Editing Apps/ImamuraHadleyCollaboration (section)

==Background==
===Imamura &amp; Hadley Collaboration===
Motivated especially by the numerical study of unstable nonaxisymmetric modes provided in [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H Paper I] and [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II] of the [[#See_Also|Imamura &amp; Hadley collaboration]], we have embarked on an effort to better understand what sparks the development of a wide range of complex eigenvectors in self-gravitating tori.  Of particular interest are "P-", "J-", and "E-modes," such as the ones whose eigenfunctions are illustrated here in Table 1.  (Click on a small image in the right-hand column to see a larger image.)  Our initial attempts to construct fits to these eigenvectors empirically have been described in [[Appendix/Ramblings/AzimuthalDistortions#Analyzing_Azimuthal_Distortions|a separate chapter]].  We begin, here, a more quantitative analysis of these structures.

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="7"><font size="+1"><b>Table 1:</b></font>&nbsp; J-, P- and E-mode Model Parameters Highlighted in Paper II
<p></p>[http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H K. Z. Hadley, P. Fernandez, J. N. Imamura, E. Keever, R. Tumblin, &amp; W. Dumas (2014, ''Astrophysics and Space Science'', 353, 191-222)]</td>
</tr>
<tr>
  <td align="center">Name</td>
  <td align="center"><math>~M_*/M_d</math></td>
  <td align="center"><math>~(n, q)</math><sup>&dagger;</sup></td>
  <td align="center"><math>~R_-/R_+</math></td>
  <td align="center"><math>~r_\mathrm{outer} \equiv \frac{R_+}{R_\mathrm{max}}</math></td>
  <td align="center"><math>~R_\mathrm{max}</math></td>
  <td align="center">Eigenfunction</td>
</tr>
<tr>
  <td align="center" colspan="6">Extracted from Table 2 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
  <td align="center" colspan="1">Extracted from Fig. 4 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
</tr>
<tr>
  <td align="center">E1</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.101</math></td>
  <td align="center"><math>~5.52</math></td>
  <td align="center"><math>~0.00613</math></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center">E2</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.202</math></td>
  <td align="center"><math>~2.99</math></td>
  <td align="center"><math>~0.0229</math></td>
  <td align="center">[[File:ImamuraPaper2Fig4.png|175px|Model E2]]</td>
</tr>
<tr>
  <td align="center">E3</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.402</math></td>
  <td align="center"><math>~1.74</math></td>
  <td align="center"><math>~0.159</math></td>
  <td align="center">[[File:ImamuraPaper2Fig4ModelE3.png|175px|Model E3]]</td>
</tr>
<tr>
  <td align="center" colspan="6">Extracted from Table 2 or Table 4 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
  <td align="center" colspan="1">Extracted from Fig. 3 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
</tr>
<tr>
  <td align="center">P1</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.452</math></td>
  <td align="center"><math>~1.60</math></td>
  <td align="center"><math>~0.254</math></td>
  <td align="center">[[File:ImamuraPaper2Fig3ModelP1.png|175px|Model P1]]</td>
</tr>
<tr>
  <td align="center">P2</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.500</math></td>
  <td align="center"><math>~1.49</math></td>
  <td align="center"><math>~0.403</math></td>
  <td align="center">[[File:ImamuraPaper2Fig3ModelP2.png|175px|Model P2]]</td>
</tr>
<tr>
  <td align="center">P3</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.600</math></td>
  <td align="center"><math>~1.33</math></td>
  <td align="center"><math>~1.09</math></td>
  <td align="center">&nbsp;</td>
</tr>
<tr>
  <td align="center">P4</td>
  <td align="center"><math>~100</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, 2)</math></td>
  <td align="center"><math>~0.700</math></td>
  <td align="center"><math>~1.21</math></td>
  <td align="center"><math>~3.37</math></td>
  <td align="center">[[File:ImamuraPaper2Fig3ModelP4.png|175px|Model P4]]</td>
</tr>
<tr>
  <td align="center" colspan="6">Extracted from Table 1 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
  <td align="center" colspan="1">Extracted from Fig. 2 of [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Paper II]</td>
</tr>
<tr>
  <td align="center">J1</td>
  <td align="center"><math>~0.01</math></td>
  <td align="center"><math>~(\tfrac{3}{2}, \tfrac{3}{2})</math></td>
  <td align="center"><math>~0.402</math></td>
  <td align="center"><math>~1.51</math></td>
  <td align="center"><math>~6.47</math></td>
  <td align="center">[[File:ImamuraPaper2Fig2ModelJ1B.png|175px|Model J1]]</td>
</tr>
<tr>
  <td align="left" colspan="7"><sup>&dagger;</sup>In all three papers from the [[#See_Also|Imamura &amp; Hadley collaboration]], <math>~q = 2</math> means, "[[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|uniform specific angular momentum]]."</td>
</tr>
</table>


[[File:ImamuraPaper2Fig4Eigenfunction.png|right|250px|Model E2 Radial Eigenfunction]]Here, for example, are a couple of questions guiding our study: 
*  Can we understand why the radial eigenfunction of, for example, model E2 &#8212; re-displayed here, on the right &#8212; exhibits a series of sharp dips whose spacing gets progressively smaller and smaller as the outer edge of the torus is approached?
*  Does the spatial structure of the unstable eigenvectors that appear in numerical simulations of geometrically thin, self-gravitating tori &#8212; such as model "P4" highlighted here in Table 1 &#8212; resemble the analytically defined eigenvector that is predicted by linear stability analyses to be unstable in ''massless'' Papaloizou-Pringle tori?

===Radial Modes in Homogeneous Spheres===

Before attempting to analyze natural modes of oscillation in a polytropic torus, it is useful to review what is known about radial oscillations in the geometrically simpler, uniform-density sphere. As we have reviewed in [[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|a separate chapter]], [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne (1937)] was the first to recognize that the set of eigenvectors that describe radial modes of oscillation in a homogeneous, self-gravitating sphere can be determined analytically.  In the present context, it is advantageous for us to pull Sterne's solution from a discussion of the same problem presented by [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)].  As we have summarized in [[SSC/PerspectiveReconciliation#Eulerian_Reformulation|a separate chapter]], the relevant eigenvalue problem is defined by the following one-dimensional, 2<sup>nd</sup>-order ODE:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
( 1-\chi_0^2 ) \frac{\partial^2 \xi}{\partial \chi_0^2}
+ \frac{2}{\chi_0}\biggl[ 1 - 2\chi_0^2 \biggr] \frac{\partial \xi}{\partial \chi_0}
+ \biggl(4 - \frac{2}{\chi_0^2} \biggr) \xi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \mathfrak{F} \xi \, ,
</math>
  </td>
</tr>
</table>
</div>
where, it is understood that the expression for the (spatially and temporally varying) radial location of each spherical shell is,
<div align="center">
<math>~r(r_0,t) = r_0 + \delta r(r_0)e^{i\sigma t} \, ,</math>
</div>
and in the present context we are adopting the variable notation, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_0</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{r_0}{R} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\xi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{\delta r}{R} \, ,</math>
  </td>
</tr>
</table>
</div>
and <math>~R</math> is the initial (unperturbed) radius of the sphere.  Here, the eigenvalue is related to the physical properties of the homogeneous sphere via the relation,
<div align="center">
<math>~\mathfrak{F}  \equiv \frac{2}{\gamma_\mathrm{g}} \biggl[\biggl(\frac{3\sigma^2}{4\pi G\rho_0}\biggr) + (4 - 3\gamma_\mathrm{g}) \biggr]  \, .</math>
</div>


Drawing from [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland's (1964)] presentation &#8212; see his p. 29, and [[SSC/PerspectiveReconciliation#Eulerian_Reformulation|our related discussion]] &#8212; the following table details the eigenvectors (radial eigenfunction and associated eigenfrequency) for the three lowest radial modes (m = 2, 4, 6) that satisfy this wave equation; the figure displayed in the right-most column has been extracted directly from p. 29 of Rosseland and shows the behavior of the lowest five radial modes (m = 2, 4, 6, 8, 10, as labeled) over the interval <math>~0 \le \chi_0 \le 1</math>.

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr>
  <th align="center" colspan="4">
<font size="+1"><b>Table 2</b></font><p></p>
Rosseland's (1964) Eigenfunctions for Homogeneous Sphere<p></p>
Figure in the right-most column extracted from p. 29 of [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)]<p></p>
"''The Pulsation Theory of Variable Stars''" (New York:  Dover Publications, Inc.)
  </th>
</tr>
<tr>
  <td align="center">Mode</td>
  <td align="center" colspan="1">Eigenfunction</td>
  <td align="center">Square of Eigenfrequency:<p></p><math>~3\sigma^2/(4\pi G\rho)</math></td>
  <td align="center" rowspan="5">[[File:RosselandModesFigure3.png|350px|center|Rosseland (1937)]]</td>
</tr>
<tr>
  <td align="center"><math>~m</math></td>
  <td align="center">As Published</td>
  <td align="center"><math>~\frac{m}{2}(m+1)\gamma - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="left"><math>~\xi = -2\chi_0</math></td>
  <td align="center"><math>~3\gamma - 4</math></td>
</tr>
<tr>
  <td align="center"><math>~4</math></td>
  <td align="left"><math>~\xi = -\frac{20}{3}\chi_0 + \frac{28}{3}\chi_0^3</math></td>
  <td align="center"><math>~10\gamma-4</math></td>
</tr>
<tr>
  <td align="center"><math>~6</math></td>
  <td align="left"><math>~\xi = -14\chi_0 + \frac{252}{5} \chi_0^3 - \frac{198}{5} \chi_0^5</math></td>
  <td align="center"><math>~21\gamma-4</math></td>
</tr>
</table>
</div>

We should point out that, except for the lowest (m = 2) mode, each of the radial eigenfunctions crosses zero at least once at some location(s) that resides between the center <math>~(\chi_0=0)</math>  of and the surface <math>~(\chi_0 = 1)</math> of the sphere.  More specifically, for each mode, <math>~m</math>, the number of such radial "nodes" is <math>~(m-2)/2</math>.  The locations of these nodes is apparent from even a casual inspection of the figure presented in the right-most column of Table 2.

When these radial modes of oscillation are discussed in the astrophysics literature, the conditions that give rise to a dynamical instability are often emphasized.  Specifically, each mode becomes unstable when <math>\sigma^2</math> becomes negative, which translates into a value of <math>~\gamma < \gamma_\mathrm{crit}(m)</math> &#8212; see our [[SSC/Stability/UniformDensity#Properties_of_Eigenfunction_Solutions|related discussion of the properties of eigenfunction solutions]] in the context of [http://adsabs.harvard.edu/abs/1937MNRAS..97..582S Sterne's (1937)] analysis of this stability problem.  Here we want to emphasize that all of natural modes of oscillation ''exist'' even when the configuration is dynamically stable.

===Singular Sturm-Liouville Problem===

[[Apps/Blaes85SlimLimit#Singular_Sturm-Liouville_Problem|As we have discussed in a separate chapter]], there is a class of eigenvalue problems in the mathematical physics literature that is of the "Singular Sturm-Liouville" type.  These problems are governed by a one-dimensional, 2<sup>nd</sup>-order ODE of the form,
<div align="center" id="SingularSturmLiouville">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\lambda \Upsilon(x) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-(1-x)^{-\alpha}(1+x)^{-\beta} \cdot \frac{d}{dx} \biggl[ (1-x)^{\alpha+1}(1+x)^{\beta+1} \cdot \frac{d\Upsilon(x)}{dx} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(x^2-1)\frac{d^2\Upsilon(x)}{dx^2} + [\alpha - \beta + (\alpha+\beta+2)x]\frac{d\Upsilon(x)}{dx} 
\, ,
</math>
  </td>
</tr>
</table>
</div>
where the pair of exponent values, <math>~(\alpha, \beta) </math>, is set by the specific physical problem, while the eigenfunction, <math>~\Upsilon(x)</math>, and associated eigenfrequency, <math>~\lambda</math>, are to be determined.  For any choice of the pair of exponents, there is an infinite number <math>~(j = 0 \rightarrow \infty)</math> of ''analytically known'' eigenvectors that satisfy this governing ODE; they are referred to as ''Jacobi Polynomials.''  Specifically, the j<sup>th</sup> eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Upsilon_j(x) = J_j^{\alpha,\beta}(x)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(1-x)^{-\alpha}(1+x)^{-\beta} \biggl\{
\frac{(-1)^j}{2^j j!} \cdot \frac{d^j}{dx^j}\biggl[ (1-x)^{j+\alpha}(1+x)^{j+\beta} \biggr] 
\biggr\} \, ;</math>
  </td>
</tr>
</table>
</div>
and the corresponding j<sup>th</sup> eigenvalue is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_j^{\alpha,\beta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~j(j+\alpha+\beta + 1) \, .</math>
  </td>
</tr>
</table>
</div>

Table 3 presents the first three eigenfunctions (j = 0, 1, 2), along with each corresponding eigenfrequency; the figure displayed in the right-most column has been extracted directly from &sect;22 (p. 773) of [http://people.math.sfu.ca/~cbm/aands/page_773.htm Abramowitz and Stegun's (1964)] ''Handbook of Mathematical Functions'' (tenth printing, December 1972, with corrections) and shows the behavior of the lowest five Jacobi polynomials (j = 1, 2, 3, 4, 5, as labeled) over the interval <math>~-1 \le x \le +1</math> and for the specific case of <math>~(\alpha,\beta) = (1.5, - 0.5)</math>.


<div align="center" id="Table3">
<table align="center" border="1" cellpadding="5">
<tr>
  <th align="center" colspan="4"><font size="+1">Table 3:  Example Eigenvector Solutions to the Singular Sturm-Liouville Problem</font></th>
</tr>
<tr>
  <td align="center"><math>~j</math></td>
  <td align="center"><math>~J_j^{\alpha,\beta}(x)</math></td>
  <td align="center"><math>~\lambda_j^{\alpha,\beta}</math></td>
  <td align="center" rowspan="5">[[File:JacobiPolynomialsAS.png|200px|Jacobi Polynomials]]</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>~\tfrac{1}{2}(\alpha+\beta+2)x + \tfrac{1}{2}(\alpha-\beta)</math>
  </td>
  <td align="center">
<math>~(\alpha+\beta+2)</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~2</math>
  </td>
  <td align="center">
<math>~
\tfrac{1}{8}(12+7\alpha + \alpha^2 + 7\beta+\beta^2+ 2\alpha\beta ) x^2
+ \tfrac{1}{4}(3\alpha + \alpha^2 - 3\beta-\beta^2) x
+ \tfrac{1}{8}(-4 - \alpha + \alpha^2-\beta + \beta^2 - 2\alpha\beta) 
</math>
  </td>
  <td align="center">
<math>~2(\alpha+\beta+3)</math>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">See also, eqs. (35)-(37) of [http://mathworld.wolfram.com/JacobiPolynomial.html Wolfram MathWorld]; and &sect;22 (p. 773) of  [http://people.math.sfu.ca/~cbm/aands/page_773.htm Abramowitz and Stegun's (1964)] ''Handbook of Mathematical Functions,'' from which the illustration on the right has been extracted. 
</tr>
</table>
</div>


Here we highlight the qualitative similarities between the behavior of the set of Jacobi polynomials [figure extracted from [http://people.math.sfu.ca/~cbm/aands/page_773.htm Abramowitz and Stegun (1964)] and reprinted here in Table 3] and the set of eigenfunctions that describe normal modes of oscillation in homogeneous, self-gravitating spheres [figure extracted from [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)] and reprinted here in Table 2].  As we shall presently see, Jacobi polynomials play an important role in illuminating the structure of normal modes of oscillation in Papaloizou-Pringle tori.