The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline
[tohline@rouge.phys.lsu.edu]
Kimberly C. B. New
John Cazes
Monika Lee
Pete Nelson

HSCF Technique:
General Method

Following Hachisu's published explanation (The Astrophysical Journal Supplement Series, Vol. 61, p. 479; Vol. 62, p. 461), we outline here the steps that should be followed when implementing the HSCF technique.

H + F + ho2 y = Co
2F = 4p Gr

[Eqs. III.F.24 and I.D.1]

Hachisu Self-Consistent-Field Technique
EXAMPLES
1.
Choose a particular barotropic equation of state. More specifically, define the functional behavior of the density-enthalpy relationship r(H), and identify what value Hsurface the enthalpy will have at the surface of your configuration. (Often, Hsurface = 0.)
Maclaurin
Spheroids

2.
Specify the functional form of the centrifugal potential y that will define the radial distribution of specific angular momentum in your equilibrium configuration.
3.
On your chosen computational lattice (e.g., on a cylindrical coordinate 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 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 HA = HB = Hsurface.
Differentially
Rotating
Incompressible
Tori

4.
Throughout the volume of your computational lattice, "guess" a trial distribution of the mass density r 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.)
5.
Via some accurate numerical algorithm, solve the Poisson equation [I.D.1] to determine the gravitational potential F(x) throughout the computational lattice that corresponds to the trial mass density distribution that was specified in Step 4 (or in Step 9).
Binary
White
Dwarfs

6.
From the gravitational potential determined in Step 5, identify the values of FA and FB at the two boundary points that were selected in Step 3.
7.
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 Co and ho that appear in the algebraic equation [III.F.24] that defines hydrostatic equilibrium.
Compressible
Spheroids
with
Uniform
Rotation

8.
From the most recently determined values of the gravitational potential F(x) (Step 5) and the values of the two constants Co and ho just determined (Step 7), determine the enthalpy distribution throughout the computational lattice.
9.
From H(x) and the selected barotropic equation of state (Step 1), calculate an "improved guess" of the density distribution r(x) throughout the computational lattice.
10.
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 Co and ho -- change very little between successive iterations and the virial error is sufficiently small.)

Click here
for other
Implementations
YES ?

==> Stop iteration.

NO ?

==> Repeat steps 5 through 10.


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