Editing Apps/PapaloizouPringleTori (section)

===Model as Described by Kojima===
From the above expression for <math>~\chi_\pm</math>, we see that,
<div align="center">
<math>
\sqrt{1-2C_\mathrm{B}^'} = \frac{1}{\chi_-} - 1 = 1 - \frac{1}{\chi_+} \, .
</math>
</div>
Hence,
<div align="center">
<math>
\chi_+ = \biggl[ 2 - \frac{1}{\chi_-}\biggr]^{-1} = \frac{\chi_-}{2\chi_- - 1} \, .
</math>
</div>
This agrees with the relation between <math>~\chi_+</math> (written as <math>~r_+</math>) and <math>~\chi_-</math> (written as <math>~r_-</math>) presented as equation (7) by [http://http://ptp.ipap.jp/link?PTP/75/251/ Kojima] (1986, Progress of Theoretical Physics, 75, 251-261), who also has examined the properties of Papaloizou-Pringle tori.  In his equation (8), Kojima points out that the inner edge of the torus lies within the range, <math>~\tfrac{1}{2} < \chi_- < 1</math>, which is consistent with stating that the value of PB84's Bernoulli constant falls within the range, <math> ~\tfrac{1}{2} \geq C_\mathrm{B}^' \geq 0</math>.  

Using the just-derived relationship between <math>~C_\mathrm{B}^'</math> and <math>~\chi_-</math>, we deduce that the enthalpy at the pressure maximum depends only on the choice of <math>~\chi_-</math> as follows:
<div align="center">
<math>
H_0 = (n+1)\biggl[ \frac{P}{\rho} \biggr]_0 = \frac{(v_\varphi |_0)^2}{2} \biggl[ \frac{1}{\chi_-} - 1 \biggr]^2 \, .
</math>
</div> 
Under adiabatic conditions, the sound speed of a gas is given by the algebraic relation,
<div align="center">
<math>
c_s^2 = \gamma \biggl[\frac{P}{\rho}\biggr] \, ,
</math>
</div>
where <math>~\gamma</math> is the specified adiabatic exponent.  Hence, the expression just derived for <math>~H_0</math> also gives us the sound speed at the pressure maximum of the torus, namely,
<div align="center">
<math>
(c_s |_0)^2 = \frac{\gamma}{(n + 1)}  \frac{(v_\varphi |_0)^2}{2} \biggl[ \frac{1}{\chi_-} - 1 \biggr]^2 \, ,
</math>
</div> 
that is,
<div align="center">
<math>
\frac{c_s|_0}{v_\varphi |_0} = \biggl[ \frac{\gamma}{2(n + 1)} \biggr]^{1/2} \biggl[ \frac{1}{\chi_-} - 1 \biggr] \, .
</math>
</div> 
This matches Kojima's equation (11) with <math>~\chi_-</math> written in place of <math>~r_-</math>, namely, 
<div align="center">
<math>
\biggl( \frac{c_s}{v_\varphi} \biggr)_0 = \frac{1}{\sqrt{6}} \biggl(\frac{1}{\chi_-} - 1 \biggr) \, ,
</math>
</div>
when one adopts Kojima's choice of <math>~\gamma = 4/3</math> and <math>~n = (\gamma - 1)^{-1} = 3</math>.