Editing Apps/PapaloizouPringleTori (section)

===Boundary Conditions===

In the equatorial plane, the locations of the inner and outer edges of the torus are obtained by setting both <math>~H</math> and <math>~\zeta</math> to zero, that is, they are obtained from the roots of the equation,
<div align="center">
<math>
~\chi^{-2} - 2\chi^{-1} + 2C_\mathrm{B}^' = 0  ,
</math>
</div>
that is, the inner <math>~(\chi_-)</math> and outer <math>~(\chi_+)</math> edges are located at,
<div align="center">
<math>
~\chi_\pm  =  \frac{1}{1 \mp \sqrt{1-2C_\mathrm{B}^'}} .
</math>
</div>
Clearly, the relative cross-sectional size &#8212; or, as defined in PP84, the dimensionless distortion parameter <math>~\delta \equiv (\chi_+ + \chi_-)/2 = (2C_\mathrm{B}^')^{-1}</math> &#8212; is determined by the choice of the dimensionless Bernoulli constant, which must lie in the range, <math>~\tfrac{1}{2} \geq C_\mathrm{B}^' \geq 0</math>.  For later reference, we will also note that the ratio of the inner edge to the outer edge is,
<div align="center">
<math>
~\frac{\chi_{-}}{\chi_{+}}  =  \frac{ 1 - \sqrt{1-2C_\mathrm{B}^'} }{1 + \sqrt{1-2C_\mathrm{B}^'} } .
</math>
</div>

Indeed, more generally, at all vertical locations above and below the equatorial plane, the surface of the torus occurs where the enthalpy goes to zero.  The meridional <math>~(\chi,\zeta)</math> cross-section through three such surfaces are displayed in the left panel of Figure 1.  The displayed surfaces correspond to <math>~C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (green curve).  Carrying forward the above discussion, we note that in all three cases the pressure maximum is located at <math>(\chi,~\zeta) = (1,~0)</math>.  Properties of these tori are described in more detail in an accompanying discussion entitled, [[Appendix/Ramblings/ToroidalCoordinates#Papaloizou-Pringle_Tori|''Toroidal Configurations and Related Coordinate Systems'']].  

We should perhaps emphasize that the shape of the surface of each torus, as shown in the left-hand panel of Figure 1, is independent of the chosen polytropic index.


<div align="center">
<table border="1" cellpadding="8" align="center" width="80%">
<tr><th align="center" colspan="2">Figure 1:  Papaloizou-Pringle Torus</th></tr>
<tr><td align="center">
[[File:PP84Tori.jpg|350px|Meridional contours of constant C<sub>B</sub>'.]]  
</td>
<td align="center" bgcolor="#D0FFFF">
[[File:TorusMovie1.gif|350px|Pivoting PP Torus]]
</td></tr>
<tr>
  <td align="left">''Left Panel'' &#8212; A meridional cross-section outlining the surface of three separate PP tori:  <math>C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (green curve).</td>
  <td align="left">''Right Panel'' &#8212; A three-dimensional, cutaway rendering illustrating the geometric structure of one PP torus having <math>C_\mathrm{B}^' = 0.3239,</math> <math>x_\mathrm{in} = \chi_{-}/\chi_{+} = 0.2551,</math> &nbsp;<math>x_0 = 0.4065.</math> The separately colored contours illustrate four surfaces of constant enthalpy; the red contour identifies the toroidal surface.</td>
</tr>
</table>
</div>