Editing Apps/ImamuraHadleyCollaboration (section)

=====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>,
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<math>~\Delta \varphi|_\mathrm{tot}</math>
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<math>~=</math>
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<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} \, .
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