Editing Apps/ImamuraHadleyCollaboration (section)

=====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>