User:Tohline/AxisymmetricConfigurations/PoissonEq

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Solving the Poisson Equation Numerically

Whitworth's (1981) Isothermal Free-Energy Surface
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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 + \Psi = C_0</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

Figure 1 extracted from I. Hachisu (1986)

"A Versatile Method for Obtaining Structures of Rapidly Rotating Stars"

ApJS, vol. 61, pp. 479-507 © American Astronomical Society

HSCF Meridional Plane
Two points A and B on the surface specify a model uniquely: (a) spheroidal structure; (b) toroidal structure.
  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. 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.
  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 computational lattice corresponding to the trial mass-density distribution that was specified in Step #4 (or in Step #9).
  6. 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.
  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, <math>~C_0</math> and <math>~\omega_0</math>, that appear in the algebraic equation that defines hydrostatic equilibrium.
  8. 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.
  9. 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.
  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 <math>~C_0</math> and <math>~\omega_0</math> — 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

Solution Methods

Early Eriguchi Applications

Other Example Applications

Henyey Technique for Nonrotating Stars


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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