Editing Apps/Ostriker64 (section)

==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>