User:Tohline/AxisymmetricConfigurations/HSCF

Hachisu Self-Consistent-Field Technique

I. Hachisu

In 1986, Izumi Hachisu published two papers in The Astrophysical Journal Supplement Series (vol. 61, pp. 479-507, and 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 an (ca. 1999) outline of the HSCF technique that appeared in our 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 web page associated with Professor Hachisu's current faculty appointment at the University of Tokyo, Komaba.

Constructing Two-Dimensional, Axisymmetric Structures

As has been explained in an accompanying discussion, our objective is to solve an algebraic expression for hydrostatic balance,

<math>~H + \Phi_\mathrm{eff} = C_\mathrm{B}</math> ,

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

<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>

Steps to Follow

1. Choose a particular barotropic equation of state.   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.
2. Choosing from, for example, a list of astrophysically relevant 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.
3. On your chosen computational lattice — for example, 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 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>.
4. 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.
5. Via some accurate numerical algorithm, solve the Poisson equation to determine the gravitational potential, <math>~\Phi(\varpi,z)</math>, throughout the computation lattice that corresponds to the trial mass-density distribution that was specified in Step #4 (or in Step #9).

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© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation