User:Tohline/Apps/GoldreichWeber80
Homologously Collapsing Stellar Cores
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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
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<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,
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<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,
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<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.
Related Discussions
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© 2014 - 2021 by Joel E. Tohline |