User:Tohline/Apps/GoldreichWeber80

Homologously Collapsing Stellar Cores

Introduction

This is principally a review of the dynamical model that Peter Goldreich & Stephen Weber (1980, ApJ, 238, 991) developed to describe the near-homologous collapse of stellar cores.

Governing Equations

Goldreich & Weber begin with the identical set of principal governing equations that serves as the foundation for all of the discussions throughout this H_Book. In particular, as is documented by their equation (1), their adopted equation of state is adiabatic/polytropic,

<math>~P = \kappa \rho^\gamma \, ,</math>

— where both <math>~\kappa</math> and <math>~\gamma</math> are constants — and therefore satisfies what we have referred to as the

Adiabatic Form of the
First Law of Thermodynamics
(Specific Entropy Conservation)

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math> .

their equation (2) is what we have referred to as the

Eulerian Representation
or
Conservative Form
of the Continuity Equation,

<math>~\frac{\partial\rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0</math>

their equation (3) is what we have referred to as the

Euler Equation
in terms of the Vorticity,

<math>~\frac{\partial\vec{v}}{\partial t} + \vec\zeta \times \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \biggl[\Phi + \frac{1}{2}v^2 \biggr] </math>

where, <math>~\vec\zeta \equiv \nabla\times \vec{v}</math> is the fluid vorticity; and their equation (4) is the

Poisson Equation

<math>\nabla^2 \Phi = 4\pi G \rho</math>

Tweaking the set of principal governing equations, as we have written them, to even more precisely match equations (1) - (4) in Goldreich & Weber (1980), we should replace the state variable <math>~P</math> (pressure) with <math>~H</math> (enthalpy), keeping in mind that, <math>~\gamma = 1 + 1/n</math>, and, as presented in our introductory discussion of barotropic supplemental relations,

<math>~H = \biggl( \frac{\gamma}{\gamma-1} \biggr) \kappa \rho^{\gamma-1} \, ,</math>

and,

<math>~\nabla H = \frac{\nabla P}{\rho} \, .</math>

Imposed Constraints

Goldreich & Weber (1980) specifically choose to examine the spherically symmetric collapse of a <math>~\gamma = 4/3</math> fluid. With this choice of adiabatic index, the equation of state becomes,

<math>~H = 4 \kappa \rho^{1/3} \, .</math>

And because a strictly radial flow-field exhibits no vorticity (i.e., <math>\vec\zeta = 0</math>), the Euler equation can be rewritten as,

<math>~\frac{\partial v_r}{\partial t} </math>

<math>~=</math>

<math>~-~ \nabla_r \biggl[ H + \Phi + \frac{1}{2}v^2 \biggr] \, .</math>

Goldreich & Weber also realize that, because the flow is vorticity free, the velocity can be obtained from a stream function, <math>~\Psi</math>, via the relation,

<math>~\vec{v} = \nabla\Psi \, .</math>

Hence, the Euler equation becomes,

<math>~\frac{\partial \Psi}{\partial t} </math>

<math>~=</math>

<math>~-~ \biggl[ H + \Phi + \frac{1}{2}\biggl( \nabla \Psi \biggr)^2 \biggr] \, ,</math>

where, <math>~H</math>, <math>~\Phi</math>, and <math>~\Psi</math> are each functions only of the radial coordinate.

Dimensionless Normalization

In their investigation, Goldreich & Weber (1980) chose the same length scale for normalization that is used in deriving the Lane-Emden equation, which governs the hydrostatic structure of a polytrope of index <math>~n</math>, that is,

<math> a_\mathrm{n} \equiv \biggl[\frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr)\biggr]^{1/2} \, , </math>

where the subscript, "c", denotes central values. In this case <math>~(n = 3)</math>, substitution of the equation of state expression for <math>~H_c</math> leads to,

<math> a(t) = \rho_c^{-1/3} \biggl(\frac{\kappa}{\pi G}\biggr)^{1/2} \, , </math>

and, quite significantly, Goldreich & Weber (see their equation 6) allow the normalizing scale length to vary with time in order for the governing equations to accommodate a self-similar dynamical solution. [This, in turn, will mean that either the central density varies with time, or the specific entropy of all fluid elements (captured by the value of <math>~\kappa</math>) varies with time, or both. In practice, Goldreich & Weber ultimately assume that <math>~\kappa</math> is held fixed, so the time-variation in the scale length, <math>~a</math>, reflects a time-varying central density.]

Next, Goldreich & Weber (1980) (see their equation 10) choose to normalize the density by the central density, specifically defining a dimensionless function,

<math>f \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/3} \, .</math>

Keeping in mind that <math>~n = 3</math>, this is also in line with the formulation and evaluation of the Lane-Emden equation, where the primary dependent structural variable is the dimensionless polytropic enthalpy,

<math>\Theta_H \equiv \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>

Finally, Goldreich & Weber (1980) (see their equation 11) normalize the gravitational potential to the square of the central sound speed,

<math>c_s^2 = \frac{\gamma P_c}{\rho_c} = \, .</math>

Specifically, the time-varying dimensionless gravitational potential is,

<math>~\psi</math>

<math>~\equiv</math>

<math>~\biggl[ \frac{3}{4} \biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} a \biggr] \Phi \, .</math>

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