User:Tohline/AxisymmetricConfigurations/HSCF - Revision history

Tohline: /* Introduction */

2018-05-02T22:13:19Z

Introduction

← Older revision		Revision as of 22:13, 2 May 2018
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	~~Light~~ teal-colored text has been extracted ''verbatim'' from §II.b of [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker and J. W.-K. Mark (1968, ApJ, 151, 1075 - 1088)]:  <font color="#009999">… we shall alternately solve each of the two problems as exactly as numerical techniques permit, and then iterate to obtain self-consistency.		In this paragraph, teal-colored text has been extracted ''verbatim'' from §II.b of [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker and J. W.-K. Mark (1968, ApJ, 151, 1075 - 1088)]:  <font color="#009999">… 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">		<div align="center">

Tohline at 23:54, 1 May 2018

2018-05-01T23:54:19Z

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	=Introduction=		=Introduction=
	~~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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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>
			{{LSU_HBook_header}}

			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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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>


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

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	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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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.		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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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). -->		<!-- 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 ~~immediately above~~, on the left, dates from the mid-1980s — 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].		The photo of Professor Izumi Hachisu shown here, on the left, dates from the mid-1980s — 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].


	~~{{LSU_HBook_header}}~~

	==Constructing Two-Dimensional, Axisymmetric Structures==		==Constructing Two-Dimensional, Axisymmetric Structures==

Tohline at 23:40, 1 May 2018

2018-05-01T23:40:29Z

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	__FORCETOC__		__FORCETOC__
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			=Introduction=
			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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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:  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">
			{{User:Tohline/Math/EQ_Poisson01}}
			</td>
			</tr>
			</table>
			</div>


			Light teal-colored text has been extracted ''verbatim'' from §II.b of [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker and J. W.-K. Mark (1968, ApJ, 151, 1075 - 1088)]:  <font color="#009999">… 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)    <math>~\rho(\vec{x}) ~~\xrightarrow{~~~~~\mathrm{potential ~theory}~~~~~~} ~ \Phi(\vec{x})</math><br />
			ii)    <math>~\Phi(\vec{x}) ~\xrightarrow{~~~\mathrm{equilibrium ~condition}~~~} ~ \rho(\vec{x})</math>
			</div>

			The integral relation between potential and density</font> — as opposed to the ''differential relation'', see Table 1 — <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>.

	=Hachisu Self-Consistent-Field Technique=		=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>		<table border="0" cellpadding="10" align="left"><tr><th>[[Image:Hai.gif\|90px\|center\|Izumi Hachisu]]<br />I. Hachisu</th></tr></table>
	~~Extracted ''verbatum'' 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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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>

	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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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.		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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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). -->		<!-- 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). -->

Tohline: /* Hachisu Self-Consistent-Field Technique */

2018-04-08T21:07:37Z

Hachisu Self-Consistent-Field Technique

← Older revision		Revision as of 21:07, 8 April 2018
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	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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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>		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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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>

	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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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).		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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 immediately above, on the left, dates from the mid-1980s — 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].		The photo of Professor Izumi Hachisu shown immediately above, on the left, dates from the mid-1980s — 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].

Tohline: /* Hachisu Self-Consistent-Field Technique */

2018-04-08T21:02:40Z

Hachisu Self-Consistent-Field Technique

← Older revision		Revision as of 21:02, 8 April 2018
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	=Hachisu Self-Consistent-Field Technique=		=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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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).		<table border="0" cellpadding="10" align="left"><tr><th>[[Image:Hai.gif\|90px\|center\|Izumi Hachisu]]<br />I. Hachisu</th></tr></table>
			Extracted ''verbatum'' 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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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>

			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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 — 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].		The photo of Professor Izumi Hachisu shown immediately above, on the left, dates from the mid-1980s — 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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	</math>		</math>
	</div>		</div>

	~~<table border="0" cellpadding="5" align="center" width="85%">~~
	~~<tr><td align="left">~~
	~~Extracted ''verbatum'' 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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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>~~
	~~</td></tr>~~
	~~</table>~~

	===Steps to Follow===		===Steps to Follow===

Tohline: /* Hachisu Self-Consistent-Field Technique */

2018-04-08T20:57:37Z

Hachisu Self-Consistent-Field Technique

← Older revision		Revision as of 20:57, 8 April 2018
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	=Hachisu Self-Consistent-Field Technique=		=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.) 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 — 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].		<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 — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 — 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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	</td></tr>		</td></tr>
	</table>		</table>

	The HSCF technique has been built upon — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & J. W.-K. Mark (1968)].

	===Steps to Follow===		===Steps to Follow===

Tohline: /* Constructing Two-Dimensional, Axisymmetric Structures */

2018-03-30T02:06:59Z

Constructing Two-Dimensional, Axisymmetric Structures

← Older revision		Revision as of 02:06, 30 March 2018
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	</td></tr>		</td></tr>
	</table>		</table>

			The HSCF technique has been built upon — and has further improved — the SCF method introduced by [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & J. W.-K. Mark (1968)].

	===Steps to Follow===		===Steps to Follow===

Tohline at 01:56, 30 March 2018

2018-03-30T01:56:29Z

← Older revision		Revision as of 01:56, 30 March 2018
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	</div>		</div>

	<table border="0" cellpadding="5" align="center" width="90%">		<table border="0" cellpadding="5" align="center" width="85%">
	<tr><td align="left">		<tr><td align="left">
	Extracted ''verbatum'' from a relatively early review by Strittmatter (1969):  <font color="darkgreen">		Extracted ''verbatum'' 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 … 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 James (1964) in considering uniformly rotating polytopes and by 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 …		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 … 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 … Ostriker and his collaborators (cf. [http://adsabs.harvard.edu/abs/1968ApJ...151.1075O J. P. Ostriker & 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 … 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>		</font>
	</td></tr>		</td></tr>

Tohline: /* Constructing Two-Dimensional, Axisymmetric Structures */

2018-03-30T01:44:49Z

Constructing Two-Dimensional, Axisymmetric Structures

← Older revision		Revision as of 01:44, 30 March 2018
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	</math>		</math>
	</div>		</div>

			<table border="0" cellpadding="5" align="center" width="90%">
			<tr><td align="left">
			Extracted ''verbatum'' from a relatively early review by 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 … 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 James (1964) in considering uniformly rotating polytopes and by 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 …
			</font>
			</td></tr>
			</table>

	===Steps to Follow===		===Steps to Follow===

Tohline: /* Reviews */

2018-03-30T01:33:08Z

Reviews

← Older revision		Revision as of 01:33, 30 March 2018
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	==Reviews==		==Reviews==

			* [http://adsabs.harvard.edu/abs/1969ARA%26A...7..665S P. A. Strittmatter (1969, Annual Review of Astronomy and Astrophysics, 7, 665 - 684)] — ''Stellar Rotation''
	* [http://adsabs.harvard.edu/abs/1967ARA%26A...5..465L N. R. Lebovitz (1967, Annual Review of Astronomy and Astrophysics, 5, 465 - 480)] — ''Rotating Fluid Masses''		* [http://adsabs.harvard.edu/abs/1967ARA%26A...5..465L N. R. Lebovitz (1967, Annual Review of Astronomy and Astrophysics, 5, 465 - 480)] — ''Rotating Fluid Masses''