User:Tohline/AxisymmetricConfigurations/HSCF
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Text colored dark green has been extracted verbatim from a relatively early review by P. A. Strittmatter (1969): 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 largecapacity, fast computers it has become possible to tackle the problem by direct numerical integration of the appropriate partialdifferential equations. This method has been adopted by R. A. James (1964) in considering uniformly rotating polytopes and by R. Stoeckly (1965) who treated the case when the rotation velocity varies as a Gaussian function of the distance from the rotation axis … Ostriker and his collaborators (cf. J. P. Ostriker & J. W.K. Mark 1968) have developed an approach (called, by them, the selfconsistent 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.
In this paragraph, tealcolored text has been extracted verbatim from §II.b of J. P. Ostriker and J. W.K. Mark (1968, ApJ, 151, 1075  1088): … we shall alternately solve each of the two problems as exactly as numerical techniques permit, and then iterate to obtain selfconsistency.
i)
ii)
The integral relation between potential and density — as opposed to the differential relation, see Table 1 — 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. It should be noted that an SCF approach is commonly used for determining molecular structure under the name HartreeFock.
Hachisu SelfConsistentField Technique
I. Hachisu 

In 1986, Izumi Hachisu published two papers in The Astrophysical Journal Supplement Series (vol. 61, pp. 479507, and vol. 62, pp. 461499) describing "A Versatile Method for Obtaining Structures of Rapidly Rotating Stars." (Henceforth, we will refer to this method as the Hachisu SelfConsistentField, or HSCF, technique.) The HSCF technique has been built upon — and has further improved — the SCF method introduced by 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 selfgravitating fluid systems under a wide variety of different circumstances. The photo of Professor Izumi Hachisu shown here, on the left, dates from the mid1980s — about the time he developed this remarkably useful numerical technique; a more recent photo can be found on the web page associated with Professor Hachisu's current faculty appointment at the University of Tokyo, Komaba.
Constructing TwoDimensional, Axisymmetric Structures
As has been explained in an accompanying discussion, our objective is to solve an algebraic expression for hydrostatic balance,
,
in conjunction with the Poisson equation in a form that is appropriate for twodimensional, axisymmetric systems, namely,
Steps to Follow

 Choose a particular barotropic equation of state. More specifically, functionally define the densityenthalpy relationship, , and identify what value, , the enthalpy will have at the surface of your configuration. For example, if a polytropic equation of state is adopted, is a physically reasonable prescription.
 Choosing from, for example, a list of astrophysically relevant simple rotation profiles, specify the corresponding functional form of the centrifugal potential, , that will define the radial distribution of specific angular momentum in your equilibrium configuration. If the choice is uniform rotation, then where is a constant to be determined.
 On your chosen computational lattice — for example, on a cylindricalcoordinate mesh — 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, .
 Throughout the volume of your computational lattice, guess a trial distribution of the mass density, , 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.
 Via some accurate numerical algorithm, solve the Poisson equation to determine the gravitational potential, , throughout the computational lattice corresponding to the trial massdensity distribution that was specified in Step #4 (or in Step #9).
 From the gravitational potential determined in Step #5, identify the values of and at the two boundary points that were selected in Step #3.
 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, and , that appear in the algebraic equation that defines hydrostatic equilibrium.
 From the most recently determined values of the gravitational potential, (Step #5), and the values of the two constants, and just determined (Step #7), determine the enthalpy distribution throughout the computational lattice.
 From and the selected barotropic equation of state (Step #1), calculate an "improved guess" of the density distribution, , throughout the computational lattice.
 Has the model converged to a satisfactory equilibrium solution? (Usually a satisfactory solution has been achieved when the derived model parameters — for example, the values of and — change very little between successive iterations and the viral error is sufficiently small.)
 If the answer is, "NO": Repeat steps 5 through 10.
 If the answer is, "YES": Stop iteration.
Related Discussions
Reviews
 P. A. Strittmatter (1969, Annual Review of Astronomy and Astrophysics, 7, 665  684) — Stellar Rotation
 N. R. Lebovitz (1967, Annual Review of Astronomy and Astrophysics, 5, 465  480) — Rotating Fluid Masses
Solution Methods
 Y. Eriguchi & E. Mueller (1985, A&A, 146, 260  268) — A General Computational Method for Obtaining Equilibria of SelfGravitating and Rotating Gases
 S. W. Stahler (1983, ApJ, 268, 155  184) — The Equilibria of Rotating, Isothermal Clouds. I.  Method of Solution
 Y. Eriguchi (1978, PASJ, 30, 507  518) — Hydrostatic Equilibria of Rotating Polytropes
 S. I. Blinnikov (1975, Soviet Astronomy, 19, 151  156) — SelfConsistent Field Method in the Theory of Rotating Stars
 M. J. Clement (1974, ApJ, 194, 709  714) — On the Solution of Poisson's Equation for Rapidly Rotating Stars
 S. Jackson (1970, ApJ, 161, 579  585) — Rapidly Rotating Stars. V. The Coupling of the Henyey and the SelfConsistent Methods
 J. P. Ostriker & J. W.K. Mark (1968, ApJ, 151, 1075  1088) — Rapidly Rotating Stars. I. The SelfConsistentField Method
 R. A. James (1964, ApJ, 140, 552  582) — The Structure and Stability of Rotating Gas Masses
Early Eriguchi Applications
 Y. Eriguchi & E. Mueller (1985, A&A, 147, 161  168) — Equilibrium Models of Differentially Rotating Polytropes and the Collapse of Rotating Stellar Cores
 I. Hachisu & Y. Eriguchi (1984, PASJ, 36, 497  503) — Bifurcation Points on the Maclaurin Sequence
 I. Hachisu & Y. Eriguchi (1984, PASJ, 36, 259  276) — Binary Fluid Star
 I. Hachisu & Y. Eriguchi (1984, PASJ, 36, 239  257) — Fission of Dumbbell Equilibrium and Binary State of Rapidly Rotating Polytropes
 I. Hachisu & Y. Eriguchi (1983, MNRAS, 204, 583  589) — Bifurcations and Phase Transitions of SelfGravitating and Uniformly Rotating Fluid
 I. Hachisu & Y. Eriguchi (1982, Prog. Theor. Phys., 68, 206  221) — Bifurcation and Fission of Three Dimensional, Rigidly Rotating and SelfGravitating Polytropes
 I. Hachisu, Y. Eriguchi, & D. Sugimoto (1982, Prog. Theor. Phys., 68, 191  205) — Rapidly Rotating Polytropes and Concave Hamburger Equilibrium
 Y. Eriguchi & D. Sugimoto (1981, Prog. Theor. Phys., 65, 1870  1875) — Another Equilibrium Sequence of SelfGravitating and Rotating Incompressible Fluid
 T. Fukushima, Y. Eriguchi, D. Sugimoto, G. S. BisnovatyiKogan (1980, Prog. Theor. Phys., 63, 1957  1970) — Concave Hamburger Equilibrium of Rotating Bodies
Other Example Applications
 E. Mueller & Y. Eriguchi (1985, A&A, 152, 325  335) — Equilibrium Models of Differentially Rotating, Completely Catalyzed, ZeroTemperature Configurations With Central Densities Intermediate to White Dwarf and Neutron Star Densities
 J. R. Ipser & R. A. Managan (1981, ApJ, 250, 362  372) — On the Existence and Structure of Inhomogeneous Analogs of the Dedekind and Jacobi Ellipsoids
 C. T. Cunningham (1977, ApJ, 211, 568  578) — Rapidly Rotating Spheroids of Polytropic Index n = 1
 R. H. Durisen (1975, ApJ, 199, 179  183) — Upper Mass Limits for Stable Rotating White Dwarfs
 P. Bodenheimer & J. P. Ostriker (1973, ApJ, 180, 159  170) — Rapidly Rotating Stars. VIII. ZeroViscosity Polytropic Sequences'
 P. Bodenheimer (1971, ApJ, 167, 153  163) — Rapidly Rotating Stars. VII. Effects of Angular Momentum on UpperMainSequence Models
 P. Bodenheimer & J. P. Ostriker (1970, ApJ, 161, 1101  1113) — Rapidly Rotating Stars. VI. PreMainSequence Evolution of Massive Stars
 R. Kippenhahn & H.C. Thomas (1970) in Proceedings of the 4th IAU Colloquium, held at the Ohio State University, Columbus, Ohio, September 8  11, 1969, Dordrecht: Riedel Publishing Co., edited by A. Slettebak — Stellar Rotation ==> Purchase proceedings from Springer, from Australia, or from Google
 In the introductory section of his paper, S. Jackson (1970) references this article by Kippenhahn & Thomas in the context of uniformly rotating, and therefore only mildly distorted, structures.
 J. P. Ostriker & J. L. Tassoul (1969, ApJ, 155, 987  ) — On the Oscillations and Stability of Rotating Stellar Models. II. Rapidly Rotating White Dwarfs
 M. J. Clement (1969, ApJ, 156, 1051  1068) — Differential Rotation in Stars on the Upper Main Sequence
 J. W.K. Mark (1968, ApJ, 154, 627  ) — Rapidly Rotating Stars. III. Massive MainSequence Stars
 J. P. Ostriker & P. Bodenheimer (1968, ApJ, 151, 1089  ) — Rapidly Rotating Stars. II. Massive White Dwarfs
 J. Faulkner, I. W. Roxburgh, & P. A. Strittmatter (1968, ApJ, 151, 203  216) — Uniformly Rotating MainSequence Stars
 R. Stoeckly (1965, ApJ, 142, 208  228) — Polytropic Models with Fast, NonUniform Rotation
 In the introductory section of his paper, S. Jackson (1970) states that a differentially rotating polytropic structure with a rotationally induced extreme distortion was first illustrated in this article by Stoeckly.
Henyey Technique for Nonrotating Stars
 L. G. Henyey, L. Wilets, K. H. Böhm, R. Lelevier, & R. D. Levee (1959, ApJ, 129, 628  ) — A Method for Automatic Computation of Stellar Evolution
 L. G. Henyey, J. E. Forbes, & N. L. Gould (1964, ApJ, 139, 306  ) — A New Method of Automatic Computation of Stellar Evolution
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