Editing AxisymmetricConfigurations/SolutionStrategies (section)

=Technique=

To solve the above-specified set of simplified governing equations we will essentially adopt [[SSCpt2/SolutionStrategies#Technique_3|''Technique 3'']] as presented in our construction of spherically symmetric configurations.  Using a barotropic equation of state &#8212; in which case <math>~dP/\rho</math> can be replaced by <math>~dH</math> &#8212; we can combine the two components of the Euler equation shown above back into a single vector equation of the form,

<div align="center">
<math>
\nabla \biggl[ H + \Phi_\mathrm{eff} \biggr] = 0 ,
</math> 
</div>
where it is understood that here, [[AxisymmetricConfigurations/PGE|as displayed earlier]], the gradient represents a two-dimensional operator written in cylindrical coordinates that is appropriate for axisymmetric configurations, namely,
<div align="center">
<math>
\nabla f = {\hat{e}}_\varpi \biggl[ \frac{\partial f}{\partial\varpi} \biggr] +  {\hat{e}}_z \biggl[ \frac{\partial f}{\partial z} \biggr] \, .
</math>
</div>

This means that, throughout our configuration, the functions {{Math/VAR_Enthalpy01}}({{Math/VAR_Density01}}) and <math>~\Phi_\mathrm{eff}</math>({{Math/VAR_Density01}}) must sum to a constant value, call it <math>~C_\mathrm{B}</math>.  That is to say, the statement of hydrostatic balance for axisymmetric configurations reduces to the ''algebraic'' expression,
<div align="center">
<math>~H + \Phi_\mathrm{eff} = C_\mathrm{B}</math> .
</div>
This relation must be solved in conjunction with the Poisson equation,
<div align="center">
<math>~
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho , 
</math><br />
</div>

giving us two equations (one algebraic and the other a two-dimensional <math>2^\mathrm{nd}</math>-order elliptic PDE) that relate the three unknown functions, {{Math/VAR_Enthalpy01}}, {{Math/VAR_Density01}}, and {{Math/VAR_NewtonianPotential01}}.