Editing Apps/ImamuraHadleyCollaboration (section)

====Tori with Small but Finite &beta;====
Using perturbation theory with <math>~\beta</math> serving effectively as an order parameter, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] extended his analytic analysis of slender tori to, what he refers to as, "more distorted thick tori."  We prefer to describe them as tori with still small, but finite, values of <math>~\beta</math>.  Blaes found that, for each ''m'', only the zeroth order co-rotation mode, [[#CorotationMode|described above]], becomes unstable at higher order.  To leading order in <math>~\beta</math>, Blaes shows that (see his equations 1.10 and 1.11) the, now complex, eigenfunction is,
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<math>~\delta W_{0,0,m}</math>
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<math>~\approx</math>
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<math>~C ~\exp[i(m\varphi + \sigma_{0,0,m} t)]  \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>
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and, to leading order in <math>~\beta</math>, the associated ''complex'' eigenfrequency is,
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<math>~\sigma_{0,0,m}  </math>
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<math>~\approx</math>
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<math>~-m \Omega_0 - ~i~m\Omega_0 \biggl( \frac{3}{2n+2}  \biggr)^{1/2}\beta \, .</math>
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In an [[Appendix/Ramblings/PPToriPt2A#Stability_Analyses_of_PP_Tori_.28Part_2.29|accompanying chapter]] that we have relegated to our [[Appendix/Ramblings#Ramblings|Ramblings Appendix]], we demonstrate in detail that this pair of ''complex'' expressions does provide a (leading order) solution to the "thick torus" eigenvalue problem.  Notice that if <math>~\beta</math> is set to zero, these two expressions reduce to the (purely real) expressions for the ''j'' = ''k'' = 0, slender torus mode.