Editing Apps/PapaloizouPringleTori (section)

===Hachisu Self-Consistent Field Technique===
For the sake of contextual continuity, it is useful to relate the above result to the Hachisu self-consistent field (HSCF) technique, which we rely upon heavily to derive the equilibrium structure of self-gravitating systems.  The HSCF technique begins by identifying the relevant algebraic expression for the enthalpy as derived above, namely,
<div align="center">
<math>
H = \frac{GM_\mathrm{pt}}{(\varpi^2 + z^2)^{1/2}} - \frac{j_0^2}{2\varpi^2}  + C_\mathrm{B} .
</math>
</div> 
In the HSCF technique, it is customary to normalize all lengths to the distance <math>\varpi_{+}</math> from the origin to the point where the outer edge of the torus touches the equatorial plane.  From this perspective, then, the relevant dimensionless coordinates are,
<div align="center">
<math>
x_\mathrm{HSCF} \equiv \frac{\varpi}{\varpi_{+}} ~~~~~\mathrm{and}~~~~~ z_\mathrm{HSCF} \equiv \frac{z}{\varpi_{+}} ,
</math>
</div>
and the equation for the dimensionless enthalpy becomes,
<div align="center">
<math>
H_\mathrm{HSCF} \equiv \frac{H}{(GM_\mathrm{pt}/\varpi_{+})} = (x_\mathrm{HSCF}^2 + z_\mathrm{HSCF}^2)^{-1/2} - \frac{1}{2} (j_\mathrm{HSCF}^2) x_\mathrm{HSCF}^{-2}  - C_\mathrm{HSCF} ,
</math>
</div> 
where the normalized Bernoulli constant,
<div align="center">
<math>
C_\mathrm{HSCF} \equiv - \frac{C_\mathrm{B} \varpi_{+}}{GM_\mathrm{pt}} ,
</math>
</div>
and the normalized specific angular momentum is,
<div align="center">
<math>
j_\mathrm{HSCF} \equiv \frac{j_0}{(GM_\mathrm{pt} \varpi_{+})^{1/2}} .
</math>
</div>
This expression for the dimensionless enthalpy has two unknowns:  <math>~j_\mathrm{HSCF}</math> and <math>~C_\mathrm{HSCF}</math>.  Hence, two boundary points must be specified before a solution for <math>~H_\mathrm{HSCF}(\varpi,z)</math> can be obtained.  In the HSCF technique, it is customary to specify the location of two points on the surface of the configuration, where by design <math>~H=0</math>.  For toroidal configurations, it is furthermore customary for that specification to be where the inner and outer edges of the torus touch the equatorial plane.  By design, the outer edge is at <math>~(x_\mathrm{HSCF}, ~z_\mathrm{HSCF}) = (1, ~0)</math>; the inner edge will be at <math>(x_\mathrm{HSCF}, ~z_\mathrm{HSCF}) = (x_{in}, ~0)</math>.  Setting <math>~H_\mathrm{HSCF}=0</math> at both of these locations gives the following two algebraic relations,
<div align="center">
<math>
C_\mathrm{HSCF} = 1 - \frac{1}{2} (j_\mathrm{HSCF}^2) ,
</math>
</div> 
and
<div align="center">
<math>
C_\mathrm{HSCF} = x_{in}^{-1} - \frac{1}{2} (j_\mathrm{HSCF}^2) x_{in}^{-2}  ,
</math>
</div> 
which can be used in combination to solve for the two unknowns. The result is,
<div align="center">
<math>
C_\mathrm{HSCF} = \frac{1}{1 + x_{in}} ,
</math>
</div> 
and,
<div align="center">
<math>
j_\mathrm{HSCF} = \biggl[ \frac{2x_{in}}{1 + x_{in}} \biggr]^{1/2} .
</math>
</div>

At what radial position in the equatorial plane,  <math>~x_0</math>, is the pressure maximum located?  The pressure maximum is also the enthalpy maximum, so the answer is given by looking in the equatorial plane <math>~(z_\mathrm{HSCF} = 0)</math> for the location where <math>~dH_\mathrm{HSCF}/dx_\mathrm{HSCF} = 0</math>.  Since,
<div align="center">
<math>
\frac{dH_\mathrm{HSCF}}{dx_\mathrm{HSCF}} = - x_\mathrm{HSCF}^{-2} + (j_\mathrm{HSCF}^2) x_\mathrm{HSCF}^{-3} \, ,
</math>
</div>
the pressure maximum must be located at,
<div align="center">
<math>
x_\mathrm{HSCF} = x_0 = j_\mathrm{HSCF}^2 \, .
</math>
</div>
Hence, in terms of <math>~x_{in}</math>,
<div align="center">
<math>
x_0 =  \frac{2x_{in}}{1 + x_{in}} .
</math>
</div>