Editing AxisymmetricConfigurations/HSCF (section)

=Introduction=
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! style="height: 150px; width: 150px; background-color:white;" |[[H_BookTiledMenu#Two-Dimensional_Configurations_.28Axisymmetric.29|<b>Hachisu Self-<br />Consistent-Field<br />[HSCF]<br />Technique</b>]]
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Text colored dark green has been extracted ''verbatim'' from a relatively early review by [http://adsabs.harvard.edu/abs/1969ARA%26A...7..665S P. A. Strittmatter (1969)]: 

<font color="darkgreen">The study of the internal structure of rotating stars is complicated by the necessity to introduce circulation currents and to solve Poisson's equation in two dimensions with an unknown boundary to the density distribution &hellip; With the advent of large-capacity, fast computers it has become possible to tackle the problem by direct numerical integration of the appropriate partial-differential equations.  This method has been adopted by [http://adsabs.harvard.edu/abs/1964ApJ...140..552J R. A. James (1964)] in considering uniformly rotating polytopes and by [http://adsabs.harvard.edu/abs/1965ApJ...142..208S R. Stoeckly (1965)] who treated the case when the rotation velocity <math>~\Omega</math> varies as a Gaussian function of the distance from the rotation axis &hellip; Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker &amp; J. W.-K. Mark 1968]) have developed an approach (called, by them, the self-consistent field or SCF method) in which Poisson's equation is replaced by its formal integral solution and an iterative procedure established in which the potential is derived from a guessed density distribution.  A new density distribution is then obtained from the equation of hydrostatic support &hellip; While the present author has no experience in using the SCF method, it would appear to combine considerable power with reasonable speed of application and must probably be considered the most practical method of solution so far developed.</font>

<div align="center">
<table border="1" cellpadding="8" align="center" width="80%">
<tr><th align="center" colspan="2"><font size="+0">Table 1: &nbsp;Poisson Equation</font></th></tr>
<tr>
  <th align="center">Integral Relation</th>
  <th align="center">Differential Relation </th>
</tr>
<tr>
  <td align="center">
<table border="0" align="center">
<tr>
  <td align="right">
<math>~ \Phi(\vec{x})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
  </td>
</tr>
</table>
  </td>
  <td align="center">
{{Math/EQ_Poisson01}}
  </td>
</tr>
</table>
</div>


In this paragraph, teal-colored text has been extracted ''verbatim'' from &sect;II.b of [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker and J. W.-K. Mark (1968, ApJ, 151, 1075 - 1088)]: &nbsp;<font color="#009999">&hellip;  we shall alternately solve each of the two problems as exactly as numerical techniques permit, and then iterate to obtain self-consistency.  

<div align="center">
i) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <math>~\rho(\vec{x})</math>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <math>~\xrightarrow{~~~~~\mathrm{potential ~theory}~~~~~~} ~ \Phi(\vec{x})</math><br />
ii) &nbsp; &nbsp;<math>~\Phi(\vec{x})+\Psi(\varpi) ~\xrightarrow{~~~\mathrm{equilibrium ~condition}~~} ~ \rho(\vec{x})</math>
</div>

The integral relation between potential and density</font> &#8212; as opposed to the ''differential relation'', see Table 1 &#8212; <font color="#009999">encourages us to think that a "bad" density will, in step (i), lead to a "good" potential, which, by step (ii), will yield a "good" density which when substituted in (i) produces a "very good" potential, etc.   The method works.</font>  It should be noted that <font color="#009999">an SCF approach is commonly used for determining molecular structure under the name [https://en.wikipedia.org/wiki/Hartree–Fock_method Hartree-Fock]</font>.