Editing Apps/Ostriker64 (section)

===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>
-->