Editing Apps/ImamuraHadleyCollaboration (section)

=====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>