Homologously Collapsing Polytropic Spheres

Review of Goldreich and Weber (1980)

In an accompanying discussion, we have reviewed the self-similar, 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 that are already written in terms of a stream function and a time-varying radial coordinate, <math>~\vec\mathfrak{x} \equiv \vec{r}/a(t)</math>, as developed in the accompanying discussion of Goldreich & Weber's (1980) work. The continuity equation, the Euler equation, and the Poisson equation are, 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>

Following Goldreich & Weber, we allow the normalizing scale length to vary with time and 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}</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>~\cancelto{0}{\frac{d\ln f^n}{dt}} + \frac{d\ln \rho_c}{dt} </math>

<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>

Homologous Solution

Goldreich & Weber (1980) discovered that the governing equations admit to an homologous, self-similar solution if they adopted a stream function of the form,

<math>~\frac{1}{2}a \dot{a} \mathfrak{x}^2 \, ,</math>

which, when acted upon by the various relevant operators, gives,

<math>~\nabla_\mathfrak{x}\psi</math>	<math>~=</math>	<math>~a \dot{a} \mathfrak{x} \, ,</math>
<math>~\nabla^2_\mathfrak{x}\psi</math>	<math>~=</math>	<math>~ \biggl( \frac{1}{2}a \dot{a} \biggr) \frac{1}{\mathfrak{x}^2} \frac{d}{d\mathfrak{x}} \biggl[\mathfrak{x}^2 \frac{d}{d\mathfrak{x}} \mathfrak{x}^2 \biggr] = 3 a \dot{a} \, , </math>
<math>~\frac{d\psi}{dt}</math>	<math>~=</math>	<math>~\mathfrak{x}^2 \biggl[ \frac{1}{2}\dot{a}^2 + \frac{1}{2}a\ddot{a} \biggr] \, .</math>

This stream-function appears to be a reasonable choice, as well, in the more general case that we are considering, in which case the radial velocity profile is, as in the Goldreich & Weber derivation,

<math>~v_r = a^{-1}\nabla_\mathfrak{x} \psi</math>

<math>~\dot{a}\mathfrak{x} \, . </math>

Also, in our more general case, the continuity equation gives,

<math>~\frac{d\ln \rho_c}{dt} + \frac{d\ln a^3}{dt} </math>	<math>~=</math>	<math>~0 </math>
<math>\Rightarrow~~~ a^3\rho_c</math>	<math>~=</math>	constant,

independent of time. But the Euler equation becomes,

<math>~ - n f - \sigma </math>	<math>~=</math>	<math>~ n \biggl\{ \biggl[\frac{4\pi G}{(n+1)^n \kappa^n}\biggr] a^2(t) \biggr\}^{1/(n-1)} \biggl\{ \mathfrak{x}^2 \biggl[ \frac{1}{2}\dot{a}^2 + \frac{1}{2}a\ddot{a} \biggr] - \frac{1}{2} ( \dot{a} \mathfrak{x} )^2 \biggr\} </math>
	<math>~=</math>	<math>~ \frac{n}{2} \biggl[\frac{4\pi G}{(n+1)^n \kappa^n}\biggr]^{1/(n-1)} a^{(n+1)/(n-1)} \mathfrak{x}^2 \ddot{a} </math>
<math>~\Rightarrow~~~~ \frac{(f + \sigma/n)}{\mathfrak{x}^2} </math>	<math>~=</math>	<math>~ -~\frac{1}{2} \biggl[\frac{4\pi G}{(n+1)^n \kappa^n}\biggr]^{1/(n-1)} a^{(n+1)/(n-1)} \ddot{a} \, . </math>

Because everything on the lefthand side of this scaled Euler equation depends only on the dimensionless spatial coordinate, <math>~\mathfrak{x}</math>, while everything on the righthand side depends only on time — via the parameter, <math>~a(t)</math> — both expressions must equal the same (dimensionless) constant. We will follow the lead of Goldreich & Weber (1980) and call this constant, <math>~\lambda/6</math>. From the terms on the lefthand side, we conclude that the dimensionless gravitational potential is,

<math>~\sigma</math>

<math>~\frac{n}{6} \lambda ~\mathfrak{x}^2 - nf \, .</math>

Inserting this expression into the Poisson equation gives,

<math>~\nabla^2_\mathfrak{x} \biggl( \frac{n}{6} \lambda ~\mathfrak{x}^2 - nf \biggr)</math>	<math>~=</math>	<math>~nf^n </math>
<math>~\Rightarrow~~~\frac{1}{\mathfrak{x}^2} \frac{d}{d\mathfrak{x}} \biggl[ \mathfrak{x}^2 \frac{d}{d\mathfrak{x}} \biggl(f - \frac{\lambda}{6} ~\mathfrak{x}^2 \biggr)\biggr]</math>	<math>~=</math>	<math>~-f^n </math>
<math>~\Rightarrow~~~\frac{1}{\mathfrak{x}^2} \frac{d}{d\mathfrak{x}} \biggl[ \mathfrak{x}^2 \frac{df}{d\mathfrak{x}} \biggr] </math>	<math>~=</math>	<math>~\lambda - f^n \, ,</math>

which becomes the familiar Lane-Emden equation for arbitrary n when <math>~\lambda = 0 \, .</math>

From the terms on the righthand side we conclude, furthermore, that the nonlinear differential equation governing the time-dependent variation of the scale length, <math>~a</math>, is,

<math>~-~\frac{\lambda}{3} \biggl[ \frac{(n+1)^n \kappa^n}{4\pi G} \biggr]^{1/(n-1)} \, .</math>

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Contents

Homologously Collapsing Polytropic Spheres

Review of Goldreich and Weber (1980)

Governing Equations

Homologous Solution

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