Editing Apps/Ostriker64 (section)

====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>
-->