User:Tohline/Apps/GoldreichWeber80

Homologously Collapsing Stellar Cores

Review of Goldreich and Weber (1980)

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>

We will insert the stream function into the Euler equation, below, after introducing the radial normalization factor used by Goldreich & Weber.

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>

Most 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. In doing this, they effectively adopted an accelerating coordinate system with a time-dependent dimensionless radial coordinate,

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

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Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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ASIDE: It wasn't immediately obvious to me how the set of differential governing equations should be modified in order to accommodate a radially contracting (accelerating) coordinate system. I did not understand the transformed set of equations presented by Goldreich & Weber as equations (7) and (8), for example. I turned to Poludnenko & Khokhlov (2007, Journal of Computational Physics, 220, 678) — hereafter, PK07 — for guidance. PK07 develop a set of governing equations that allows for coordinate rotation as well as expansion or contraction; here we will ignore any modifications due to rotation. We note, first, that PK07 (see their equation 4) adopt an accelerated radial coordinate of the same form as Goldreich & Weber, <math>~\tilde{r} \equiv \biggl[ \frac{1}{a(t)} \biggr] \vec{r} \, ,</math> but the PK07 time-dependent scale factor is dimensionless, whereas the scale factor adopted by Goldreich & Weber — denoted here as <math>~a_{GW}(t)</math> — has units of length. To transform from the KP07 notation, we ultimately will set, <math>~\mathfrak{x} = \frac{1}{a_0} \tilde{r} ~~~~~\Rightarrow ~~~~~ a_{GW}(t) = a_0 a(t) \, ,</math> where, <math>~a_0</math> is understood to be the Goldreich & Weber scale length at the onset of collapse, that is, at <math>~t = 0</math>. According to PK07, this leads to a new "accelerated" time (see, again, their equation 4 with the exponent, <math>~\beta = 0</math>) <math>~\tau \equiv \int_0^t \frac{dt}{a(t)} \, .</math> According to equation (7) of PK07 — again, setting their exponent <math>~\beta=0</math> — the relationship between the fluid velocity in the inertial frame, <math>~\vec{v}</math>, to the fluid velocity measured in the accelerated frame, <math>~\tilde{v}</math>, is <math>~\vec{v} = \tilde{v} + \biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde{r} \, .</math> We note that, according to equation (8) of PK07, the first derivative of <math>~a(t)</math> with respect to physical time is, <math>~\dot{a} = \frac{d\ln a}{d\tau} \, ,</math> so the transformation between velocities may equally well be written as, <math>~\vec{v} = \tilde{v} + \dot{a} \tilde{r} \, ;</math> and we note that (see equation 9 of PK07), <math>~\ddot{a} = \frac{1}{a} \biggl[ \frac{d^2\ln a}{d\tau^2} \biggr] \, .</math> Next, we note that Goldreich & Weber introduce a variable to track the dimensionless density, <math>~f^3</math> <math>~=</math> <math>~\biggl( \frac{\rho}{\rho_c} \biggr) = \biggl( \frac{\pi G}{\kappa} \biggr)^{3/2} [a_{GW}(t)]^3 \rho \, .</math> Comparing this to equation (10) of PK07, which introduces a density field, <math>~\tilde\rho</math>, as viewed in the accelerated frame of reference of the form, <math>~\tilde\rho = [a(t)]^\alpha \rho \, ,</math> we see that, by setting the exponent <math>~\alpha = 3</math>, the Goldreich & Weber dimensionless density can be retrieved from the PK07 work by setting, <math>~f^3= \frac{\tilde\rho}{\rho_0} \, ,</math> where, <math>~\rho_0 \equiv \biggl( \frac{\kappa}{\pi G a_0^2} \biggr)^{3/2} \, .</math> PK07 then claim that, in the accelerating reference frame, the continuity equation and Euler equation become, respectively, <math>~\frac{\partial \tilde\rho}{\partial \tau} + \tilde{\nabla}\cdot(\tilde\rho \tilde{v})</math> <math>~=</math> <math>~(3-\nu)\biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde\rho \, ,</math> <math>~\frac{\partial \tilde\rho \tilde{v} }{\partial \tau} + \tilde{\nabla} \cdot(\tilde\rho \tilde{v} \tilde{v}) </math> <math>~=</math> <math>~(2-\nu)\biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde\rho \tilde{v} - \biggl[ \frac{d^2\ln a}{d\tau^2} \biggr] \tilde\rho \tilde{r} - \tilde{\nabla}\tilde{P} \, ,</math> where PK07 have introduced <math>~\nu</math> as a "dimensionality parameter of the problem." In an effort to rewrite the left-hand-side of PK07's Euler equation in a form that matches Goldreich & Weber's Euler equation, we note that, <math>~\nabla\cdot [(\tilde\rho \tilde{v}) \tilde{v}]</math> <math>~=</math> <math>~\tilde\rho(\tilde{v}\cdot \tilde\nabla) \tilde{v} + \tilde{v}[\tilde\nabla \cdot (\tilde\rho \tilde{v})] \, ,</math> and, with the help of the PK07 continuity equation, <math>~\frac{\partial (\tilde\rho \tilde{v})}{\partial\tau}</math> <math>~=</math> <math>~\tilde\rho \frac{\partial \tilde{v}}{\partial\tau} + \tilde{v} \frac{\partial \tilde\rho}{\partial\tau} </math>   <math>~=</math> <math>~\tilde\rho \frac{\partial \tilde{v}}{\partial\tau} + \tilde{v} \biggl[ (3-\nu)\biggl( \frac{d\ln a}{d\tau} \biggr) \tilde\rho - \tilde{\nabla}\cdot(\tilde\rho \tilde{v}) \biggr] \, .</math> Hence, the Euler equation becomes, <math>~ \tilde\rho \frac{\partial \tilde{v}}{\partial\tau} + \tilde\rho(\tilde{v}\cdot \tilde\nabla) \tilde{v} </math> <math>~=</math> <math>~-~\biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde\rho \tilde{v} - \biggl[ \frac{d^2\ln a}{d\tau^2} \biggr] \tilde\rho \tilde{r} - \tilde{\nabla}\tilde{P} </math> <math>~ \Rightarrow ~~~ \frac{\partial \tilde{v}}{\partial\tau} + (\tilde{v}\cdot \tilde\nabla) \tilde{v} </math> <math>~=</math> <math>~-~\dot{a} \tilde{v} - a \ddot{a} \tilde{r} - \tilde{\nabla}\tilde{H} \, .</math> <math>~ \Rightarrow ~~~ \frac{\partial \tilde{v}}{\partial\tau} + \frac{1}{2} \tilde\nabla({\tilde{v}} \cdot \tilde{v} ) + \tilde{\zeta}\times \tilde{v} </math> <math>~=</math> <math>~-~\dot{a} \tilde{v} - a \ddot{a} \tilde{r} - \tilde{\nabla}\tilde{H} \, ,</math> where the vector identity that has been used to obtain this last expression has been drawn from our separate presentation of the Euler equation written in terms of the fluid vorticity, <math>~\tilde\zeta \equiv \tilde\nabla \times \tilde{v}</math>. Now, let's shift to physical parameters — or example, <math>~\tilde{v}</math> <math>~~~\rightarrow~~~</math> <math>~\vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} = \vec{v} - \dot{a} \tilde{r} ~~ \, ;</math> <math>~\frac{\partial}{\partial\tau}</math> <math>~~~\rightarrow~~~</math> <math>~\frac{\partial t}{\partial\tau} \frac{\partial}{\partial t} = a \frac{\partial}{\partial t} </math> — and, following Goldreich & Weber, set the vorticity to zero. The Euler equation becomes, <math>~\frac{\partial }{\partial t} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] + \frac{1}{2}a^{-1} \tilde\nabla \biggl[ \biggl( \vec{v} - \dot{a} \tilde{r} \biggr) \cdot \biggl( \vec{v} - \dot{a} \tilde{r} \biggr) \biggr] </math> <math>~=</math> <math>~-~\frac{\dot{a}}{a} \biggl( \vec{v} - \dot{a} \tilde{r} \biggr) - \ddot{a} \tilde{r} - a^{-1}\tilde{\nabla}\tilde{H} </math> <math>~\Rightarrow ~~~~ \frac{\partial \vec{v} }{\partial t} - \biggl[ \biggl(\frac{\dot{a}}{a} \biggr)\frac{\partial\vec{r}}{\partial t} + \frac{\ddot{a}}{a} \vec{r} - \biggl( \frac{\dot{a}}{a}\biggr)^2 \vec{r} \biggr] + \frac{1}{2}a^{-1} \tilde\nabla \biggl[ \vec{v} \cdot \vec{v} -2\dot{a} \vec{v} \tilde{r} + (\dot{a} \tilde{r} )^2 \biggr] </math> <math>~=</math> <math>~-~\frac{\dot{a}}{a} \biggl( \vec{v} - \frac{\dot{a}}{a} \vec{r} \biggr) - \frac{\ddot{a}}{a} \vec{r} - a^{-1}\tilde{\nabla}\tilde{H} </math> <math>~\Rightarrow ~~~~ \frac{\partial \vec{v} }{\partial t} + \frac{1}{2}a^{-1} \tilde\nabla \biggl[ \vec{v} \cdot \vec{v} -2\dot{a} \vec{v} \tilde{r} + (\dot{a} \tilde{r} )^2 \biggr] </math> <math>~=</math> <math>\frac{\dot{a}}{a} \biggl(\frac{\partial\vec{r}}{\partial t} - \vec{v} \biggr) - a^{-1}\tilde{\nabla}\tilde{H}</math> <math>~\Rightarrow ~~~~ \frac{\partial \vec{v} }{\partial t} + a^{-1} \tilde\nabla \biggl[ \frac{1}{2}(\vec{v} \cdot \vec{v}) - \dot{a} \vec{v} \tilde{r} + \tilde{H} \biggr] + \biggl( \frac{\dot{a}}{a} \biggr)^2 \vec{r} </math> <math>~=</math> <math>\frac{\dot{a}}{a} \biggl(\frac{\partial\vec{r}}{\partial t} - \vec{v} \biggr) \, .</math> Now, let's tackle the continuity equation: <math>~\frac{\partial \tilde\rho}{\partial \tau} + \tilde\rho \tilde{\nabla}\cdot \tilde{v} + \tilde{v} \cdot \tilde\nabla \tilde\rho </math> <math>~=</math> <math>~(3-\nu)\biggl[ \frac{d\ln a}{d\tau} \biggr] \tilde\rho </math> <math>~\Rightarrow~~~~\frac{a}{\tilde\rho}\frac{\partial \tilde\rho}{\partial t} + \tilde{\nabla}\cdot \tilde{v} + \tilde{v} \cdot \frac{\tilde\nabla \tilde\rho}{\tilde\rho} </math> <math>~=</math> <math>~(3-\nu) \dot{a} </math> <math>~\Rightarrow~~~~ \frac{a}{\tilde\rho}\frac{\partial \tilde\rho}{\partial t} + (\vec{v} - \dot{a}\tilde{r})\cdot \frac{\tilde\nabla \tilde\rho}{\tilde\rho} + \tilde{\nabla}\cdot (\vec{v} - \dot{a}\tilde{r}) </math> <math>~=</math> <math>~(3-\nu) \dot{a} </math> <math>~\Rightarrow~~~~ \frac{1}{a^3\rho}\frac{\partial (a^3\rho)}{\partial t} + a^{-1}(\vec{v} - \dot{a}\tilde{r})\cdot \frac{\tilde\nabla \rho}{\rho} + a^{-1}\tilde{\nabla}\cdot \vec{v} </math> <math>~=</math> <math>~(3-\nu) \frac{\dot{a}}{a} + a^{-1}\tilde{\nabla}\cdot (\dot{a}\tilde{r}) </math> <math>~\Rightarrow~~~~ \frac{1}{\rho}\frac{\partial \rho}{\partial t} + a^{-1}(\vec{v} - \dot{a}\tilde{r})\cdot \frac{\tilde\nabla \rho}{\rho} + a^{-1}\tilde{\nabla}\cdot \vec{v} </math> <math>~=</math> <math>~(3-\nu) \frac{\dot{a}}{a} + a^{-1}\tilde{\nabla}\cdot (\dot{a}\tilde{r}) -3 \frac{\dot{a}}{a} </math>   <math>~=</math> <math>~\frac{\dot{a}}{a} (\tilde{\nabla}\cdot \tilde{r}-\nu) </math> If we set <math>~\nu = 3</math>, this last expression appears to match equation (7) of Goldreich & Weber. With the aid of the continuity equation, the left-hand-side of the Euler equation can be rewritten as, <math>~\frac{\partial \tilde\rho \tilde{v} }{\partial \tau} + \tilde{\nabla} \cdot(\tilde\rho \tilde{v} \tilde{v}) </math> <math>~=</math> <math> \biggl[ \tilde\rho \frac{\partial \tilde{v} }{\partial \tau} + \tilde{v} \frac{\partial \tilde\rho }{\partial \tau} \biggr] + \biggl[ (\tilde{v} \cdot \tilde{\nabla} ) \tilde\rho \tilde{v} + (\tilde\rho \tilde{v} \cdot \tilde{\nabla})\tilde{v} \biggr] </math>   <math>~=</math> <math> \tilde\rho \frac{\partial \tilde{v} }{\partial \tau} + \biggl[(3-\nu)\biggl( \frac{d\ln a}{d\tau} \biggr) \tilde\rho \tilde{v} - (\tilde{v} \cdot \tilde{\nabla} ) \tilde\rho \tilde{v} \biggr] + \biggl[ (\tilde{v} \cdot \tilde{\nabla} ) \tilde\rho \tilde{v} + (\tilde\rho \tilde{v} \cdot \tilde{\nabla})\tilde{v} \biggr] </math>   <math>~=</math> <math> \tilde\rho \biggl[ \frac{\partial \tilde{v} }{\partial \tau} + (3-\nu)\biggl( \frac{d\ln a}{d\tau} \biggr) \tilde{v} + (\tilde{v} \cdot \tilde{\nabla})\tilde{v} \biggr] \, . </math> Hence, the Euler equation becomes, <math> \frac{\partial \tilde{v} }{\partial \tau} + (\tilde{v} \cdot \tilde{\nabla})\tilde{v} + \biggl( \frac{d\ln a}{d\tau} \biggr) \tilde{v} + \biggl( \frac{d^2\ln a}{d\tau^2} \biggr) \tilde{r} </math> <math>~=</math> <math>~ - \frac{\tilde{\nabla}\tilde{P}}{\tilde\rho} \, .</math> Now, let's shift to physical parameters. For example, <math>~\tilde{v}</math> <math>~~~\rightarrow~~~</math> <math>~\vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \, ;</math> <math>~\frac{\partial}{\partial\tau}</math> <math>~~~\rightarrow~~~</math> <math>~\frac{\partial t}{\partial\tau} \frac{\partial}{\partial t} = a \frac{\partial}{\partial t} \, .</math> Hence, the Euler equation becomes, <math>~ - \tilde{\nabla}\tilde{H} </math> <math>~=</math> <math> a\frac{\partial}{\partial t} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] + (\tilde{v} \cdot \tilde{\nabla})\tilde{v} + \dot{a} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] + \ddot{a} \vec{r} </math>   <math>~=</math> <math>a\frac{\partial \vec{v} }{\partial t} - a \biggl[ \biggl(\frac{\ddot{a}}{a} \biggr) \vec{r} - \biggl(\frac{\dot{a}}{a} \biggr)^2 \vec{r} + \biggl(\frac{\dot{a}}{a} \biggr) \frac{\partial \vec{r} }{\partial t} \biggr] + (\tilde{v} \cdot \tilde{\nabla})\tilde{v} + \dot{a} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] + \ddot{a} \vec{r} </math>   <math>~=</math> <math>a\frac{\partial \vec{v} }{\partial t} + (\tilde{v} \cdot \tilde{\nabla})\tilde{v} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr] </math>   <math>~=</math> <math>a\frac{\partial \vec{v} }{\partial t} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr] + \biggl\{\biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \tilde{\nabla} \biggr\} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] </math>   <math>~=</math> <math>a\frac{\partial \vec{v} }{\partial t} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr] + (\vec{v} \cdot \tilde\nabla)\biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \cdot \tilde{\nabla} \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] </math>   <math>~=</math> <math> \biggl\{ a\frac{\partial \vec{v} }{\partial t} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr] + (\vec{v} \cdot \tilde\nabla)\vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \cdot \tilde{\nabla} \vec{v} \biggr\} - (\vec{v} \cdot \tilde\nabla)\biggl[ \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] + \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \cdot \tilde{\nabla} \biggl[\biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] </math>   <math>~=</math> <math> \biggl\{ a\frac{\partial \vec{v} }{\partial t} + \dot{a} \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr] + (\vec{v} \cdot \tilde\nabla)\vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \cdot \tilde{\nabla} \vec{v} \biggr\} - \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \tilde{\nabla} \biggl[\dot{a} \tilde{r} \biggr] </math>   <math>~=</math> <math> \biggl\{ a\frac{\partial \vec{v} }{\partial t} + (\vec{v} \cdot \tilde\nabla)\vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \cdot \tilde{\nabla} \vec{v} \biggr\} + \dot{a} \biggl\{ \biggl[ \vec{v} - \frac{\partial \vec{r} }{\partial t} \biggr] - \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \tilde{\nabla} \biggl[\tilde{r} \biggr] \biggr\} </math> And the continuity equation becomes, <math>~(3-\nu) \dot{a} </math> <math>~=</math> <math>~\frac{a}{\tilde\rho} \frac{\partial \tilde\rho}{\partial t} + \tilde{\nabla}\cdot \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] + \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \frac{\tilde{\nabla} \tilde\rho}{\tilde\rho} </math>   <math>~=</math> <math>~\frac{a}{\tilde\rho} \frac{\partial \tilde\rho}{\partial t} + \tilde{\nabla}\cdot \vec{v} - \tilde{\nabla}\cdot \biggl[ \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] + \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \frac{\tilde{\nabla} \tilde\rho}{\tilde\rho} </math> <math>~\Rightarrow ~~~ (3-\nu) \dot{a} + \tilde{\nabla}\cdot \biggl[ \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] </math> <math>~=</math> <math>~\frac{1}{a^2 \rho} \frac{\partial (a^3\rho)}{\partial t} + \tilde{\nabla}\cdot \vec{v} + \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \frac{\tilde{\nabla} \rho}{\rho} </math>   <math>~=</math> <math>~\frac{a}{\rho} \frac{\partial \rho}{\partial t} + 3\dot{a} + \tilde{\nabla}\cdot \vec{v} + \biggl[ \vec{v} - \biggl(\frac{\dot{a}}{a} \biggr) \vec{r} \biggr] \cdot \frac{\tilde{\nabla} \rho}{\rho} </math> <math>~\Rightarrow ~~~ \frac{1}{\rho} \frac{\partial \rho}{\partial t} + a^{-1}\tilde{\nabla}\cdot \vec{v} + \biggl[ \vec{v} - \dot{a} \biggl( \frac{\vec{r}}{a}\biggr) \biggr] \cdot \frac{a^{-1}\tilde{\nabla} \rho}{\rho} </math> <math>~=</math> <math> \frac{\dot{a}}{a} \biggl[\tilde{\nabla}\cdot \biggl( \frac{\vec{r}}{a} \biggr) -\nu \biggr] </math> <math>~\Rightarrow ~~~ \frac{1}{\rho} \frac{\partial \rho}{\partial t} + a^{-1}\tilde{\nabla}\cdot \vec{v} + a^{-1} \biggl[ \vec{v} - \dot{a} \vec{\mathfrak{x}} \biggr] \cdot \frac{\tilde{\nabla} \rho}{\rho} </math> <math>~=</math> <math> \frac{\dot{a}}{a} \biggl[\tilde{\nabla}\cdot \vec{\mathfrak{x}} -\nu \biggr] </math>

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 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; specifically,

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

Defining the dimensionless radial coordinate,

<math>~\vec{x} \equiv \frac{1}{a} \vec{r} \, ,</math>

inserting the following replacements for the spatial operators,

<math>~\nabla_r ~\rightarrow~ a^{-1} \nabla_x</math>        and        <math>~\nabla_r^2 ~\rightarrow~ a^{-2} \nabla_x^2 \, ,</math>

and writing the velocity in terms of the appropriately scaled stream function,

<math>~\vec{v} = a^{-1}\nabla_x \psi \, ,</math>

the Euler equation becomes,

<math>~\frac{\partial }{\partial t} \biggl[\frac{1}{a} \nabla_x\psi \biggr]</math>	<math>~=</math>	<math>~-~ \frac{1}{a}\nabla_x \biggl[ H + \Phi + \frac{1}{2}\biggl(\frac{1}{a} \nabla_x\psi \biggr)^2 \biggr] </math>
<math>~\Rightarrow~~~~\frac{1}{a} \frac{\partial }{\partial t} \biggl(\nabla_x\psi \biggr) - \biggl( \frac{\dot{a}}{a^2} \biggr)\nabla_x\psi </math>	<math>~=</math>	<math>~-~ \frac{1}{a}\nabla_x \biggl[ H + \Phi + \frac{1}{2}\biggl(\frac{1}{a} \nabla_x\psi \biggr)^2 \biggr] \, .</math>

Now, because (by design) the dimensionless "<math>~x</math>" coordinate is independent of time and the scaling parameter, <math>~a(t)</math>, is not a function of space, the <math>~\nabla_x</math> operator in the first term on the left-hand-side can be brought outside of the time derivative and, in the second term on the left-hand-side, the coefficient involving the scale length can be brought inside the spatial operator. If we also multiply through by <math>~a</math>, the Euler equation becomes,

<math>~\nabla_x \biggl[ \frac{\partial \psi}{\partial t} - \biggl( \frac{\dot{a}}{a} \biggr)\psi + H + \Phi + \frac{1}{2}\biggl(\frac{1}{a} \nabla_x\psi \biggr)^2 \biggr] </math>

<math>~=</math>

<math>~0 \, .</math>

As Goldreich & Weber (1980) point out, because all terms in this equation are inside the gradient operator, the sum of the terms inside the square brackets must equal a constant — that is, the sum must be independent of spatial position throughout the spherically symmetric configuration. If, following Goldreich & Weber's lead, we simply fold this integration constant into the potential, the Euler equation becomes (see their equation 8),

<math>~\frac{\partial \psi}{\partial t} - \biggl( \frac{\dot{a}}{a} \biggr)\psi + H + \Phi + \frac{1}{2}\biggl(\frac{1}{a} \nabla_x\psi \biggr)^2 </math>

<math>~=</math>

<math>~0 \, .</math>

<math>~\frac{\partial \rho}{\partial t} + \rho \nabla_r \cdot \vec{v} + \vec{v}\cdot \nabla_r \rho</math>	<math>~=</math>	<math>~0</math>
<math>~\Rightarrow ~~~~ \frac{1}{\rho} \frac{\partial \rho}{\partial t} + \nabla_r \cdot \vec{v} + \vec{v}\cdot \frac{\nabla_r \rho}{\rho}</math>	<math>~=</math>	<math>~0</math>
<math>~\Rightarrow ~~~~ \frac{1}{\rho} \frac{\partial \rho}{\partial t} + a^{-1} \nabla_x \cdot \biggl[ a^{-1} \nabla_x \psi \biggr] + a^{-1} \nabla_x \psi \cdot \frac{a^{-1}\nabla_x \rho}{\rho}</math>	<math>~=</math>	<math>~0</math>
<math>~\frac{1}{\rho} \frac{\partial \rho}{\partial t} + a^{-1}(a^{-1} \nabla_x\psi - \dot{a} \vec{x}) \cdot \frac{\nabla_x\rho}{\rho} + a^{-2} \nabla_x^2\psi </math>	<math>~=</math>	<math>~0</math>

Goldreich & Weber's (1980) Governing Equations After Initial Length Scaling (yet to be demonstrated)

<math>~\frac{1}{\rho} \frac{\partial \rho}{\partial t} + a^{-1}(a^{-1} \nabla_x\psi - \dot{a} \vec{x}) \cdot \frac{\nabla_x\rho}{\rho} + a^{-2} \nabla_x^2\psi </math> <math>~=</math> <math>~0</math> <math>~\frac{\partial \psi}{\partial t} - \frac{\dot{a}}{a} \vec{x}\cdot \nabla_x\psi + \frac{1}{2} a^{-2} | \nabla_x\psi|^2 + H + \Phi</math> <math>~=</math> <math>~0</math> <math>~ a^{-2} \nabla_x^2\Phi - 4\pi G \rho </math> <math>~=</math> <math>~0</math> where, <math>~\vec{x} \equiv \frac{\vec{r}}{a} \, ,</math> and it is understood that derivatives in the <math>~\nabla_x</math> and <math>~\nabla_x^2</math> operators are taken with respect to the dimensionless radial coordinate, <math>~x</math>.

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

Specifically, their dimensionless gravitational potential is,

<math>~\sigma</math>

<math>~\equiv</math>

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

With these additional scalings, the continuity equation becomes,

<math>~\frac{\partial}{\partial t} \biggl[ \ln \biggl(\frac{f}{a} \biggr)^3 \biggr]</math>

<math>~=</math>

<math>~-~ a^{-1}(a^{-1} \nabla_x\psi - \dot{a} \vec{x}) \cdot \nabla_x(\ln f^3) - a^{-2} \nabla_x^2\psi \, ;</math>

the Euler equation becomes,

<math>~\frac{\partial \psi}{\partial t} - \frac{\dot{a}}{a} \vec{x}\cdot \nabla_x\psi + \frac{1}{2} a^{-2} | \nabla_x\psi|^2</math>

<math>~=</math>

<math>~ - a^{-1} \biggl[ \frac{4}{3} \biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2} \biggr] (3f + \sigma) \, ;</math>

and the Poisson equation becomes,

<math>~\frac{4}{3} \biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2} a^{-3} \nabla_x^2\sigma</math>	<math>~=</math>	<math>~4\pi G\biggl( \frac{\kappa}{\pi G} \biggr)^{3/2} a^{-3} f^3 </math>
<math>~\Rightarrow~~~~\nabla_x^2\sigma</math>	<math>~=</math>	<math>~3 f^3 \, .</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>~\psi</math>

<math>~=</math>

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

which generates a radial velocity profile,

<math>~\vec{v} = a^{-1}\nabla_x \psi</math>

<math>~=</math>

<math>~\hat{e}_x a^{-1} \biggl[ \frac{\partial}{\partial x} \biggl( \frac{1}{2}a \dot{a} x^2 \biggr)\biggr] = \dot{a} \vec{x} \, . </math>

Recognizing, as well, that,

<math>~a^{-2} \nabla_x^2 \psi </math>	<math>~=</math>	<math>~\frac{1}{(ax)^2} \frac{\partial}{\partial x} \biggl[ x^2\frac{\partial }{\partial x} \biggl( \frac{1}{2}a \dot{a} x^2 \biggr)\biggr] </math>
	<math>~=</math>	<math>~ \biggl( \frac{\dot{a}}{a} \biggr) \frac{1}{x^2} \frac{\partial}{\partial x} \biggl[ x^3\biggr] = \frac{3\dot{a}}{a} = \frac{d\ln a^3}{dt} \, ,</math>

the continuity equation becomes,

<math>~\frac{\partial \ln f^3}{\partial t} - \frac{d \ln a^3}{dt} </math>	<math>~=</math>	<math>~- \frac{d \ln a^3}{dt} </math>
<math>~\Rightarrow ~~~ \frac{\partial \ln f^3}{\partial t} </math>	<math>~=</math>	<math>~0 \, ,</math>

that is, the dimensionless density profile, <math>~f</math>, is independent of time. With the adopted stream function, the Euler equation becomes,

<math>~ - a^{-1} \biggl[ 4\biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2} \biggr] \biggl(f + \frac{\sigma}{3} \biggr) </math>	<math>~=</math>	<math>~\frac{\partial }{\partial t} \biggl( \frac{1}{2}a \dot{a} x^2 \biggr) - \dot{a}^2 x^2 + \dot{a}^2 x^2</math>
	<math>~=</math>	<math>~\frac{x^2}{2} \frac{d }{dt} \biggl( a \dot{a} \biggr) </math>
<math>~\Rightarrow~~~~ \frac{1}{x^2} \biggl(f + \frac{\sigma}{3} \biggr) </math>	<math>~=</math>	<math>~-~\biggl[ \frac{1}{8}\biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} \biggr] a \frac{d }{dt} \biggl( a \dot{a} \biggr) </math>
	<math>~=</math>	<math>~-~\biggl[ \frac{1}{8}\biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} \biggr] a ( \dot{a}^2 + a \ddot{a}) \, .</math>

Goldreich & Weber's (1980) Euler Equation after all Scaling (yet to be demonstrated)

<math>~ \frac{1}{x^2} \biggl(f + \frac{\sigma}{3} \biggr) </math> <math>~=</math> <math>~-~\biggl[ \frac{1}{8}\biggl( \frac{\pi G}{\kappa^3} \biggr)^{1/2} \biggr] a^2 \ddot{a} </math> Note that the right-hand-side of this expression differs from ours, so we need to identify and correct the discrepency.

Because everything on the left-hand-side of Goldreich & Weber's scaled Euler equation depends only on the dimensionless spatial coordinate, <math>~x</math>, while everything on the right-hand-side depends only on time — via the parameter, <math>~a(t)</math> — both expressions must equal the same constant. Goldreich & Weber (1980) (see their equation 12) call this constant, <math>~\lambda/6</math>. They conclude, therefore, (see their equation 13) that the dimensionless gravitational potential is,

<math>~\sigma</math>

<math>~=</math>

<math>~\frac{\lambda x^2}{2} - 3f \, .</math>

Also, the nonlinear differential equation governing the time-dependent variation of the scale length, <math>~a</math>, is,

<math>~ a^2 \ddot{a} </math>

<math>~=</math>

<math>~-~\frac{4\lambda}{3} \biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2} \, .</math>

As Goldreich & Weber (1980) point out, this nonlinear differential equation can be integrated twice to produce an algebraic relationship between <math>~a</math> and time, <math>~t</math>. First, rewrite the equation as, <math>~ \frac{d \dot{a} }{dt} </math> <math>~=</math> <math>~-\frac{B}{2a^2} \, , </math> where, <math> ~B \equiv \frac{8\lambda}{3} \biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2} \, . </math> Then, multiply both sides by <math>~2\dot{a} = 2da/dt</math> to obtain, <math>~ 2\dot{a} \frac{d\dot{a}}{dt} </math> <math>~=</math> <math>~-B \biggl( a^{-2} \frac{da}{dt} \biggr) </math> <math>~\Rightarrow~~~~ \frac{d\dot{a}^2}{dt} </math> <math>~=</math> <math>~B \frac{d}{dt} \biggl( \frac{1}{a} \biggr) </math> which integrates once to give, <math> ~\dot{a}^2 = \frac{B}{a} + C \, , </math> or, <math> ~dt = \biggl( \frac{B}{a} + C \biggr)^{-1/2} da \, . </math> For the case, <math>~C = 0</math>, this differential equation can be integrated straightforwardly to give (see Goldreich & Weber's equation 15), For the cases when <math>~C \ne 0</math>, Wolfram Mathematica's online integrator can be called upon to integrate this equation and provide the following closed-form solution, <math>~t</math> <math>~=</math> <math> \frac{a}{C} \biggl( \frac{B}{a} + C \biggr)^{1/2} - \frac{B}{2C^{3/2}} \ln \biggl[2aC^{1/2} \biggl( \frac{B}{a} + C \biggr)^{1/2} + B + 2aC \biggr] \, . </math>

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© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation