Homologously Collapsing Polytropic Spheres

Review of Goldreich and Weber (1980)

In an accompanying discussion, we review the dynamical model that Peter Goldreich & Stephen Weber (1980, ApJ, 238, 991) developed to describe the near-homologous collapse of stellar cores that obey an equation of state having an adiabatic index, <math>~\gamma=4/3</math> — and polytropic index, <math>~n=3</math>. Here we examine whether or not this self-similar collapse model can be extended to configurations with arbitrary polytropic index.

Governing Equations

We begin with the set of principal governing equations already written in terms of a stream function, as developed in the accompanying discussion of Goldreich & Weber's (1980) work. Specifically, the continuity equation, the Euler equation, and the Poisson equation become, respectively,

<math>~\frac{1}{\rho} \frac{d\rho}{dt} </math>	<math>~=</math>	<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi \, ;</math>
<math>~\frac{d\psi}{dt} </math>	<math>~=</math>	<math>~\frac{1}{2} a^{-2} ( \nabla_\mathfrak{x} \psi )^2 - H - \Phi \, ;</math>
<math>~a^{-2}\nabla_\mathfrak{x}^2 \Phi </math>	<math>~=</math>	<math>~4\pi G \rho \, .</math>

Following Goldreich & Weber (1980), the normalization length scale, <math>~a</math>, that appears in this set of equations is the same length scale that is used in deriving the Lane-Emden equation, namely,

<math> a \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 and, as presented in our introductory discussion of barotropic supplemental relations,

<math>~H = (n+1) \kappa \rho^{1/n} \, .</math>

But, unlike Goldreich & Weber, we will leave the polytropic index unspecified. Inserting this equation of state expression into the definition of the normalization length scale leads to,

<math> a = \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \rho_c^{-(n-1)/(2n)} \, . </math>

Again, following Goldreich & Weber, we allow the normalizing scale length to vary with time an adopt an accelerating coordinate system with a time-dependent dimensionless radial coordinate,

<math>~\vec\mathfrak{x} \equiv \frac{1}{a(t)} \vec{r} \, .</math>

(The spatial operators in the above set of principal governing equations exhibit a subscript, <math>~\mathfrak{x}</math>, indicating the adoption of this accelerating coordinate system.) This, in turn, reflects a time-varying central density; specifically,

<math> \rho_c = \biggl\{ \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \frac{1}{a(t)} \biggr\}^{2n/(n-1)} \, . </math>

Next, we normalize the density by the central density, defining a dimensionless function,

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

which is 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>

And we normalize both the gravitational potential and the enthalpy to the square of the central sound speed,

<math>c_s^2 = \frac{\gamma P_c}{\rho_c} = \frac{H_c}{n} = \biggl( \frac{n+1}{n} \biggr) \kappa \rho_c^{1/n} = \biggl( \frac{n+1}{n} \biggr) \kappa \biggl\{ \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \frac{1}{a(t)} \biggr\}^{2/(n-1)} = \, .</math>

<math>~c_s^2 = \frac{\gamma P_c}{\rho_c} = \frac{H_c}{n}</math>	<math>~=</math>	<math>~\biggl( \frac{n+1}{n} \biggr) \kappa \rho_c^{1/n}</math>
	<math>~=</math>	<math>~\biggl( \frac{n+1}{n} \biggr) \kappa \biggl\{ \biggl[\frac{(n+1)\kappa}{4\pi G}\biggr]^{1/2} \frac{1}{a(t)} \biggr\}^{2/(n-1)}</math>
	<math>~=</math>	<math>~\frac{1}{n} \biggl\{ \biggl[\frac{(n+1)^n \kappa^n}{4\pi G}\biggr] \frac{1}{a^2(t)} \biggr\}^{1/(n-1)} \, ,</math>

giving,

<math>~\sigma</math>

<math>~\equiv</math>

<math>~\frac{\Phi}{c_s^2} = n \biggl\{ \biggl[\frac{4\pi G}{(n+1)^n \kappa^n}\biggr] a^2(t) \biggr\}^{1/(n-1)} \Phi \, ,</math>

and,

<math>~n \biggl( \frac{\rho}{\rho_c} \biggr)^{1/n} = nf \, .</math>

With these additional scalings, the continuity equation becomes,

<math>~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi \, ,</math>

the Euler equation becomes,

<math>~ n \biggl\{ \biggl[\frac{4\pi G}{(n+1)^n \kappa^n}\biggr] a^2(t) \biggr\}^{1/(n-1)} \biggl[ \frac{d\psi}{dt} - \frac{1}{2a^2} ( \nabla_\mathfrak{x} \psi )^2 \biggr] </math>

<math>~ - n f - \sigma \, ;</math>

and the Poisson equation becomes,

<math>\nabla_\mathfrak{x}^2 \sigma = n f^n \, .</math>

User:Tohline/Apps/HomologousCollapse

Homologously Collapsing Polytropic Spheres

Review of Goldreich and Weber (1980)

Governing Equations

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