the fractional perturbation exhibits its maximum amplitude, <math>~|g_m|</math>, is identified by setting the exponent of the exponential to zero.  That is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varphi = \varphi_\mathrm{max}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~-\frac{1}{m}\biggl[\alpha(\eta,\theta) \biggr] 
= -\frac{1}{m}\biggl\{ \tan^{-1}\biggl[ \frac{\mathrm{Im}(g_m)}{\mathrm{Re}(g_m)} \biggr] \biggr\}
\, .</math>
  </td>
</tr>
</table>
</div>

An equatorial-plane plot of <math>~\varphi_\mathrm{max}(\eta)</math> should produce the "constant phase locus" referenced, for example, in recent papers from the [[Appendix/Ramblings/ToHadleyAndImamura#Summary_for_Hadley_.26_Imamura|Imamura &amp; Hadley collaboration]].

<!--  COMMENT OUT
It should be noted that the leading (negative) sign that appears on the right-hand side of this expression for <math>~\phi_\mathrm{max}</math> is rather arbitrary, as is the additional <math>~\pi/2</math> phase shift that appears in that right-hand side expression.  Henceforth, for simplicity, we will omit both and use, instead,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~m\phi_\mathrm{max}(\varpi)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl[ \frac{\mathrm{Re}(f_m)}{\mathrm{Im}(f_m)} \biggr] 
\, ,</math>
  </td>
</tr>
</table>
</div>
unless and until the sign and/or a global phase shift is needed to adjust the orientation of a "constant phase locus" plot to facilitate comparison with published figures.
-->

====Density Fluctuations in Tori with Small but Finite &beta;====
The eigenvector that describes density fluctuations in tori with small but finite <math>~\beta</math> is obtained by combining the [[#DensityPerturbation2|above expression]] for <math>~\rho^'/\rho_0</math> in terms of <math>~\delta W</math>, with the expressions for <math>~\delta W_{0,0,m}</math> and <math>~\sigma_{0,0,m}</math> derived by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] and [[#Tori_with_Small_but_Finite_.CE.B2|summarized above]].  Keeping in mind that the Blaes85 analysis targeted structures with uniform specific angular momentum, which means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\Omega}{\Omega_0}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi^{-2}
= (1 - x\cos\theta)^{-2} \approx (1 + 2\eta\beta\cos\theta ) \, ,
</math>
  </td>
</tr>
</table>
</div>

we obtain,

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

<tr>
  <td align="right">
<math>~\biggl[\frac{\rho^'}{\rho_0}\biggr]_{0,0,m} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{n}{n+1}\biggl[\biggl(\frac{\sigma }{\Omega_0 }\biggr)_{0,0,m} + m\cdot \frac{\Omega}{\Omega_0}\biggr] \frac{\delta W_{0,0,m}}{f} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{C}{(1-\eta^2)}\biggl(\frac{n}{n+1}\biggr)\biggl\{
\biggl[-m  - ~i~m\biggl( \frac{3}{2n+2}  \biggr)^{1/2}\beta\biggr] + m (1 + 2\eta\beta\cos\theta ) 
\biggr\} e^{[i(m\varphi + \sigma_{0,0,m} t)]}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~</math>
  </td>
  <td align="left">
<math>~
\times ~ \biggl\{
1 + \beta^2 m^2\biggl[
2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} ~\pm ~4i \biggl( \frac{3}{2n+2}  \biggr)^{1/2}\eta\cos\theta
\biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{C}{(1-\eta^2)}\biggl(\frac{n}{n+1}\biggr)e^{[i(m\varphi + \sigma_{0,0,m} t)]}  \biggl[
\mathrm{Re}(\Delta) + i~\mathrm{Im}(\Delta)
\biggr] \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}(\Delta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
2\eta(\beta m)\cos\theta\biggl\{ 1 + (\beta m)^2\biggl[
2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~</math>
  </td>
  <td align="left">
<math>~\pm
(\beta m)\biggl[\frac{3}{2(n+1)}\biggr]^{1/2} \biggl\{4(\beta m)^2\biggl[\frac{3}{2(n+1)}\biggr]^{1/2} \eta\cos\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\eta(\beta m)\cos\theta \pm \mathcal{O}(\beta^3) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathrm{Im}(\Delta)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
-~(\beta m) \biggl[\frac{3}{2(n+1)}\biggr]^{1/2}\biggl\{ 1 + (\beta m)^2\biggl[
2\eta^2 \cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} \biggr] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~</math>
  </td>
  <td align="left">
<math>~
\pm~2\eta (\beta m)\cos\theta \biggl\{ 4(\beta m)^2 \biggl[\frac{3}{2(n+1)}\biggr]^{1/2} \eta\cos\theta\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>-~(\beta m) \biggl[\frac{3}{2(n+1)}\biggr]^{1/2} \pm \mathcal{O}(\beta^3) \, .</math>
  </td>
</tr>
</table>
</div>

Therefore, at any instant in time, this density eigenfunction can be written in the [[#DensityEigenfunction|form discussed above]], namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[ \frac{\rho^'}{\rho_0} \biggr]_{0,0,m}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~|g_{0,0,m}(\eta,\theta)|\exp\{i[\alpha_{0,0,m}(\eta,\theta)+m\varphi]\}  \, ,</math>
  </td>
</tr>
</table>
</div>
if we set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~|g_{0,0,m}(\eta,\theta)|</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{C}{(1-\eta^2)}\biggl(\frac{n}{n+1}\biggr)|\Delta| \, ,</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~|\Delta|^2</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl[ 2\eta(\beta m)\cos\theta\biggr]^2 + \biggl\{ -~(\beta m) \biggl[\frac{3}{2(n+1)}\biggr]^{1/2}  \biggr\}^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4(\beta m)^2 \biggl[ \eta^2\cos^2\theta + \frac{3}{8(n+1)} \biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
and if we simultaneously set,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha_{0,0,m}(\eta,\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tan^{-1}\biggl[ \frac{\mathrm{Im}(\Delta)}{\mathrm{Re}(\Delta)} \biggr] 
\approx \tan^{-1}\biggl\{ \frac{-\sqrt{3/[2(n+1)]}}{2\eta\cos\theta} \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

We note that, in the case of non-self-gravitating PP tori <math>~(\delta \Phi = 0)</math>, the amplitude of the "perturbed enthalpy" as defined by equation (38) of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley et al. (2014; Paper II)] is, to within a leading scale factor, just <math>~|\Delta|</math>.  Specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{W}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{P_0}{\rho_0}\biggl(\frac{n+1}{n}\biggr)  |g_{0,0,m}| + \cancelto{0}{\delta\Phi}
= \biggl(\frac{1-\eta^2}{n}\biggr)  |g_{0,0,m}|
= \biggl(\frac{C}{n+1}\biggr)  |\Delta| \, .
</math>
  </td>
</tr>
</table>
</div>


<div align="center" id="Table5">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="8">
<font size="+1">Table 5:&nbsp;&nbsp;Density Fluctuations in Tori with <math>~n = \tfrac{3}{2}</math> and Small but Finite</font> <math>~\beta</math>
<p></p>
<font size="+1">&hellip; &nbsp;Equatorial plane</font>&nbsp;&nbsp; &nbsp;&nbsp; <math>~~\Rightarrow~~~\cos\theta = \pm 1</math> 
</th></tr>
<tr>
  <td align="center" rowspan="2"><math>~\eta\cos\theta</math>
  <td align="center" rowspan="2"><math>~\frac{|\Delta|}{(\beta m)}</math>
  <td align="center" rowspan="2"><math>~\frac{|g_{0,0,m}|}{C(\beta m)}</math>
  <td align="center" colspan="3"><math>~\alpha_{0,0,m}+\varphi_\mathrm{shift}</math></td>
  <td align="center" colspan="2"><font size="+1"><math>~x\cos\theta</math>
<p></p>
(</font>assuming <math>~\beta = 0.12</math><font size="+1">)</font></td>
</tr>
<tr>
  <td align="center">Quadrant</td>
  <td align="center"><math>~\varphi_\mathrm{shift}=0</math></td>
  <td align="center"><math>~\varphi_\mathrm{shift}=\pi</math></td>
  <td align="center" rowspan="1">inner<p></p>
<math>~\cos\theta =-1</math>
  </td>
  <td align="center" rowspan="1">outer<p></p>
<math>~\cos\theta =+1</math>
  </td>
</tr>
<tr>
  <td align="center"><math>~-1.00</math></td>
  <td align="center"><math>~2.145</math></td>
  <td align="center"><math>~\infty</math></td>
  <td align="center">3<sup>rd</sup></td>
  <td align="center">---</td>
  <td align="center"><math>~+3.5111</math></td>
  <td align="center"><math>~- 0.1088</math></td>
  <td align="center"></td>
</tr>
<tr>
  <td align="center"><math>~-0.75</math></td>
  <td align="center"><math>~1.688</math></td>
  <td align="center"><math>~2.315</math></td>
  <td align="center">3<sup>rd</sup></td>
  <td align="center">---</td>
  <td align="center"><math>~+3.6183</math></td>
  <td align="center"><math>~- 0.0833</math></td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center"><math>~-0.50</math></td>
  <td align="center"><math>~1.265</math></td>
  <td align="center"><math>~1.012</math></td>
  <td align="center">3<sup>rd</sup></td>
  <td align="center">---</td>
  <td align="center"><math>~+3.8006</math></td>
  <td align="center"><math>~- 0.0569</math></td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center"><math>~-0.25</math></td>
  <td align="center"><math>~0.922</math></td>
  <td align="center"><math>~0.5901</math></td>
  <td align="center">3<sup>rd</sup></td>
  <td align="center">---</td>
  <td align="center"><math>~+4.1392</math></td>
  <td align="center"><math>~-0.0292</math></td>
  <td align="center">---</td>
</tr>
<tr>
  <td align="center"><math>~\mp 0.00</math></td>
  <td align="center"><math>~0.775</math></td>
  <td align="center"><math>~0.4648</math></td>
  <td align="center">---</td>
  <td align="center"><math>~-\tfrac{\pi}{2}</math></td>
  <td align="center"><math>~\tfrac{3\pi}{2}</math></td>
  <td align="center"><math>~0.0</math></td>
  <td align="center"><math>~0.0</math></td>
</tr>
<tr>
  <td align="center"><math>~+0.25</math></td>
  <td align="center"><math>~0.922</math></td>
  <td align="center"><math>~0.5901</math></td>
  <td align="center">4<sup>th</sup></td>
  <td align="center"><math>~-0.9976</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~0.0310</math></td>
</tr>
<tr>
  <td align="center"><math>~+0.50</math></td>
  <td align="center"><math>~1.265</math></td>
  <td align="center"><math>~1.012</math></td>
  <td align="center">4<sup>th</sup></td>
  <td align="center"><math>~-0.6591</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~0.0643</math></td>
</tr>
<tr>
  <td align="center"><math>~+0.75</math></td>
  <td align="center"><math>~1.688</math></td>
  <td align="center"><math>~2.315</math></td>
  <td align="center">4<sup>th</sup></td>
  <td align="center"><math>~-0.4767</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~0.1007</math></td>
</tr>
<tr>
  <td align="center"><math>~+1.00</math></td>
  <td align="center"><math>~2.145</math></td>
  <td align="center"><math>~\infty</math></td>
  <td align="center">4<sup>th</sup></td>
  <td align="center"><math>~-0.3695</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~0.1416</math></td>
</tr>
</table>
</div>


The second and third columns of [[#Table5|Table 5]] detail how, respectively, <math>~|\Delta|</math> and <math>~|g_{0,0,m}|</math> vary with location, <math>~-1 \le \eta\cos\theta \le +1</math>, in the equatorial plane <math>~(\cos\theta = \pm 1)</math> of a slim, <math>~n=\tfrac{3}{2}</math> PP torus.  Table 5 also contains an evaluation of the phase angle, <math>~\alpha_{0,0,m}</math>, across the equatorial plane of the same torus.  As indicated, in making this evaluation, care has been taken to place the phase angle in the proper quadrant of the equatorial plane.  Specifically &#8212; keeping in mind that, according to Blaes' analytic solution, the numerator of the arctangent argument, <math>~\mathrm{Im}(\Delta)</math>, is always negative &#8212; the phase angle should land in either the 3<sup>rd</sup> or 4<sup>th</sup> quadrant depending on whether the denominator is, respectively, negative <math>~(\eta\cos\theta < 0)</math> or positive <math>~(\eta\cos\theta > 0)</math>.  Because a standard evaluation of the arctangent function returns an angle that lies either in the 1<sup>st</sup> quadrant (positive argument) or the 4<sup>th</sup> quadrant (negative argument), we have added <math>~\varphi_\mathrm{shift} = \pi</math> to the value returned by the arctangent function in order to push the phase angle from the 1<sup>st</sup> to the 3<sup>rd</sup> quadrant wherever the denominator is negative  &#8212; that is to say, this phase shift has been implemented, throughout the range, <math>-1 \le \eta\cos\theta < 0</math>.

====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>