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Homologously Collapsing Stellar Cores
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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
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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>
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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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>
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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,
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<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> |
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<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,
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<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),
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<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> |
Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
| Go Home |
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<math>~\frac{\partial \rho}{\partial t} + \rho \nabla_r \cdot \vec{v} + \vec{v}\cdot \nabla_r \rho</math> |
<math>~=</math> |
<math>~0</math> |
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<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> |
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<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> |
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<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>
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<math>~=</math> |
<math>~0</math> |
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Goldreich & Weber's (1980) Governing Equations After Initial Length Scaling (yet to be demonstrated) | ||||||||||||
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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,
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<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,
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<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>
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the Euler equation becomes,
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<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,
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<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> |
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<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,
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<math>~\psi</math> |
<math>~=</math> |
<math>~\frac{1}{2}a \dot{a} x^2 \, ,</math> |
which generates a radial velocity profile,
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<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,
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<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> |
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<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,
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<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> |
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<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,
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<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> |
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<math>~=</math> |
<math>~\frac{x^2}{2} \frac{d }{dt} \biggl( a \dot{a} \biggr) </math> |
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<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> |
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<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> |
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Goldreich & Weber's (1980) Euler Equation after all Scaling (yet to be demonstrated) | |||
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,
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<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,
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<math>~ a^2 \ddot{a} </math> |
<math>~=</math> |
<math>~-~\frac{4\lambda}{3} \biggl( \frac{\kappa^3}{\pi G} \biggr)^{1/2} \, .</math> |
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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,
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,
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,
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