Editing Apps/ImamuraHadleyCollaboration (section)

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Fig. 1 extracted without modification from p. 554 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>
"''Oscillations of Slender Tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, <br />vol. 216, Issue 3, October 1985, pp. 553-563<br />&copy; Royal Astronomical Society<br />
https://doi.org/10.1093/mnras/216.3.553
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[[File:Blaes85Fig1.png|center|300px|Figure 1 from Blaes (1985)]]
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This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society. 
<div align="center">[[File:PermissionsRectYellow.png|75px|link=Appendix/CopyrightPermissions#Blaes1985]]<br />Original publication website:  https://academic.oup.com/mnras/article/216/3/553/1005987</div>
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As is illustrated in his Figure 1 &#8212; which we have reprinted for convenience here, on the right &#8212; [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] shifted from cylindrical coordinates to a (dimensionless) polar-coordinate <math>~(x,\theta)</math> system whose origin sits at the pressure-maximum of the initial, unperturbed Papaloizou-Pringle torus, a distance, <math>~\varpi_0</math>, from the symmetry axis of the cylindrical coordinate system.  Mapping between these two coordinate systems is accomplished via the relations,

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<math>~x^2 = \biggl(1-\frac{\varpi}{\varpi_0}\biggr)^2 + \biggl(\frac{z}{\varpi_0}\biggr)^2</math>
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&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
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<math>~\theta = \tan^{-1}\biggl[\frac{\zeta}{1-\chi}\biggr] \, ;</math>
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<tr><td align="center" colspan="3">&nbsp; &nbsp; or &nbsp; &nbsp; &nbsp; &nbsp;</td></tr>
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<math>~\frac{\varpi}{\varpi_0} = 1 - x\cos\theta</math>
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&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
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<math>~\frac{z}{\varpi_0} = x\sin\theta \, .</math>
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Furthermore, he set <math>~\Gamma = (n+1)/n</math>, and rewrote the (initial, unperturbed) equilibrium pressure and density distributions in terms of the dimensionless enthalpy distribution in the PP torus, namely,

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<math>~p_0 </math>
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<math>~=</math>
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<math>~p_\mathrm{max} f^{n+1}\, ,</math>
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<math>~\rho_0 </math>
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<math>~=</math>
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<math>~\rho_\mathrm{max} f^{n}\, ,</math>
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where, the two-dimensional dimensionless enthalpy distribution is,

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<math>~f(x,\theta) </math>
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<math>~\approx</math>
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<math>~1 - \frac{x^2}{\beta^2}\biggl[ 
1 + x(3\cos\theta -\cos^3\theta) \biggr] \, ,</math>
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<math>~\beta^2</math>
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<math>~\equiv</math>
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<math>~\frac{2n}{\mathfrak{M}_0^2} \, ,</math>
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and <math>~\mathfrak{M}_0</math> is the Mach number of the circular, azimuthal flow at the pressure and density maximum.  It is important to appreciate that <math>~\beta</math> is a dimensionless parameter whose value dictates the relative thickness of the equilibrium torus; slim tori have <math>~\beta \ll 1</math>. 

<span id="DensityPerturbation2">Finally, Blaes replaced the perturbation variable,</span> <math>~W</math>, preferred by Papaloizou &amp; Pringle (1985) with an equivalent but ''dimensionless'' perturbation variable,
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<math>~\delta W \equiv \biggl[ \frac{\Omega_0 \rho_\mathrm{max}}{p_\mathrm{max}} \biggr]W </math>
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&nbsp; &nbsp; &nbsp; 
<math>~\Rightarrow</math>
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<math>~\frac{\rho^'}{\rho_0} 
= \biggl(\frac{ \bar\sigma }{\gamma_g \Omega_0 } \biggr) \frac{\delta W}{f} 
= \frac{n}{n+1}\biggl(\frac{\sigma }{\Omega_0 } + m\cdot \frac{\Omega}{\Omega_0}\biggr) \frac{\delta W}{f} 
\, ,</math>
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where <math>~\Omega_0</math> is the angular frequency at the pressure and density maximum.  [Actually, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] calls this dimensionless variable <math>~W</math>, rather than <math>~\delta W</math>, so care must be taken when published equations from these separate studies are compared.]  After working carefully through these modifications &#8212; again, see our [[Apps/PapaloizouPringle84#Equivalent_Dimensionless_Expression|accompanying discussion]] for details &#8212; Blaes arrives at the governing PDE (his equation 3.2) that is highlighted in the following bordered box.  Notice that, in this published expression, <math>~\nu \equiv \sigma/\Omega_0</math>, represents the azimuthal-mode eigenfrequency, normalized to the system's orbital frequency at the origin of the Blaes85 coordinate system.

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Equation (3.2)  extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>
"''Oscillations of Slender Tori''"<p></p>
Monthly Notices of the Royal Astronomical Society, vol. 216, Issue 3, October 1985, pp. 553-563<br />&copy; Royal Astronomical Society<br />
https://doi.org/10.1093/mnras/216.3.553
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[[File:Blaes85Eq3.2.png|600px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
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This digital image has been posted by permission of author, O.M. Blaes, and by permission of Oxford University Press on behalf of the Royal Astronomical Society.<div align="center">[[File:PermissionsRectYellow.png|75px|link=Appendix/CopyrightPermissions#Blaes1985]]<br />Original publication website:  https://academic.oup.com/mnras/article/216/3/553/1005987</div>
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In a direct analogy with [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne's (1937)] analysis of normal modes of oscillation in homogeneous spheres &#8212; [[#Radial_Modes_in_Homogeneous_Spheres|discussed above]] &#8212; the ultimate objective here is to determine what two-dimensional eigenfunction(s), <math>~\delta W_j(x,\theta)</math>, and corresponding eigenfrequency(ies), <math>~\nu_j</math>,  satisfy this governing PDE for arbitrarily thick/thin PP tori.  In general, both the eigenfunction and corresponding eigenfrequency should be treated as complex functions/numbers.  As we summarize, below, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] derived analytic expressions that provide ''one approximate'' solution for tori with small, but finite, values of <math>~\beta</math>.  But, first, we will briefly review how he derived an entire spectrum of analytically specifiable normal modes in the limit of "slender tori," that is, tori for which <math>~\beta</math> is effectively zero.