Editing Appendix/Ramblings/PPToriPt1A (section)

===Equilibrium Configuration===
In our [[Apps/PapaloizouPringleTori#Solution|separate discussion of PP84]], we showed that the equilibrium structure of a PP-torus is defined by the enthalpy distribution,
<div align="center">
<math>
H = \frac{GM_\mathrm{pt}}{\varpi_0} \biggl[ (\chi^2 + \zeta^2)^{-1/2} - \frac{1}{2}\chi^{-2}  - C_B^' \biggr] .
</math>
</div> 
Normalizing this expression by the enthalpy at the "center" &#8212; ''i.e.,'' at the pressure maximum &#8212; of the torus which, as we have [[Apps/PapaloizouPringleTori#Pressure_Maximum|already shown]], is 
<div align="center">
<math>
H_0 = \frac{GM_\mathrm{pt}}{2\varpi_0} [1-2C_B^' ] \,
</math>
</div> 
gives,
<div align="center">
<math>
[1-2C_B^' ]\biggl(\frac{H}{H_0}\biggr) = 2(\chi^2 + \zeta^2)^{-1/2} - \chi^{-2} -1  + [1 - 2C_B^' ].
</math>
</div> 

Now, in our review of [[Apps/PapaloizouPringleTori#Model_as_Described_by_Kojima|Kojima's (1986)]] work, we showed that the square of the Mach number at the "center" of the torus is,

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

<tr>
  <td align="right">
<math>~\mathfrak{M}_0^2 \equiv \frac{(v_\varphi|_0)^2}{(c_s|_0)^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2(n+1)}{\gamma}\biggl[ \frac{1}{\chi_-} - 1 \biggr]^{-2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2n [ 1- 2C_B^' ]^{-1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ [1 - 2C_B^'] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2n}{\mathfrak{M}_0^2} \, , </math>
  </td>
</tr>
</table>
</div>
where, in obtaining this last expression we have related the adiabatic exponent to the polytropic index via the relation, <math>~\gamma = (n+1)/n</math>.  Instead of specifying the system's Mach number, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] defines the dimensionless parameter,

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

<tr>
  <td align="right">
<math>~\beta^2 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{2n}{\mathfrak{M}_0^2} \, .</math>
  </td>
</tr>

</table>
</div>

Implementing this parameter swap, the equilibrium expression becomes,
<div align="center">
<math>
\beta^2 \biggl(\frac{H}{H_0}\biggr) = 2(\chi^2 + \zeta^2)^{-1/2} - \chi^{-2} -1  + \beta^2 \, ,
</math>
</div> 
or,

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

<tr>
  <td align="right">
<math>~\frac{H}{H_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{\beta^2}\biggl[\chi^{-2} - 2(\chi^2 + \zeta^2)^{-1/2} + 1 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Looking at Figure 1 of Blaes85 &#8212; see also the coordinate definitions given in his equation (2.1) &#8212; I conclude that,

<div align="center">
<math>~\chi = 1 - x\cos\theta</math> 
&nbsp; &nbsp;&nbsp;&nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>\zeta = x\sin\theta \, .</math>
</div>

Hence,

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

<tr>
  <td align="right">
<math>~\frac{H}{H_0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{\beta^2}\biggl\{ [1 - x\cos\theta]^{-2} - 2[(1 - x\cos\theta)^2 + x^2\sin^2\theta]^{-1/2} + 1 \biggr\} </math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{\beta^2}\biggl[ (1 - x\cos\theta)^{-2} - 2(1 - 2x\cos\theta + x^2)^{-1/2} + 1 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

This matches equation (2.2) of Blaes85, if we acknowledge that Blaes uses <math>~f</math> &#8212; instead of the parameter notation, <math>~\Theta_H</math>, found in [[SSC/Structure/Polytropes#Governing_Relations|our discussion of equilibrium polytropic configurations]] &#8212; to denote the normalized enthalpy; that is,

<div align="center">
<math>~f_\mathrm{Blaes85} = \Theta_H \equiv \frac{H}{H_0} \, .</math>
</div>

This expression for the enthalpy throughout a Papaloizou-Pringle torus is valid for tori of arbitrary thickness <math>~(0 < \beta < 1)</math>.  When considering only slim tori, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] points out that this expression can be written in terms of the following power series in <math>~x</math> (see his equation 1.3):
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Theta_H</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 - \frac{1}{\beta^2}\biggl[x^2 + x^3(3\cos\theta - \cos^3\theta) + \mathcal{O}(x^4)  \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

Blaes then adopts a related parameter that is constant on isobaric surfaces, namely,
<div align="center">
<math>\eta^2 \equiv 1 - \Theta_H \, ,</math>
</div>
which is unity at the surface of the torus and which goes to zero at the cross-sectional center of the torus.  Notice that <math>~\eta</math> tracks the "radial" coordinate that measures the distance from the center of the torus; in particular, keeping only the leading-order term in <math>~x</math>, there is a simple linear relationship between  <math>~\eta</math> and <math>~x</math>, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\eta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~[1 - \Theta_H]^{1/2} \approx \frac{x}{\beta} \, .</math>
  </td>
</tr>
</table>
</div>