Editing Apps/ImamuraHadleyCollaboration (section)

=====Analytic Solution=====
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! style="height: 125px; width: 125px; background-color:#ffeeee;" |[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like_2|<b>Analytic Analysis<br />by <br />Blaes<br />(1985)</b>]]
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Piecing this together &#8212; including, as well, the time and azimuthal mode, ''m'', dependence &#8212; we therefore ultimately conclude that,
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<math>~\delta W_{j,k,m}^{(0)}</math>
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<math>~=</math>
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<math>~\eta^{|k|} \exp[i(k\theta + m\varphi + \sigma_{j,k,m} t)]  ~J_j^{n-1,|k|}(2\eta^2-1)  \, ,</math>
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and,
&nbsp; <br />

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<math>~\biggl( \frac{\sigma}{\Omega_0} \biggr)_{j,k,m}  </math>
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<math>~=</math>
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<math>~-m \pm \biggl[\frac{2j^2+2jn + 2j|k| + n|k|}{n}\biggr]^{1/2} \, .</math>
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This is a fantastic result, as it provides a totally analytic description of the eigenvectors that define a full spectrum of normal-mode oscillations in ''slender'' tori that have uniform specific angular momentum and a range of reasonable polytropic indexes. As [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] states &#8212; see the discussion immediately following his equation (1.8) &#8212; "<font color="green">The three parameters ''j'' (a non-negative integer), ''k'' (an integer) and ''m'' (an integer) completely describe the solution.</font>"


Pulling from the [[#DensityPerturbation2|expression developed, above]], this means that the corresponding density perturbation is,
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<math>~\biggl(\frac{\rho^'}{\rho_0} \biggr)_{j,k,m}^{(0)}</math>
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<math>~=</math>
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<math>~\frac{n}{n+1}\biggl(\frac{\sigma }{\Omega_0 } + m\biggr)_{j,k,m}^{(0)} \frac{\delta W_{j,k,m}^{(0)}}{f} </math>
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&nbsp;
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<math>~=</math>
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<math>~\pm ~\frac{n^{1/2}}{(n+1)} \cdot \frac{\eta^{|k|}}{1-\eta^2}\biggl[2j^2+2jn + 2j|k| + n|k|\biggr]^{1/2}  
\exp[i(k\theta + m\varphi + \sigma_{j,k,m} t)]  ~J_j^{n-1,|k|}(2\eta^2-1)  \, .
</math>
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We should keep in mind that some restrictions accompany the ''slender'' torus approximation.
* Each eigenvector represents a solution of an eigenvalue problem that is simpler than the eigenvalue problem defined by equation (2.19) of [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou &amp; Pringle (1985)] &#8212; [[#Papaloizou_.26_Pringle_.281985.29|see above]] &#8212; or, equivalently, equation (3.2) of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; [[#Setup|see above]].
* In the ''slender'' torus, all isobaric surfaces have meridional cross-sections that are perfect circles; and the surface, in particular, has a cross-sectional radius whose value is obtained by setting <math>~\eta = 1 ~~\Rightarrow~~r_\mathrm{torus} = \beta\varpi_0</math>.
And the analytically specified eigenvector exhibits the following simplified attributes:
* As pointed out by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; see the comments immediately following his equation (1.8) &#8212; for all values of <math>~(j,k,m)</math>, the eigenfrequency, <math>~\sigma/\Omega_0</math>, is real, hence all of the identified oscillation modes are stable.
<!--* As specified by the relevant Jacobi polynomial, the radial component of each eigenfunction is insensitive to the sign of <math>~\eta</math>, so each eigenvector is symmetric about the center of the circular cross-section.-->
* For all values of <math>~(j,k)</math>, the eigenfunction is real and, as a result, the constant phase locus of each eigenvector will exhibit no azimuthal structure; see more discussion of this attribute, below.