User:Tohline/Apps/ReviewStahler83

Stahler's (1983) Rotationally Flattened Isothermal Configurations

Consider the collapse of an isothermal cloud (characterized by isothermal sound speed, <math>~c_s</math>) that is initially spherical, uniform in density, uniformly rotating <math>~(\Omega_0)</math>, and embedded in a tenuous intercloud medium of pressure, <math>~P_e</math>. Now suppose that the cloud maintains perfect axisymmetry as it collapses and that <math>~c_s</math> never changes at any fluid element. To what equilibrium state will this cloud collapse if the specific angular momentum of every fluid element is conserved? In a paper titled, The Equilibria of Rotating, Isothermal Clouds. I. - Method of Solution, S. W. Stahler (1983a, ApJ, 268, 155 - 184) describes a numerical scheme — a self-consistent-field technique — that he used to construct such equilibrium states.

In what follows, lines of text that appear in a dark green font have been extracted verbatim from Stahler (1983a).

Governing Equations

Stahler (1983a) states that the equilibrium configuration is found by solving the equation for momentum balance together with Poisson's equation for the gravitational potential, <math>~\Phi_g</math>. Working in cylindrical coordinates <math>~(\varpi, z)</math> (the assumption of axisymmetry eliminates the azimuthal angle), the momentum equation is (see Stahler's equation 2):

<math>~\frac{\nabla P}{\rho} + \nabla\Phi_g + \nabla\Phi_c</math>

<math>~=</math>

<math>~0 \, ,</math>

where, <math>~\nabla \equiv (\partial/\partial\varpi, \partial/\partial z)</math>, and the centrifugal potential is given by (see Stahler's equation 3):

<math>~\Phi_c(\varpi)</math>

<math>~\equiv</math>

<math>~ \int_0^\varpi \frac{j^2(\varpi^') d\varpi^'}{(\varpi^')^3} \, , </math>

where <math>~j</math> is the z-component of the angular momentum per unit mass. Except for the overall sign, this last expression is precisely the same expression for the centrifugal potential that we have defined in the context of our discussion of simple rotation profiles and, as Stahler stresses, it implicitly assumes that <math>~j</math> is not a function of <math>~z</math>; this builds in the physical constraint enunciated by the Poincaré-Wavre theorem, which guarantees that rotational velocity is constant on cylinders for the equilibrium of any barotropic fluid.

Stahler (1983a) chooses to use the integral form of Poisson's equation to define the gravitational potential, namely (see his equation 10),

<math>~ \Phi_g(\vec{x})</math>

<math>~=</math>

<math>~ G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>

Except for the overall sign, this matches the expression for the Scalar Gravitational Potential that is widely used in astrophysics.

As we have demonstrated in our overview discussion of axisymmetric configurations, the equations that govern the equilibrium properties of axisymmetric structures are,

<math>~ \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3} </math> <math>~=</math> <math>~0 \, ,</math> <math>~ \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math> <math>~=</math> <math>~0 \, ,</math> <math>~ \frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} </math> <math>~=</math> <math>~4\pi G \rho \, .</math>

Let's compare this set of governing equations with the ones used by Stahler (1983a).

Solution Technique

Following exactly along the lines of the HSCF technique that has been described in an accompanying chapter,

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