Editing Apps/Ostriker64

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=Polytropic &amp; Isothermal Tori=
{| class="Ostriker64" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
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! style="height: 125px; width: 125px; background-color:white;" |[[H_BookTiledMenu#Toroidal_.26_Toroidal-Like|<b>Ostriker<br />(1964)</b>]]
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Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible &#8212; specifically, polytropic and isothermal &#8212; fluids as presented in a series of papers by Jeremiah P. Ostriker:
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. Ostriker (1964, ApJ, 140, 1056)] &#8212; ''The Equilibrium of Polytropic and Isothermal Cylinders''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. Ostriker (1964, ApJ, 140, 1067)] &#8212; ''The Equilibrium of Self-Gravitating Rings''
* [http://adsabs.harvard.edu/abs/1964ApJ...140.1529O J. Ostriker (1964, ApJ, 140, 1529)] &#8212; ''On the Oscillations and the Stability of a Homogeneous Compressible Cylinder''
* [http://adsabs.harvard.edu/abs/1965ApJS...11..167O J. Ostriker (1965, ApJ Supplements, 11, 167)] &#8212; ''Cylindrical Emden and Associated Functions''
I believe that much, if not all, of this material was drawn from Ostriker's doctoral dissertation research at the University of Chicago (and Yerkes Observatory) under the guidance of [https://en.wikipedia.org/wiki/Subrahmanyan_Chandrasekhar S. Chandrasekhar].

==Coordinate System==

===Basics===
In &sect;IIa of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker defines a set of orthogonal coordinates, <math>~(r,\phi,\theta)</math>, that is related to the traditional Cartesian coordinate system, <math>~(x,y,z)</math>, via the relations,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(R+r\cos\phi)\cos\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(R+r\cos\phi)\sin\theta \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r\sin\phi \, .</math>
  </td>
</tr>
</table>
As Ostriker states, <font color="darkgreen">"The coordinate <math>~r</math> is the distance from a reference circle of radius <math>~R</math> (later chosen to be the major radius of the ring) &hellip;"</font>  The angle, <math>~\theta</math>, plays the role of the azimuthal angle, as is familiar in both cylindrical and spherical coordinates, while, here, <math>~\phi</math> is a meridional-plane polar angle measured counterclockwise from the equatorial plane.  For axisymmetric systems, there will be no dependence on the azimuthal angle, so the pair of relevant coordinates in the meridional plane are, 

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\varpi \equiv (x^2+y^2)^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R+r\cos\phi \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r\sin\phi \, .</math>
  </td>
</tr>
</table>

<div align="center" id="THH12Figure4">
<table border="1" cellpadding="8">

<tr><td align="center">
Figure 1 extracted without modification from p. 1077 of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O J. P. Ostriker (1964; Paper II)]<p></p>
"''The Equilibrium of Self-Gravitating Rings''"<p></p>
ApJ, vol. 140, pp. 1067-1087 &copy; American Astronomical Society
</td>
</tr>

<tr>
<td align="center">
[[File:Ostriker64PaperIIFig1.png|600px|Figure 1 from Ostriker (1964) Paper II]]
</td>
</tr>
</table>
</div>
For later reference, we note that (see eq. 3 of Paper II) the corresponding line element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\delta s^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\delta r^2 + r^2 \delta\phi^2 + (R+r\cos\phi)^2\delta\theta^2 \, ,
</math>
  </td>
</tr>
</table>
which means that the relevant scale factors for the adopted coordinate system, <math>~(r,\phi,\theta)</math>, are
<div align="center">
<math>~h_1 = 1 \, ,</math> &nbsp; &nbsp; &nbsp; <math>~h_2 = r \, ,</math> &nbsp; &nbsp; &nbsp; <math>~h_3 = (R+r\cos\phi) \, ,</math>
</div>
and the relevant differential volume element is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^3 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~h_1 h_2 h_3 dr d\phi d\theta
= r(R+r\cos\phi) dr d\phi d\theta\, .
</math>
  </td>
</tr>
</table>

===Relationship to Toroidal Coordinate===
Referring back to our separate discussion of the [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|basic elements of a toroidal coordinate system]], we know that, the meridional-plane toroidal coordinates <math>~(\eta,\theta)</math> are related to traditional meridional-plane cylindrical coordinate pair <math>~(\varpi,z)</math> via the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\varpi}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sinh\eta}{\cosh\eta - \cos\theta} \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\frac{z}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sin\theta}{\cosh\eta - \cos\theta} \, ,</math>
  </td>
</tr>
</table>

assuming that the cylindrical-coordinate location of the ''anchor ring'' is <math>~(\varpi,z) = (R,0)</math>.  Let's determine how to transform between these two sets of coordinate pairs.  

====Independent Exploration====
First, eliminating reference to Ostriker's "polar angle" <math>~\phi</math>, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r^2}{R^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{\varpi}{R} - 1 \biggr)^2 + \biggl(\frac{z}{R}\biggr)^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{\sinh\eta}{\cosh\eta - \cos\theta} - 1 \biggr]^2 + \biggl[ \frac{\sin\theta}{\cosh\eta - \cos\theta} \biggr]^2</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(\sinh\eta - \cosh\eta + \cos\theta)^2 + \sin^2\theta}{(\cosh\eta - \cos\theta)^2} \biggr] \, .</math>
  </td>
</tr>
</table>
Then, eliminating reference to Ostriker's radial coordinate <math>~r</math>, we find,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cot\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi/R - 1}{z/R}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sinh\eta - \cosh\eta + \cos\theta}{\sin\theta}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cot\theta  + \frac{\sinh\eta - \cosh\eta }{\sin\theta} \, .</math>
  </td>
</tr>
</table>
Now let's try to derive the alternate transformation.  We'll start by eliminating the "polar angle" in toroidal coordinates.
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta - \cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sinh\eta}{\varpi/R}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cosh\eta - \frac{\sinh\eta}{\varpi/R} \, .</math>
  </td>
</tr>
</table>
The same relation also implies that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{z}{R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\varpi}{R}\biggr) \frac{\sin\theta}{\sinh\eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \sin\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{z}{R}\biggl( \frac{\varpi}{R}\biggr)^{-1} \sinh\eta \, .</math>
  </td>
</tr>
</table>
Together, then, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 = \sin^2\theta + \cos^2\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{z}{R}\biggl( \frac{\varpi}{R}\biggr)^{-1} \sinh\eta \biggr]^2
+ \biggl[ \cosh\eta - \frac{\sinh\eta}{\varpi/R} \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{\varpi}{R}\biggr)^{-2} \biggl[ \frac{z}{R} \cdot \sinh\eta \biggr]^2
+ \biggl(\frac{\varpi}{R}\biggr)^{-2} \biggl[ \frac{\varpi}{R}\cdot \cosh\eta - \sinh\eta \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl( \frac{\varpi}{R}\biggr)^{2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{z}{R} \cdot \sinh\eta \biggr]^2
+ \biggl[ \frac{\varpi}{R}\cdot \cosh\eta - \sinh\eta \biggr]^2 \, .
</math>
  </td>
</tr>
</table>
Alternatively, in an attempt to eliminate <math>~\eta</math>, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\sinh\eta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\varpi}{R}\biggl( \frac{z}{R}\biggr)^{-1} \sin\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \cosh\eta = \biggl[ 1 + \sinh^2\eta\biggr]^{1 / 2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 + \biggl(\frac{\varpi}{R}\biggr)^2 \biggl( \frac{z}{R}\biggr)^{-2} \sin^2\theta \biggr]^{1 / 2} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{z}{R}\biggr)^{-1} \biggl[ \biggl( \frac{z}{R}\biggr)^{2}  + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>
But, also,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{z}{R} \biggr)^{-1} \sin\theta + \cos\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[ \biggl( \frac{z}{R}\biggr)^{2}  + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta \biggr]^{1 / 2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin\theta +  \biggl( \frac{z}{R}\biggr) \cos\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl( \frac{z}{R}\biggr)^{2}  + \biggl(\frac{\varpi}{R}\biggr)^2 \sin^2\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin^2\theta + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta +  \biggl( \frac{z}{R}\biggr)^2 \cos^2\theta</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl( \frac{z}{R}\biggr)^{2}\biggl[1-\cos^2\theta  \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin^2\theta\biggl[1 -  \biggl(\frac{\varpi}{R}\biggr)^2  \biggr] + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\sin^2\theta\biggl[1 -  \biggl(\frac{\varpi}{R}\biggr)^2  - \biggl( \frac{z}{R}\biggr)^{2}\biggr] + 2\biggl( \frac{z}{R}\biggr)\sin\theta \cos\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\biggl[1 -  \biggl(\frac{\varpi}{R}\biggr)^2  - \biggl( \frac{z}{R}\biggr)^{2}\biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - 2\biggl( \frac{z}{R}\biggr) \cot\theta </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \cot\theta </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \frac{1}{2}\biggl( \frac{z}{R}\biggr)^{-1} \biggl[1 -  \biggl(\frac{\varpi}{R}\biggr)^2  - \biggl( \frac{z}{R}\biggr)^{2}\biggr] \, .</math>
  </td>
</tr>
</table>
Now that I think about it, this is all a bit silly because from the [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|basic elements of a toroidal coordinate system]] we already know how to shift from cylindrical to toroidal coordinates.

====Back to Basics====

Mapping the other direction [see equations 2.13 - 2.15 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] ], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(r_1^2 + r_2^2 - 4R^2)}{2r_1 r_2} \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_1^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[\varpi + R]^2 + z^2 \, ,</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  <td align="right">
<math>~r_2^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~[\varpi - R]^2 + z^2 \, ,</math>
  </td>
</tr>
</table>
</div>
and <math>~\theta</math> has the same sign as <math>~z</math>.  Now, given that Ostriker's <math>~(r,\phi)</math> coordinates are related to cylindrical coordinates via the expressions,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\varpi </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R+r\cos\phi \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r\sin\phi \, ,</math>
  </td>
</tr>
</table>
we can write,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_1^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[2R + r\cos\phi]^2 + r^2\sin^2\phi </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4R^2 + 4Rr\cos\phi + r^2 \, ;</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r_2^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~r^2 \, .</math>
  </td>
</tr>
</table>
Hence,

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

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2r[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} \biggl[ 4Rr\cos\phi + 2r^2  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2R\cos\phi + r }{[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~e^{2\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ 4R^2 + 4Rr\cos\phi + r^2 }{r^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1 \, . 
</math>
  </td>
</tr>
</table>

===Summary===

<table border="1" cellpadding="10" align="center" width="85%">
<tr>
  <td align="left" width="50%">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r^2}{R^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  <td align="left">
<math>~\biggl[ \frac{(\sinh\eta - \cosh\eta + \cos\theta)^2 + \sin^2\theta}{(\cosh\eta - \cos\theta)^2} \biggr] </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\cot\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cot\theta  + \frac{\sinh\eta - \cosh\eta }{\sin\theta} </math>
  </td>
</tr>
</table>

  </td>
  <td align="left">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{2\eta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{2R\cos\phi + r }{[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} 
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>

===Second Attempt===
====Single Offset Circle====
Now an [[Appendix/Ramblings/ToroidalCoordinates#Off-center_Circle|off-center circle]] whose major and minor radii are, respectively, <math>~(\varpi_0,d)</math>, will be described by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(\varpi - \varpi_0)^2 + z^2 \, .
</math>
  </td>
</tr>
</table>

<span id="Dsquared">where both <math>~d</math> and <math>~\varpi_0</math> are held constant while mapping out the variation of <math>~z</math> with <math>~\varpi</math>.  If we acknowledge that, in general, <math>~\varpi_0 \ne R_\mathrm{JPO}</math>, then we know how <math>~r</math> varies with <math>~\phi</math> via the relation,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ R_\mathrm{JPO} + r\cos\phi - \varpi_0\biggr]^2 + r^2\sin^2\phi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(R_\mathrm{JPO}-\varpi_0)^2 
+ 2\biggl[ (R_\mathrm{JPO}-\varpi_0) r\cos\phi \biggr]
+r^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r^2 + 2r\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr] + \biggl[(R_\mathrm{JPO}-\varpi_0)^2 - d^2\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ r </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{
- 2\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr] \pm 
\sqrt{ 4\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr]^2 - 4\biggl[(R_\mathrm{JPO}-\varpi_0)^2 - d^2\biggr] }
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl\{
2\biggl[ (\varpi_0 - R_\mathrm{JPO}) \cos\phi \biggr] \pm 
\sqrt{ 4\biggl[ (\varpi_0 - R_\mathrm{JPO}) \cos\phi \biggr]^2 - 4\biggl[(\varpi_0 - R_\mathrm{JPO})^2 - d^2\biggr] }
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{r}{ (\varpi_0 - R_\mathrm{JPO}) }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos\phi  \pm 
\sqrt{ \cos^2\phi  - 1 + d^2 (\varpi_0 - R_\mathrm{JPO})^{-2} }
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos\phi  \pm \sqrt{ d^2 (\varpi_0 - R_\mathrm{JPO})^{-2}-\sin^2\phi }
</math>
  </td>
</tr>
</table>

In order to align this expression with the terminology (and variable labels) that we use in the context of a toroidal coordinate system, we associate the radius of the ''anchor ring'' as <math>~R_\mathrm{JPO}\leftrightarrow a</math>, and we associate the major radius of each circular torus as <math>~\varpi_0 \leftrightarrow R_0</math>.  We therefore have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r}{ (R_0-a) }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\cos\phi  \pm \sqrt{ d^2 (R_0-a)^{-2}-\sin^2\phi } 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{r}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{R_0}{a}-1 \biggr)
\biggl[ \cos\phi  \pm \sqrt{ \biggl(\frac{d}{a}\biggr)^2 \biggl(\frac{R_0}{a}-1 \biggr)^{-2}-\sin^2\phi } \biggr]
</math>
  </td>
</tr>
</table>

and, the coordinates of points along the surface of the torus <math>~(\varpi,z)</math> are provided by the expressions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\varpi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a + (R_0 - a)\cos\phi \biggl[
\cos\phi  \pm \sqrt{ d^2 (R_0 - a)^{-2}-\sin^2\phi }
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(R_0 - a)\sin\phi \biggl[
\cos\phi  \pm \sqrt{ d^2 (R_0 - a)^{-2}-\sin^2\phi }
\biggr]
</math>
  </td>
</tr>
</table>
We have tested this pair of expressions using Excel and have successfully demonstrated that they do, indeed, trace out a circle of radius, <math>~d</math>, whose center is offset from the symmetry axis by a distance, <math>~R_0</math>.

====Set of Circles Whose Offset Increases With Circle Diameter====

A set of nested off-center circles will be described by allowing <math>~R_0 = R_0(d)</math>, that is, by having the off-set distance, <math>~R_0</math>, vary with the size of the circle, <math>~d</math>.  The above prescription for the normalized "coordinate" <math>~r/a</math> will work for ''any'' prescribed <math>~R_0(d)</math> function.

But a ''particular'' <math>~R_0(d)</math> function is demanded if we want this derived prescription to represent the behavior of toroidal coordinates.  In a [[Apps/DysonWongTori#Introducing_Toroidal_Coordinates|toroidal coordinate system]], a specification of the value of the "radial" coordinate, <math>~\eta</math>, automatically dictates the ratio <math>~R_0/d</math>; but we are not at liberty to separately define the value of the ''difference,'' <math>~(R_0 - d)</math>.  Instead, we must enforce the toroidal-coordinate relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~a^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~R_0^2 - d^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{R_0}{a}-1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 + \delta^2\biggr]^{1 / 2} -1 \, ,</math>
  </td>
</tr>
</table>
where we have adopted the shorthand notation, <math>~\delta\equiv d/a</math>.  Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ \sqrt{1+\delta^2} -1 ]
\{ \cos\phi  \pm [\delta^2 ( \sqrt{1+\delta^2} -1 )^{-2}-\sin^2\phi ]^{1 / 2} \}
</math>
  </td>
</tr>
</table>

Now, in a [[Apps/DysonWongTori#Introducing_Toroidal_Coordinates|toroidal coordinate system]], there is a similar "radial" coordinate, <math>~\eta</math>, whose value varies with distance from the ''anchor ring'' of radius, <math>~a</math>.  Its value depends on both <math>~R_0</math> and <math>~d</math> via the relation,
<div align="center">
<math>~R_0 = d\cosh\eta \, .</math>
</div>
This means that, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\cosh\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\delta}\biggl(\frac{R_0}{a}\biggr) = \frac{\sqrt{1+\delta^2}}{\delta} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \delta^2 \cosh^2\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + \delta^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \delta^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\cosh^2\eta - 1} = \frac{1}{\sinh^2\eta} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \sqrt{1  + \delta^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[1 + \frac{1}{\sinh^2\eta} \biggr]^{1 / 2}
= \coth\eta
\, ,</math>
  </td>
</tr>
</table>
which also means that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{r}{a}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[ \coth\eta -1 ]
\biggl\{ \cos\phi  \pm \biggl[ ( \cosh\eta -\sinh\eta )^{-2} -\sin^2\phi \biggr]^{1 / 2}
\biggr\} \, .
</math>
  </td>
</tr>
</table>

====Case of Small Offset====

Another way to look at this issue is to go [[#Dsquared|back to the expression]],
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~d^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~
(R_\mathrm{JPO}-\varpi_0)^2 
+ 2\biggl[ (R_\mathrm{JPO}-\varpi_0) r\cos\phi \biggr]
+r^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \delta^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
 <td align="left">
<math>~\biggl(\frac{r}{a}\biggr)^2
+ \frac{r}{a}\biggl[ 2\biggl(1 - \frac{R_0}{a}\biggr)\biggr] \cos\phi 
+ \biggl(1 - \frac{R_0}{a}\biggr)^2 
</math>
  </td>
</tr>
</table>
and assume that, while still dependent on the radial coordinate, the dimensionless offset is small.  That is, assume that,
<div align="center">
<math>~\Delta(\delta) \equiv 1 - \frac{R_0(\delta)}{a} \ll 1 \, .</math>
</div>
In this case, we can write,

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

<tr>
  <td align="right">
<math>~ \delta^2</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
 <td align="left">
<math>~\biggl(\frac{r}{a}\biggr)^2
+ 2\Delta(\delta) \biggl( \frac{r}{a} \biggr)  \cos\phi 
+\cancelto{0}{\Delta^2(\delta)} \, .
</math>
  </td>
</tr>
</table>
And differentiating both sides of the expression with respect to <math>~r/a</math> gives,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0 </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~2\biggl(\frac{r}{a}\biggr) + 2\Delta(\delta) \cos\phi</math>
  </td>
</tr>
</table>

<font color="red">'''COMMENT by Tohline'''</font> (15 August 2018):  I'm not sure that this is leading where I had hoped.  I am gearing up to draw a comparison between these last expressions and eq. (74) in [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Ostriker's (1964) Paper II].

<!--
===First Attempt===
Based on my (initial, casual) study of this paper, Figure 1 appears to illustrate a configuration in which the density is constant on various nested toroidal surfaces such that, working from the highest density location, outward, the location <math>~R</math> of the center of each torus shifts to larger and larger values.  It would therefore appear as though Ostriker's <math>~R</math> must be a function of the density-marker.  Using the subscript, <math>~i</math>, as the marker, we represent the density as a function, <math>~\rho(r_i)</math>, and recognize that <math>~R = R(r_i)</math> as well.

We recognize that the radial coordinate, <math>~\eta</math>, in a toroidal-coordinate system behaves in this same manner.  Each <math>~\eta = ~ \mathrm{const}</math> surface is a circle of radius, <math>~d</math>, whose center is located a distance from the symmetry axis, <math>~R_0 = \sqrt{a^2 + d^2}</math>.  And, holding <math>~a</math> fixed, the accompanying definition is,
<div align="center">
<math>~\cosh\eta = \frac{R_0}{d} =\biggl[ 1 + \frac{a^2}{d^2} \biggr]^{1 / 2} = \frac{1}{\delta}\biggl[1 + \delta^{2} \biggr]^{1 / 2} \, ,</math>
</div>
where, <math>~\delta \equiv d/a</math>.  Comparing Ostriker's notation with a [[2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System|toroidal coordinate system]] whose ''anchor ring'' is at the meridional-plane location <math>~(\varpi,z) = (a,0)</math>, we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R+r\cos\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)} \, ,</math>&nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~r\sin\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>
  </td>
</tr>
</table>
It appears that we can make the following direct associations: &nbsp; <math>~R_0  \leftrightarrow R_\mathrm{JPO}</math> and <math>~d \leftrightarrow r_\mathrm{JPO}</math>.  Hence, we have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\sin\phi}{R_0+d\cos\phi}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sin\theta}{\sinh\eta}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\sin\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d \sinh\eta \sin\phi}{R_0+d\cos\phi} \, .</math>
  </td>
</tr>
</table>
And,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_0+d\cos\phi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a\sinh\eta}{(\cosh\eta - \cos\theta)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\cos\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\cosh\eta - \frac{a\sinh\eta}{R_0+d\cos\phi} \, .</math>
  </td>
</tr>
</table>
Putting these together we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 = \sin^2\theta + \cos^2\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{d \sinh\eta \sin\phi}{R_0+d\cos\phi}  \biggr]^2 + \biggl[ \cosh\eta - \frac{a\sinh\eta}{R_0+d\cos\phi} \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (R_0 + d\cos\phi)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[d \sinh\eta \sin\phi]^2 + [ \cosh\eta(R_0 + d\cos\phi) - a\sinh\eta]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl[ \frac{R_0}{d} + \cos\phi\biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[\sinh\eta \sin\phi]^2 + \biggl[ \cosh\eta \biggl(\frac{R_0}{d} + \cos\phi \biggr) - \frac{a}{d} \sinh\eta \biggr]^2
</math>
  </td>
</tr>
</table>
-->

==Gravitational Potential==

===Potential of a Thin Hoop===
In &sect;IIb of his [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II],  Ostriker (1964) derives an expression for the gravitational potential of a torus in the ''Thin Ring'' approximation, beginning specifically with the [[SR/PoissonOrigin#Step_1|integral form of the Poisson equation]] that is widely referred to in the astrophysics community as an expression for the,
<div align="center" id="GravitationalPotential">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center" colspan="3">
<font color="#770000">'''Scalar Gravitational Potential'''</font>
  </td>
</tr>
<tr>
  <td align="right">
<math>~ \Phi(\vec{x})</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 31, Eq. (2-3)<br />
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], &sect;10, p. 17, Eq. (11)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;4.2, p. 77, Eq. (12)
  </td>
</tr>
</table>
</div>

(Note: &nbsp; Consistent with the usage favored by his doctoral dissertation advisor in [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], throughout his collection of 1964 papers Ostriker adopts a ''different sign convention'' as well as a different variable name to represent the gravitational potential.)  Employing [[#Coordinate_System|Ostriker's adopted coordinate system]], and recognizing that, <font color="darkgreen">"the distance between the point of integration <math>~(0,0,\theta^')</math> and the point of observation <math>~(r,\phi,0)</math>"</font> is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~|\vec{x}^{~'} - \vec{x}|</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (21)
  </td>
</tr>
</table>

this expression for the gravitational potential becomes,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~ \Phi(r,\phi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left" colspan="2">
<math>~ -G \int \int \rho(r^',\phi^') r^' (R+r^'\cos\phi^')  dr^' d\phi^' \int \frac{d\theta^'}{[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} }  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -G (2\sigma R) \int_0^\pi \frac{d\theta^'}{[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} }  </math>
  </td>
  <td align="right" rowspan="4">[[File:WolframAlphaResult.png|300px|WolframAlpha result]]</td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{4G \sigma R}{r} \int_0^\pi \frac{\tfrac{1}{2}d\theta^'}{[1 +n^2\sin^2(\tfrac{1}{2}\theta^')]^{1 / 2} }  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{4G \sigma R}{r} \biggl[ \frac{K(k)}{\sqrt{n^2+1}}  \biggr]  \, ,</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (22)
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~n^2 \equiv \frac{4R(R+r\cos\phi)}{r^2}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~k \equiv \biggl[ \frac{n^2}{n^2+1} \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1070, Eq. (23)
  </td>
</tr>
</table>

<table border="1" cellpadding="10" align="center" width="85%"><tr><td align="left">
Mapping back to cylindrical coordinates, for the moment, we recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\varpi - R)^2 + z^2</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ n^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4R\varpi}{(\varpi - R)^2 + z^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ n^2 + 1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{4R\varpi + (\varpi - R)^2 + z^2}{(\varpi - R)^2 + z^2} = \frac{(\varpi + R)^2 + z^2}{(\varpi - R)^2 + z^2} \, .</math>
  </td>
</tr>
</table>
Acknowledging as well that the mass of Ostriker's "thin hoop" is, <math>~M = 2\pi \sigma R</math>, his expression for the potential becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi(\varpi,z)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{2G M}{\pi} \biggl[ \frac{K(k)}{\sqrt{(\varpi + R)^2 + z^2}}  \biggr] \, ,</math>
  </td>
</tr>
</table>
where,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~k</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{4R\varpi}{(\varpi + R)^2 + z^2} \biggr]^{1 / 2} \, .</math>
  </td>
</tr>
</table>

After adopting the variable association, <math>~R \leftrightarrow a</math>, it is clear that Ostriker's derived expression is identical to the Key Equation that we have [[Apps/DysonWongTori#Thin_Ring_Approximation|identified elsewhere]] as providing the,
<table border="0" cellpadding="5" align="center">
<tr>
<td align="center" colspan="1"><font color="#770000">'''Gravitational Potential in the Thin Ring (TR) Approximation'''</font></td>
<td align="center" colspan="1" rowspan="2">[[File:FlatColorContoursCropped.png|220px|Contours for Thin Ring Approximation]]</td>
</tr>
<tr>
  <td align="center">
{{ Math/EQ TRApproximation }}
  </td>
</tr>
</table>

</td></tr></table>

===Series Expansion===
In the context of Ostriker's expression for the potential, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(k')^{-2} \equiv \biggl[ \frac{1}{1-k^2}\biggr]= n^2 + 1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4R(R+r\cos\phi)}{r^2} + 1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2R}{r}\biggr)^2 \biggl[ 1 + \frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2\biggr] \, .
</math>
  </td>
</tr>
</table>
Hence, in the vicinity of the ring where <math>~r/R \ll 1</math> and <math>~k'</math> is a "small parameter," we can draw on the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] and write, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(k')^m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl[ 1 + \frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2\biggr]^{-m / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl\{ 1 -\frac{m}{2} \biggl[\frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2 \biggr] 
+ \frac{1}{2}\biggl[ -\frac{m}{2}\biggl( -\frac{m}{2}-1\biggr) \biggr]\biggl[\frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2 \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl\{ 1 - \biggl(\frac{m}{2}\biggr) \frac{r}{R}\cos\phi - \biggl(\frac{m}{2^3}\biggr)  \biggl(\frac{r}{R}\biggr)^2 
+ \frac{m}{4}\biggl( \frac{m}{2} + 1\biggr) \biggl[\frac{r}{R}\cos\phi  \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\} \, .
</math>
  </td>
</tr>
</table>
Note, in particular, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl( \frac{R}{r}\biggr) \biggl\{ 1 + \biggl(\frac{1}{2}\biggr) \frac{r}{R}\cos\phi + \biggl(\frac{1}{2^3}\biggr)  \biggl(\frac{r}{R}\biggr)^2 
- \frac{1}{2^3} \biggl[\frac{r}{R}\cos\phi  \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2R}{r} \biggl\{ 1 + \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~k'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{r}{2R} \biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr] 
\, ;
</math> &nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~(k')^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^2}\biggl( \frac{r}{R}\biggr)^{2} \biggl[ 1 - \frac{r}{R}\cos\phi + \frac{1}{2^2}  \biggl(\frac{r}{R}\biggr)^2 (4\cos^2\phi - 1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr] 
\, .
</math>
  </td>
</tr>
</table>

Next we recognize that the following series expansion for the ''complete elliptic integral of the first kind'' &#8212; written in terms of the small parameter, <math>~k'</math> &#8212; appears, for example, as eq. (8.113.3) in the Fourth Edition of [http://www.mathtable.com/gr/ Gradshteyn &amp; Ryzhik (1965)]:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K(k)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \frac{4}{k^'}  + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} \biggr){k'}^2
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} \biggr){k'}^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl( \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} - \frac{2}{5\cdot 6} \biggr){k'}^6 + \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \frac{4}{k^'}  + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2
+ \frac{3^2}{2^6} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4
+ \frac{5^2}{2^8} \biggl( \ln\frac{4}{k^'} - \frac{37}{30}  \biggr){k'}^6 + \cdots
</math>
  </td>
</tr>
</table>
[This series expansion &#8212; up through the term <math>~\mathcal{O}(k'^4)</math> &#8212; appears as equation 24 in Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II].]  Put together, then, Ostriker's expression for the gravitational potential in the ''thin ring'' approximation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{2GM}{\pi r}  k' K(k)  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2GM}{\pi r}  \biggl[
k' \ln \frac{4}{k^'}  + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^3
+  \cdots
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2GM}{\pi r}  \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr] - \frac{1}{4} k'^3 + \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2GM}{\pi r}  \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr] 
- \frac{1}{2^5}\biggl( \frac{r}{R}\biggr)^{3} \biggl[ 1 - \biggl(\frac{3}{2}\biggr) \frac{r}{R}\cos\phi - \biggl(\frac{3}{2^3}\biggr)  \biggl(\frac{r}{R}\biggr)^2 
+ \frac{3}{4}\biggl( \frac{3}{2} + 1\biggr) \biggl(\frac{r}{R}\cos\phi  \biggr)^2 + \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr) \biggr] 
+ \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2GM}{\pi R}  \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr] \frac{R}{r}
- \frac{1}{2^5}\biggl( \frac{r}{R}\biggr)^{2} \biggl[ 1 - \biggl(\frac{3}{2}\biggr) \frac{r}{R}\cos\phi + \mathcal{O}\biggl( \frac{r^2}{R^2}\biggr) \biggr] 
+ \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}\, ,
</math>
  </td>
</tr>
</table>
where, again, we have recognized that the mass of the thin hoop is, <math>~M = 2\pi\sigma R</math>.  Now,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ k' \biggl[ 1 + \frac{k'^2}{2^2} \biggr] \frac{R}{r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl\{1 +  
\frac{1}{2^4} \biggl[ \biggl( \frac{r}{R}\biggr)^{2} - \biggl(\frac{r}{R}\biggr)^3\cos\phi 
+ \mathcal{O}\biggl(\frac{r^4}{R^4} \biggr) \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl\{\frac{1}{2} +  
\frac{1}{2^5} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr)  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr] \, ;
</math>
  </td>
</tr>
</table>
and, given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln[a (1+x)] = \ln a + \ln(1+x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln a + x - \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 - \tfrac{1}{4}x^4 + \cdots 
</math>
  </td>
</tr>
</table>

we also have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln \frac{4}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \ln\biggl[ 1 + \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
- \frac{1}{2} \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{3}\biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]^3 + \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi \biggr]
- \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2\cos^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi ) + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \, .
</math>
  </td>
</tr>
</table>

So our series expansion for Ostriker's "thin ring" potential becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{ \ln \frac{4}{k^'} \biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}  
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{ \ln \frac{8R}{r} \biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi ) + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}  
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{ \ln \frac{8R}{r} \biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^4} \biggl(\frac{r}{R}\biggr)^2 ( 6 \cos^2\phi  - 1) 
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi ) 
- \frac{1}{2^2} \biggl(\frac{r}{R}\biggr)^2\cos^2\phi 
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}  
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Finally, dropping the explicit mention of all terms <math>~\mathcal{O}(r^3/R^3)</math> and smaller gives the series expansion formulation presented by Ostriker, namely,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{\ln \frac{8R}{r}
- \frac{r}{2R}\biggl[ \ln \frac{8R}{r} - 1\biggr]\cos\phi
~+~ \frac{r^2}{2^4R^2} \biggl[ \ln \frac{8R}{r}  ( 6 \cos^2\phi  - 1) + (1 - 8\cos^2\phi ) 
\biggr] ~+ ~\cdots \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{\ln \frac{8R}{r}
- \frac{r}{2R}\biggl[ \ln \frac{8R}{r} - 1\biggr]\cos\phi
~+~ \frac{r^2}{2^4R^2} \biggl[ \biggl(2\ln\frac{8R}{r} - 3 \biggr) + \biggl( 3\ln\frac{8R}{r} - 4 \biggr)\cos 2\phi
\biggr] ~+ ~\cdots \biggr\} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1071, Eq. (25)
  </td>
</tr>
</table>

===The Dimensionless Radial Coordinate, &xi;, and Smallness Parameter, &beta;===

As we have [[SSC/Structure/Polytropes#Polytropic_Spheres|reviewed separately]], when researchers in the astrophysics community discuss the structure of ''spherical'' polytropes, the 
<div align="center">
<span id="LaneEmdenEquation"><font color="#770000">'''Lane-Emden Equation'''</font></span>
<br />
{{Math/EQ_SSLaneEmden01}}
</div>
invariably arises, as it is the governing 2<sup>nd</sup>-order ODE whose solution, <math>~\Theta_H(\xi)</math>, defines the internal structure of spherically symmetric equlibrium configurations.  Traditionally, as well, the dimensionless radial coordinate,
<div align="center">
<math>~\xi \equiv \frac{r}{a_n} \, ,</math>
</div>
is defined in terms of <math>~a_n</math>, which is a natural length scale of the (spherical) problem.  Equation (42) of Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II] provides the traditional definition of <math>~a_n</math>.  It is therefore not surprising that, even though Ostriker's set of 1964 papers deal largely with the equilibrium and stability of ''ring-like'' configurations, he adopts a similar definition for the dimensionless radial coordinate; specifically, eq. (5) of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II] states that,
<div align="center">
<math>~\alpha \xi \equiv r \, .</math>
</div>
But, of course, in the context of Ostriker's presentation, <math>~r</math> is not a spherical radial coordinate but is, rather, as [[#Coordinate_System|defined above]]; and <math>~\alpha</math> is of the same order as the minor, cross-sectional radius of the torus.

[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 17 August 2018:  There appears to be a typographical error in the definition of &beta; that is provided by equation (6) in &sect;IIa of Ostriker's ''Paper II''.  The published equation defines &beta; as the ratio of &alpha; to ''r'' rather than, as we have indicated here, as the ratio of &alpha; to ''R''.  Equation (77) on p. 1078 of Paper II confirms this suspicion.]]In eq. (6) of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker also defines the dimensionless parameter,
<div align="center">
<math>~\beta \equiv \frac{\alpha}{R} \, ,</math>
</div>
where <math>~R</math> is associated with the major radius of the ring.  Then he states that, <font color="darkgreen">"&hellip; since <math>~\alpha \ll R</math> (by hypothesis), we may be sure that <math>~\beta \ll 1</math> &hellip;"</font>

With the definitions of these two dimensionless parameters in hand &#8212; and, more specifically, after appreciating that,
<div align="center">
<math>~\frac{R}{r} = \frac{1}{\beta\xi}  ~~~\Rightarrow ~~~ \ln\frac{8R}{r} = \biggl[ \ln\frac{8}{\beta} - \ln\xi \biggr] </math>
</div>
&#8212; we can follow Ostriker's lead and rewrite his derived expression for <math>~\Phi_\mathrm{TR}</math> in the form,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{ \ln\frac{8}{\beta} - \ln\xi 
+ \frac{\beta\xi}{2}\biggl[  - \biggl( \ln\frac{8}{\beta} -1 \biggr) + \ln\xi  \biggr]\cos\phi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+~ \frac{\beta^2\xi^2}{2^4} \biggl[ \biggl(2 \ln\frac{8}{\beta} - 3 - 2\ln\xi \biggr) + \biggl( 3 \ln\frac{8}{\beta} - 4 - 3\ln\xi \biggr)\cos 2\phi
\biggr] ~+ ~\cdots \biggr\} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1071, Eq. (26)
  </td>
</tr>
</table>

=See Also=

* [http://adsabs.harvard.edu/abs/2016AJ....152...35F T. Fukushima (2016, AJ, 152, article id. 35, 31 pp.)] &#8212; ''Zonal Toroidal Harmonic Expansions of External Gravitational Fields for Ring-like Objects''
* [http://adsabs.harvard.edu/abs/2016ApJ...829...45K W.-T. Kim &amp; S. Moon (2016, ApJ, 829, article id. 45, 22 pp.)] &#8212; ''Equilibrium Sequences and Gravitational Instability of Rotating Isothermal Rings''
* [http://adsabs.harvard.edu/abs/2011MNRAS.411..557B E. Y. Bannikova, V. G. Vakulik &amp; V. M. Shulga (2011, MNRAS, 411, 557 - 564)] &#8212; ''Gravitational Potential of a Homogeneous Circular Torus: a New Approach''
* [http://adsabs.harvard.edu/abs/2008MNRAS.389..156P D. Petroff &amp; S. Horatschek (2008, MNRAS, 389,156 - 172)] &#8212; ''Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity''
<table border="0" cellpadding="8" align="center" width="90%">
<tr><td align="left">
The following quotes have been taken from [http://adsabs.harvard.edu/abs/2008MNRAS.389..156P Petroff &amp; Horatschek (2008)]:<p></p>
<b>&sect;1:</b> &nbsp; "The problem of the self-gravitating ring captured the interest of such renowned scientists as Kowalewsky (1885), Poincar&eacute; (1885a,b,c) and Dyson (1892, 1893).  Each of them tackled the problem of an axially symmetric, homogeneous ring in equilibrium by expanding it about the thin ring limit.  In particular, Dyson provided a solution to fourth order in the parameter <math>~\sigma = a/b</math>, where <math>~a = r_t</math> provides a measure for the radius of the cross-section of the ring and <math>~b = \varpi_t</math> the distance of the cross-section's centre of mass from the axis of rotation."<p></p>
<b>&sect;7:</b> &nbsp; "In their work on homogeneous rings, Poincar&eacute; and Kowalewsky, whose results disagreed to first order, both had made mistakes as Dyson has shown.  His result to fourth order is also erroneous as we point out in Appendix B."
</td></tr>
</table>
* [http://adsabs.harvard.edu/abs/2006IJMPB..20.3113C P. H. Chavanis (2006, International Journal of Modern Physics B, 20, 3113 - 3198)] &#8212; ''Phase Transitions in Self-Gravitating Systems''
* [http://adsabs.harvard.edu/abs/2003MNRAS.339..515A M. Ansorg, A. Kleinw&auml;chter &amp; R. Meinel (2003, MNRAS, 339, 515)] &#8212; ''Uniformly Rotating Axisymmetric Fluid Configurations Bifurcating from Highly Flattened Maclaurin Spheroids''
* [http://adsabs.harvard.edu/abs/2001A%26A...375.1091L M. Lombardi &amp; G. Bertin (2001, Astronomy &amp; Astrophysics, 375, 1091 - 1099)] &#8212; ''Boyle's Law and Gravitational Instability''
* [http://adsabs.harvard.edu/abs/1996MNRAS.282..234K W. Kley (1996, MNRAS, 282, 234)] &#8212; ''Maclurin Discs and Bifurcations to Rings''
* [http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline, &amp; I. Hachisu (1994, ApJ, 420, 247 - 267)] &#8212; ''The Stability of Thick, Self-Gravitating Disks in Protostellar Systems''
* [http://adsabs.harvard.edu/abs/1991ApJ...374..610B I. Bonnell &amp; P. Bastien (1991, ApJ, 374, 610 - 622)] &#8212; ''The Collapse of Cylindrical Isothermal and Polytropic Clouds with Rotation''
* [http://adsabs.harvard.edu/abs/1990ApJ...361..394T J. E. Tohline &amp; I. Hachisu (1990, ApJ, 361, 394 - 407)] &#8212; ''The Breakup of Self-Gravitating Rings, Tori, and Thick Accretion Disks''
* [http://adsabs.harvard.edu/abs/1988A%26A...200..127S F. Schmitz (1988, Astronomy &amp; Astrophysics, 200, 127 - 134)] &#8212; ''Equilibrium Structures of Differentially Rotating Self-Gravitating Gases''
* [http://adsabs.harvard.edu/abs/1985Ap%26SS.109...45V P. Veugelen (1985, Astrophysics &amp; Space Science, 109, 45 - 55)] &#8212; ''Equilibrium Models of Differentially Rotating Polytropic Cylinders''
* [http://adsabs.harvard.edu/abs/1984MNRAS.208..279A M. A. Abramowicz, A. Curir, A. Schwarzenberg-Czerny, &amp; R. E. Wilson (1984, MNRAS, 208, 279 - 291)] &#8212; ''Self-Gravity and the Global Structure of Accretion Discs''
* [http://adsabs.harvard.edu/abs/1983A%26A...119..109B P. Bastien (1983, Astronomy &amp; Astrophysics, 119, 109 - 116)] &#8212; ''Gravitational Collapse and Fragmentation of Isothermal, Non-Rotating, Cylindrical Clouds''
* [http://adsabs.harvard.edu/abs/1981PThPh..65.1870E Y. Eriguchi &amp; D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875)] &#8212; ''Another Equilibrium Sequence of Self-Gravitating and Rotating Incompressible Fluid''
* [http://adsabs.harvard.edu/abs/1980ApJ...236..160T J. E. Tohline (1980, ApJ, 236, 160 - 171)] &#8212; ''Ring Formation in Rotating Protostellar Clouds''
* [http://adsabs.harvard.edu/abs/1980PThPh..63.1957F T. Fukushima, Y. Eriguchi, D. Sugimoto, &amp; G. S. Bisnovatyi-Kogan (1980, Progress of Theoretical Physics, 63, 1957 - 1970)] &#8212; ''Concave Hamburger Equilibrium of Rotating Bodies''
* [http://adsabs.harvard.edu/abs/1978MNRAS.184..709K J. Katz &amp; D. Lynden-Bell (1978, MNRAS, 184, 709 - 712)] &#8212; ''The Gravothermal Instability in Two Dimensions''
* [http://adsabs.harvard.edu/abs/1977ApJ...214..584M P. S. Marcus, W. H. Press, &amp; S. A. Teukolsky (1977, ApJ, 214, 584- 597)] &#8212; ''Stablest Shapes for an Axisymmetric Body of Gravitating, Incompressible Fluid'' (includes torus with non-uniform rotation)
<ol type="a"><li>Shortly after their equation (3.2), Marcus, Press &amp; Teukolsky make the following statement: "&hellip; we know that an ''equilibrium'' incompressible configuration must rotate uniformly on cylinders (the famous "Poincar&eacute;-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) &hellip;"</li></ol>
* [http://adsabs.harvard.edu/abs/1976ApJ...207..736H C. J. Hansen, M. L. Aizenman, &amp; R. L. Ross (1976, ApJ, 207, 736 - 744)] &#8212; ''The Equilibrium and Stability of Uniformly Rotating, Isothermal Gas Cylinders''
* [http://adsabs.harvard.edu/abs/1974ApJ...190..675W C.-Y. Wong (1974, ApJ, 190, 675 - 694)] &#8212; ''Toroidal Figures of Equilibrium''
* [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279 - 353)] &#8212; ''Toroidal and Spherical Bubble Nuclei''
* [http://adsabs.harvard.edu/abs/1942ApJ....95...88R Gunnar Randers (1942, ApJ, 95, 88)] &#8212; ''The Equilibrium and Stability of Ring-Shaped 'barred SPIRALS'.''
* [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential William Duncan MacMillan (1958)], ''The Theory of the Potential'', New York:  Dover Publications
* [https://archive.org/details/foundationsofpot033485mbp Oliver Dimon Kellogg (1929)], ''Foundations of Potential Theory'', Berlin:  Verlag Von Julius Springer
* [http://adsabs.harvard.edu/abs/1917RSPSA..93..148R Lord Rayleigh (1917, Proc. Royal Society of London. Series A, 93, 148-154)] &#8212; ''On the Dynamics of Revolving Fluids''
* [http://adsabs.harvard.edu/abs/1893RSPTA.184.1041D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106)] &#8212; ''The Potential of an Anchor Ring. Part II.'' <ol type="a"><li>In this paper, Dyson derives the gravitational potential ''inside'' the ring mass distribution</li></ol>
* [http://adsabs.harvard.edu/abs/1893RSPTA.184...43D F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95)] &#8212; ''The Potential of an Anchor Ring. Part I.''<ol type="a"><li>In this paper, Dyson derives the gravitational potential ''exterior to'' the ring mass distribution</li></ol>
* [http://adsabs.harvard.edu/abs/1885AN....111...37K S. Kowalewsky (1885, Astronomische Nachrichten, 111, 37)] &#8212; ''Zus&auml;tze und Bemerkungen zu Laplace's Untersuchung &uuml;ber die Gestalt der Saturnsringe''
* Poincar&eacute; (1885a, C. R. Acad. Sci., 100, 346), (1885b, Bull. Astr., 2, 109), (1885c, Bull. Astr. 2, 405). &#8212; references copied from paper by [http://adsabs.harvard.edu/abs/1974ApJ...190..675W Wong (1974)]


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