Editing Apps/OstrikerBodenheimerLyndenBell66 (section)

===Our Approach===

When the stated objective is to construct steady-state equilibrium models of rotationally flattened, axisymmetric configurations, the [[AxisymmetricConfigurations/Equilibria#Axisymmetric_Configurations_.28Steady-State_Structures.29|accompanying introductory chapter]] shows how the overarching set of [[PGE#Principal_Governing_Equations|principal governing equations]] can be reduced in form to the following set of three coupled PDEs (expressed either in terms of cylindrical or spherical coordinates):


<table align="center" border="1" cellpadding="10">
<tr><th align="center" colspan="2"><font size="+0">Table 1: &nbsp; Simplified Set of Three Coupled PDEs</font></th></tr>
<tr>
  <th align="center" width="50%">Cylindrical Coordinate Base</th>
  <th align="center">Spherical Coordinate Base</th>
</tr>
<tr>
  <td align="center">
<font color="#770000">'''Poisson Equation'''</font><br />

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

<tr>
  <td align="right">
<math>~
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho </math>
  </td>
</tr>
</table>

The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
<br />

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~{\hat{e}}_\varpi</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3}  
</math>
  </td>
</tr>
<tr>
  <td align="right"><math>~{\hat{e}}_z</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">
<font color="#770000">'''Poisson Equation'''</font><br />

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

<tr>
  <td align="right">
<math>~
\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr] 
+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G\rho</math>
  </td>
</tr>
</table>

The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
<br />

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right"><math>~{\hat{e}}_r</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
~0
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho} \frac{\partial P}{\partial r}+  \frac{\partial \Phi }{\partial r} \biggr]  - \biggl[ \frac{j^2}{r^3 \sin^2\theta} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~{\hat{e}}_\theta</math>: &nbsp; &nbsp;</td>
  <td align="right">
<math>
~0
</math>
  </td>
  <td align="center">
=
  </td>
  <td align="left">
<math>
\biggl[ \frac{1}{\rho r}  \frac{\partial P}{\partial\theta} +  \frac{1}{r} \frac{\partial \Phi}{\partial\theta}  \biggr] -  \biggl[ \frac{j^2}{r^3 \sin^3\theta} \biggr] \cos\theta
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>


This set of simplified governing relations must then be supplemented by a specification of: (a) a barotropic equation of state, <math>P(\rho)</math>; and (b) the equilibrium configurations's radial specific angular momentum profile <math>j(\varpi)</math>.  How does this recommended modeling approach compare to the approach outlined by [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract Ostriker, Bodenheimer &amp; Lynden-Bell (1966)] and further detailed and executed by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract J. P. Ostriker &amp; P. Bodenheimer (1968)]?