Editing AxisymmetricConfigurations/HSCF (section)

=Hachisu Self-Consistent-Field Technique=

<table border="0" cellpadding="10" align="left"><tr><th>[[Image:Hai.gif|90px|center|Izumi Hachisu]]<br />I. Hachisu</th></tr></table> 
In 1986, Izumi Hachisu published two papers in '''The Astrophysical Journal Supplement Series''' ([http://adsabs.harvard.edu/abs/1986ApJS...61..479H vol. 61, pp. 479-507], and [http://adsabs.harvard.edu/abs/1986ApJS...62..461H vol. 62, pp. 461-499]) describing ''"A Versatile Method for Obtaining Structures of Rapidly Rotating Stars."''  (Henceforth, we will refer to this method as the Hachisu Self-Consistent-Field, or HSCF, technique.)  The HSCF technique has been built upon &#8212; and has further improved &#8212; the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker &amp; J. W.-K. Mark (1968)].  We have found the HSCF technique to be an extremely powerful tool for constructing equilibrium configurations of self-gravitating fluid systems under a wide variety of different circumstances.  
<!-- This chapter has been built upon [http://www.phys.lsu.edu/astro/H_Book.current/Applications/Structure/HSCF_Code/HSCF.outline.html an (''ca.'' 1999) outline of the HSCF technique] that appeared in our [http://www.phys.lsu.edu/astro/H_Book.current/H_Book.html original version of this HyperText Book] (H_Book).  -->
The photo of Professor Izumi Hachisu shown here, on the left, dates from the mid-1980s &#8212; about the time he developed this remarkably useful numerical technique.
<!-- A more recent photo can be found on the [http://gpes.c.u-tokyo.ac.jp/faculty-staff/materials-systems-and-dynamics/izumi-hachisu.html web page associated with Professor Hachisu's current faculty appointment at the University of Tokyo, Komaba].
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==Constructing Two-Dimensional, Axisymmetric Structures==
As has been explained in [[AxisymmetricConfigurations/SolutionStrategies#Technique|an accompanying discussion]], our objective is to solve an ''algebraic'' expression for hydrostatic balance,
<div align="center">
<math>~H + \Phi + \Psi = C_0</math> ,
</div>

in conjunction with the Poisson equation in a form that is appropriate for two-dimensional, axisymmetric systems, namely,
<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>
</div>

===Steps to Follow===
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<table border="1" cellpadding="5" align="center">
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Figure 1 extracted from [http://adsabs.harvard.edu/abs/1986ApJS...61..479H I. Hachisu (1986)]<p></p>
"''A Versatile Method for Obtaining Structures of Rapidly Rotating Stars''"<p></p>
ApJS, vol. 61, pp. 479-507 &copy; American Astronomical Society
</td></tr>
<tr><td>
[[File:HSCF MeridionalPlane.png|center|250px|HSCF Meridional Plane]]</td></tr>
<tr><td align="left">Two points A and B on the surface specify a model uniquely: (a) spheroidal structure; (b) toroidal structure.</td></tr></table>
</td></tr></table>
<ol>
  <li>'''Choose a particular [[SR#Barotropic_Structure|barotropic equation of state]].''' &nbsp; More specifically, functionally define the density-enthalpy relationship, <math>~\rho(H)</math>, and identify what value, <math>~H_\mathrm{surface}</math>, the enthalpy will have at the surface of your configuration.  For example, if a ''polytropic'' equation of state is adopted, <math>~H_\mathrm{surface} = 0</math> is a physically reasonable prescription.</li>
  <li>Choosing from, for example, a list of astrophysically relevant ''[[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|simple rotation profiles]],'' '''specify the corresponding functional form of the centrifugal potential''', <math>~\Psi(\varpi)</math>, that will define the radial distribution of specific angular momentum in your equilibrium configuration.  If the choice is uniform rotation, then <math>~\Psi = - \varpi^2 \omega_0^2/2 \, ,</math> where <math>~\omega_0</math> is a constant to be determined.</li>
  <li>On your chosen computational lattice &#8212; for example, on a cylindrical-coordinate mesh &#8212; '''identify two boundary points''', A and B, that will lie on the surface of your equilibrium configuration.  These two points should remain fixed in space during the HSCF iteration cycle and ultimately will confine the volume and define the geometry of the derived equilibrium object.  Note that, by definition, the enthalpy at these two points is, <math>~H_A = H_B = H_\mathrm{surface}</math>.</li>
  <li>Throughout the volume of your computational lattice, ''' ''guess'' a trial distribution of the mass density,''' <math>~\rho(\varpi,z)</math>, such that no material falls outside a volume defined by the two boundary points, A and B, that were identified in Step #3.  Usually an initially uniform density distribution will suffice to start the SCF iteration.</li>
  <li>Via ''some'' accurate numerical algorithm, '''solve the Poisson equation''' to determine the gravitational potential, <math>~\Phi(\varpi,z)</math>, throughout the computational lattice corresponding to the trial mass-density distribution that was specified in Step #4 (or in Step #9).</li>
  <li>From the gravitational potential determined in Step #5, '''identify the values of <math>~\Phi_A</math> and <math>~\Phi_B</math>''' at the two boundary points that were selected in Step #3.</li>
  <li>From the "known" values of the enthalpy (Step #3) and the gravitational potential (Step #6) at the two selected surface boundary points A and B, '''determine the values of the constants, <math>~C_0</math> and <math>~\omega_0</math>,''' that appear in the algebraic equation that defines hydrostatic equilibrium.</li>
  <li>From the most recently determined values of the gravitational potential, <math>~\Phi(\varpi,z)</math> (Step #5), and the values of the two constants, <math>~C_0</math> and <math>~\omega_0</math> just determined (Step #7), '''determine the enthalpy distribution throughout the computational lattice.'''</li>
  <li>From <math>~H(\varpi,z)</math> and the selected barotropic equation of state (Step #1), '''calculate an "improved guess" of the density distribution,''' <math>~\rho(\varpi,z)</math>, throughout the computational lattice.</li>
  <li>'''Has the model converged to a satisfactory equilibrium solution?'''  (Usually a satisfactory solution has been achieved when the derived model parameters &#8212; for example, the values of <math>~C_0</math> and <math>~\omega_0</math> &#8212; change very little between successive iterations and the viral error is sufficiently small.)
  <ul>
<li>If the answer is, "NO": &nbsp; Repeat steps 5 through 10.</li>
<li>If the answer is, "YES": &nbsp; Stop iteration.</li>
  </ul>
  </li>
</ol>