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Homologously Collapsing Polytropic Spheres

Whitworth's (1981) Isothermal Free-Energy Surface
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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, ~\gamma=4/3 — and polytropic index, ~n=3. 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, ~\vec\mathfrak{x} \equiv \vec{r}/a(t), as developed in the accompanying discussion of Goldreich & Weber's (1980) work. The continuity equation, the Euler equation, and the Poisson equation are, respectively,

~\frac{1}{\rho} \frac{d\rho}{dt}

~=

~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi  \, ;

~\frac{d\psi}{dt}

~=

~\frac{1}{2} a^{-2} ( \nabla_\mathfrak{x} \psi )^2 - H - \Phi  \, ;

~a^{-2}\nabla_\mathfrak{x}^2 \Phi

~=

~4\pi G \rho \, .

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


a \equiv \biggl[\frac{1}{4\pi G}~ \biggl( \frac{H_c}{\rho_c} \biggr)\biggr]^{1/2}  \, ,

where the subscript, "c", denotes central values and, as presented in our introductory discussion of barotropic supplemental relations,

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

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,


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

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,

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

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


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

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

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

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,

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

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

~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)}

 

~=

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

giving,

~\sigma

~\equiv

~\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 \, ,

and,

~\frac{H}{c_s^2}

~=

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

With these additional scalings, the continuity equation becomes,

~\cancelto{0}{\frac{d\ln f^n}{dt}} + \frac{d\ln \rho_c}{dt}

~=

~-~ a^{-2} \nabla_\mathfrak{x}^2 \psi  \, ;

the Euler equation becomes,

~
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]

~=

~ - n f - \sigma  \, ;

and the Poisson equation becomes,

\nabla_\mathfrak{x}^2 \sigma = n f^n \, .

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,

~\psi

~=

~\frac{1}{2}a \dot{a} \mathfrak{x}^2 \, ,

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

~\nabla_\mathfrak{x}\psi

~=

~a \dot{a} \mathfrak{x} \, ,

~\nabla^2_\mathfrak{x}\psi

~=

~
\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} \, ,

~\frac{d\psi}{dt}

~=

~\mathfrak{x}^2 \biggl[ \frac{1}{2}\dot{a}^2 + \frac{1}{2}a\ddot{a} \biggr] \, .

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,

~v_r = a^{-1}\nabla_\mathfrak{x} \psi

~=

~\dot{a}\mathfrak{x} \, .

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

~\frac{d\ln \rho_c}{dt}  + \frac{d\ln a^3}{dt}

~=

~0

\Rightarrow~~~ a^3\rho_c

~=

constant,

independent of time. But the Euler equation becomes,

~ - n f - \sigma

~=

~
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\}

 

~=

~
\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}

~\Rightarrow~~~~ \frac{(f + \sigma/n)}{\mathfrak{x}^2}

~=

~
-~\frac{1}{2} \biggl[\frac{4\pi G}{(n+1)^n \kappa^n}\biggr]^{1/(n-1)}   a^{(n+1)/(n-1)} \ddot{a} \, .


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

~\sigma

~=

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

Inserting this expression into the Poisson equation gives,

~\nabla^2_\mathfrak{x} \biggl( \frac{n}{6} \lambda ~\mathfrak{x}^2 - nf  \biggr)

~=

~nf^n

~\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]

~=

~-f^n

~\Rightarrow~~~\frac{1}{\mathfrak{x}^2} \frac{d}{d\mathfrak{x}} \biggl[ \mathfrak{x}^2 \frac{df}{d\mathfrak{x}} 
\biggr]

~=

~\lambda - f^n \, ,

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

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

~
a^{(n+1)/(n-1)} \ddot{a}

~=

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



Whitworth's (1981) Isothermal Free-Energy Surface

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