Editing Apps/ImamuraHadleyCollaboration (section)

===Blaes (1985)===

====Setup====

<table border="1" cellpadding="5" align="right" width="30%">
<tr><td align="center" bgcolor="lightgreen">
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
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Fig1.png|center|300px|Figure 1 from Blaes (1985)]]
  </td>
</tr>
<tr>
  <td align="left">
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>
  </td>
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</table>
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,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x^2 = \biggl(1-\frac{\varpi}{\varpi_0}\biggr)^2 + \biggl(\frac{z}{\varpi_0}\biggr)^2</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\theta = \tan^{-1}\biggl[\frac{\zeta}{1-\chi}\biggr] \, ;</math>
  </td>
</tr>
<tr><td align="center" colspan="3">&nbsp; &nbsp; or &nbsp; &nbsp; &nbsp; &nbsp;</td></tr>
<tr>
  <td align="right">
<math>~\frac{\varpi}{\varpi_0} = 1 - x\cos\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{z}{\varpi_0} = x\sin\theta \, .</math>
  </td>
</tr>
</table>
</div>

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,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~p_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~p_\mathrm{max} f^{n+1}\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\rho_\mathrm{max} f^{n}\, ,</math>
  </td>
</tr>
</table>
</div>
where, the two-dimensional dimensionless enthalpy distribution is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~f(x,\theta) </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~1 - \frac{x^2}{\beta^2}\biggl[ 
1 + x(3\cos\theta -\cos^3\theta) \biggr] \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\beta^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{2n}{\mathfrak{M}_0^2} \, ,</math>
  </td>
</tr>

</table>
</div>
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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta W \equiv \biggl[ \frac{\Omega_0 \rho_\mathrm{max}}{p_\mathrm{max}} \biggr]W </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; 
<math>~\Rightarrow</math>
&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>
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.

<div align="center" id="EigenvalueBlaes85">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="lightgreen">
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
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Eq3.2.png|600px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
<tr>
  <td align="left">
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>
  </td>
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</table>
</div>

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.

====Normal Modes in Slender Tori====
=====Establishing the Simpler Eigenvalue Problem=====

In what he termed the "slender torus approximation," [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] found it advantageous to introduce a function, <math>~\eta</math>, defined in terms of the equilibrium enthalpy distribution, <math>~f</math>, such that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta^2 \equiv 1 - f</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{x^2}{\beta^2}\biggl[ 
1 + x(3\cos\theta -\cos^3\theta) \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

One nice feature of this parameter is that, for all PP tori, its value varies from zero at the density maximum (also the origin of the Blaes85 polar coordinate system) to unity at the surface of the torus.  Also, in the thin torus limit <math>~(\beta \ll 1)</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta^2</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{x^2}{\beta^2} \, ,</math>
  </td>
</tr>
</table>
</div>
so <math>~\eta</math> can effectively be used as the independent radial coordinate in place of <math>~x</math>.  As we demonstrate in detail in an [[Apps/PapaloizouPringle84#Slender_Torus_Approximation|accompanying discussion]], in the Blaes85 "slender torus approximation," some of the terms in his equation (3.2) governing PDE dominate over others, facilitating simplification.  The result &#8212; equation (1.6) in [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], also displayed in the following bordered box &#8212; is a well-defined eigenvalue problem whose solution(s) provide approximate descriptions of normal mode(s) of oscillation in slender PP tori.  Notice that Blaes attaches the superscript, (0), to denote eigenvector solutions to this governing PDE are only approximate solutions valid in the slender torus approximation.

<div align="center" id="EigenvalueBlaes85">
<table border="1" cellpadding="5" width="80%">
<tr><td align="center" bgcolor="lightgreen">
Equation (1.6) &#8212; identical to Eq. (3.5) &#8212; extracted without modification from p. 555 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
</td></tr>
<tr>
  <td align="center">
[[File:Blaes85Eq1.6.png|600px|center|Blaes (1985, MNRAS, 216, 553)]]
<!-- [[Image:AAAwaiting01.png|400px|center|Whitworth (1981) Eq. 5]] -->
  </td>
</tr>
<tr>
  <td align="left">
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>
  </td>
</tr>
</table>
</div>

=====Trial Eigenfunction=====
Following [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] &#8212; see our [[Apps/Blaes85SlimLimit#Oscillations_of_PP_Tori_in_the_Slim_Torus_Limit|accompanying discussion]] for details &#8212; If we assume that the eigenfunction is of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta W^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\eta^{|k|} \exp(ik\theta) \biggr] \Upsilon(\eta) \, ,</math>
  </td>
</tr>
</table>
</div>
we find that the function, <math>~\Upsilon(\eta)</math>, must satisfy the one-dimensional, 2<sup>nd</sup>-order ODE,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~(1-\eta^2)
\frac{d^2\Upsilon}{d\eta^2} + \frac{1}{\eta}\biggl[(2|k|+1) - (2|k|+1+2n)\eta^2\biggr]\frac{d\Upsilon}{d\eta} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2n \biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon  \, ,
</math>
  </td>
</tr>
</table>
</div>
which is a fairly standard looking eigenvalue problem.  If, furthermore, we adopt an independent variable given by the expression,
<div align="center">
<math>\Lambda \equiv 2\eta^2 - 1 \, ,</math>
</div>
in which case, 
<div align="center">
<math>\frac{d}{d\eta} \rightarrow 2^{3/2} (\Lambda+1 )^{1/2}\frac{d}{d\Lambda}</math>
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
<math>\frac{d^2}{d\eta^2} \rightarrow 
\biggl[ 4\frac{d}{d\Lambda} + 8(\Lambda+1)\frac{d^2}{d\Lambda^2} \biggr] \, ,
</math>
</div>
<!-- DETAILS
<div align="center">
<math>\frac{d}{d\eta} \rightarrow \frac{d\Lambda}{d\eta} \cdot \frac{d}{d\Lambda} = 4\eta \frac{d}{d\Lambda} 
= 2^{3/2}(\Lambda+1)^{1/2}\frac{d}{d\Lambda}</math>
</div>
and, 
<div align="center">
<math>\frac{d^2}{d\eta^2} \rightarrow 2^{3/2}\biggl[\Lambda+1\biggr]^{1/2}\cdot 
\frac{d}{d\Lambda} \biggl\{ 2^{3/2}\biggl[\Lambda+1\biggr]^{1/2}\frac{d}{d\Lambda}\biggr\}
=2^{3}\biggl[\Lambda+1\biggr]^{1/2} \biggl\{ 
\frac{1}{2}\biggl[\Lambda+1\biggr]^{-1/2}\frac{d}{d\Lambda} + \biggl[\Lambda+1\biggr]^{1/2}\frac{d^2}{d\Lambda^2}\biggr\}
=
4\frac{d}{d\Lambda} + 8 (\Lambda+1 )\frac{d^2}{d\Lambda^2}
</math>
</div>
END DETAILS -->
the statement of the eigenvalue problem becomes,

<div align="center">
<table border="0" cellpadding="5" align="center">
<!--
<tr>
  <td align="right">
<math>~
- 2n \biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1-\tfrac{1}{2}(\Lambda+1) \biggr]
\biggl[ 4\frac{d\Upsilon}{d\Lambda} + 8(\Lambda+1)\frac{d^2\Upsilon}{d\Lambda^2} \biggr]
+ 2^{1/2}(\Lambda+1)^{-1/2} \biggl[(2|k|+1) - (2|k|+1+2n)\tfrac{1}{2}(\Lambda+1)\biggr] 2^{3/2} (\Lambda+1 )^{1/2}\frac{d\Upsilon}{d\Lambda} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2(1-\Lambda)
\biggl[ \frac{d\Upsilon}{d\Lambda} + 2(\Lambda+1)\frac{d^2\Upsilon}{d\Lambda^2} \biggr]
+ 2\biggl[(4|k|+2) - (2|k|+1+2n)(\Lambda+1)\biggr] \frac{d\Upsilon}{d\Lambda} 
</math>
  </td>
</tr>
-->
<tr>
  <td align="right">
<math>~
\frac{n}{2} \biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)^2 -|k|\biggr]  \Upsilon
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\Lambda^2-1)\frac{d^2\Upsilon}{d\Lambda^2} 
+ \biggl[(n - |k|-1 ) + (|k|+1+n)\Lambda\biggr] \frac{d\Upsilon}{d\Lambda} \, .
</math>
  </td>
</tr>
</table>
</div>

This definition of the eigenvalue problem is exactly of the [[#SingularSturmLiouville|singular Sturm-Liouville form, as described above]], from which we draw the following coefficient associations:


<div align="center">
<math>~(\alpha - \beta) \leftrightarrow (n-|k|-1)</math>&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>~(\alpha+\beta+2) \leftrightarrow (|k|+1+n) \, ,</math>
</div>

that is,

<div align="center">
<math>~\alpha \leftrightarrow (n-1)</math>&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>~\beta\leftrightarrow |k|\, .</math>
</div>

Hence the j<sup>th</sup> solution to this eigenvalue problem is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Upsilon_j(\Lambda)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~J_j^{n-1,|k|}(\Lambda) \, ,</math>
  </td>
</tr>
</table>
</div>
with the associated eigenvalue,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{n}{2}\biggl[\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)_j^2 -|k|\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~j(j+n + |k|)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \biggl( \frac{\sigma}{\Omega_0} + m  \biggr)_j^2  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{n}\biggl[2j^2+2jn + 2j|k| + n|k|\biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Table 4 presents the polynomial expressions that are relevant to the first three (j = 0, 1, 2) eigenfunctions of "slender" PP-tori, along with the associated eigenvalues.

<div align="center" id="Table4">
<table align="center" border="1" cellpadding="5">
<tr>
  <th align="center" colspan="3"><font size="+1">Table 4:  Example Jacobi Polynomials Relevant to "Slender" PP-Tori Eigenvectors</font></th>
</tr>
<tr>
  <td align="center"><math>~j</math></td>
  <td align="center"><math>~J_j^{n-1,|k|}(2\eta^2-1)</math></td>
  <td align="center"><math>~\biggl( \frac{\sigma}{\Omega_0} + m  \biggr)_j^2</math></td>
</tr>
<tr>
  <td align="center">
<math>~0</math>
  </td>
  <td align="center">
<math>~1</math>
  </td>
  <td align="center">
<math>~0</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~1</math>
  </td>
  <td align="center">
<math>~(n + 1 + |k|)\eta^2 - (1 + |k|)</math>
  </td>
  <td align="center">
<math>~\tfrac{1}{n}[2+2n+(2+n)|k|]</math>
  </td>
</tr>
<tr>
  <td align="center">
<math>~2</math>
  </td>
  <td align="center">
<math>~
\tfrac{1}{2}[(6 + 5|k|+k^2) +n(5+2|k|) +n^2 ]\eta^4
- [4 + 4|k|+k^2 +n(2+|k|)]\eta^2 
+  \tfrac{1}{2} [2 + 3|k| + k^2] 
</math>
  </td>
  <td align="center">
<math>~\tfrac{1}{n}[8+4n + (4+n)|k|] </math>
  </td>
</tr>
</table>
</div>

=====Analytic Solution=====
{| class="BlaesAnalytic" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! 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>]]
|}
Piecing this together &#8212; including, as well, the time and azimuthal mode, ''m'', dependence &#8212; we therefore ultimately conclude that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta W_{j,k,m}^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\eta^{|k|} \exp[i(k\theta + m\varphi + \sigma_{j,k,m} t)]  ~J_j^{n-1,|k|}(2\eta^2-1)  \, ,</math>
  </td>
</tr>
</table>
</div>
and,
&nbsp; <br />

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{\sigma}{\Omega_0} \biggr)_{j,k,m}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-m \pm \biggl[\frac{2j^2+2jn + 2j|k| + n|k|}{n}\biggr]^{1/2} \, .</math>
  </td>
</tr>
</table>
</div>

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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{\rho^'}{\rho_0} \biggr)_{j,k,m}^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>


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.

=====Plots of a Few Example Eigenvectors=====

[http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] makes the following brief comments regarding the role of the indices, ''j'' and ''k'', in defining the geometric distortions associated with each normal mode.  "<font color="green">The significance of ''k'' is obvious &#8212; it is simply the number of wavelengths on a given isobaric surface</font> [''i.e.'', varying <math>~\theta</math> while holding <math>~\eta</math> constant] <font color="green">so that there are 2''k'' nodal lines radiating from the torus centre at</font> <math>~\eta = 0</math>."  Alternatively, moving radially through the torus at a fixed <math>~\theta</math>, "<font color="green">In addition to the</font> [radial] <font color="green">node at <math>~\eta = 0</math> which occurs for <math>~k \ne 0</math>, there are nodal surfaces at ''j'' other distinct values of <math>~\eta</math> inside the torus</font>."


The plots presented below in Figures 1 and 2 are intended to further illustrate the radial structure of various normal modes, and to begin to draw a connection between the analytically describable attributes of ''slender'' PP tori and the broad range of self-gravitating tori whose stability has been investigated numerically by the [[#See_Also|Imamura &amp; Hadley collaboration]].  For nine separate "slender torus" normal modes, panels A, B, &amp; C of Figure 1 display the radial variation of:  (Middle) <math>~\delta W_{j,k}^{(0)}</math> versus <math>~\eta</math>; (Left) the associated Jacobi polynomial, <math>~J_{j}^{n-1,|k|}</math> versus <math>~x</math>, where <math>~x = (2\eta^2-1)</math>; and (Right) <math>\log_{10}|\delta W_{j,k}^{(0)}|</math> versus <math>~\eta</math>.  In panel A, all three displayed modes have ''j'' = 0; in panel B, all modes have ''j'' = 1; and in panel C, all modes have ''j'' = 2.  In all three of these panels (A, B, C), blue curves are associated with ''k'' = 0; red curves are associated with ''k'' = 1; and green curves are associated with ''k'' = 2.  


The left-most segment of each panel (A, B, C) displays the specific Jacobi polynomial that seeds the oscillatory behavior of that panel's associated radial eigenvector; they have been shown here in order to emphasize overlap with solutions to the singular Sturm-Liouville problem as illustrated in [[#Table3|Table 3, above]].  In reality, the right-most segment of each panel (A, B, C) presents the same information as is presented in the middle segment, but by plotting the log of the absolute value of the radial eigenfunction we are displaying that information in a manner that aligns with the means of presentation used by the [[#See_Also|Imamura &amp; Hadley collaboration]].  Because the log of this function goes to minus infinity whenever the eigenfunction crosses zero, it is particularly easy to identify the number and location of radial nodes in the right-most segment of each panel.  Referencing again the comment by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)], there is a node at <math>~\eta = 0</math>, except when ''k = 0'' (the blue curves in each panel of our Figure 1); in addition, there are nodal surfaces at ''j'' other distinct values of <math>~\eta</math> inside the torus &#8212; that is, between <math>~\eta = 0</math> and  <math>~\eta = 1</math> or between <math>~\eta = 0</math> and  <math>~\eta = -1</math>.


<div align="center">
<table border="1" cellpadding="5" align="center">
<tr><th><font size="+1">Figure 1:  Blaes85 Eigenfunctions for Slender Tori with <math>~(n,q) = (\tfrac{3}{2},2)</math></font></th></tr>
<tr><th><font size="+1">Panel A: &nbsp;&nbsp; j = 0 and (k = 0, 1, 2)</font></th></tr>
<tr><td align="center">
[[File:N1.5j0_Combinedsmall.png|750px|center|j0 Eigenfunction from Blaes85]]
</td></tr>
<tr><th><font size="+1">Panel B: &nbsp;&nbsp; j = 1 and (k = 0, 1, 2)</font></th></tr>
<tr><td align="center">
[[File:N1.5j1_Combinedsmall.png|750px|center|j1 Eigenfunction from Blaes85]]
</td></tr>
<tr><th><font size="+1">Panel C: &nbsp;&nbsp; j = 2 and (k = 0, 1, 2)</font></th></tr>
<tr><td align="center">
[[File:N1.5j2_Combinedsmall.png|750px|center|j2 Eigenfunction from Blaes85]]
</td></tr>
</table>
</div>


<span id="CorotationMode">
<font color="maroon"><b>COROTATION MODE:</b></font> As [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] has pointed out, for any azimuthal number, ''m'', the simplest mode occurs for ''j'' = ''k'' = 0. </span> In this case, the [[#Analytic_Solution|analytic expression]] for the ''slender torus'' eigenfrequency is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma^{(0)}_{0,0,m}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-m \Omega_0 \, ,</math>
  </td>
</tr>
</table>
</div>
and, the associated analytic expression for the eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta W_{0,0,m}^{(0)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~C ~\exp[i(m\varphi + \sigma^{(0)}_{0,0,m} t)]  ~J_0^{n-1,0}(2\eta^2-1) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>C~\exp[im(\varphi - \Omega_0 t)] \, ,</math>
  </td>
</tr>
</table>
</div>

where, <math>~C</math> is an arbitrary, overall scale factor.  A plot showing the radial structure of this "simplest mode" &#8212; assuming <math>~C=1</math> &#8212; is provided by the blue curve in the middle segment and in the right-hand segment of panel A.  The perturbation, <math>~\delta W_{0,0,m}^{(0)}</math>, has a constant amplitude throughout the configuration.  Note, however, that as the left-most segment of panel D in Figure 2 shows, the fractional ''density'' perturbation is not uniform throughout the configuration; this is primarily because an extra factor of <math>~(1-\eta^2)^{-1}</math> appears in the expression for <math>~\rho^'/\rho_0</math> &#8212; [[#DensityPerturbation2|see above]].

While this <font color="maroon"><b>COROTATION</b></font> mode exhibits a rather boring structure relative to other modes, it plays a key role in the Blaes (1985) publication.  As his analysis is expanded to include the examination of oscillations in tori with finite &#8212; but still small &#8212; equilibrium values of <math>~\beta</math>, he finds that, for all ''m'', both the eigenfunction and the eigenfrequency of the ''j'' = ''k'' = 0 mode exhibit nonzero imaginary components.  (More on this, [[#Tori_with_Small_but_Finite_.CE.B2|immediately below]].)


<div align="center">
<table border="1" cellpadding="5" align="center" width="750px">
<tr><th colspan="3"><font size="+1">Figure 2:  Mass Density Eigenfunctions for Slender Tori with <math>~(n,q) = (\tfrac{3}{2},2)</math></font></th></tr>
<tr><th colspan="3"><font size="+1">Panel D: &nbsp;&nbsp; j = 0 (left), j = 1 (middle), and j = 2 (right) and, in each case, (k = 0, 1, 2)</font></th></tr>
<tr><td align="right" colspan="3">
[[File:LogDensityCombinedSmall01.png|700px|right|j0 Eigenfunction from Blaes85]]
</td></tr>
<tr><th colspan="3"><font size="+1">Panel E: &nbsp;&nbsp; Models J1 (left), E3 (middle) and E2 (right) from Figs. 2 &amp; 4 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)]</font><p></p>
(blue curves show log10 of mass-density fluctuation amplitudes)</th></tr>
<tr>
  <td align="right" width="250px">[[File:ImamuraPaper2Fig2ModelJ1.png|right|250px|Model J1 Radial Eigenfunction]]</td>
  <td align="right" width="250px">[[File:ImamuraPaper2Fig4ModelE3Revised.png|right|220px|Model E3 Radial Eigenfunction]]</td>
  <td align="right" width="250px">[[File:ImamuraPaper2Fig4Eigenfunction.png|right|200px|Model E2 Radial Eigenfunction]]</td>
</tr>
</table>
</div>


<font color="maroon"><b>MODEL E3 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)]:</b></font> In the middle segment of Figure 2's panel E, we have re-printed the semi-log plot of the magnitude of the radial eigenfunction that developed in model E3 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)].  Both curves in the plot exhibit two radial nodes:  One associated with the co-rotation radius, which we have purposely aligned with the node at <math>~\eta = 0</math> that appears in the middle segment of panel D; and one that lies between the center of the torus and the surface.  This association strongly suggests that the unstable mode found in model E3 displays an underlying ''linear'' radial eigenfunction akin to the "''j'' = 1" Jacobi polynomial. Also, because the model E3 eigenfunction exhibits a node at co-rotation, we conclude that <math>~k \ne 0</math>; and, while the spacing between the two E3 nodes suggests a value of the index ''k'' greater than unity, identifying the precise value of ''k'' may require an examination of the node count above/below the equatorial plane along one or more isobaric surfaces.


<font color="maroon"><b>MODEL E2 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)]:</b></font>  In the right-most segment of Figure 2's panel E, we have re-printed the semi-log plot of the magnitude of the radial eigenfunction that developed in model E2 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)].  In addition to the radial node aligned with co-rotation, both curves in the plot exhibit (at least) four radial nodes.  This strongly suggests that this unstable mode displays an underlying radial eigenfunction that is a polynomial of degree, ''j'' = 4 (or higher).


<font color="maroon"><b>MODEL J1 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)]:</b></font>  In the left-most segment of Figure 2's panel E, we have re-printed the semi-log plot of the magnitude of the radial eigenfunction that developed in model J1 from [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, et al. (2014)].  In all respects, the blue curve (associated with the nonaxisymmetric structure of the density fluctuation) resembles the red curve displayed in the left-most segment of panel D.  Because this <font color="maroon"><b>MODEL J1</b></font> eigenfunction exhibits a single radial node that is closely aligned with the center of the torus strongly suggests that the unstable mode corresponds to an oscillation mode having <math>~j = 0</math>, but <math>~k \ne 0</math>.  Given its similarity to the red curve in left-most panel of panel D, we suspect that <math>~k = 1</math>.

====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,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta W_{0,0,m}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<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>
  </td>
</tr>
</table>
</div>
and, to leading order in <math>~\beta</math>, the associated ''complex'' eigenfrequency is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sigma_{0,0,m}  </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~-m \Omega_0 - ~i~m\Omega_0 \biggl( \frac{3}{2n+2}  \biggr)^{1/2}\beta \, .</math>
  </td>
</tr>
</table>
</div>

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.