Editing Apps/PapaloizouPringleTori (section)

===Pressure Maximum===
It is instructive to examine at what location inside the torus the enthalpy (and density and gas pressure) assumes its maximum value.  At any radial location, <math>~\chi_i</math>, <math>~H</math> has a vertical maximum where  <math>~\partial H/\partial\zeta = 0.</math>  That is, the maximum occurs where,
<div align="center">
<math>
\frac{\partial}{\partial\zeta} \biggl[ (\chi_i^2 + \zeta^2)^{-1/2} - \frac{1}{2}\chi_i^{-2}  - C_\mathrm{B}^' \biggr] 
= -\zeta (\chi_i^2 + \zeta^2)^{-3/2} = 0 .
</math>
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For all values of <math>~\chi_i</math>, this relation will be satisfied only at <math>~\zeta = 0</math>.  Hence, vertically, the enthalpy maximum will be located in the equatorial plane of the torus.  Examining the radial enthalpy profile in the equatorial plane, we realize that the enthalpy maximum will occur where,
<div align="center">
<math>
\frac{\partial}{\partial\chi} \biggl[ \chi^{-1} - \frac{1}{2}\chi^{-2}  - C_\mathrm{B}^' \biggr] 
= -\chi^{-2} + \chi^{-3} = 0 .
</math>
</div> 
This relation will be satisfied only at <math>~\chi = \varpi/\varpi_0 = 1</math>.  Hence, <math>~\varpi_0</math> is not only the radial location (in the equatorial plane) where the angular frequency of the torus equals the Keplerian frequency, it is also the radial location of the pressure maximum.  

Note that the enthalpy at the pressure maximum &#8212; that is, at the location <math>~(\chi,\zeta) = (1,0)</math> &#8212; is,
<div align="center">
<math>
H_0 = (n+1)\biggl[ \frac{P}{\rho} \biggr]_0 = \frac{GM_\mathrm{pt}}{\varpi_0} \biggl[ \frac{1}{2}  - C_\mathrm{B}^' \biggr]  = \frac{(v_\varphi |_0)^2}{2} [1 - 2C_\mathrm{B}^' ] \, .
</math>
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