Editing Appendix/Ramblings/ToroidalCoordinates (section)

==Papaloizou-Pringle Tori==
===Summary of Structure===
As derived [[Apps/PapaloizouPringleTori|elsewhere]], the accretion tori constructed by Papaloizou &amp; Pringle (1984; hereafter PP84) have the following surface properties.  For a given choice of the dimensionless Bernoulli constant, <math>C_\mathrm{B}^'</math>,

<div align="center">
<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>
~\chi_\mathrm{inner}
</math>
  </td>
  <td align="center">
<math>~ =</math> 
  </td>
  <td align="left"><math>
~\frac{1}{1 + \sqrt{1-2C_\mathrm{B}^'}} \, ;
  </math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\chi_\mathrm{outer} 
</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left"><math>
~\frac{1}{1 - \sqrt{1-2C_\mathrm{B}^'}} \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\chi_0 
</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left"><math>
~\frac{1}{2} (\chi_\mathrm{outer} + \chi_\mathrm{inner}) = \frac{1}{2C_\mathrm{B}^'} \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
~\delta 
</math>
  </td>
  <td align="center">
<math>~\equiv</math> 
  </td>
  <td algin="left">
<math>
~\frac{\chi_\mathrm{outer} - \chi_\mathrm{inner}}{\chi_0} = 2\sqrt{1-2C_\mathrm{B}^'} \, .
</math>
  </td>
</tr>
</table>
</div>

So if I want to construct PP84 tori that are approximately the same size/shape as the MF53 tori illustrated above, I should choose values of the dimensionless Bernoulli constant as follows:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
~{\chi_0}\biggr|_\mathrm{PP84} 
</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left"><math>
~{\chi_0}\biggr|_\mathrm{MF53}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\Rightarrow~~~~ \frac{1}{2C_\mathrm{B}^'}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\xi_1}{(\xi_1^2-1)^{1/2}}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>
\Rightarrow~~~~ C_\mathrm{B}^'
</math>
  </td>
  <td align="center">
<math>~=</math> 
  </td>
  <td align="left">
<math>~\frac{(\xi_1^2-1)^{1/2}}{2\xi_1} \, .
</math>
  </td>
</tr>
</table>
</div>

In the accompanying figure labeled "Papaloizou-Pringle Tori," we've drawn three different <math>~C_\mathrm{B}^' = \mathrm{constant}</math> meridional contours for the PP84 tori where the values of the dimensionless Bernoulli constants have been chosen to produce values of <math>~\chi_0</math> that are identical to the values displayed by the three MF53 tori shown above.  The following table details properties of these three PP84 tori that have been constructed in an effort to facilitate comparison with the table shown above for MF53 tori.

<table align="center" border="1" cellpadding="8">
<tr>
  <th align="center" colspan="6">
<font color="maroon">
Properties of <math>C_\mathrm{B}^' = \mathrm{constant}</math> PP84 Toroidal Surfaces
</font>
  </th>
</tr>
<tr>
  <td align="center">
Curve in<br />Figure
  </td>
  <td align="center">
<math>C_\mathrm{B}^'</math>
  </td>
  <td align="center">
<math>\chi_\mathrm{inner}</math>
  </td>
  <td align="center">
<math>\chi_\mathrm{outer}</math>
  </td>
  <td align="center">
<math>\chi_0</math>
  </td>
  <td align="center">
<math>\delta</math>
  </td>
</tr>

<tr>
  <td align="center">
Red
  </td>
  <td align="center">
0.208
  </td>
  <td align="center">
0.567
  </td>
  <td align="center">
4.234
  </td>
  <td align="center">
2.400
  </td>
  <td align="center">
1.528
  </td>
</tr>

<tr>
  <td align="center">
Blue
  </td>
  <td align="center">
0.276
  </td>
  <td align="center">
0.599
  </td>
  <td align="center">
3.019
  </td>
  <td align="center">
1.809
  </td>
  <td align="center">
1.338
  </td>
</tr>

<tr>
  <td align="center">
Gold
  </td>
  <td align="center">
0.373
  </td>
  <td align="center">
0.665
  </td>
  <td align="center">
2.019
  </td>
  <td align="center">
1.342
  </td>
  <td align="center">
1.009
  </td>
</tr>
</table>

===Advantageous Coordinate System===

According to [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima's (1986)] review of the [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] discussion &#8212; see his equation (14) &#8212; surfaces of constant density can be defined by the coordinate, <math>~\chi_{PP}</math>, where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\chi_{PP}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{z}{ \frac{z^2}{\sqrt{\varpi^2 + z^2}} + \sqrt{\varpi^2 + z^2} - 1}
</math>
  </td>
</tr>
</table>
</div>

Indeed, equation (6.6) of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P PP84] defines the coordinate, <math>~\chi_{PP}</math>, via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\chi_{PP}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{\cos\theta}{ \cos^2\theta + 1 - \varpi_0/r} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~z = r\cos\theta</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\varpi = r\sin\theta \, .</math>
  </td>
</tr>
</table>
</div>

Let's see if these match.  Starting from the Kojima expression, we have,

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

<tr>
  <td align="right">
<math>~\tan\chi_{PP}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{r\cos\theta}{ \frac{r^2\cos^2\theta}{r} + r - 1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\cos\theta}{ \cos^2\theta + 1 - 1/r} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, they are the same, as long as we appreciate that Kojima assumes all length scales are normalized to <math>~\varpi_0</math>.  Let's express this coordinate in terms of the <math>~(\xi_1, \xi_2)</math> toroidal coordinates as defined by MF53, namely,

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

<tr>
  <td align="right">
<math>
~\frac{z}{a}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{\varpi}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 \biggr]^{1/2}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\frac{r}{a} \equiv \biggl[ \biggl(\frac{x}{a}\biggr)^2 + \biggl(\frac{y}{a}\biggr)^2 + \biggl(\frac{z}{a}\biggr)^2\biggr]^{1/2}
</math>
  </td>
  <td align="center">
<math>
~=
</math>
  </td>
  <td align="left">
<math>
~\biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} \, .
</math>
  </td>
</tr>
</table>

Kojima's expression becomes:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\tan\chi_{PP}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~z \biggl[ \frac{z^2}{\sqrt{\varpi^2 + z^2}} + \sqrt{\varpi^2 + z^2} - 1 \biggr]^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}  \biggl\{ \biggl[\frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)^2}  \biggr] \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
+ \biggl[ \frac{\xi_1 + \xi_2}{\xi_1 - \xi_2} \biggr]^{1/2} - 1 \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\xi_2^2)^{1/2}  \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)}   \biggl[ \frac{\xi_1 - \xi_2}{\xi_1 + \xi_2} \biggr]^{1/2}
+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - (  \xi_1 - \xi_2 ) \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\xi_2^2)^{1/2}  \biggl\{ \frac{(1-\xi_2^2)}{ (\xi_1 - \xi_2)^{1/2} (\xi_1 + \xi_2)^{1/2}}  
+ (\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2} - (  \xi_1 - \xi_2 ) \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\xi_2^2)^{1/2} (\xi_1 - \xi_2)^{1/2} (\xi_1 + \xi_2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1 - \xi_2)(\xi_1 + \xi_2)
- (  \xi_1 - \xi_2 )(\xi_1 + \xi_2)^{1/2} (\xi_1 - \xi_2)^{1/2}  \biggr\}^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\xi_2^2)^{1/2} (\xi_1^2 - \xi_2^2)^{1/2} \biggl\{ (1-\xi_2^2) + (\xi_1^2 - \xi_2^2)
- (  \xi_1 - \xi_2 )(\xi_1^2 - \xi_2^2)^{1/2} \biggr\}^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{(1-\xi_2^2)}{ (\xi_1^2 - \xi_2^2) } \biggr]^{1/2} \biggl\{ \biggl[ \frac{(1-\xi_2^2)}{(\xi_1^2 - \xi_2^2)}\biggr] + 1
- \biggl[ \frac{ (  \xi_1 - \xi_2 )}{(\xi_1 + \xi_2)} \biggr]^{1/2}\biggr\}^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
This does not appear to be very useful or productive!